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Electronic correlation effects in multi-band systems

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Electronic correlation effects in multi-band systems

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Content ELECTRONIC CORRELATION EFFECTS IN MULTI-BAND SYSTEMS
by
Kok Wee Song
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
May 2014
Copyright 2014 Kok Wee Song
Dedication
To my beloved wife, parents, and family.
ii
Acknowledgments
I would like to express my deepest gratitude to my research advisor, Professor Stephan
Haas. The support and guidance from him undoubtedly bring me toward the achievement
of my PhD degree. For sure, he has tailored a very conformable research environment
for me, and given me a lot of freedom in the research topics based on my interests. It is
impossible to forget those nice working experiences with him. In addition, I would like
to thanks Prof. Itzhak Bars for offering an opportunity to work in his group at the early
stage of the program. Also, I would like to thank Prof. Kingman Cheung for his long-term
support in many aspects since my Master program in Taiwan.
Furthermore, I would also like to thank Professors Gene Bickers, Werner D¨ appen,
Christoph Haselwandter, Edmond Jonckheere, and Chi Mak for serving on my advisory
committee. Particularly, I truly thank for Professors Gene Bickers and Werner D¨ appen for
many encouragements and help in my academic career.
Finally, many thanks to my collaborators, Yung-Ching Liang and Hokiat Lim for many
inspiring discussions and participation in my research. I want to thank Yi-Chen Chang
who has motivated me to initiate research in graphene. Moveover, I would like to thank
my colleagues and friends, Min-Chak Ho, Rodrigo Muniz, Siddhartha Santra, and Isaiah
Yoo for many fruitful discussions and encouragements.
iii
Table of Contents
Dedication ii
Acknowledgments iii
List of Tables vi
List of Figures vii
Abstract x
Chapter 1: Introduction 1
1.1 New materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Topological insulator . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Iron-based superconductors . . . . . . . . . . . . . . . . . . . . 7
1.2 Many-body effects in multi-band system . . . . . . . . . . . . . . . . . 10
1.2.1 Competing orders beyond Fermi liquid criticality . . . . . . . . 11
1.2.2 Electronic nematicity . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 2: Theoretical background and methods 14
2.1 Landau’s Fermi-liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Quasiparticle lifetime and Fermi-Dirac statistic . . . . . . . . . 15
2.1.2 Thermal equilibrium properties . . . . . . . . . . . . . . . . . . 17
2.1.3 Universality class . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.4 Pomeranchuk instabilities . . . . . . . . . . . . . . . . . . . . . 20
2.2 Fields theoretical formalism . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Functional path integrations . . . . . . . . . . . . . . . . . . . 26
2.3 Renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . 29
iv
2.3.1 Critical phenomena, scale invariant, and power-law . . . . . . . 29
2.3.2 RG: general idea and parameters space . . . . . . . . . . . . . . 31
2.4 Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Hartree-Fock variational methods . . . . . . . . . . . . . . . . 37
2.4.2 Hubbard-Stratonovich transfomation . . . . . . . . . . . . . . . 44
Chapter 3: Competing orders in Bilayer graphene 51
3.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Bilayer Graphene and the model Hamiltonian . . . . . . . . . . . . . . 54
3.2.1 Coupling Constants and the Interacting Hamiltonian . . . . . . 56
3.2.2 Interactions in the BLG Hamiltonian . . . . . . . . . . . . . . . 58
3.2.3 Coupling Constant Expansion . . . . . . . . . . . . . . . . . . 60
3.3 Renormalization Group Analysis of The BLG Model . . . . . . . . . . 62
3.3.1 Action of the Model Hamiltonian . . . . . . . . . . . . . . . . . 63
3.3.2 Scaling Properties and Effective Action at the Tree Level . . . . 64
3.4 Susceptibilities and Possible Ground States . . . . . . . . . . . . . . . 70
3.4.1 Case I:g
0
=g
1
=g
2
= 0 . . . . . . . . . . . . . . . . . . . . . 70
3.4.2 Case II:g
0
,g
1
,g
2
6= 0 . . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Mean Field Analysis of the Ground States . . . . . . . . . . . . . . . . 77
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Chapter 4: Iron pnictide and electronic nematicity 84
4.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Instabilities of the isotropic phase . . . . . . . . . . . . . . . . . . . . . 88
4.3.1 Hubbard-Stratonovich Transformation . . . . . . . . . . . . . . 89
4.3.2 Isotropic metallic phase . . . . . . . . . . . . . . . . . . . . . . 91
4.3.3 Instabilities due to magnetic fluctuations . . . . . . . . . . . . . 93
4.4 Nematic ordered phase . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.1 Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.2 Nematic ordering phase transition . . . . . . . . . . . . . . . . 97
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.5.1 The control parameter . . . . . . . . . . . . . . . . . . . . . . . 99
4.5.2 Comparison with other work . . . . . . . . . . . . . . . . . . . 99
4.5.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 5: Conclusion 102
Bibliography 103
v
List of Tables
3.1 Bare coupling parameters that marginal at tree level. . . . . . . . . . . . 66
3.2 Pairing susceptibilities corresponding to the competing ground states
in the presence of interactions. The corresponding ordered states are
anti-ferromagntism (AFM), excitonic insulator (EI), spin-density-
wave (SDW ), charge-density-wave (CDW ), s-wave superconduc-
tivity (SC), and and extended s-wave superconductivity (SC
0
). The
prime inAFM
0
andEI
0
indicates the order state with valley symme-
try breaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
vi
List of Figures
1.1 (a) 2D-Topological insulator is indicated as the yellow shaded region.
The blue (green) arrowed line indicate the right (left) propagating edge
states. Note that the left and right propagating states must has opposite
spin. (b) energy spectrum in the of the topological insulator which has
projected all the crystal momentumk
y
states ontok
x
. Note that, there
is a gapless edge state in the spectrum.[35] . . . . . . . . . . . . . . . . 2
1.2 The left panel shows the schematic honeycomb lattice structure of
graphene. The blue and yellow dot represent the carbon atom locate at
sublattice A and B respectively, where
1
=
2
=
3
represent lattice
constants, anda
1
anda
2
represent the Bravais lattice primitive vectors.
The right panel shows the reciprocal (momentum) space of the Bravais
lattice. The region inside the hexagon is the first Brillouin zone. b
1
andb
2
represent the reciprocal lattice primitive vectors.K andK
0
are
the Fermi momenta. [13] . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Energy dispersion of the two lowest energy bands of single layer graphene.
Furthermore, the dispersion is linear in the vicinity of the Fermi momenta
K andK
0
shown in Fig. 1.2. [13] . . . . . . . . . . . . . . . . . . . . . 6
1.4 (a) The family of the iron-based superconductor lattice structure, where
the shaded region indicate the FeAs layer. (b) The upper panel show
the FeAs layer from the side view, with iron atoms show in red and
As (or Se) show in gold. The lower panel show the top view of FeAs
layer, and the black dashed square indicate the primitive unit cell of
the layer. The blue square is the primitive unit cell when the As atoms
are ignored. The arrows on the iron atoms show the spin configuration
in the magnetic phase. [75] . . . . . . . . . . . . . . . . . . . . . . . . 8
vii
1.5 In (a), The energy dispersion of the effective five-band model. The
Fe ion active orbital are indicated in (b), whered
xz
d
yz
are a linear
combination ofd
xz
andd
xy
orbitals (strongly hybridized). In addition,
the color line indicates the main orbital component in the band, and
the gray line in (a) is the result from first principle calculation [12]. In
(b), which show the Fermi surface of the parent compound (x = 0),
where
1
,
2
are the hole pockets, and
1
,
2
are the electron pockets.
[34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 The left and the right is the phase diagram of CeFeAsO
1x
F
x
[103]
and Ba(Fe
1x
Co
x
)
2
As
2
[67] respectively. AFM is the antiferromag-
netic phase, SC is the superconducting phase. In the region between
the normal metal phase and AFM exhibit a 90

rotational symme-
try breaking. In some of the iron-based superconductors, the super-
conductivity and magnetic order can coexist. For instance, the green
region in the left diagram indicates the coexistence. [39] . . . . . . . . 9
2.1 The cartoon show the elementary excitations of Fermi liquid as the
oscillation of Fermi surface. . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 The parameters in (2.26) form an N-Dimensional parameters space.
The dashed line represent the trajectory of the parameters after each
RG transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Bilayer graphene with AB-stacking: a
1
,b
1
are the two sublattice sites
in the upper layer, a
2
, b
2
are the two sublattice sites in the lower
layer.
0
is the tight-binding hopping constant between a
1
; b
1
,
1
is
the hopping betweena
1
anda
2
;
3
is the hopping betweenb
1
andb
2
.
a
1
=
a
2
(3;
p
3) and a
2
=
a
2
(3;
p
3) are the primitive lattice vec-
tors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 (a)K =
2
3a
(1;
1
p
3
) andK
0
=
2
3a
(1;
1
p
3
) are the two points, constitut-
ing the Fermi surface of the non-interacting system. is the momen-
tum cutoff of the theory,d is a thin shell contain high energy modes
to be integrated out. (b)
c
(K) and
f
(K) are the dispersion energy of
the conduction and the valence band respectively. The other two bands
are gapped by
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
viii
3.3 Feynman Diagrams: 1; 2; 3; 4 represent the low energy modes with
momentum K
1;2;3;4
, band index I, valley index , and spin . The
momentum inside the loop,K, must lie within the shelld, andQ =
K
3
K
1
, Q
0
= K
4
K
1
, P = K
1
+K
2
. Note that the interaction
lines are suppressed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Flow of the susceptibilities: Here, we seth
0
= u
0
= v
0
,h
1
= u
1
=
v
1
= 0:8h
0
,h
2
= u
2
= v
2
= 0:1h
0
, andg
0
= g
1
= g
2
= 0 (h
0
> 0)
. TheFM instabilities occupy a large region in parameter space, and
fine-tuning is not necessary. . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Phase Diagram for a representative choice of bare couplings h
0
=
u
0
=v
0
= 1,h
1
=v
1
=u
1
= 0:8h
0
, andh
2
=u
2
=v
2
=g
2
= 0:1h
0
.
The phase diagram is determined by monitoring which channels diver-
gence first during the RG flow. . . . . . . . . . . . . . . . . . . . . . . 76
3.6 The flow ofg
0
as the bare value ofg
1
varies. We setg
0
=g
2
= 0:1h
0
,
and the remaining bare couplings are same as Fig. 3.4 . . . . . . . . . . 77
4.1 A schematic diagram showing the FS structure of a typical FeAS in
the unfolded Brillioun Zone: The bold curves represent the FS of the
model in the isotropic metallic phase. the dotted curves represent the
unequally renormalized FS pockets in the nematic phase withh

i>
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 The nematic order phase transition occurs at some finite temperature
where

S
nem
=S
nem
(2)
2
=(m
2
0
). . . . . . . . . . . . . . . . . . . . . . 98
ix
Abstract
The recent dominant trends in condensed matter physics research can be roughly summa-
rized into three newly discovered materials: topological insulators, graphene, and iron-
based superconductors. All these materials exhibit many intriguing properties which are
fundamentally related to their electronic band structure. Therefore, this lead to many
intense investigations on multi-band electronic system to explore new physics.
The physics of multi-band electronic structure is fascinating in several aspects. With-
out many-body effects, because of the gauge freedom of Bloch states, topological insula-
tors can give rise a robust metallic behavior at its boundaries. In graphene, the touching
between conduction and valence band at Fermi level yields a new criticality class which
exhibit many unconventional electronic properties, especially its quasi-relativistic behav-
ior. Turning to the many-body effects, for instance, the iron-based superconductors can
sustain an superconducting ground state despite of no attractive interactions in the sys-
tem. Therefore, a deeper understanding for the conventional notions in condensed matter
physics has put forward by many of these experimental observations.
In this thesis, the many-body effects in multi-band systems are the main focus, espe-
cially the study of graphene and iron-based superconductors which can be compared to
experiments. These theoretical studies intend to understand how the underlying electronic
x
bands degree of freedom can give rise to Fermi-liquid instabilities, and how these effects
can be related to intriguing physical properties.
We first study the electrons correlation effects in bilayer graphene by a renormalization
group technique. In this study, we build a microscopic model of bilayer graphene from a
tight-binding approach. In our finding, the peculiar Fermi surface configuration leads to
critical behavior which is beyond the Fermi-liquid paradigm. Furthermore, due to the
electron-electron interactions between different bands, excitonic instabilities are found in
many different scattering channels. This analysis suggest a collection of competing orders
in the system ground states. This result is consistent with the experimental observation
that bilayer graphene is an insulator.
Next, we study nematic order in the metallic phase of iron pnictides. In contrast to
graphene, the density of states is finite at the Fermi surface. By careful investigating
the scattering processes near these Fermi surface, and then identifying the most relevant
collective modes from these processes, we find that a Pomeranchuk instability can be
driven by magnetic fluctuations. This instability eventually leads to the break down of
the isotropic metallic phase which electronic system exhibit broken crystalline rotational
symmetry but preserve translation symmetry. As the experiment suggests, this can be a
candidate for nematic order in the metallic phase.
xi
Chapter 1
Introduction
One of the most commonly found phases of matter at low temperatures is solid. More-
over, many of the solids are favorable to form crystalline structure on its ions. Therefore,
in condensed matter physics, how the electrons behave under the effect of this lattice
translational symmetry is the very first step to understand the physics of solids. For sure,
electron quantum states with a periodic potential are best described by Bloch’s theorem.
From this theorem, the quantum numbers of the electrons are completely determined, and
introduces the concept of bands. This concept is first successfully applied to metals, and
explain many of the basic properties in metal which are not captured by the classical pic-
ture (Drude model)[4]. Despite the concept of bands has been known for a long time,
until today, there are still many new experimental results which show surprises when band
degrees of freedoms are important in the material. In the following sections, we will give
a brief overview of these new materials.
1.1 New materials
1.1.1 Topological insulator
To answer why a material is an insulator may not be a trivial task. Although there are many
different mechanism responsible for insulating behavior, one of the best known class of
insulators are band insulator. In the band insulator, all the bands are fully filled or empty,
1
but none of them is partially filled. Therefore, this implies there that exists a minimum
energy gap for exciting a electron. In order to conduct current, the applying electric field
must be large enough to overcome the energy gap such that the electrons can be excited
to break the ground state time reversal symmetry. However, there is one missing piece
from this conventional simple idea, until the recent discovery of topological insulators
(TI). [41, 42]
By carefully examining the solution space of band electron, the conventional band
insulators can actually be generalized to the so call topological band insulator. One of the
remarkable properties of these insulators is the existence of robust metallic edges states,
but with a finite energy gap in the bulk just like conventional band insulator. In addition
the spin of this edges states is ‘locked’ to their propagating direction (see Fig. 1.1). This
intriguing property is the consequence from the topology configurations of Bloch’s wave
functional space.
Theabovediscussionwaspredicatedontheconserva-
tion of spin S
z
. This is not a fundamental symmetry,
though, and spin nonconserving processes—present in
any real system—invalidate the meaning of !
xy
s
. This
brings into question theories that relied on spin conser-
vation to predict an integer quantized !
xy
s
!Volovik and
Yakovenko, 1989; Bernevig and Zhang, 2006; Qi, Wu,
and Zhang, 2006", as well as the influential theory of the
!nonquantized"spinHallinsulator !Murakami,Nagaosa,
and Zhang, 2004". Kane and Mele !2005a" showed that
due to T symmetry the edge states in the quantum spin
Hall insulator are robust even when spin conservation is
violated because their crossing at k=0 is protected by
the Kramers degeneracy discussed in Sec. II.C. This es-
tablished the quantum spin Hall insulator as a topologi-
cal phase.
ThequantumspinHalledgestateshavetheimportant
“spin filtered” property that the up spins propagate in
one direction, while the down spins propagate in the
other. Such edge states were later termed “helical” !Wu,
Bernevig, and Zhang, 2006", in analogy with the corre-
lation between spin and momentum of a particle known
as helicity. They form a unique 1D conductor that is
essentially half of an ordinary 1D conductor. Ordinary
conductors, which have up and down spins propagating
in both directions, are fragile because the electronic
states are susceptible to Anderson localization in the
presence of weak disorder !Anderson, 1958; Lee and
Ramakrishnan, 1985". By contrast, the quantum spin
Hall edge states cannot be localized even for strong dis-
order. To see this, imagine an edge that is disordered in
a finite region and perfectly clean outside that region.
The exact eigenstates can be determined by solving the
scattering problem relating incoming waves to those re-
flected from and transmitted through the disordered re-
gion. Kane and Mele !2005a" showed that the reflection
amplitude is odd under T—roughly because it involves
flipping the spin. It follows that unless T symmetry is
broken, an incident electron is transmitted perfectly
across the disordered region. Thus, eigenstates at any
energy are extended, and at temperature T=0 the edge
state transport is ballistic. For T"0 inelastic back-
scattering processes are allowed, which will, in general,
lead to a finite conductivity.
The edge states are similarly protected from the ef-
fects of weak electron interactions, though for strong
interactions the Luttinger liquid effects lead to a mag-
neticinstability !Wu,Bernevig,andZhang,2006;Xuand
Moore,2006".Thisstronginteractingphaseisinteresting
because it will exhibit charge e/2 quasiparticles similar
to solitons in the model of Su, Schrieffer, and Heeger
!1979". For sufficiently strong interactions similar frac-
tionalization could be observed by measuring shot noise
in the presence of magnetic impurities !Maciejko et al.,
2009" or at a quantum point contact !Teo and Kane,
2009".
B. HgTeÕCdTe quantum well structures
Graphene is made out of carbon—a light element
with a weak spin-orbit interaction. Though there is dis-
agreement on its absolute magnitude !Huertas-
Hernando, Guinea, and Brataas, 2006; Min et al., 2006;
Boettger and Trickey, 2007; Yao et al., 2007; Gmitra et
al., 2009", the energy gap in graphene is likely to be
small. Clearly, a better place to look for this physics
wouldbeinmaterialswithstrongspin-orbitinteractions,
made from heavy elements near the bottom of the Peri-
odic Table. To this end, Bernevig, Hughes, and Zhang
!2006" !BHZ" had the idea to consider quantum well
structuresofHgCdTe.Thispavedthewaytotheexperi-
mental discovery of the quantum spin Hall insulator
phase.
Hg
1−x
Cd
x
Te is a family of semiconductors with strong
spin-orbit interactions !Dornhaus and Nimtz, 1983";
CdTe has a band structure similar to other semiconduc-
tors. The conduction-band-edge states have an s-like
symmetry, while the valence-band-edge states have a
p-like symmetry. In HgTe, the p levels rise above the s
levels, leading to an inverted band structure. BHZ con-
sidered a quantum well structure where HgTe is sand-
wiched between layers of CdTe. When the thickness of
theHgTelayeris d#d
c
=6.3 nmthe2Delectronicstates
bound to the quantum well have the normal band order.
For d"d
c
, however, the 2D bands invert. BHZ showed
that the inversion of the bands as a function of increas-
ing d signals a quantum phase transition between the
trivial insulator and the quantum spin Hall insulator.
Thiscanbeunderstoodsimplyintheapproximationthat
thesystemhasinversionsymmetry.Inthiscase,sincethe
s and p states have opposite parity the bands will cross
each other at d
c
without an avoided crossing. Thus, the
energy gap at d=d
c
vanishes. From Eq. !12", the change
in the parity of the valence-band-edge state signals a
phase transition in which the Z
2
invariant$ changes.
Within a year of the theoretical proposal the
Würzburg group, led by Laurens Molenkamp, made the
devices and performed transport experiments that
showed the first signature of the quantum spin Hall in-
sulator. König et al. !2007" measured the electrical con-
ductance due to the edge states. The low-temperature
ballistic edge state transport can be understood within a
simple Landauer-Büttiker !Büttiker, 1988" framework in
E
E
F
Conduction Band
Valence Band
Quantum spin
Hall insulator
ν=1
Conventional
Insulator
ν=0
(a) (b)
k 0 /a −π /a −π
FIG. 5. !Color online" Edge states in the quantum spin Hall
insulator !QSHI". !a" The interface between a QSHI and an
ordinary insulator. !b" The edge state dispersion in the
graphene model in which up and down spins propagate in op-
posite directions.
3053
M. Z. Hasan and C. L. Kane: Colloquium: Topological insulators
Rev. Mod. Phys., Vol. 82, No. 4, October–December 2010
Figure 1.1: (a) 2D-Topological insulator is indicated as the yellow shaded region. The
blue (green) arrowed line indicate the right (left) propagating edge states. Note that the
left and right propagating states must has opposite spin. (b) energy spectrum in the of the
topological insulator which has projected all the crystal momentumk
y
states ontok
x
. Note
that, there is a gapless edge state in the spectrum.[35]
2
To understand this topological nature, one may need to note that, a Bloch’s wave solu-
tion for a given quantum numbers (crystal momentum and band index) which satisfies the
Bloch wave equation is only unique up to a phase. Namely, given that the eigenstate of a
Bloch Hamiltonian isj
nk
i (Bloch wave) wheren is the band index, andk is the crystal
momentum in the Brillouin zone. The gauge transformation is defined by
j
0
nk
i =e
i
nk
j
nk
i; (1.1)
where
nk
is gauge parameter. Bothj
nk
i andj
0
nk
i are the solution and which cannot be
distingusihed locally in k-space. Hence, this is the gauge symmetry of the Bloch’s wave
functions. Now suppose, we have found one Bloch’s wave solution for each momentum
and bands which is a smooth function over the entire Brillouin zone. Then, one may
think that all the other possible smooth solutions can be obtained by performing a gauge
transformations by a single gauge parameter which are smoothly defined in the entire
Brillouin zone. However, in general, this is not the case.
In fact, there exist other smooth solutions, and their smooth gauge transformations are
composed by multiple gauge parameters. Each of these parameters can only be smoothly
defined on certain patches of the Brillouin zone (the patches must have some overlapping
region with others), but impossible to be defined smoothly in the entire Brillouin zone.
Therefore, the smooth solutions over the Brillouin zone break into different topological
classes, since they cannot be connected to each other by any smooth gauge transforma-
tion. In mathematics, this is basically a topology problem to find the possible distinct
continuous mapping from a topological space (Brillouin zone, d-dimensional torus) to
3
other topological space (N-dimensional complex vector space, since there areN Bloch’s
wave functions forN valance bands) under the homotopic theory. [5, 65]
In the differential geometry point of view, this topology problem is well described
by the fiber bundle theory that classifies aN-dimensional complex vector bundle over a
d-dimensional torus. [42] In the theory, each different topological class can be assigned
by a different topological invariant – Chern number. In terms of physics, Chern invari-
ant is equivalent to the TKNN invariant [90] that arises from Berry’s phase, and which is
ultimately related to quantum Hall physics. For TIs, its Bloch’s wave solution is topolog-
ical distinct from the conventional band insulator, and the TI Chern number is nonzero.
Therefore, this implies that the system Hall conductance is not zero, and guarantees the
existence of gapless edge states.
Nevertheless, this non-trivial topological property is tightly related to bands effects,
and the number of relevant bands in the material must be at least greater than one.
Although the concept of TI was first realized in a theoretical context, it has been con-
firmed by experiments for some materials with strong spin-orbital coupling.[3, 35, 78] TI
is currently one of the fast moving-pace research. In particular, the extension of this idea
to superconductors is very interesting in the theoretical and applications aspect. We will
not go further in the discussion of TI, since it is not the main subject of this thesis.
1.1.2 Graphene
Graphene is a quasi-2D carbon material with a honeycomb lattice (see Fig. 1.2). This is
the first real 2D lattices can be produced in laboratory. Because of the structural simplicity,
this provides an enormous freedom to engineer the graphene lattice structure. Furthermore,
4
this also makes direct manipulation of its electrons feasible. Therefore, this feature is very
attractive for many applications and experimental interests.
trino” billiards !Berry and Modragon, 1987; Miao et al.,
2007". It has also been suggested that Coulomb interac-
tions are considerably enhanced in smaller geometries,
such as graphene quantum dots !Milton Pereira et al.,
2007", leading to unusual Coulomb blockade effects
!Geim and Novoselov, 2007" and perhaps to magnetic
phenomena such as the Kondo effect. The transport
properties of graphene allow for their use in a plethora
of applications ranging from single molecule detection
!Schedin et al., 2007; Wehling et al., 2008" to spin injec-
tion !Cho et al., 2007; Hill et al., 2007; Ohishi et al., 2007;
Tombros et al., 2007".
Because of its unusual structural and electronic flex-
ibility, graphene can be tailored chemically and/or struc-
turally in many different ways: deposition of metal at-
oms !Calandra and Mauri, 2007; Uchoa et al., 2008" or
molecules !Schedin et al., 2007; Leenaerts et al., 2008;
Wehling et al., 2008" on top; intercalation #as done in
graphite intercalated compounds !Dresselhaus et al.,
1983; Tanuma and Kamimura, 1985; Dresselhaus and
Dresselhaus, 2002"$; incorporation of nitrogen and/or
boron in its structure !Martins et al., 2007; Peres,
Klironomos, Tsai, et al., 2007"#in analogy with what has
beendoneinnanotubes !Stephan et al.,1994"$;andusing
different substrates that modify the electronic structure
!Calizo et al., 2007; Giovannetti et al., 2007; Varchon et
al., 2007; Zhou et al., 2007; Das et al., 2008; Faugeras et
al., 2008". The control of graphene properties can be
extended in new directions allowing for the creation of
graphene-based systems with magnetic and supercon-
ducting properties !Uchoa and Castro Neto, 2007" that
are unique in their 2D properties. Although the
graphene field is still in its infancy, the scientific and
technological possibilities of this new material seem to
be unlimited. The understanding and control of this ma-
terial’s properties can open doors for a new frontier in
electronics. As the current status of the experiment and
potential applications have recently been reviewed
!Geim and Novoselov, 2007", in this paper we concen-
trate on the theory and more technical aspects of elec-
tronic properties with this exciting new material.
II. ELEMENTARY ELECTRONIC PROPERTIES OF
GRAPHENE
A. Single layer: Tight-binding approach
Graphene is made out of carbon atoms arranged in
hexagonal structure, as shown in Fig. 2. The structure
can be seen as a triangular lattice with a basis of two
atoms per unit cell. The lattice vectors can be written as
a
1
=
a
2
!3,
%
3", a
2
=
a
2
!3,−
%
3", !1"
where a&1.42 Å is the carbon-carbon distance. The
reciprocal-lattice vectors are given by
b
1
=
2!
3a
!1,
%
3", b
2
=
2!
3a
!1,−
%
3". !2"
Ofparticularimportanceforthephysicsofgrapheneare
the two points K and K
!
at the corners of the graphene
Brillouin zone !BZ". These are named Dirac points for
reasons that will become clear later. Their positions in
momentum space are given by
K=
'
2!
3a
,
2!
3
%
3a
(
, K!=
'
2!
3a
,−
2!
3
%
3a
(
. !3"
The three nearest-neighbor vectors in real space are
given by
!
1
=
a
2
!1,
%
3" !
2
=
a
2
!1,−
%
3" "
3
=−a!1,0"!4"
while the six second-nearest neighbors are located at
"
1
!=±a
1
, "
2
!=±a
2
, "
3
!=±!a
2
−a
1
".
The tight-binding Hamiltonian for electrons in
graphene considering that electrons can hop to both
nearest- and next-nearest-neighbor atoms has the form
!we use units such that #=1"
H=−t
)
*i,j+,$
!a
$,i
†
b
$,j
+H.c."
−t!
)
**i,j++,$
!a
$,i
†
a
$,j
+b
$,i
†
b
$,j
+H.c.", !5"
where a
i,$
!a
i,$
†
" annihilates !creates" an electron with
spin $ !$=↑,↓ " on site R
i
on sublattice A !an equiva-
lent definition is used for sublattice B", t!&2.8 eV" is the
nearest-neighbor hopping energy !hopping between dif-
ferent sublattices", and t
!
is the next nearest-neighbor
hopping energy
1
!hopping in the same sublattice". The
energy bands derived from this Hamiltonian have the
form !Wallace, 1947"
E
±
!k"=±t
%
3+f!k"−t!f!k",
1
The value of t
!
is not well known but ab initio calculations
!Reich et al., 2002" find 0.02t%t
!
%0.2t depending on the tight-
binding parametrization. These calculations also include the
effect of a third-nearest-neighbors hopping, which has a value
of around 0.07 eV. A tight-binding fit to cyclotron resonance
experiments !Deacon et al., 2007" finds t
!
&0.1 eV.
a
a
1
2
b
b
1
2
K
Γ
k
k
x
y
1
2
3
M
δ
δ
δ
AB
K’
FIG. 2. !Color online" Honeycomb lattice and its Brillouin
zone. Left: lattice structure of graphene, made out of two in-
terpenetrating triangular lattices !a
1
anda
2
are the lattice unit
vectors, and "
i
, i=1,2,3 are the nearest-neighbor vectors".
Right: corresponding Brillouin zone. The Dirac cones are lo-
cated at the K and K
!
points.
112
Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009
Figure 1.2: The left panel shows the schematic honeycomb lattice structure of graphene.
The blue and yellow dot represent the carbon atom locate at sublattice A and B respec-
tively, where
1
=
2
=
3
represent lattice constants, anda
1
anda
2
represent the Bravais
lattice primitive vectors. The right panel shows the reciprocal (momentum) space of the
Bravais lattice. The region inside the hexagon is the first Brillouin zone. b
1
andb
2
repre-
sent the reciprocal lattice primitive vectors.K andK
0
are the Fermi momenta. [13]
Since for each sublattice (A or B), the carbon atomp
z
-orbital provides one electron, the
electronic structure of graphene is effectively depicted by two-bands model. Its valence
band is fully filled by electrons, and the conduction band is totally empty. Just right at
the Fermi level, these two bands become degenerate and touch each other at the high
symmetry points in the Brillouin zone. Therefore, graphene is a semi-metal rather than
an insulator. In addition, in the vicinity of the Fermi points, the dispersion relation of the
band can be linearized (Fig. 1.3) i.e. in the continuum limit, the effective Hamiltonian of
graphene is exactly a 2D massless Dirac equation, giving rise quasi-relativistic behavior of
the electrons. These special properties lead to many intriguing phenomena in integer and
5
fractional quantum Hall effects, anti-localization, no screening effect on charge carriers,
[13], Kondo physics [32], etc.
f!k"=2cos!
#
3k
y
a"+4cos$
#
3
2
k
y
a%cos
$
3
2
k
x
a
%
, !6"
where the plus sign applies to the upper !!
*
" and the
minus sign the lower !!" band. It is clear from Eq. !6"
that the spectrum is symmetric around zero energy if t
!
=0. For finite values of t
!
, the electron-hole symmetry is
broken and the ! and !
*
bands become asymmetric. In
Fig. 3, we show the full band structure of graphene with
both t and t
!
.Inthesamefigure,wealsoshowazoomin
ofthebandstructureclosetooneoftheDiracpoints !at
the K or K
!
point in the BZ". This dispersion can be
obtained by expanding the full band structure, Eq. !6",
close to the K !or K
!
" vector, Eq. !3", as k=K+q, with
&q&"&K&!Wallace, 1947",
E
±
!q"' ±v
F
&q&+O(!q/K"
2
), !7"
where q is the momentum measured relatively to the
Dirac points and v
F
is the Fermi velocity, given by v
F
=3ta/2, with a value v
F
*1#10
6
m/s. This result was
first obtained by Wallace !1947".
The most striking difference between this result and
the usual case, $!q"=q
2
/!2m", where m is the electron
mass, is that the Fermi velocity in Eq. !7" does not de-
pend on the energy or momentum: in the usual case we
have v=k/m=
#
2E/m and hence the velocity changes
substantiallywithenergy.Theexpansionofthespectrum
around the Dirac point including t
!
up to second order
in q/K is given by
E
±
!q"*3t!±v
F
&q&−
$
9t!a
2
4
±
3ta
2
8
sin!3%
q
"
%
&q&
2
, !8"
where
%
q
=arctan
$
q
x
q
y
%
!9"
is the angle in momentum space. Hence, the presence of
t
!
shifts in energy the position of the Dirac point and
breaks electron-hole symmetry. Note that up to order
!q/K"
2
the dispersion depends on the direction in mo-
mentum space and has a threefold symmetry. This is the
so-called trigonal warping of the electronic spectrum
!Ando et al., 1998, Dresselhaus and Dresselhaus, 2002".
1. Cyclotron mass
The energy dispersion !7" resembles the energy of ul-
trarelativistic particles; these particles are quantum me-
chanically described by the massless Dirac equation !see
Sec. II.B for more on this analogy". An immediate con-
sequence of this massless Dirac-like dispersion is a cy-
clotronmassthatdependsontheelectronicdensityasits
square root !Novoselov, Geim, Morozov, et al., 2005;
Zhangetal.,2005".Thecyclotronmassisdefined,within
the semiclassical approximation !Ashcroft and Mermin,
1976", as
m
*
=
1
2!
+
!A!E"
!E
,
E=E
F
, !10"
with A!E" the area in k space enclosed by the orbit and
given by
A!E"=!q!E"
2
=!
E
2
v
F
2
. !11"
Using Eq. !11" in Eq. !10", one obtains
m
*
=
E
F
v
F
2
=
k
F
v
F
. !12"
The electronic density n is related to the Fermi momen-
tum k
F
as k
F
2
/!=n !with contributions from the two
DiracpointsKandK
!
andspinincluded",whichleadsto
m
*
=
#
!
v
F
#
n. !13"
Fitting Eq. !13" to the experimental data !see Fig. 4"
provides an estimation for the Fermi velocity and the
FIG. 3. !Color online" Electronic dispersion in the honeycomb
lattice. Left: energy spectrum !in units of t" for finite values of
t and t
!
, with t=2.7 eV and t
!
=−0.2t. Right: zoom in of the
energy bands close to one of the Dirac points.
FIG. 4. !Color online" Cyclotron mass of charge carriers in
graphene as a function of their concentration n. Positive and
negative n correspond to electrons and holes, respectively.
Symbols are the experimental data extracted from the tem-
perature dependence of the SdH oscillations; solid curves are
the best fit by Eq. !13". m
0
is the free-electron mass. Adapted
from Novoselov, Geim, Morozov, et al., 2005.
113
Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009
Figure 1.3: Energy dispersion of the two lowest energy bands of single layer graphene.
Furthermore, the dispersion is linear in the vicinity of the Fermi momenta K and K
0
shown in Fig. 1.2. [13]
Moreover, in contrast to a normal metal, graphene has zero density of states at the
Fermi level. Due to these peculiar electronic properties, the critical behavior of graphene
is different from normal metals [82] and does not belong to the universality class of
Fermi-liquid in the theory of critical phenomea. Furthermore, the criticality properties
of graphene can be engineered by stacking up several layers of graphene. In pristine
single layer graphene, by the renormalization group argument, the interactions between
electrons are not important in determining the ground state properties. However, in few
layer graphene, the dispersion relation at the Fermi point will eventually go beyond lin-
ear. Therefore, the scaling properties is fundamentally different from the single layer case
6
and many-body effects are no longer negligible. Stacking up graphene to manipulate its
criticality, this can lead many new physical phenomena.
1.1.3 Iron-based superconductors
The physics of the strongly correlated cuprates is one of the big puzzle in condensed matter
physics. In the research of cuprates, many debates among the theories and experiments still
remain unresolved today. Nevertheless, an interesting superconducting state in doped iron
pnictides (FeAs) which has been discovered recently adds one more new member to the
family of unconventional superconductors. Traditionally, it may not be very intuitive to
search for superconductivity in iron compounds, because ferromagnetic properties tend
to break the formation of Cooper pair in the spin singlet channel. On contrary, magnetic
fluctuations are believed to played a central role for the pairing mechanism in iron-based
superconductors.
The Iron-based superconductor is also a layered material. The electronic properties are
dominated by the electrons from the FeAs layer (Fig. 1.4a). Furthermore, the supercon-
ducting condensate is formed in this layer as well. In each FeAs layer, the iron atoms are
arranged into a simple 2D square lattice, and the arsenide atoms are located at the center
of these unit square (blue line which is shown in Fig. 1.4b) but outside the iron plane.
Despite the fact that iron-based superconductors and cuprates share several similarities,
their electronic structures are fundamentally different. In contrast to cuprates, iron-based
superconductors are effectively depicted by a multi-band system rather than just one band.
The current widely accepted effective model is the five-band model, and only the Fe ions
orbitals play the principle role in the model. This model has been checked consistently
with first principle calculations (Fig. 1.5).
7
NATUREPHYSICS DOI:10.1038/NPHYS1759
REVIEWARTICLE
Box1 |Theiron-basedsuperconductorfamily.
Iron, one of the most common metals on earth, has been known
as a useful element since the aptly named Iron Age. However,
it was not until recently that, when combined with elements
from the group 15 and 16 columns of the periodic table (named,
respectively, the pnictogens, after the Greek verb for choking,
and chalcogens, meaning ‘ore formers’), iron-based metals were
shown to readily harbour a new form of high-temperature su-
perconductivity. This general family of materials has quickly
grown to be large in size, with well over 50 different compounds
identified that show a superconducting transition that occurs
at temperatures approaching 60K, and includes a plethora of
differentvariationsofiron-andnickel-basedsystems.Sofar,five
unique crystallographic structures have been shown to support
superconductivity. As shown in Fig. B1a, these structures all
possess tetragonal symmetry at room temperature and range
from the simplest↵ -PbO-type binary element structure to more
complicated quinternary structures composed of elements that
spantheentireperiodictable.
Thekeyingredientisaquasi-two-dimensionallayerconsisting
of a square lattice of iron atoms with tetrahedrally coordinated
bondstoeitherphosphorus,arsenic,seleniumortelluriumanions
that are staggered above and below the iron lattice to form a
chequerboard pattern that doubles the unit-cell size, as shown
in Fig. B1b. These slabs are either simply stacked together, as in
FeSe, or are separated by spacer layers using alkali (for example,
Li), alkaline-earth (for example, Ba), rare-earth oxide/fluoride
(for example, LaO or SrF) or more complicated perovskite-type
combinations (for example, Sr
3
Sc
2
O
5
). These so-called blocking
layers provide a quasi-two-dimensional character to the crystal
becausetheyformatomicbondsofmoreioniccharacterwiththe
FeAs layer, whereas the FeAs-type layer itself is held together by
a combination of covalent (that is, Fe–As) and metallic (that is,
Fe–Fe)bonding.
Intheiron-basedmaterials,thecommonFeAsbuildingblockis
considered a critical component to stabilizing superconductivity.
Because of the combination of strong bonding between Fe–Fe
and Fe–As sites (and even interlayer As–As in the 122-type
systems),thegeometryoftheFeAs
4
tetrahedraplaysacrucialrole
in determining the electronic and magnetic properties of these
systems. For instance, the two As–Fe–As tetrahedral bond angles
seem to play a crucial role in optimizing the superconducting
transition temperature (see the main text), with the highest T
c
valuesfoundonlywhenthisgeometryisclosesttotheidealvalue
of⇠ 109.47
.
Long-range magnetic order also shares a similar pattern
in all of the FeAs-based superconducting systems. As shown
in the projection of the square lattice in Fig. B1b, the iron
sublattice undergoes magnetic ordering with an arrangement
consisting of spins ferromagnetically arranged along one
chain of nearest neighbours within the iron lattice plane,
and antiferromagnetically arranged along the other direc-
tion. This is shown on a tetragonal lattice in the figure,
but actually only occurs after these systems undergo an
orthorhombic distortion as explained in the main text. In
the orthorhombic state, the distance between iron atoms with
ferromagnetically aligned nearest-neighbour spins (highlighted
in Fig. B1b) shortens byapproximately 1% as compared withthe
perpendiculardirection.
FeSe
LiFeAs
SrFe
2
As
2
Sr
3
Sc
2
O
5
Fe
2
As
2
LaFeAsO/
SrFeAsF
a b
FigureB1 |Crystallographicandmagneticstructuresoftheiron-basedsuperconductors.a,Thefivetetragonalstructuresknowntosupport
superconductivity.b,Theactiveplanarironlayercommontoallsuperconductingcompounds,withironionsshowninredandpnictogen/chalcogen
anionsshowningold.ThedashedlineindicatesthesizeoftheunitcelloftheFeAs-typeslab,whichincludestwoironatomsowingtothestaggered
anionpositions,andtheorderedspinarrangementforFeAs-basedmaterialsisindicatedbyarrows(thatis,notshownforFeTe).
of structural parameters, disorder location, chemical bonding and
density. This is one of the key properties that has led to a
rapid but in-depth understanding of these materials. In due time,
controlledexperimentalcomparisons— forinstanceofHalleffect
(carrierdensity)underpressureversusdoping,ofdifferentchemical
substitutionseriesandfurtherunderstandingofthelocalnatureof
chemical substitution — will help pinpoint the important tuning
parametersforthesesystems.
NATUREPHYSICS | VOL 6 | SEPTEMBER 2010 | www.nature.com/naturephysics 647
Figure 1.4: (a) The family of the iron-based superconductor lattice structure, where the
shaded region indicate the FeAs layer. (b) The upper panel show the FeAs layer from the
side view, with iron atoms show in red and As (or Se) show in gold. The lower panel show
the top view of FeAs layer, and the black dashed square indicate the primitive unit cell of
the layer. The blue square is the primitive unit cell when the As atoms are ignored. The
arrows on the iron atoms show the spin configuration in the magnetic phase. [75]
7
Γ Γ X M
–2
–1
0
1
2
3
0
k /
x
a
0
k
y
a
α α
12
β
1
β
2
d
xz
d
yz
d
xz
–d
yz
d
x2–y2
d
xy
d
3z2–r2
(a)
(b)
/
Figure 5. (a) The backfolded band structure for the five-band model with 0,
X and M denoting the symmetry points in the real BZ corresponding to the two
Fe unit cell. The main orbital contributions are shown by the following colors:
d
xz
(red),d
yz
(green),d
xy
(yellow),d
x
2
y
2 (blue),d
3z
2
r
2 (magenta)andastrongly
hybridized d
xz
d
yz
band (brown). The gray lines show the correct DFT band
structure calculated by Cao et al. (b) The FS sheets of the five-band model for
the undoped compound (x=0).
The Hamiltonian for the five band model takes the following form:
H
0
=
X
k X
mn
(⇠ mn
(k)+✏
m
mn
)d
†
m (k)d
n (k). (6)
Here d
†
m, (k) creates a particle with momentum k and spin in the orbital m. The kinetic
energy terms ⇠ mn
(k) together with the parameters for a five-band tight-binding fit of the DFT
band structure by Cao et al are listed in the appendix. A diagonalization of this Hamiltonian
yields the eigenenergies and the matrix elements analogously to the two-band case discussed
above.
In figure 5(a), we have plotted the resulting band structure in the backfolded ‘small’ BZ
whereas the FS sheets for zero doping are shown in figure 5(b). The colors correspond to the
dominant orbital weight of each band in momentum space. The gray lines represent the DFT
New Journal of Physics 11 (2009) 025016 (http://www.njp.org/)
7
Γ Γ X M
–2
–1
0
1
2
3
0
k /
x
a
0
k
y
a
α α
12
β
1
β
2
d
xz
d
yz
d
xz
–d
yz
d
x2–y2
d
xy
d
3z2–r2
(a)
(b)
/
Figure 5. (a) The backfolded band structure for the five-band model with 0,
X and M denoting the symmetry points in the real BZ corresponding to the two
Fe unit cell. The main orbital contributions are shown by the following colors:
d
xz
(red),d
yz
(green),d
xy
(yellow),d
x
2
y
2 (blue),d
3z
2
r
2 (magenta)andastrongly
hybridized d
xz
d
yz
band (brown). The gray lines show the correct DFT band
structure calculated by Cao et al. (b) The FS sheets of the five-band model for
the undoped compound (x=0).
The Hamiltonian for the five band model takes the following form:
H
0
=
X
k X
mn
(⇠ mn
(k)+✏
m
mn
)d
†
m (k)d
n (k). (6)
Here d
†
m, (k) creates a particle with momentum k and spin in the orbital m. The kinetic
energy terms ⇠ mn
(k) together with the parameters for a five-band tight-binding fit of the DFT
band structure by Cao et al are listed in the appendix. A diagonalization of this Hamiltonian
yields the eigenenergies and the matrix elements analogously to the two-band case discussed
above.
In figure 5(a), we have plotted the resulting band structure in the backfolded ‘small’ BZ
whereas the FS sheets for zero doping are shown in figure 5(b). The colors correspond to the
dominant orbital weight of each band in momentum space. The gray lines represent the DFT
New Journal of Physics 11 (2009) 025016 (http://www.njp.org/)
Figure 1.5: In (a), The energy dispersion of the effective five-band model. The Fe ion
active orbital are indicated in (b), where d
xz
d
yz
are a linear combination of d
xz
and
d
xy
orbitals (strongly hybridized). In addition, the color line indicates the main orbital
component in the band, and the gray line in (a) is the result from first principle calculation
[12]. In (b), which show the Fermi surface of the parent compound (x = 0), where
1
,
2
are the hole pockets, and
1
,
2
are the electron pockets. [34]
8
6
0
0 0.04 0.08 0.12 0.16 0.20
40
AFM
80
120
160
Temperature (K)
SC
x
CeFeAsO
1–x
F
x
T
N
(Fe)
T
N
(Ce)
T
c
T
s
(P4/nmm to Cmma)
0
0
0.4
0.8
0.04
x
T = 40 K
Fe moment
Moment (
B
/Fe) µ
d
FIG. 5: (Color online) Temperature T versus composition x phase diagrams of illustrative 1111-type electron-doped polycrys-
talline CeFeAsO1−xFx (Ref. 40) and 122-type electron-doped single crystalline Ba(Fe1−xCox)2As2.
46
The regions of the phase
diagrams are: paramagnetic tetragonal (high-temperature regions) forT>TS,Tc;orthorhombicstructuraldistortionbetween
temperaturesTS andTN(Fe), Ort; orthorhombicdistortion andlong-rangeantiferromagnetic orderingoccurringtogether(AFM,
Ort); and superconductivity (SC). The inset of the phase diagram for CeFeAsO1−xFx shows the low-temperature ordered Fe
moment μ versus x;the μ(x)dataforBa(Fe1−xCox)2As2 are very similar.
45
In both systems, superconductivity can coexist
with the orthorhombically distorted structure. In CeFeAsO1−xFx,SCandAFdonotcoexist. InBaFe2−xCoxAs2,SCandAF
coexist at low temperatures over the restricted compositionrange0.035
<
∼
x
<
∼
0.06; note the re-entrant behaviors on the right
side of this region (see Fig. 10 in Sec. IID and Fig. 78 in Sec. IVB6 below). Inboth systems, the optimum superconducting
transition temperatures are not reached until the long-range structural and magnetic transitions are both completely sup-
pressed. The highest Tc occurs at “optimum” doping x.Thelower Tc region at smaller x is called the “underdoped” region and
the lower Tc region at larger x is called the “overdoped” region. See also Refs. 45, 47 and 48 for very similar Ba(Fe1−xCox)2As2
phase diagrams derived from single crystal studies. Reprinted with permission from Refs. 40 and 46. Figure from Ref. 40:
Reprinted by permission from Macmillan Publishers Ltd, Copyright (2008). Figure from Ref. 46: Copyright (2010) by the
American Physical Society.
containing a spin glass phase between the antiferromag-
netic and superconducting phases, as shown in Fig. 6. A
similar phase diagram was obtained in Ref. 69. Long-
range antiferromagnetic ordering ceases for x
>
∼
0.1, and
bulk superconductivity does not set in until x exceeds
∼ 0.3 to 0.4. Structural studies indicate that the low-
temperature long-range structural distortion ceases with
increasingxatthesamecompositionatwhichlong-range
antiferromagnetic ordering ceases, and further suggest
that the spin glass phase at low temperatures is accom-
panied by the onset of lattice disorder of some kind.
68
As discussed in Sec. IIIE3 below, it appears that the
SDW/AFM region corresponds to local moment, rather
than itinerant, antiferromagnetism. Additional evidence
for short-range antiferromagnetic ordering was observed
inotherinvestigations.
70–74
ThephasediagramofRef.72
shows the short-range ordering regime extending all the
waytox=0.45,overlappingtheregionofbulksupercon-
ductivity. On the other hand, Ref. 74 also finds short-
range ordering up to x=0.45, but with no bulk su-
perconductivity. At x=0.50, bulk superconductivity is
found, but with no static magnetic ordering. The au-
thors suggest that bulk static magnetic order and bulk
superconductivity may be mutually exclusive in this sys-
tem; when they do appear to occur simultaneously, they
may occur in different spatial regions.
74
The emergence of high T
c
upon destruction of long-
range AF order as illustrated in Fig. 5 is qualitatively
similar to observations in the layered cuprate high T
c
superconductors.
75
The close association of AF order-
ing and superconductivity in both types of materials
suggests that the superconductivity may have an elec-
tronic/magnetic mechanism. However, the cuprate par-
ent compounds are antiferromagnetic insulators rather
than metals, which is an important distinction between
these two classes of high T
c
superconductors.
Many research papers have been written on the prop-
erties of the above Fe-based and related materials and
their theoretical interpretations since the spring of 2008.
About 2000 experimental papers and 500 theoretical pa-
pers have been published in journals and/or posted on
the arXiv
76
so far. Much of this researchis driven by the
following questions: What is the mechanism for T
c
?Is
new physics involved in the properties? What is the up-
perlimitofT
c
forthisclassofmaterials? Whatmaterials
propertiescontrolT
c
? Whereshouldwelook nextto find
new superconductors with high T
c
? Should we consider
the materials to be strongly correlated electron systems?
Figure 1.6: The left and the right is the phase diagram of CeFeAsO
1x
F
x
[103] and
Ba(Fe
1x
Co
x
)
2
As
2
[67] respectively. AFM is the antiferromagnetic phase, SC is the
superconducting phase. In the region between the normal metal phase and AFM exhibit a
90

rotational symmetry breaking. In some of the iron-based superconductors, the super-
conductivity and magnetic order can coexist. For instance, the green region in the left
diagram indicates the coexistence. [39]
Nevertheless, the common feature in iron pnictides electronic structure from the effec-
tive model or first principle calculation, the Fermi surface break into several disconnected
pockets. Furthermore, the the hole and electron pockets are nested. Therefore, its normal
metal phase is highly unstable against interactions between electrons. This instability is
currently believed to be the main reason for producing the rich phase structure in FeAs
(Fig. 1.6)
Due to this nesting effect, magnetic fluctuations are one of the most important collec-
tive modes in the system. In fact, the parent compound of iron pnictide is magnetically
ordered. Weakening the nesting effects by doping, the system magnetic fluctuations can
9
be suppressed, and it eventually evolves into superconducting state. From many supercon-
ducting gap measurements which have been reported so far, favor the pairing symmetry
that closely tied to spin fluctuations scenario. Although spin fluctuation theory seems
to be the plausible explanation for the pairing mechanism, there are still several debates
that remain unsettled up to date. First, superconducting states persist in some of these
iron-based superconductors while nesting effects only play a minor in the entire doping
range. Furthermore, some of the materials have a nodal superconducting gap which is not
expected from the spin fluctuations scenario.
Except for the superconducting state, the physics of these materials in other phases
are not yet fully understood and still under active investigations. In particular, increasing
experimental evidence shows an unexpected crystalline rotational symmetry breaking in
the metallic phase of certain iron pnictides. There is a widespread believe that magnetic
fluctuations play an essential role in affecting its physical properties. This is an interesting
electronic nematicity phenomena in the system, which is one of the main themes of this
thesis.
1.2 Many-body effects in multi-band system
These new materials are the current dominant trends of research in condensed matter
physics. To best summarize these materials, they all have multi-bands but with different
Fermi surface configurations. Topological insulators are gapped in the bulk, no electronic
states at the Fermi level, and many-body effects may not play an essential role in the low
temperature limit. In contrast to graphene and iron-based superconductors, they are both
semi-metal and have hole-like and electron-like Fermi surfaces. The difference is that
10
the graphene Fermi surface just has two discrete points with zero density of states, while
iron-based superconductors is has few pockets with finite density of states. Depending
on their critical properties, many-body effects could play a crucial role in determining the
system ground state. Therefore, in this thesis, graphene and iron-based superconductors
are our main subject to study. Specifically, we focus on the topics of competing orders in
bilayer graphene and nematic order in iron pnictides. This is outlined is in the following
two sections.
1.2.1 Competing orders beyond Fermi liquid criticality
Motivation:
The ground state of bilayer graphene is a debated topic in theories and experiments, and
particularly, whether the ground state is gapped or gapless. In this thesis, the phase dia-
gram of bilayer graphene is analyzed by a microscopic model. In the analysis, the major
instability of the system is the main focus. The result from the analysis can actually narrow
down the possible phase for the bilayer ground state.
Method and conclusion:
In this study, bilayer honeycomb lattice with nearest neighbor hopping is used to model
the bilayer graphene. In the absence of interactions, the Fermi surface of this model at
half-filling consists of two nodal points with momentaK,K
0
, where the conduction band
and valence band touch each other, yielding a semi-metal. Since near these two points the
energy dispersion is quadratic with perfect particle-hole symmetry, excitonic instabilities
are inevitable if inter-band interactions are present. Using a perturbative renormalization
11
group analysis up to the one-loop level, the critical behavior of semi-metallic phase can
be defined. Furthermore, different competing ordered ground states are found, including
anti-ferromagnetism, superconductivity, spin and charge density wave states with ordering
vector Q = KK
0
, and excitonic insulator states. In addition, two states with valley
symmetry breaking are found in the excitonic insulating and ferromagnetic phases. This
analysis suggests that the ground state of bilayer graphene should be gapped, and with the
exception of superconductivity, all other possible ground states are insulating.
1.2.2 Electronic nematicity
Motivation:
Nematic order is a phase of a system which breaks rotational symmetry but preserves
translational symmetry. It was first found in liquid crystals. The origin of this ordering is
due to the anisotropic molecular shape and the anisotropic interactions between molecules
in classical fluids. In the past decade, similar ordering has been found in some strongly
correlated electronic systems. In contrast to classical systems, electrons are point-like par-
ticle and they interact with each other by the isotropic Coulomb potential. Therefore, the
physics of nematicity in a quantum system is different from classical fluids. Neverthe-
less, the theory of this novel quantum phases is still under development. There are many
strong indications suggesting that iron pnictides exhibit this intriguing electronic nematic-
ity. Therefore, another main theme of the thesis is the origin of nematic order in FeAs
i.e. to yield insight for understanding the novel quantum phase as well as the electronic
properties of FeAs.
12
Method and conclusion:
In this thesis, in order to identify the nematic order, we first simplify the minimum effec-
tive five-bands model in the low energy limit. In this limit, the fermionic relevant degree
of freedoms are constrained to the vicinity of the Fermi surface. Further using the phase
space argument, the relevant scattering prosseses between electrons can be identified. The
next step is to use Hubbard-Stratonovich transformation to decouple these relevant scatter-
ing terms by introduceing several bosonic fields which describe the low-energy collective
modes of the system. After this procedure, the nematic order parameter can be identified.
In addition, this yields a Ginzburg-Landau energy functional which allows us to check
the stability of the isotropic phase explicitly. By studying the behavior of these collective
modes, in the isotropic metal phase, a possible instability which is driven by the mag-
netic fluctuations is found in the forward scattering channel (Pomeranchuk instabilities).
Using mean field analysis, we obtain a static and homogeneous metallic ground state,
but the electron Fermi pockets are distorted unequally at different pockets in momentum
space. This results in a nematic ordering which breaks the latticeC
4
symmetry but pre-
serves translational symmetry and may explain several experimental observations. Most
importantly, we have demonstrated that the nematic order is closely related to the system
magnetic fluctuations.
13
Chapter 2
Theoretical background and methods
This chapter will discuss the essential concepts and techniques that are used in thesis. The
outline is the following. In Sec 2.1, the theory of Landau-Fermi liquid will be introduced,
since many of the discussion in the thesis are based on this basis concept. In Sec 2.2, a
review of fields theory will be given. In Sec 2.3, we briefly discuss the idea of renormal-
ization group. In Sec 2.4, we explain how to implement the Mean field theory by using
Hubbard model as an example.
2.1 Landau’s Fermi-liquid
Landau’s Fermi-liquid theory is one of the cornerstones in condensed matter physics. This
theory first successfully established a phenomenological model for Helium-3 and then for
the electronic system in metal. However, Fermi-liquid phenomena are quite universal and
prevail in most of the fermion systems. Therefore, this important theoretical concept is also
a foundation of condensed matter physics. The statement of the theory is the following.
Landau’s Fermi-liquid theory: If an interacting Fermi system is a Fermi liquid, the
system possesses the following properties: (1) The quantum number of the system ele-
mentary excitations near the system ground state has a one to one correspondence to its
non-interacting case quantum number (momentum, etc.). (2) These excitations are NOT
the exact eigenstates of the system, but it has a finite lifetime / 1=
2

, where

is the
14
excitation energy of state. (3) The mean occupation number of these excited states sat-
isfy Fermi-Dirac distribution. Therefore, the elementary excitations of this system can be
treated as fermion-like particle (quasiparticle) with an infinite at

= 0 which define
the Fermi surface.
The implication of the theory simply allows us to treat the excitation as a ‘particle’ just
like those in free fermi system. At the first sight, one may think that this statement may
look pausible in the weakly interacting regime. However, for many real systems such as
alkaline metals, their interacting Coulomb potential energy are generally comparable to
the kinetic energy term. Thus, although the consequences of the theory is simple, it should
not be treated like a trivial argument. The underlying principle of the theory is adiabatic
continuation [2] which it is basically a non-perturbative result.
In the following section, we will discuss some details of the theory from the phe-
nomenological approach to further illustrate the idea. We first demonstrate the relationship
between quasiparticle lifetime and Fermi-Dirac statistic.
2.1.1 Quasiparticle lifetime and Fermi-Dirac statistic
In the theory statements, the quasiparticles lifetime become infinitely long as it approaches
the Fermi energy. The origin of this property is actually from the Fermi-Dirac statistic of
quasiparticles. A quick way to illustrate this property, we exploit Fermi Golden rule. To
be specific, considering two quasiparticles initially has the states with momentum and spin
fkg andfk
0

0
g respectively, and denoted byjk;k
0

0
i. The transition rate of the two
particles in tojk +q;k
0
q
0
i is

k;k
0
;q
= 2jhk +q;k
0
q
0
jV (q)jk;k
0

0
ij
2
(
k+q
+
k
0
q

k
0
k
); (2.1)
15
where V (q) is just some 2-body interaction strength between quasiparticles with
exchanged momentumq.
Hence, applying the Golden rule, the decay rate of the state injk;k
0

0
i is
1

k
'
2
V
2
X
k
0
q

k;k
0
;q
[n
k
n
k
0(1n
k+q
)(1n
k
0
q
) (1n
k
)(1n
0
k
)n
k+q
n
k
0
q
]; (2.2)
wheren
k
is the mean occupation number of quasiparticle with momentumk. The factor
of two accounts the spin degeneracy. Note that the sum over all k
0
is to account the
final available states for a scattered quasiparticle, and the sum over all q is to account
all the available ‘channel’ (intermediate states) for the quasiparticle before achieving to a
particular available final state. The weighting factor in the sum is because of the Fermi-
Dirac statistical behavior of quasiparticles. The first term in (2.2) is book-keeping how
many state can states is available to scattering ‘out’ from k, while the second term is
scattering ‘in’ tok.
We omit the calculation of (2.2) here, and refer it to other excellent texts [7, 33]. One
can also utilize simple phase space argument [7], which immediately yields the rough
estimate
1

2
k
. This show that Fermi-Dirac statistic of quasiparticle result in the
decay rate vanishes as the square of it energy. On the other hand, from the microscopic
point of view, Luttinger theorem [53] shows that quasiparticles decaying rate vanishes at
Fermi surface imply Fermi-Dirac statistic. Therefore, the basic assumptions in the theory,
statement (2) and (3), are consistent and not independent to each other.
16
2.1.2 Thermal equilibrium properties
It is desirable to present the Fermi liquid theory into some concrete form for demonstrating
the concept. Keeping in mind that quasiparicle is just a good approximated excitations, the
statement of the theory can actually be translated into the following mathematical formula,
E[n
k
]'E
0
+
X
k
"
k
n
k
+
1
2
X
kk
0

0
f

0(k;k
0
)n
k
n
k
0

0; (2.3)
where E[n
k
] represent the total energy of the Fermi-liquid, which is the functional of
n
k
. E
0
is the ground state energy with no quasiparticle. n
k
= 1 (n
k
= 1)
represent a quasiparticle (quasihole) occupy the statefkg,n
k
= 0 represent no quasi-
particle."
k
is the quasiparticle energy, andf

0(k;k
0
) is Landau parameters which model
the interaction energy between quasiparticles.
Note that the interacting term in the energy functional only up to second order. This
can be rigorously justified by using RG argument. However, this can be understood by
a simple phase space argument. The other higher order terms correspond to n-body (
n > 2). In general, in the condensed matter system, the interaction is Coulomb, which
is a 2-body scattering potential. Forn-body scattering, it can only be achieved, if several
quantum virtual transitions states are involved. Due to the conservation of momentum, this
lead to a stringent constraint for the available phase space. Hence, then-body interaction
scattering events are basically negligible as compare to 2-body interaction. Therefore, it is
not desirable to includen> 2 interaction terms.
17
The partition function
The partition function can be evaluated by
Z = Tr[e
E
] =
X
fn
k
g
exp[(
X
k
("
k
)n
k
+
1
2
X
kk
0

0
f

0(k;k
0
)n
k
n
k
0

0)];
where the unimportant constantE
0
is dropped. Because of the interaction term, to evaluate
the sum over all configuration offn
k
g is formidable. Therefore, we use mean field
approximation. First, let
n
k
=hn
k
i + (n
k
hn
k
i); where,hn
k
i = Tr(n
k
e
E
):
Mean field approximation simply assume that n
k
fluctuation is small and its value is
closed to its averagehn
k
i i.e. (n
k
hn
k
i)
2
term is neglected. Thus, the partition
function become
Z'
X
fn
k
g
exp[
X
k
(
k
+
X
k
0

0
f

0(k;k
0
)hn
k
0

0i)n
k
+=2
X
kk
0

0
f

0(k;k
0
)hn
k
ihn
k
0

0i]:
(2.4)
Now the sum can be done straightforwardly. The last term is independent ofn
k
and can
be factored out. This immediately yields
Z'
Y
kk
0

0
e
=2f

0(k;k
0
)hn
k
ihn
k
0

0i
Y
k
(1 +e
(~ "
k
)
k
);
(2.5)
18
where ~ "
k
= "
k
+
P
k
0

0
f

0(k;k
0
)hn
k
0

0i and
k
=1, the upper (lower) sign is for
quasiparticle with"
k
> 0 ( quasihole with"
k
< 0). From the partition function in
(2.5), many other thermodynamical quantities can be obtained, and the qualitative result is
indeed identical to the non-interacting case. [57] We therefore show that, the Fermi-liquid
exhibit the physical properties of non-interacting fermion system.
2.1.3 Universality class
The physical meaning of Landau’s Fermi-liquid theory can be further elaborated by the
renormalization group (RG) approach. Under the RG language, which can naturally
explain why the Landau-Fermi liquid phenomena is quite universal in many interacting
Fermi system. Instead of presenting the full calculation, we sketch only those essential
points to understand the universal behavior from the RG perspective in this section. We
refer the rigorous RG calculation to the excellent reference in [81].
The concept of universality class is first introduced by the theory of critical phenomena.
In the theory, the physical properties of a system near its phase transition critical point,
exhibit a universal power-laws scaling in their thermodynamical quantities. These power-
laws depend only on the dimensionality, symmetry, and interaction range, although these
system share no common microscopical details. Based on the system scaling property,
such like the power exponent at the critical point, they can be classified into numbers of
different universality classes. Furthermore, the physics of the system near the critical point
is governed by these universal power-laws only, regardless their microscopic details. This
concept has a very depth insight and precise interpretation under the RG treatment.
From the RG point of view [81], Fermi-liquid is a universality class under the theory
of critical phenomena. In the low temperature limit, the physical system in this class can
19
be treated like a free system, and only the interactions between particles in the forward
scattering (direct channel) are important. This special scattering processes only allow
small momentum transfer between electron, and basically keeping the final momenta of the
two scattered electrons remain approximately equal to their original momenta. Moreover,
forward scatterings determine the interactions between quasiparticles [the f

0(k;k
0
) in
(2.3)], and also define the interaction between the collective (charge-density) modes.
In graphene, from the RG point of view, graphene (single and few layers) is not con-
sider as the same universality class as Fermi-liquid, since graphene universal scaling-laws
is different. Furthermore, the collective behavior in a single layer graphene, for instance,
particles density acoustic modes (zero sound, first sound, etc [57]) are not expected to be
identical to Fermi-liquid. In graphene, Short range interactions between electrons are not
important.
2.1.4 Pomeranchuk instabilities
Instabilities of Fermi-liquid is an interesting topic, because this implies a phase transition
in a system. As the instabilities set in, the system do drastic changes to search for a stable
and lower energy phases. The mechanism are many, but generally, to produce instabilities,
the system should have some special scattering processes, such like attractive interaction
in BCS channel or repulsive interaction between nested Fermi surfaces and so on. In this
section, we will demonstrate an instability which is purely given rise by the forward scat-
tering (direct channel) in Fermi-liquid, because this yields a essential concept to explain
the nematicity in iron pnictide.
The instability in direct channel of a Fermi-liquid was first discussed by Pomeranchuk.
Following Ref. [77], and considering a rotational invariant system with a spherical Fermi
20
Figure 2.1: The cartoon show the elementary excitations of Fermi liquid as the oscillation
of Fermi surface.
surface, the general phenomenological energy functional of the system is simply given by
(2.3). In the vicinity of the Fermi surface, if the energy dispersion can be linearized, this
yields
"
k
'v
F
(kk
F
); (2.6)
where k
F
is the Fermi momentum and the Fermi velocity v
F
is a constant. Note that,
because of rotational invariant the energy dispersion only depend on k =jkj. Now we
may expandk atk
F
(Fermi surface) into spherical modes. Namely, the radial momentum
k can be expressed as
k(;) =k
F
+
X
lm

lm
Y
lm
(;); (2.7)
where
lm
is the expansion constant andY
lm
(;) =
q
(lm)!
(l+m)!
P
m
l
(cos)e
im
,P
m
l
is the
associated Legendre polynomials. Therefore, the excitations of the Fermi liquid can be
described by the oscillations of Fermi surface (see Fig. 2.1).
By using (2.7), the first term in equation (2.3) yields
X
k
"
k
n
k
=
Z
k
2
F
d

Z
kk
F
0
d(kk
F
)
(2)
3
v
F
(kk
F
); (2.8)
21
where
R
dkn

(k) =
R
kk
F
0
d(kk
F
) at low temperature limit. Similary, the second term,
1
2
P
kk
0

0
f

0(k;k
0
)n
k
n
k
0

0 is
1
2
v
4
F
(2)
3
ZZ
d

1
d

2
(k
1
k
F
)(k
2
k
F
)
X

0
f

0(cos
12
):
(2.9)
Note that, we have assumed that the Landau parameters is a slow varying function ofk and
approximates it to the value atk
F
. In addition, rotational invariant is used in (2.9), so the
Landau parameters only depend of the cos
12
= cos
1
cos
2
+ sin
1
sin
2
cos(
1

2
),
where
12
is the angle betweenk andk
0
. Following the conventional notation, the Landau
parameters are represented in the functionsf andg.
f
""
(cos
12
) =f
##
(cos
12
) =f(cos
12
) +g(cos
12
);
f
"#
(cos
12
) =f
#"
(cos
12
) =f(cos
12
)g(cos
12
):
(2.10)
Expanding the Landau parameters in the Legendre polynomial,
f(cos
12
) =
X
l
f
l
P
l
(cos
12
); g(cos
12
) =
X
l
g
l
P
l
(cos
12
):
(2.11)
Combining (2.7), (2.8), (2.9), and (2.11), the energy functional
EE
0
=
k
2
F
v
F
2(2)
3
X
lm

2
lm
4
2l + 1
(lm)!
(l +m)!
+
k
4
F
2(2)
6
X
lm

2
lm

4
2l + 1

2
(lm)!
(l +m)!
g
l
:
Therefore, the stability condition for the Fermi liquid is
1 +
k
2
F
v
F
(2)
3
4
2l + 1
g
l
> 0; 8l> 0: (2.12)
22
Equation (2.12) is the consequence by imposingEE
0
> 0, since this ensure thatE
0
is
the lowest energy state which represents the ground state of the system. IfEE
0
< 0,
this imply thatE
0
is not the true ground state, the excitation energy is lower thanE
0
. This
is the hallmark of instability and lead to a phase transition of the system.
In a rotational invariant system, Pomeranchuk instabilities can be produced by some
peculiar two-body scattering processes in the direct channel which can give rise to large
negative values of someg
l
. The instability inl 1 imply that isotropic configuration of
Fermi surface is not the true ground state. The system is tend to deform its Fermi surface,
i.e. the system free energy can be minimized in some anisotropic configurations. This
give rise to nematic order of Fermi-liquid which rotational symmetry are broken but not
the translational symmetry. This similar idea is exploited in iron pnictide to explain its
nematic order phase. Although a slight variation in iron pnictide, due to the configuration
of Fermi surface ( which breaks into several pockets), the concept is identical.
2.2 Fields theoretical formalism
The many-body Shr¨ odinger equations can be expressed in the second quantization lan-
guage in Fock’s space [1, 7, 29]. The representation of a many-body Hamiltonian is
H =
Z
d
3
r
y

(r)

1
2m
r
2
r
+V
ext
(r
i
)

(r)
+
1
2
ZZ
d
3
rd
3
r
0

y

0
(r
0
)
y

(r)
e
2
jrr
0
j

(r)

0(r
0
);
(2.13)
where
y

(r) (

(r)) is the local field operator that create (annihilate) a electron at position
r, andV
ext
is the background potential that is generated by the ions.
23
In most of the cases, the one-particle Hamiltonian [the first term of equation (2.13)]
yields a natural basis for the problem. In particular, if the one-particle Hamiltonian is a
Bloch Hamiltonian,

1
2m
r
2
r
+V
ext
(r
i
)

'
nk
(r) =
nk
'
nk
(r); (2.14)
where'
nk
(r) are Bloch’s wave. Therefore, expanding the local fields operator into Bloch’s
wave, this yields

y
(r) =
X
nk
'

nk
(r)a
y
nk
; (r) =
X
nk
'
nk
(r)a
nk
: (2.15)
By substituting (2.15) into (2.13), we obtain
H =
X
nk

nk
a
y
nk
a
nk
+
1
2
X
n
3
n
4
n
2
n
1
X
k
3
k
4
k
2
k
1

n
3
n
4
;n
2
n
1
(k
3
k
4
;k
2
k
1
)a
y
n
3
k
3
a
y
n
4
k
4
a
n
2
k
2
a
n
1
k
1
;
(2.16)
where the coupling constants

n
3
n
4
;n
2
n
1
(k
3
k
4
;k
2
k
1
) =
ZZ
d
3
rd
3
r
0
'

n
3
k
3
(r
0
)'

n
4
k
4
(r)
e
2
jrr
0
j
'
n
2
k
2
(r)'
n
1
k
1
(r
0
):
This general formula will be used in our study in the bilayer graphene and iron-pnictides
for describing the electron-electron interaction terms.
24
2.2.1 Tight-binding model
Sometime, in certain material, the one-particle behavior of the electrons are well defined
by local basis. In this case, the full Hamiltonian is more convenient to work in the local
atomic orbital basis.

y

(r) =
X
i

R
i

(r)a
y
i
;

(r) =
X
i

R
i

(r)a
i
; (2.17)
where
R
i

(r) represent the local atomic orbital state within the unit cell, and the center
of the unit cell is atR
i
.
Considering that, only one quantum state is available for the local atomic orbital, by
using (2.17), we immediately obtain,
H =
X
ij
t
ij
a
y
i
a
j
+
X
ijkl
U
ijkl
a
y
i
0
a
y
j
a
k
a
l
0; (2.18)
where the hopping constant and the coupling constant
t
ij
=
Z
d
3
r

R
i
(r)

~
2
2m
r
2
r
+V
ext
(r)

R
j
(r);
U
ijkl
=
1
2
Z
d
3
r
Z
d
3
r
0

R
i
(r
0
)

R
j
(r)
e
2
jrr
0
j

R
k
(r)
R
l
(r
0
):
While the orbitals are well localized, it is valid to approximate (2.18) by keeping only
the nearest neighbor hopping and onsite interaction, since the tight-binding constant and
coupling constant strength depend on the overlapping between the local orbitals. This
yields the well known Hubbard model. This basis will be used in the later study of bilayer
graphene for its one-particle electronic structure.
25
2.2.2 Functional path integrations
In second quantization, field operators are used explicitly in the formalism. Although this
formalism is sufficient for most of the analysis, sometime, another equivalent path integral
formalism for field theory is more convenient. This formalism utilize the functional
integration method, and turn the field operators into numbers. We omit the lengthy
derivation how to obtain the path integral form of (2.13) and refer to Ref. [1]. In this
section, we outline the essential points need for the derivation and the result.
Coherent states
The field operators (a

ora
y

) are not hermitian and its eigenstates of these operators
are a coherent states,
ji =e

P

a
y

j0i; and a

ji =

ji; (2.19)
where is a complex number, if the field operators is bosonic, and complex Grassmann
number, if the field operators is fermionic. [Note that, the commuting or anti-commuting
property of is inherent from the field operator.]
Completeness of coherent states
The coherent states form a overcomplete basis in the Fock space. The completeness
equation is
Z
Y

d

d

1=2
e

P

jihj =1
F
; (2.20)
where1
F
is the identity operator in Fock space. The proof can be easily given by Schur’s
lemma.[1]
26
Functional integral representation
Now we are ready to construct the path integral formalism. Given that the (Gibbs)
partition function
Z Tr(e
H
) =
X
En
hE
n
je
(HN)
jE
n
i; (2.21)
where thejE
n
i are the energy eigenstates of the Hamiltonian H. Inserting the identity
operator (2.20) into the partition function, this yields
Z =
Z
Y

d

d

e

P

hje
(HN)
ji; (2.22)
where the factor of =1 is because the Grassmannian property of.
Note that, the density operatore
(HN)
in the expansion series of field operators is
not in normal order. [Normal order: all the creation operators are stay to the left of all
annihilation operators.] In this circumstance, the eigenvalues of density matrix cannot be
easily evaluated, since we need to pass all the annihilating operators to the right.
Normal order approximation
Therefore, breaking down the interval [0;] intoN pieces by repeatedly inserting1
F
and labeling the inserted coherent states byj
n

i. Namely,
Z =
Z
Y
n
d
n

d

n

e

P
n

n

n

h
N
jj
n1
ih
n1
je

N
(HN)
j
n
ih
n
jj
N
i:
Imposing normal order ine

N
(HN)
for every breaking down pieces, this operation give
rise to an errorO((=N )
2
). Namely,e

N
(HN) =:e

N
(HN) : +O((=N )
2
),
27
where : : is the symbol for normal order. Ignoring the error terms, we can re-exponentiate
the normal order terms. This lead to
Z
Y
n
d
n

d

n

e
P
n

n

(
n1

n

)=
e

P
n
(H[

n1

;
n

]+N[

n1

;
n

])
+O((=N )
2
);
where ==N , and we have set

0

=

N

,
0

=
N

.
Continuum limit and the result
The errorO((=N )
2
) vanishes asN!1. This yields the desirable result
Z =
Z
D[

; ]e
S[

; ]
; (2.23)
whereD[

; ] =
Q

D[

;

] is the functional integration of each individual

and

fields, and the actionS[

; ] is
S[

; ] =
Z

0
d
X

()@

() +H[

();

()]N[

();

()]

: (2.24)
This is the principle result of path integral formalism. The final result can be easily
obtained by just replacing the creation and annihilation operators by its coherent fields.
Finally, we remark the underlying mathematics of path integral formalism. First, one may
think that the functional space of fermionic field is just like a real or complex space in the
sense of mathematical analysis (sequences, concept of convergences, continuity proper-
ties, etc). Therefore, the meaning of derivatives and integrations on the Grassmann num-
ber function has a similar notion in real or complex function, although ordering between
Grassmann number may not exist.
28
2.3 Renormalization group
Renormalization group (RG) is one of the important theoretical concept in modern physics.
The basic idea is to systematically eliminate the high-energy modes, which averaging out
the short-wavelength degree of freedoms i.e. the long-wavelength modes the effects are
accumulated and manifest in the low energy limit.
The application of RG is very broad. For instance, there are the application to UV-
divergent problem in relativistic quantum field theory, investigating the instabilities in
many-body system, application to critical phenomena, and so on. RG approach yields
many deep insights which is very powerful and become a standard analytic tool in modern
physics. In this section, the general concept of RG will be presented.
2.3.1 Critical phenomena, scale invariant, and power-law
We have explained critical phenomena in the Sec 2.1.3. This concept will be explored
in-depth, especially the role of scale invariant, since this is the ultimate link between the
mathematics (RG method) and the physics.
For the sake of concreteness, we focus on the canonical example of Heisenberg spin
system. For this system, the correlation function of the spinsS(r) between positionsr and
r
0
has the following asymptotic form.
C(r
0
r;t) =hS(r
0
)S(r)ihS(r
0
)ihS(r)ie
jr
0
rj=(t)
;
wheret =
TT
C
T
C
,t> 0, andT
C
is the transition temperature. In addition,(t) is the corre-
lation length which depend ont. Roughly speaking, the correlation length represents the
29
typical cluster size of a ripple created by a local spin fluctuations. The value of define
the fluctuations ‘pattern’ of the system. This eventually break the system scale invariant,
since this typical cluster size is more important to the other cluster.
However, in the vicinity of critical point, the spin system long-wavelength fluctuations
are no longer negligible, because they are Goldstone modes of the system with rotational
symmetry breaking. In this case, the fluctuations from any scale are now equally impor-
tant. This lead to the manifestation of scale invariant, since the fluctuation pattern is self-
similar in any scale. Furthermore, at the critical points, must be 0 or1, since any finite
length will not invariant under rescaling, and this breaks scale invariant.
Basically, the power-law behavior can be understand as the consequence of scale
invariant property at critical points. In the scale invariant system, as one scaling up or
down the strength of fluctuations (this can be achieved by increasing or decreasing the
temperature difference betweenT andT
c
), which must not alter the correlation between
fluctuations fundamentally i.e. the fluctuations self-similarity in different length scale are
retained. This put a very strong constraint to the functional form of C(t) (the distance
rr
0
is dropped), and which must has the following scaling property
C(b
yt
t) =b

C(t) ) C(t) =b

C(b
yt
t); (2.25)
whereb is some scaling factor, and ,y
t
are some exponents. Mathematically, the function
satisfying (2.25) is a homogeneous function or equivalently,C(t) is a polynomial oft with
power . This naturally lead to power-law behavior ofC(t)t
=yt
.
In the context of RG, the power law, scaling theory and universality in the theory of
critical phenomena can be casted into a rigorous mathematical framework. This yields a
30
systematic treatment and in-depth insight to the physics of phase transition. In the next
section, we will present the general idea of RG.
2.3.2 RG: general idea and parameters space
The action of a physical system can be generally express as
S[] =
N
X
a
g
a
O
a
[]; (2.26)
where,g
a
represent the parameters in the theory, such like temperature, chemical potential,
coupling constants, etc, andO
a
[] are operators which are composed by. Therefore, the
partition function of the system is
Z =
Z
D[]e
S[]
: (2.27)
To study the physics of this system, the straightforward method is to calculate its partition
function. As we known, evaluatingZ exactly is not always possible. Therefore, a the
following different approach will be used to tackle the problem.
Step I: Fast and slow modes
First, identifying the fast (high-energy) and slow (low-energy) degree of freedom. Namely,
Z =
Z
D[
s
]
Z
D[
f
]e
S[s;
f
]
; (2.28)
where
s
(
f
) is the slow (fast) modes and they are complicatedly coupled together.
31
Step II: RG transformation
Then, eliminating the fast modes by integrated out, and only the slow modes are retained
Z'
Z
D[
s
]e
S
0
[s]
=
Z
D[
s
] exp
"
N
X
a
g
0
a
O
0
a
[
s
]
#
: (2.29)
Note that, integrating out the fast mode is not unique and may involve certain approxima-
tion scheme, like cumulant expansion, etc. This is why ‘'’ symbol is used in (2.29). After
the integration, the functional form of the action may change, and denote byS
0
[
s
]. Also
note that, some new operators may be introduced intoS
0
[
s
]. If this is the case we have to
include these new operators initially and redo the fast mode elimination again.
Step III: Rescaling and Fields renormalization
Now, the new action in (2.29) will have a smaller phase space (available physical states),
because of the modes elimination in step II. Therefore, the new theory do not describe
the same physics as the origin one. However, this can be resumed by rescaling the phase
space. For instance, the phase space of a band electron is simply those crystal momenta in
the first Brillouin zone. Therefore, the rescaling will be
k
0
!bk
0
;
where b is some scaling factor to bring the phase space back into its original size. The
associated rescaling of the frequency which is the dynamic term,
!!b
z
!:
32
After these rescaling the fields
s
, now it possess the same degree of freedom as before,
but the factor ofb appeared inS
0
[
s
] explicitly. However, this factor can be removed by
the following redefinition (renormalization) of the fields
(k;!) =b
d

s
(bk
0
;b
z
!):
Note that k represent the momentum in the whole Brillouin zone, and k
0
represent the
momentum in the smaller subset of Brillouin zone after modes elimination. With some
appropriate exponent d

(scaling dimensional of field), all the explicit factor can be
absorbed,
Z'
Z
D[] exp
"
N
X
a
g
0
a
O
a
[]
#
: (2.30)
Finally, the original theory is restored but its parameters are modified intog
0
a
.
RG flow rate equation
Comparing (2.26) and (2.30), basically, everything remain the same, except those param-
etersg =fg
a
g. For each RG step,g change intog
0
=fg
0
a
g, whereg
0
is a function ofg.
Namely,g
0
a
= f
a
(g) or express in the compact formg
0
=f(g), wheref(g) =ff
a
(g)g.
Note that, this function may be different as different RG scheme is used. However, this do
not give rise any significant effects as long as the RG transformation is performed near the
critical points. Now, let` = lnb be the RG ‘time’ or control parameter which represents
the system energy scale. It is very convenient to think that g flow in an N-dimensional
parameters space, as` is changed from 0! lnb. The schematic picture is given in Fig.
2.2.
33
g’
g
=0
= ln b
Figure 2.2: The parameters in (2.26) form an N-Dimensional parameters space. The
dashed line represent the trajectory of the parameters after each RG transformation.
` = 0 asb = 1, this mean that no modes elimination is taken. Therefore, the flow line
at` = 0 represent the parameters initial value. The RG flow rate equation
dg
d`
= lim
`!0
g
0
g
`
= lim
`!0
f(g)g
`
(g); (2.31)
where(g) =f
a
(g)g is a vector function which depend on the vectorg. This is known
as generalized -function or Gell-mann–Low equation, and this yields the information
about how the theory evolve as its energy scale change.
Fixed points
However, in most of the cases, we are only interested to those special points g

in the
parameter space
(g

) = 0;
g

is also called fixed point. These points do not change it value or flow at each RG step,
since the flow rate at these point vanishes. In other word, the theory withg

manifest scale
34
invariant, since it is invariant under RG step (energy scaling). This imply that there must
not exist any characteristic length scale at these fixed-points, and lead to the diverging or
vanishing correlation length at fixed points. All of these signatures coincide with the crit-
ical point in phase transition. Indeed, the universal power-laws naturally can be obtained
correctly by analyzing the behavior of the vicinity at fixed-points, and the other desired
result in the theory of critical phenomena.
Relevant, Marginal and irrelevant operators
Expanding the-function near a given fixed-point, and linearize the RG flow equation,
d(g
a
g

a
)
d`
'
X
b
@
a
@g
b

g

(g
b
g

b
)
X
b
W
ab
(g
b
g

b
); (2.32)
where(g

) = 0 is used.
Solving this equation, we need to diagonalize theW
ab
matrix. Suppose the eigenvalue,
eigenvectors and diagonalized matrix are

,v

andD
a
respectively, where = 1N.
Therefore,by rotating the linearized RG flow equation in (2.32),
v

=
X
a
D
a
(g
a
g

a
); and

=
X
ab
D
a
W
ab
[D
1
]
b
: (2.33)
Therefore, (2.32) become N decoupled equation, and these linearized equations can be
solved trivially. Namely,
dv

d`
'

v

) v

(`)'v
(0)

e
`
: (2.34)
We therefore know the behavior of the parameters flow near the critical points.
35
Now, considering that, we start with the fixed point theoryS

[] =
P
N
a
g

a
O
a
[] and
perturb the system from away its fixed-point by
S[] =S

[] +
N
X
a
g
a
O
a
[];
whereg
a
=g
a
g

a
. Now we may rotate the perturbed terms into the eigenvector accord-
ing to (2.33).
S[] =S

[] +
N
X

v

~
O

[]; (2.35)
where
~
O

[] =
P
a
O
a
[][D
1
]
a
. After each RG step, S

[] do not flow, but the new
perturbed operators will move toward or away from the fixed-point with regard to the
eigenvalues in (2.32).
irrelevant

< 0, these perturbative operators cannot draw the theory away from the
fixed-point. All these operators vanish as the RG transformation proceed. This is
actually why universality manifest at the critical points, and microscopic details are
not important. No matter what the initial parameters, they all eventually flow back
to the fixed-point.
marginal

= 0, in this situation, the perturbation neither vanish nor grow.
relevant

> 0, these perturbation grow as RG transformation is performed, and even-
tually become more important thanS

[], drawing the theory away the fixed-point,
and change the original theory fundamentally.
Thus, we have illustrated the general idea of RG. We also show demonstrated how RG
is related to the practical application of phase transition and critical phenomena.
36
2.4 Mean field theory
Although RG method can yield a reliable prediction about how the theory evolve into low
energy limit, to understand the low energy physics of the system further, we require other
analytic tools. Therefore, mean field (MF) theory is another theoretical method which
will be used in this thesis. MF theory is a standard treatment to study the effects from
complicated the quartic fermionic interactions terms of a many-body system. In particular,
this method is useful to build an effective Hamiltonian for a ordered ground state.
2.4.1 Hartree-Fock variational methods
This method is based on variational principle by using a trial wave functions such that the
system ground state energy is minimized. The spirit of this method is exactly parallel to
Hartree-Fock approximation.
The sketch of the idea
The general idea is to construct a trial waves function which the Hartree-Fock ground
state that is composed by a set of free quasiparticles (these quasiparticles are exactly the
Hartree-Fock states). The annihilation field operators of the quasiparticles are in the form
of

=
X

u

a
y

+v

a

; (2.36)
wherea
y

anda

are the electron field operator in state. is some other quantum number
of quasiparticle that emerges from the combination of states. Moreover,u

andv

is
the set of variational parameters to minimize the trial ground state energy. The hardest part
of MF method is to establish (2.36), since the form can be quite arbitrary. Usually, this
37
need strenuously efforts. For instance, one can base on symmetry argument to identify the
ground state relevant quantum number, from RG analysis, etc.
To construct the Hartree-Fock ground state, we required this fields satisfying Fermi-
Dirac statistic. Namely,f

;
y

0
g =

0 is required. Therefore, this determines the cre-
ation field operator

y

=
X

u
0

(u;v)a

+v
0

(u;v)a
y

;
and the Hartree-Fock ground state is
j i =
Y

y

j0i:
Expressing the field operatorsa
y

anda

in term of
y

and

by using (2.36). The ground
state energy of the Hartree-Fock state,
E[u

;v

] =h jHj i: (2.37)
The quartic terms can be explicitly evaluated by the Hartree-Fock trial ground state. Fur-
thermore, this term is just the Hartree-Fock potential. Then, solving the variational equa-
tions
E[u

;v

] = 0:
This yields the optimized values foru

andv

which minimize the ground state energy.
Mean field method is not an automatic program, since many physical reasoning for the
ground state properties may need to be established during the analysis. Since MF analysis
is case dependent, we therefore sketch the general idea by using a concrete.
38
Example: Hubbard model
For definiteness, an example [97] of mean field analysis on Hubbard model with two
dimensional square lattice is given.
H =t
X
hnmi
a
y
ms
a
m

U
4
X
n
[(a
y
ns

z
ss
0a
ns
0)
2
(a
y
ns

ss
0a
ns
0)
2
]: (2.38)
Note that the fermion quartic term is decomposed into spin and charge density, wheren =
(n
x
;n
y
) andm = (m
x
;m
y
) andn
x
;n
y
;m
x
;m
y
are integers which label the lattice sites.
To establish the relationship between the real particles and its corresponding quasiparticles
in (2.36), we first construct the MF Hamiltonian for the quasiparticles by assuming anti-
ferromagnetic nature of ground stateH.
H
MF
=
~
t
X
hnmi
a
y
ms
a
ns

X
n
(1)
nx+ny
a
y
ns

z
ss
0a
ns
0; (2.39)
where
~
t and determine the values ofu andv, so they are variational parameters. Note
that, the overall scaling ofH
MF
will not alter the result. Therefore, we set
~
t =t as the tight
binding constant in the model. The quasiparticles Hamiltonian can be solved exactly in
momentum space. By restricting the sum ofk within the Fermi sea, which is the reduced
Brillouin zone (RBZ). Therefore, we have
H
MF
=
X
k
0

k
(a
y
ks
a
ks
a
y
k+Q;s
a
k+Q;s
)

N
site
X
k
a
ks

z
ss
0a
y
k+Q;s
0
+h:c:; (2.40)
where
P
k
0
is sum over the RBZ only,
k
= 2t(cosk
x
+cosk
y
),Q = (;) are the nesting
vector and the lattice constant is set to 1. In addition, we have used (1)
nx+ny
=e
iQn
.
39
Therefore,H
MF
can be rewritten into a 2 2 symmetric matrix.
H
MF
=
X
0
k

a
y
k;s

z
ss
0a
y
k+Q;s
0

0
@

k

k
1
A
0
@
a
ks

z
ss
0a
k+Q;s
0
1
A
: (2.41)
The MF Hamiltonian can be diagonalized by an orthogonal rotational matrix. we obtains
the two different type of quasiparticles, and their field operators are

ks
=u
k
a
ks
+v
k

z
ss
0a
k+Q;s
0; (2.42)

ks
=v
k

z
ss
0a
ks
0 +u
k
a
k+Q;s
; (2.43)
where
u
k
=
1
p
2
s
1

k
p

2
k
+
2
; v
k
= sgn()
1
p
2
s
1 +

k
p

2
k
+
2
: (2.44)
It is straightforward to check that the quasiparticle satisfy the fermion properties
f
k
0
;s
0;
y
k;s
g =f
k
0
;s
0;
y
k;s
g =
kk
0
ss
0; (2.45)
Note that, momentum is still a good quantum number for the quasiparticles, but only those
define in the RBZ. The diagonalizedH
MF
is
H
MF
=
X
k
0
E
k
(
y
ks

ks

y
ks

ks
); (2.46)
where E
k
=
p

2
k
+
2
, and -field (-field) have the positive (negative) energy in the
entire spectrum in RBZ.
40
Since the total number of particles must be conserved, only the negative energy states
are fully occupied such that the energy is lowest. Therefore, the Hartree-Fock trial ground
state
j

i =
Y
ks

y
ks
j0i: (2.47)
Now, we are ready to evaluate the energy functionalE[] =h

jHj

i: Note that, is
the only variational parameter for this trial ground state, sinceu andv are depend on .
Therefore,
E[] =h

jH
MF
j

i+
h

j

U
4
X
n
(a
y
ns

z
ss
0a
ns
0)
2
+

N
site
X
k
0
a
ks

z
ss
0a
y
k+Q;s
0
+h:c:
!
j

i
(2.48)
The first term can be easily obtained,
h

jH
MF
j

i =2
X
k
0
E
k
; (2.49)
where the factor of 2 is from the spin sum. The second term can be evaluated straightfor-
wardly by Wick’s theorem but tedious. Note that, from (2.42) and (2.43)
a
ks
=u
k

ks
v
k

z
ss
0
k
0
s
; a
k+Q;s
=v
k

z
ss
0
ks
0 +u
k

ks
: (2.50)
Furthermore, in real space
a
ns
=
1
N
site
X
k
0
[(e
ikn
u
k

ss
0 +e
i(k+Q)n
v
k

z
ss
0)
ks
0
+ (e
i(k+Q)n
u
k

ss
0e
ikn
v
k

z
ss
0)
ks
0]:
(2.51)
41
Since
ks
j

i = 0, the local field operator can be separated into the standard destruction
and creation part. Therefore, Wick’s theorem can be applied [29, 76] and this yields
h

j
U
4
X
n
(a
y
ns

z
ss
0a
ns
0)
2
j

i =
U
4
X
n

z
s
1
s
0
1

z
s
2
s
0
2
[ha
y
ns
1
a
ns
0
1
a
y
ns
2
a
ns
0
2
i
+ha
y
ns
1
a
ns
0
1
a
y
ns
2
a
ns
0
2
i]:
(2.52)
Note that, the first term and the second term in the contractions is analog to the usual Fock-
and Hartree- potential term respectively. By using the following propagator,
ha
y
ns
1
a
ns
2
i =
1
N
site
X
k
0
[(u
2
k
+v
2
k
)
s
1
s
2
+ 2e
iQn

z
s
1
s
2
u
k
v
k
]
=
1
2

s
1
s
2
+ (1)
nx+ny

z
s
1
s
2
1
N
site
X
k
0

E
k
:
(2.53)
The fermion quartic term in (2.48) can be evaluated with the Wick’s contraction. Note
that, the 1=2 factor in (2.53) is because
P
k
0
sum over RBZ only. Therefore, we have
h

j
U
4
X
n
(a
y
ns

z
ss
0a
ns
0)
2
j

i =
U
4
X
n
[(
1
2
+ 2(
1
N
site
X
k
0

E
k
)
2
) + 4(
1
N
site
X
k
0

E
k
)
2
]
Similarly, the contribution from charge density can be obtained.
h

j
U
4
X
n
(a
y
ns

z
ss
0a
ns
0)
2
j

i =
U
4
X
n
[(
1
2
+ 2(
1
N
site
X
k

E
k
)
2
) + 1]: (2.54)
Also, by using (2.50), we have
h

j
X
k
a
ks

z
ss
0a
y
k+Q;s
0
j

i = 2
X
k
u
k
v
k
=
X
k

2
E
k
(2.55)
42
Therefore, we finally obtain the energy functional
h

jHj

i =2
X
k

2
k
E
k
U
1
N
site

X
k

E
k
!
2
+
N
site
U
4
(2.56)
Note that the constant is not important as we varying . Using h

jHj

i = 0, we
have
2
X
k

2
k
E
3
k

U
1
N
site
X
k

E
k
!
= 0 (2.57)
If 6= 0, this yields the self-consistent equation for the gap function ,
1
U
N
site
X
k
1
p

2
k
+
2
= 0 (2.58)
Alternatively, the gap equation can be obtained by an easier approach by approximat-
ing the quartic fermionic term as the following. We first assume that the ground state is in
antiferromagnetic nature. Namely, the average spin at siten is assumed by
S
n
= (1)
nx+ny
1
2
ha
y
ns

z
ss
0a
ns
0i

U
(2.59)
Therefore, the original Hamiltonian (2.38) can be rewritten into
H =H
MF
+ (HH
MF
) =H
MF
+
N
site

2
U

U
4
X
n

a
y
ns

z
ss
0a
ns
0
2
U

2
(2.60)
We further assume that the spins and charge density fluctuations are small ((a
y
ns

ss
0a
ns
0
2
U
) 0 anda
y
ns

ss
0a
ns
0 0), by dropping the quartic fermionic term, the final approxi-
mated Hamiltonian can be diagonalized by the quasiparticles in (2.42) and (2.43) exactly.
43
The Hartree-Fock trial ground state is the same in (2.47). The energy functional is approx-
imately given by,
h

jHj

i'2
X
k
E
k
+
N
site

2
U
(2.61)
Varying the energy functional with respect to , we immediately obtain the gap equation
(2.58).
Another method to obtain gap equation (2.58), one can substitute (2.53) into (2.59)
and immediately lead to the result. Therefore, the Hamiltonian can be diagonalized self-
consistently. No matter which method is used, variational principle is the underlying prin-
ciple to the applicability of mean field theory.
For finite temperature, variational ground state approach may not easy to perform,
since excitations also play an essential role in the system. In addition, free energy is the
function to be minimized in a thermodynamical system, but not the system total energy.
Therefore, the finite temperature mean field method will be discussed in the next section.
2.4.2 Hubbard-Stratonovich transfomation
This method utilizes path integration technique to decouple the quartic fermionic inter-
actions terms by introducing auxiliary bosonic fields which describe the low energy col-
lective modes of the system. Similarly, this is not an automatic program. Many physical
argument are required such like the identifying the relevant scattering processes i.e. to
perform sensible decoupling in the fermion quartic terms.
The central identity which is used in this method is the Gaussian integral.
e
a
2
=2
=
Z
1
1
dx
p
2
e
x
2
=2ax
(2.62)
44
In the transformation,a represent the composite quadratic fermion fields which describes
certain collective mode in the system, andx is the corresponding auxiliary bosonic fields.
After the transformation, the original fermion quartic terms are removed from the problem,
but a new bosonic fields are introduced. Similar to the Hartree-Fock variational method,
the first hardest step is to decouple the fermion quartic terms. Furthermore, for some cases,
the interactions can be complicated, and need more than one auxiliary fields to decouple
all the fermion quartic terms.
Example: Hubbard model
To be concrete, this method is also demonstrated by Hubbard model (2.38) again. Consid-
ering the grand partition function if the system in path integral formalism, by using (2.23)
and (2.24), we have
Z =
Z
D[

] exp
8
<
:

Z

0
d
2
4
X
hmni

ms
() [
mn
(@

) +t]
ns
()
+
U
4
X
n
[(

ns
()
ss
0
ns
0())
2
(

ns
()
z
ss
0
ns
0())
2
]
#)
(2.63)
By using the identity (2.62)
e
1
2
(
p
U
2

ns()
z
ss
0

ns
0())
2
=
Z
D[
n
]e

1
2
2
U

2
n
()n()

ns()
z
ss
0

ns
0()
(2.64)
e

1
2
(
p
U
2

ns()
ss
0
ns
0())
2
=
Z
D[
n
]e

1
2
2
U

2
n
()in()

ns()
ss
0
ns
0()
(2.65)
Therefore, we introduce 2N
site
auxiliary real bosonic fields, N
site
of
n
which described
the spin density fluctuations andN
site
of
n
which described the charge density fluctuations
45
to decouple the fermion quartic terms. (Note that
n
should be a vector that indicate the
direction of spin. Without loss of generality, we have assumed the spin is pointing in
z-direction.) The partition function become,
Z =
Z
D[

]
Z
D[
n
]
Z
D[
n
] exp
(

Z

0
d
"
1
U
X
n
[
2
n
() +
2
n
()]
+
X
mn

ms
() [
mn
[(@

+i
n
()) +t
nm;a
]
ss
0 +
n
()]
z
ss
0]
ns
()
#) (2.66)
wherea = (1; 0); (0;1) are the relative neighboring sites. Introducing the following
Fourier transformation in the frequency domain,

ns
() =
1

X
!n
e
i!n

ns
(!
n
);
ns
(!
n
) =
Z

0
de
i!n

ns
(!
n
) (2.67)
where!
n
is the fermionic Matsubara frequency. Using the Nambu spinor notation,

n!n
=
(

n"
(!
n
);

n#
(!
n
)). Therefore, the quadratic fermion term can be compactly expressed
into
1

X
hmni
X
!n!
n
0

n!n
G
1
n!n;m!
n
0

m;!
n
0
(2.68)
whereG
1
is a 2 2 matrix in spin space.
[G
1
n!n;m!
n
0
]
ss
0 = (G
1
n!n;m!
n
0
+i
n!n;m!
n
0
)
ss
0 +
n!n;m!
n
0

z
ss
0 (2.69)
where the Green’s function for a free particle is
G
1
n!n;m!
n
0
= [
nm
(i!
n
) +t
nm;a
](!
n
!
n
0); (2.70)
46
and the Fourier transform for the auxiliary bosonic fields are

n!n;m!
n
0
=
n
(!
n
!
n
0)
nm
=
1

Z

0
de
i(!n!
n
0)

n
()
nm
; (2.71)

n!n;m!
n
0
=
n
(!
n
!
n
0)
nm
=
1

Z

0
de
i(!n!
n
0)

n
()
nm
: (2.72)
By integrating out the fermionic fields, and writing the action into frequency-momentum
space, the partition function
Z =
Z
D[
n
]
Z
D[
n
]e
S[;]
(2.73)
The action is
S[;] =

U
X
nn
[
n
(
n
)
n
(
n
) +
n
(
n
)
n
(
n
)] Tr lnG
1
[;]
(2.74)
where
n
is the bosonic Matsubara frequency, and the Tr is sum over all lattice sites,
frequencies, and spins.
So far, this result is exact. What has just been done is only express the original fermions
problem into its collective bosonic modes. In order to extract further information from
the partition function, saddle points approximation can be applied. We first evaluate the
extremum of the action
S

n
(
n
)
=
S

n
(
n
)
= 0 (2.75)
This yields
2
U

n
(
n
)

n
(
n
)
Tr lnG
1
= 0;
2
U

n
(
n
)

n
(
n
)
Tr lnG
1
= 0 (2.76)
47
By using the identity

n(n)
Tr lnG
1
= TrG

n(n)
G
1
,

n
(
n
)
Tr lnG
1
=
X
m;!n!
n
0
[G
m!n;m!
n
0
]
ss
0
z
s
0
s

nm
(!
n
!
n
0
n
)
=
X
!n
[G
n!n;n;!nn
]
ss
0
z
s
0
s
(2.77)
where tr is the sum over the spin. Therefore, (2.75) yields
2
U

n
(
n
)
X
!n
[G
n!n;n;!nn
]
ss
0
z
s
0
s
= 0 (2.78)
2
U

n
(
n
)i
X
!n
[G
n!n;n;!nn
]
ss
0
s
0
s
= 0 (2.79)
We therefore solve the above equations to obtain the extremum. However, without know-
ing the functional form of and, invertingG
1
is impossible. In order to make progress
we need to guess the functional form of the solution i.e. G
1
can be inverted and satis-
fying (2.78) and (2.79) simultaneously. To achieve this, we argue that the ground state is
antiferromagnetic in nature. In addition, homogeneous spatial and temporal configuration
is energetically favorable. Therefore, we claim the functional form of the solution is

n
() = (1)
nx+ny
=e
iQn
;
n
() = = 0 (2.80)
whereQ = (;) is the nesting vector, and = 0 because of the ground state is charge
neutral. Thus, using the following,
[G
1
k!n;k
0
!
n
0
]
ss
0 =
X
n;m
e
i(knk
0
m)
[G
1
n!n;m!
n
0
]
ss
0: (2.81)
48
we can represent the ansatz solution in (2.80) for the bosonic fields into the momentum
space. First, we note that
n
(
n
) is not diagonalized in momentum space, since the it
stagger in real space. Therefore, in momentum space,
[G
1
k!n;k
0
!
n
0
]
ss
0 =
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
(i!
n
+
k
)
ss
0(!
n
!
n
0)(kk
0
) , k;k
0
2 RBZ

z
ss
0(!
n
!
n
0)(kk
0
Q) , k2 RBZ;k
0
= 2 RBZ

z
ss
0(!
n
!
n
0)(kk
0
+Q) , k = 2 RBZ;k
0
2 RBZ
(i!
n

k
)
ss
0(!
n
!
n
0)(kk
0
) , k;k
0
= 2 RBZ
To invertG
1
, we solve the following functional equation
X
k
00
!
n
00
[G
1
k!n;k
00
!
n
00
]
ss
00[G
k
00
!
n
00;k
0
!
n
0
]
s
00
s
0 =
ss
0(!
n
!
n
0)(kk
0
) (2.82)
for [G
k
00
!
n
00;k
0
!
n
0
]
s
00
s
0, which is the inverse ofG
1
. For allk andk
0
in the whole BZ. Note
that,G
1
is treated a matrix with some block structure in momentum space, since the
operation in (2.82) define the momentum space matrix structure. Therefore, the following
satisfy (2.82).
[G
k!n;k
0
!
n
0
]
ss
0 =
(!
n
!
n
0)
[G

k!n
]
1
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
(i!
n

k
)
ss
0(kk
0
) , k;k
0
2 RBZ

z
ss
0(kk
0
Q) , k2 RBZ;k
0
= 2 RBZ

z
ss
0(kk
0
+Q) , k = 2 RBZ;k
0
2 RBZ
(i!
n
+
k
)
ss
0(kk
0
) , k;k
0
= 2 RBZ
where [G

k!n
]
1
= (i!
n
)
2

2
k

2
. Furthermore, theG from ansatz ( = 0) also
satisfies (2.79) trivially.
49
From equation (2.78),
2
U
e
iQn

1
N
site
X
kk
0
;!n
e
i(k
0
k)n
[G
k;!n;k
0
!
n
0
]
ss
0
z
s
0
s
= 0 (2.83)
In the sum over k and k
0
, since Q andQ differ to a reciprocal lattice vector, the only
non-trivial term is whenkk
0
=Q. We then obtain
2
U

2 2
N
site
X
k!n
0

(i!
n
)
2

2
k

2
= 0 (2.84)
Note that,
P
k
0
is sum over the RBZ. Furthermore, the first factor of 2 is fromkk
0
=Q
terms, and the second factor of 2 is from the spin sum.
To further evaluate the result, we note that there are two poles in the denominator

p

2
k
+
2
. Carrying out the Matsubara sum, we obtain the gap function in finite
temperature.
1
U
N
site
X
k
tanh[=2(
p

2
k
+
2
)] tanh[=2( +
p

2
k
+
2
)]
2
p

2
k
+
2
= 0 (2.85)
By taking the zero temperature limit, !1 i.e. the tanh become unity. Only when
<
p

2
k
+
2
, we can then recover the result in the Hartree-Fock variational method.
In conclusion, the extremum in the saddle point approximation yields the mean field
solution. Hubbard-Stratonovich transformation is convenient to study finite temperature
problem. First, once the collective modes are identified, the rest of the work may be tedious
but systematic. Furthermore, the study can extended beyond the ground state by including
the fluctuations which are encapsulated in the correction terms beyond the saddle points.
50
Chapter 3
Competing orders in Bilayer graphene
3.1 Introdution
From many recent experiments, the ground state of few layers graphene is gapped, and this
results is not expected from the one-particle picture from the band structure calculations.
This insulating ground state provide a strong evidence that the instabilities of Fermi surface
driven by interactions between electrons. In contrast to single layer graphene, many-body
effects do not open a gap, and one-particle picture in fact yields a surprisingly good result.
The pristine single layer graphene band structure is captured by a tight binding model,
as illustrated in Fig. 1.2, with two interpenetrating triangular sublatticesa andb
H
A
=
0
X
hi;ji
a
y
i
b
j
+h:c:;
wherehi;ji denotes a sum over all nearest neighbor pairs. At the charge neutrality point,
this model yields a semi-metal for which the Fermi surface (FS) contains only two nodal
points. Since the energy dispersion is linear in the vicinity of these Dirac points, the cor-
responding low-energy effective Hamiltonian is given by a 2D Dirac model. This unique
electronic structure leads to many interesting phenomena.[13]
51
Although interactions between electrons are present in graphene, the one-particle pic-
ture works surprisingly well. In contrast to ordinary metals, the ground state of the elec-
trons in graphene does not behave like a Landau’s Fermi-liquid, but rather belongs to
the universality class of Dirac liquids.[82] One of the differences between these ground
states is that short-range interactions between electrons are irrelevant in Dirac liquids.[94]
This may explain why the one-particle picture is applicable, regardless of the perfect
particle-hole nesting properties in the FS. However, recent experiments have shown evi-
dence that the Dirac cone is renormalized,[23] suggesting that electron interactions are
important on some level. Recently, the interactions between electrons in graphene have
been modeled by a long-range Coulomb interaction or by using an effective (2+1)D QED
model.[19, 45, 94]
For bilayer graphene (BLG), tight-binding calculations also show that the non-
interacting ground state is a semi-metal. But in this case, the dispersion near the FS
points is quadratic rather than linear.[62] Because of this, all short-range interactions now
become relevant perturbations, and recent theories have predicted various possible sponta-
neous symmetry breaking ground states.[20, 49, 50, 64, 68, 69, 91, 92, 101]Furthermore,
recent experiments[31, 58, 61, 93, 96] have shown some evidence for FS reconstruction in
BLG. These findings contradict the simple one-particle picture for BLG, based on a tight-
binding model, and rather suggest that interactions between electrons play an important
role in breaking down the FS.
In this thesis, the instabilities in BLG will be addressed by using a perturbative renor-
malization group approach. We consider the bilayer honeycomb structure with nearest
neighbor hopping as the low-energy effective model for BLG. Particle-hole symmetry is
assumed, and RG arguments are used to identify the dominant channels and eliminate
52
a
1
γ
3
a
2
b
2
b
1
γ
0
γ
1
γ
0
γ
4
γ
4
: b
1
↔ a
2
γ
3
: b
1
↔ b
2
γ
1
:a
1
↔ a
2
γ
0
: a
1,2
↔ b
1,2
1: upper layer
2: lower layer
a
1
a
2
Figure 3.1: Bilayer graphene with AB-stacking: a
1
,b
1
are the two sublattice sites in the
upper layer, a
2
, b
2
are the two sublattice sites in the lower layer.
0
is the tight-binding
hopping constant betweena
1
;b
1
,
1
is the hopping betweena
1
anda
2
;
3
is the hopping
betweenb
1
andb
2
.a
1
=
a
2
(3;
p
3) anda
2
=
a
2
(3;
p
3) are the primitive lattice vectors.
the irrelevant channels due to the interactions in the model. Using this setup, an array of
possible ordered phases is found, which are competing with each other. In the following
sections, the details of the model and the results and implications of our calculations will
be discussed.
53
3.2 Bilayer Graphene and the model Hamiltonian
The crystal structure of BLG is given by a Bernal AB stacking of two sheets of graphene,
shown in Fig. 3.1). In the absence of interactions, its band structure is effectively described
by a tight-binding model.[13] In momentum space, the one-particle Hamiltonian with
4
'
0 is given by
H
AB
=
X
K;

y
K
H
K

K
;
whereH
K
is
0
B
B
B
B
B
B
B
@
0
0
f(K) 0
3
f

(K)

0
f

(K) 0
1
0
0
1
0
0
f(K)

3
f(K) 0
0
f

(K) 0
1
C
C
C
C
C
C
C
A
; (3.1)
where
y
K
=

b
y
1K
;a
y
1K
;a
y
2K
;b
y
2K

are the local orbital field operators, f(K) =
P
3
i=1
e
iK
i
, and
1
=
a
2
(1;
p
3),
2
=
a
2
(1;
p
3),
3
= a(1; 0) are nearest-neighbor
in-plane displacement vectors (a is the lattice constant). Fig. 3.2 shows the 1st Brillouin
zone in momentum space with reciprocal vectorsb
1
=
2
3a
(1;
p
3) andb
2
=
2
3a
(1;
p
3).
Since only low energy excitations are of interest here, we expandf(K) nearK andK
0
(up to a phase factore
i=6
),
f(K)'
3a
2
atK; f(K)'
3a
2

atK
0
;
where = k
x
+ ik
y
, k = (k
x
;k
y
) is a small momentum deviation from K, K
0
, and
jkj jKj;jK
0
j.
54
K
K
′
K
x
K
y
Γ
Λ
dΛ
Ε
c
HKL
Ε
f
HKL
K, K
'
E
Γ
1
Γ
1
Figure 3.2: (a)K =
2
3a
(1;
1
p
3
) andK
0
=
2
3a
(1;
1
p
3
) are the two points, constituting the
Fermi surface of the non-interacting system. is the momentum cutoff of the theory,d
is a thin shell contain high energy modes to be integrated out. (b)
c
(K) and
f
(K) are
the dispersion energy of the conduction and the valence band respectively. The other two
bands are gapped by
1
.
In the following discussion, the trigonal warping term
3
will be neglected. (The jus-
tification for this will be discussed in the Sec. 3.6). As shown in Fig. 3.2, the resulting
tight-binding band structure consists of 4 bands. Two of these bands are gapped by
1
from the FS, whereas the other two bands touch each other at theK andK
0
Fermi points.
This is similar to single-layer graphene, but for the bi-layer case the energy dispersion is
quadratic at the Fermi surface,

c;f
(K)'
v
2
F

1
k
2
atK;K
0
: (3.2)
In the following analysis of instabilities, the gapped bands will be ignored, because
they are not important in the low energy limit. Before writing down the model Hamilto-
nian, let us introduce the creation (annihilation) operators for electrons in bands
c
(K) and
55

f
(K) to bec
K
(c
y
K
) andf
K
(f
y
K
) respectively. c
K
andf
K
are linear combinations
of the local orbital field operators (b
1K
;a
1K
;a
2K
;b
2K
):
c
K
=C
c
K

K
; f
K
=C
f
K

K
; (3.3)
where C
I
K
= (C
I
1K
;C
I
2K
;C
I
3K
;C
I
4K
). These coefficients near the Fermi surface can be
found in Ref. [73]. Then, the non-interacting part of the model Hamiltonian under the
band represention is
H
0
=
X
K

c
(K)c
y
K
c
K
+
f
(K)f
y
K
f
K
; (3.4)
where summing over all is implicitly assumed.
3.2.1 Coupling Constants and the Interacting Hamiltonian
In this section, we show how the interacting Hamiltonian for the BLG model is con-
structed. To accomplish this, we approximate the Bloch wave function by using the p
z
orbital(r) =
p

5
=ze
r
with = 1:72a
1
0
,[95]
'
IK
(r) =
MN
X
mn;i
C
I
iK
p
N
uc
e
iKRmn
(rR
mn
t
i
); (3.5)
where N
uc
is the number of unit cells, a
0
is the Bohr radius, I = c;f bands, R
mn
=
ma
1
+na
2
denote the center of the unit cell, andt
m
are the basis in the unit cell.K is the
in-plane crystal momentum of BLG in the first Brillouin zone.i = 1 represents the siteb
2
witht
1
= (0; 0;c),i = 2 represents the sitea
2
witht
2
= (2a; 0;c),i = 3 represents
56
the sitea
1
witht
3
= (2a; 0; 0), andi = 4 represents the siteb
1
witht
4
= (a; 0; 0) (see Fig.
3.1).a is the lattice spacing 1:57
˚
A, andc is the layer separation 3:35
˚
A.
Graphene can be considered as a 2D material, but the electrons still live in 3D real
space. In order to obtain correct interaction terms between electrons, we start from the
original Hamiltonian[1](Born-Oppenheimer approximation is used), which describes the
electrons with Coulomb interaction in real space representation,
H
full
=
Z
d
3
ra
y

(r)[
~
2
2m
r
2
+V
ext
(r)]a

(r)+
1
2
Z
d
3
rd
3
r
0
a
y

(r)a
y

0
(r
0
)V
int
(rr
0
)a

0(r
0
)a

(r);
wherea
y

(r) (a

(r)) are local field operator which create (annihilate) an electron atr.V
ext
is the potential produced by the ions.V
int
is the Coulomb potential between electrons atr
andr
0
.
Now we approximate (the gapped bands are neglected) the full operator a

(r) by
expanding it into Bloch waves from (3.5). Then we have
a
y

(r)'
X
K
['

c;K
(r)c
y
K
+'

f;K
(r)f
y
K
]; a

(r)'
X
K
['
c;K
(r)c
y
K
+'
f;K
(r)f
y
K
]: (3.6)
By using
[
~
2
r
2
2m
+V
ext
(r)]'
c;f;K
(r) =
c;f
(K)'
c;f;K
(r)
and substituting the above equation intoH
full
, one can easily obtainH
0
in (3.4) andH
int
in (3.15) and (3.8). The coupling constant is determined by
U(K
3
K
4
K
2
K
1
) =
Z
d
3
rd
3
r
0
V
int
(rr
0
)'

I
3
K
3
(r)'

I
4
K
4
(r
0
)'
I
2
K
2
(r
0
)'
I
1
K
1
(r): (3.7)
57
By substituting (3.5) into (3.7), one can also verify that the valley and particle-hole symme-
tries still hold. Note that projecting out the gapped bands introduces a hard cutoff, further
modifyingU, which should screen the original long-range Coulomb interaction[81].
3.2.2 Interactions in the BLG Hamiltonian
From Eqs. (3.7) we find ten inequivalent interaction terms, which are not ruled out by
symmetry and conservation laws,[87] which are those in (3.15) and
U
4
(K
3
K
4
K
2
K
1
)f
y
K
3
c
y

0
K
4
c

0
K
2
c
K
1
U
4
(K
3
K
4
K
2
K
1
)c
y
K
3
f
y

0
K
4
f

0
K
2
f
K
1
U
5
(K
3
K
4
K
2
K
1
)c
y
K
3
c
y

0
K
4
c

0
K
2
f
K
1
U
5
(K
3
K
4
K
2
K
1
)f
y
K
3
f
y

0
K
4
f

0
K
2
c
K
1
(3.8)
Of these,U
4
andU
5
are irrelevant under RG tree level, because the first leading non-
vanishing term in the small momentum expansion at the FS is orderk. We will see later in
the RG analysis that momentum dependent interaction terms are irrelevant. In the follow-
ing, we will proveU
4
(K;K;K;K) = 0, and the proof for different valley combination and
U
5
is similar. After the proof, (3.15) will therefore conclude our interacting Hamiltonian.
Using (3.7), we have
U
4
(K;K;K;K) =
Z
d
3
xd
3
x
0
'

vK
(x)'

cK
(x
0
)V
int
(xx
0
)'
cK
(x)'
cK
(x
0
) (3.9)
Note thatC
v
K
=
1
p
2
(1; 0; 0; 1) andC
c
K
=
1
p
2
(1; 0; 0;1) atK =K;K
0
. We have exploited
the gauge freedom of the Bloch waves to ensure that we work with a Bloch wave basis set
58
that is smooth at all points inK space, i.e., well localized Wannier function can be obtained
[59, 60](This is also required to ensure that the expansion in (3.14) is well defined). Now
we use Eq. (3.5) to write out the Bloch wave function explicitly,
U
4
(K;K;K;K) =
X
m
1
n
1
X
m
0
1
n
0
1
X
m
2
n
2
X
m
0
2
n
0
2
Z
d
3
xd
3
x
0
e
iK(Rm
1
n
1
+R
m
0
1
n
0
1
Rm
2
n
2
R
m
0
2
n
0
2
)

V
int
(xx
0
)
v
(xR
m
1
n
1
)
c
(x
0
R
m
0
1
n
0
1
)
c
(x
0
R
m
0
2
n
0
2
)
c
(xR
m
2
n
2
);
(3.10)
where
v
(x) =
1
p
2
[(xt
1
)(xt
4
)],
c
(x) =
1
p
2
[(xt
1
) +(xt
4
)], and is
some constant. Applying changes of variables,x!x+(t
1
+t
4
) andx
0
!x
0
+(t
1
+t
4
),
and focusing on the
v
(xR
m
1
n
1
)
c
(xR
m
2
n
2
) product term in the integrand.

v
(xR
m
1
n
1
)
c
(xR
m
2
n
2
)
=
1
2
((xR
m
1
n
1
t
1
)(xR
m
2
n
2
t
4
))
((xR
m
1
n
1
t
1
) +(xR
m
2
n
2
t
4
))
!
1
2
((xR
m
1
n
1
+t
4
)(xR
m
2
n
2
+t
1
))
((xR
m
1
n
1
+t
4
) +(xR
m
2
n
2
+t
1
)):
Next, we perform changes of variables for m and n, m! Mm and n! Nn.
Therefore,R
mn
!R
MN
R
mn
and this does not affect anything but,

v
(xR
m
1
n
1
)
c
(xR
m
2
n
2
)!
1
2
((xR
MN
+R
m
1
n
1
+t
4
)(xR
MN
+R
m
2
n
2
+t
1
))
((xR
MN
+R
m
1
n
1
+t
4
) +(xR
MN
+R
m
2
n
2
+t
1
))
59
R
MN
can be removed by performing changes of variables x!x +R
MN
and x
0
!
x
0
+R
MN
. Notice that the
z
orbital satisfies(x) =(x). Therefore, we obtain
exactly the same expression as in (3.10), except a minus sign. This meansU
4
must be
vanish at FS. Similarly,U
5
= 0 for the same reason.
From the result of the RG tree level, all couplings with smallk dependence are irrele-
vant. The first leading non-vanishing term inU
4
,U
5
isO(k). Therefore, the interactions
in (3.8) are irrelevant.
3.2.3 Coupling Constant Expansion
Here we show how to expand the coupling constants aroundK andK
0
. First we perform
a unitary transformation by changing from Bloch wave to Wannier representation. The
Wannier function is defined as
w
I;mn
(r) =
B:Z:
X
K
e
iKRmn
p
N
uc
'
IK
(r); (3.11)
'
IK
(r) =
M;N
X
m;n=0
e
iKRmn
p
N
uc
w
I;mn
(r); (3.12)
whereN
uc
is the total number of unit cells. Also from (3.11) and (3.12), we can derive the
identity
X
mn
e
iKRmn
= (2)
2
N
uc

2
(K) (3.13)
Note that, e
iKRmn
= e
iKRmn+GRmn
, where G is a reciprocal lattice vector. Therefore

(K) in (3.13) is not exactly the Dirac delta function, but equal to a delta function up to a
reciprocal lattice vector.
60
Now, using (3.13) and (3.7), we obtain
U(K
3
K
4
K
2
K
1
) =
1
N
2
uc
X
n
3
m
3
X
n
4
m
4
X
n
2
m
2
X
n
1
m
1
e
iK
3
Rm
3
n
3
e
iK
4
Rm
4
n
4
e
iK
2
Rm
2
n
2
e
iK
1
Rm
1
n
1

Z
d
3
xd
3
x
0
w

I
3
;m
3
n
3
(x)w

I
4
;m
4
n
4
(x
0
)V
int
(xx
0
)w
I
2
;m
2
n
2
(x
0
)w
I
1
;m
1
n
1
(x)
U(K
3
K
4
K
2
K
1
) =
1
N
2
uc
X
n
3
m
3
X
n
4
m
4
X
n
2
m
2
X
n
1
m
1
e
i(K
4
+K
3
K
2
K
1
)Rm
4
n
4

e
iK
3
(Rm
3
n
3
Rm
4
n
4
)
e
iK
2
(Rm
2
n
2
Rm
4
n
4
)
e
iK
1
(Rm
1
n
1
Rm
4
n
4
)

Z
d
3
xd
3
x
0
w

I
3
;m
3
n
3
(x)w

I
4
;m
4
n
4
(x
0
)V
int
(xx
0
)w
I
2
;m
2
n
2
(x
0
)w
I
1
;m
1
n
1
(x)
by changes of variablesx!x +R
m
4
n
4
,x
0
!x
0
+R
m
4
n
4
and using (3.13),
U(K
4
K
3
K
2
K
1
) =
(2)
2
N
uc

2
(K
4
+K
3
K
2
K
1
)
X
n
0
3
m
0
3
X
n
0
2
m
0
2
X
n
0
1
m
0
1
e
iK
3
R
m
0
3
n
0
3
e
iK
2
R
m
0
2
n
0
2
e
iK
1
R
m
0
1
n
0
1
Z
d
3
xd
3
x
0
w

I
3
;m
0
3
n
0
3
(x)w

I
4
;00
(x
0
)V
int
(xx
0
)w
I
2
;m
0
2
n
0
2
(x
0
)w
I
1
;m
0
1
n
0
1
(x)
(3.14)
The coupling constant expansion can be achieved by expanding e
iK
3
R
m
0
3
n
0
3
,
e
iK
2
R
m
0
2
n
0
2
, e
iK
1
R
m
0
1
n
0
1
in Eq. (3.14).

(K
4
+K
3
K
2
K
1
) means that momentum
is conserved up to a reciprocal lattice vector. Therefore, the real conserved quantity is
not the electrons real momentum. This is expected since the the ionic period potential is
the external applied force on the electronic system. We make two remarks here. First,
the 2-body interactions allow Umklapp processes, since the momentum only conserved up
to a reciprocal lattice vector. Second, the conservation of crystal momentum is expected,
because the in-plane lattice translational invariance manifest in the system.
61
Turning to the interaction part of the Hamiltonian, we follow the approach outlined in
Ref.[87] (for more details see Sec 3.2.1), and require particle-hole symmetry of exchang-
ing the valence and conduction bands. Then the electron-electron interaction term can be
written as
H
int
=
1
2
X
K
1
;K
2
X
K
3
;K
4
f U
0
(K
3
K
4
K
2
K
1
)c
y
K
3

c
y
K
4

0
c
K
2

0c
K
1

+U
1
(K
3
K
4
K
2
K
1
)c
y
K
3

c
y
K
4

0
f
K
2

0f
K
1

+U
2
(K
3
K
4
K
2
K
1
)c
y
K
3

f
y
K
4

0
f
K
2

0c
K
1

+U
3
(K
3
K
4
K
2
K
1
)f
y
K
3

c
y
K
4

0
f
K
2

0c
K
1

g
+fexchange (c$f)g;
(3.15)
where momentum conservation is implicitly contained inU (see Sec. 3.2.1). Here, the
coupling constantU
0
denotes the intra-band interaction, whereasU
1
,U
2
andU
3
are inter-
band interactions.
So far, no explicit advantages are obvious by using the Bloch wave basis. In addition,
the momentum dependence in the coupling constants complicates the study too. However,
this complication will be removed due to the trivial topology of the FS.
3.3 Renormalization Group Analysis of The BLG Model
Here, we apply the pertubative renormalization group (RG) method to explore the low-
energy physics of the BLG model in the presence of interactions, following the standard
procedure outlined in Ref. [81].
62
3.3.1 Action of the Model Hamiltonian
The RG transformation is performed using the path integral formalism. Therefore, the
model Hamiltonian from Section 2 should be rewritten into action form. S =
R
dL =
S
0
+S
int
, whereS
0
is the free action andS
int
contains the interaction terms. The derivation
is tedious, and one needs to introduce coherent states of the creation and annihilation field
operators[81]. However, the result is simple, which can be achieved by replacing the field
creation and annihilation operator by Grassmann fields. Namely,c
y

!

,c

!

and
f
y

!

,f!

. Therefore, theS
0
andS
int
can be written as
S
0
=
Z
1
1
d!
2
Z
jKKj
jKK
0
j
d
2
K
(2)
2
2
4

(K;!)(i!
c
(K))

(K;!)
+

(K;!)(i!
v
(K))

(K;!)
3
5
and
S
int
=
1
2

4
Y
i=1
Z
1
1
d!
i
2
Z
jK
i
Kj
jK
i
K
0
j
d
2
K
i
(2)
2

2(!
1
+!
2
!
3
!
4
)

U
0
(K
3
K
2
K
2
K
1
)

(K
3
;!
3
)

0(K
4
;!
4
)

0(K
2
;!
2
)

(K
1
;!
1
)
+U
1
(K
3
K
4
K
2
K
1
)

(K
3
;!
3
)

0(K
4
;!
4
)

0(K
2
;!
2
)

(K
1
;!
1
)
+U
2
(K
3
K
4
K
2
K
1
)

(K
3
;!
3
)

0(K
4
;!
4
)

0(K
2
;!
2
)

(K
1
;!
1
)
+U
3
(K
3
K
4
K
2
K
1
)

(K
3
;!
3
)

0(K
4
;!
4
)

0(K
2
;!
2
)

(K
1
;!
1
)

+ [exchange ( $)]
(3.16)
Now we can perform the RG analysis. First, S
0
is chosen to be the fixed point in
the theory. This choice will determine the scaling properties of ! and , which will be
discussed in the following section.
63
3.3.2 Scaling Properties and Effective Action at the Tree Level
Since we interested in low energy limit, only the energy modes in the vicinity of Fermi
points are considered (see Fig. 3.2). Expanding
c;f
(K) aroundK andK
0
, and combining
with the results from (3.2),
S
0
=
X
=K;K
0
Z
1
1
d!
2
Z
jkj
d
2
k
(2)
2
2
4

(k;!)(i!
v
2
F

1
k
2
)

(k;!)
+

(k;!)(i! +
v
2
F

1
k
2
)

(k;!)
3
5
: (3.17)
We introduce a short hand notation

(k;!) =

( +k;!),

(k;!) =

( +
k;!),

(k;!) =

( +k;!),

(k;!) =

( +k;!), and =K;K
0
is known as
the ‘valley’ degree of freedom.Valley index is similar to L (left) and R (right) index in the
one dimensional case.[81]
The RG transformation is simply integrating out the high energy modes of

(k;!),

(k;!) and

(k;!),

(k;!) which lie within the thin shell,d, in Figure 3.2, and
considering how these modes affect the low energy theory. After integrating out, only
those mode withjk
0
j d = =s remain in the theory. In order to evaluate what
has changed from the original theory,k
0
must be rescaled (k
0
= sk) back to the original
available phase space such thatjkj . SinceS
0
is the fixed point, this requires that!,

;
(k;!) and
;
(k;!) must be rescaled,
!
0
=s
2
!;
0

(k
0
;!
0
) =s
3

(k
0
=s;!);
0

(k
0
;!
0
) =s
3

(k
0
=s;!):
With this scaling relation, one can now ask how the coupling constantsU
0
,U
1
,U
2
,
U
3
,U
4
andU
5
scale under the RG transformation. Again, we use Eq. (3.14) to expand
64
couplings around K and K
0
. By enforcing momentum conservation, only the constant
term in the expansion do not renormalize to zero (marginal under tree level).
Note thatS
int
remains unchanged whenK
4
$ K
3
andK
2
$ K
1
simultaneously. In
addition, using time reversal symmetry (valley symmetry) in the model, hence exchanging
K$ K
0
in (Table 3.1) will not produce another set of independent coupling constants.
Thus we obtainS
int
at the tree level,
1
2

4
Y
i=1
Z
1
1
d!
i
2
Z
jk
i
j
d
2
k
i
(2)
2

(2)
2

2
(k
1
+k
2
k
3
k
4
)2(!
1
+!
2
!
3
!
4
)

2
6
6
6
4
h
0

K
(k
3
;!
3
)

K
0(k
4
;!
4
)
K
0(k
2
;!
2
)
K
(k
1
;!
1
)
+h
1

K
(k
3
;!
3
)

K
0

0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
) +exchange (K$K
0
)
+h
2

K
0

(k
3
;!
3
)

K
0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
)
3
7
7
7
5
+
2
6
6
6
4
g
0

K
(k
3
;!
3
)

K
0(k
4
;!
4
)
K
0(k
2
;!
2
)
K
(k
1
;!
1
)
+g
1

K
(k
3
;!
3
)

K
0

0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
) +exchange (K$K
0
)
+g
2

K
0

(k
3
;!
3
)

K
0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
)
3
7
7
7
5
+
2
6
6
6
4
u
0

K
(k
3
;!
3
)
K
0(k
4
;!
4
)
K
0(k
2
;!
2
)
K
(k
1
;!
1
)
+u
1

K
(k
3
;!
3
)
K
0

0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
) +exchange (K$K
0
)
+u
2

K
0

(k
3
;!
3
)
K
0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
)
3
7
7
7
5
+
2
6
6
6
4
v
0

K
(k
3
;!
3
)

K
0(k
4
;!
4
)
K
0(k
2
;!
2
)
K
(k
1
;!
1
)
+v
1

K
(k
3
;!
3
)

K
0

0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
) +exchange (K$K
0
)
+v
2

K
0

(k
3
;!
3
)

K
0(k
4
;!
4
)
K
0

0(k
2
;!
2
)
K
(k
1
;!
1
)
3
7
7
7
5
+ [exchange ( $)]

:
(3.18)
65
In summary, from the tree level analysis, we find that only a finite set of coupling
constants are marginal. In the low energy limit, only the interacting channels which depend
on K, K
0
are not renormalized to zero. The corresponding bare coupling constants are
listed and classified into Table 3.1.
U
0
U
1
U
2
U
3
U(K,K,K,K) h
0
g
0
u
0
v
0
U(K,K
0
,K
0
,K) h
1
g
1
u
1
v
1
U(K
0
,K,K
0
,K) h
2
g
2
u
2
v
2
Table 3.1: Bare coupling parameters that marginal at tree level.
Here, the subscripts 0, 1, 2 of the coupling constants indicate the various scattering
processes between valleys. The difference between processes with 0, 1 versus 2 is that after
scattering processes with 0, 1 do not exchange valley indices between two particles, but
processes with 2 do. Therefore, the scattering processes with subscript 2 always involve
large momentum transfers.
Since these coupling constants are marginal, performing one-loop corrections to the
RG flow equations is necessary. Since the interaction is quartic, i.e. involving only two-
body scattering, there are only three distinct channels to transfer momentum. Following
the terminology of Ref. [81], these processes are named ZS, ZS’, and BCS.
The corresponding Feynman diagrams are schematically shown in Fig. 3.3. All modes
in the loop are high energy and need to be integrated out. After rescaling back to the
original phase space volume, the coupling constants are modified, i.e. they are flowing in
a 12 dimensional space of couplings.
In order to have non-vanishing one-loop corrections, in the ZS and ZS’ diagrams the
two propagators in the loop must pair up with a different band. For BCS, both propagators
must pair up within the same band. Those graphs that do not satisfy the above criteria
66
BCS
ZS’ ZS
1
2
3 4
1 2
3
4
1
2
3
4
ω,Q+K
ω,K ω,K
ω,Q
′
+K
−ω,P −K
ω,K
Figure 3.3: Feynman Diagrams: 1; 2; 3; 4 represent the low energy modes with momentum
K
1;2;3;4
, band indexI, valley index, and spin. The momentum inside the loop,K, must
lie within the shelld, andQ =K
3
K
1
,Q
0
=K
4
K
1
,P =K
1
+K
2
. Note that the
interaction lines are suppressed.
contain double poles in the frequency! contour integration. With this, many contributions
of these diagram can be eliminated, thus greatly simplifying the calculation.
In this work, we consider flow equations for the coupling constants up to the one-
loop level. Cumulant expansion and Wick’s contraction are used in the calculation. This
method is convenient to keep track of the prefactor for each different diagram.
The loop momentum integration (bubble diagram) can be evaluated,
Z
2
0
Z

d
dkdk
(2)
2
1
2j
I
(K)j
=
dt
4
0
; (3.19)
where
0
=v
2
F
=
1
, anddt =
d

is the RG running parameter. Therefore, the RG flow rate
equations under one-loop correction are given by the following. Forh
0
;h
1
;h
2
,
d
dt
2
6
6
6
4
h
0
h
1
h
2
3
7
7
7
5
=
1
4
0
2
6
6
6
4
h
2
0
g
2
0
h
2
2
h
2
1
g
2
2
g
2
1
2h
1
h
2
2g
1
g
2
3
7
7
7
5
: (3.20)
67
Foru
0
;u
1
;u
2
,
d
dt
2
6
6
6
4
u
0
u
1
u
2
3
7
7
7
5
=
1
4
0
2
6
6
6
4
u
2
0
+u
2
2
+g
2
0
+g
2
2
u
2
1
+g
2
1
2u
0
u
2
+ 2g
0
g
2
3
7
7
7
5
: (3.21)
Forv
0
;v
1
;v
2
,
d
dt
2
6
6
6
4
v
0
v
1
v
2
3
7
7
7
5
=
1
4
0
2
6
6
6
4
2(u
0
v
0
)v
0
+ 2(u
2
v
1
)v
1
+ 2(g
2
g
1
)g
1
2u
2
v
0
+ 2(u
0
2v
0
)v
1
+ 2(g
2
g
1
)g
0
2(u
1
v
2
)v
2
+ 2(g
1
g
2
)g
2
3
7
7
7
5
: (3.22)
Forg
0
;g
1
;g
2
,
d
dt
2
6
6
6
4
g
0
g
1
g
2
3
7
7
7
5
=
1
4
0
2
6
6
6
4
2g
0
(u
0
h
0
v
0
) + 2g
1
(u
2
2v
1
) + 2g
2
(v
1
+u
2
)
2g
0
(u
2
v
1
) + 2g
1
(u
0
+u
1
h
1
2v
0
) + 2g
2
(v
0
h
2
)
2g
0
u
2
+ 2g
1
(v
2
h
2
) 2g
2
(h
1
2v
2
+u
0
+u
1
)
3
7
7
7
5
:
(3.23)
Ifg
0
= g
1
= g
2
= 0, these RG flow rate equation can be solved exactly, and decoupled
into a simple result,
d
dt
2
6
6
6
4
h
0
h
1
+h
2
h
1
h
2
3
7
7
7
5
=
1
4
0
2
6
6
6
4
h
2
0
(h
1
+h
2
)
2
(h
1
h
2
)
2
3
7
7
7
5
; (3.24)
d
dt
2
6
6
6
4
u
0
+u
2
u
1
u
0
u
2
3
7
7
7
5
=
1
4
0
2
6
6
6
4
(u
0
+u
2
)
2
u
2
1
(u
0
u
2
)
2
3
7
7
7
5
; (3.25)
68
and
d
dt
2
6
6
6
4
(u
0
2v
0
) + (u
2
2v
1
)
(u
0
2v
0
) (u
2
2v
1
)
u
1
2v
2
3
7
7
7
5
=
1
4
0
2
6
6
6
4
((u
0
2v
0
) + (u
2
2v
1
))
2
((u
0
2v
0
) (u
2
2v
1
))
2
(u
1
2v
2
)
2
3
7
7
7
5
: (3.26)
Before completing this section, we need to address the effects of quadratic perturba-
tions. The two most relevant perturbations are the chemical potenal and trigonal warping,
i.e. the
3
hopping term. These perturbations are in principle relevant under the tree level,
i.e. scaling ass
2
ands respectively. The chemical potential determines the density of the
system, and the trigonal warping splits the original two Fermi points into four.
However, the divergences of the susceptibilities (see next section) emerge at some finite
energy scale, and the RG flow must be stopped at this point. This energy scale determines
the ordered state mean field transition temperatureT
c
. IfT
c
is far above the trigonal warp-
ing reconstruction energy, this quadratic perturbation is not significant. This introduces an
infrared cut-off to the validity of the analysis. [50, 92, 101]. For the chemical potential,
although one should follow the procedure in Ref. [81] to fine-tune the chemical poten-
tial to keep the system density fixed. Not carrying out this procedure does not affect the
results significantly, since the divergences imply that the original FS is unstable towards
reconstruction (which is eventually opening a gap).
Because of these reasons, we argue that trigonal warping and the chemical potential do
not play an essential role in the analysis at the one-loop level, as long as the energy scale
of the instabilities is found to be far beyond the infrared limit.
69
3.4 Susceptibilities and Possible Ground States
To gain further understanding into the physics of BLG, we introduce test vertices into
the original Hamiltonian. [18, 70] These test vertices are perturbations of the original
Hamiltionian, and we will test the relevancy of these perturbations. These test vertices
correspond to the pairing susceptibilities,

j
P
k
c
y
s;k

0

ss
0f

0
s
0
;k
; (3.27)

SC
P
k
h
c
s;k

x

0

y
ss
0
c

0
s
0
;k
+f
s;k

x

0

y
ss
0
f

0
s
0
;k
i
(3.28)

SC
0
P
k
h
c
s;k

x

0

y
ss
0
c

0
s
0
;k
f
s;k

x

0

y
ss
0
f

0
s
0
;k
i
(3.29)
where, ; = 0;x;y;z,
0
=
0
are 2 2 identity matrices,
x;y;z
and
x;y;z
are 2 2
Pauli matrices. denotes the valley degree of freedom with basis (K;K
0
), and denotes
the spin degree of freedom.j indicates the different pairings listed in Table 3.2.
Performing an RG analysis at the one-loop level with this additional new perturbed
Hamiltonian, the vertices (s) are renormalized, and the new renormalized vertices are of
the form

Ren
j
=
j
(1 +
1
4
0

j
lns); (3.30)
where the
j
are listed in Table 3.2.
3.4.1 Case I:g
0
=g
1
=g
2
= 0
Ifg
0
=g
1
=g
2
= 0, Eq. (3.24) decouples the susceptibilities, and one obtains

g=0
j
(t) =

g=0
j
(0)
1
1
4
0

g=0
j
(0)t
. (3.31)
70
j

j
AFM
0

z
u
0
+g
0
+u
2
+g
2
AFM
0

z

z
u
0
+g
0
(u
2
+g
2
)
SDW
x

z
u
1
+g
1
EI
0

0
(u
0
2v
0
) + (u
2
2v
1
) + (g
0
(g
2
2g
1
))
EI
0

z

0
(u
0
2v
0
) (u
2
2v
1
) + (g
0
+ (g
2
2g
1
))
CDW
x

0
u
1
+g
1
2(v
2
+g
2
)
SC
x

y
(h
1
h
2
+ (g
1
g
2
))
SC
0

x

y
(h
1
h
2
(g
1
g
2
))
Table 3.2: Pairing susceptibilities corresponding to the competing ground states in
the presence of interactions. The corresponding ordered states are anti-ferromagntism
(AFM), excitonic insulator (EI), spin-density-wave (SDW ), charge-density-wave
(CDW ), s-wave superconductivity (SC), and and extended s-wave superconductivity
(SC
0
). The prime inAFM
0
andEI
0
indicates the order state with valley symmetry break-
ing.
Whether and where the susceptibilities (
j
lns, wheret = lns) diverge is determined by
the bare coupling constants. Each divergence in
g=0
j
(t) indicates that the system has a
tendency toward the corresponding ordered state, labeled by ’j’. The first instability in a
given channel represents the most dominant ordered state of the system at low energy.
For the caseg
0
= g
1
= g
2
= 0, the situation is relatively simple. In order to produce
instabilities,
j
(0) must be positive, such that mean field solutions exist, and the suscep-
tibilities given in (3.31) can diverge at some finite t. If only repulsive interactions are
considered, theAFM andSDW channels are expected to represent the dominant insta-
bilities, because in the other channels some level of fine-tuning in the bare parameters is
required to ensure
j
(0)> 0. Under these stringent condition, the parameters span only a
small region in the parameter space.
More generally, to do better prediction, constraining the search is desirable in order to
make the exploration and analysis of the phase diagram meaningful. The relative strength
71
between the bare couplings can be estimated. In general, the scattering processes within
the same valleyh
0
,g
0
,u
0
,v
0
andh
1
,g
1
,u
1
,v
1
are expected to be larger thanh
2
,g
2
,u
2
,
v
2
, because intra-valley scattering processes involve only small momentum transfer.[17]
Applying these constraints, in Fig. 3.4 we show how these pairing susceptibilities compete
with each other for a representative choice of bare coupling parameters. In this example,
the dominant low-energy divergence occurs in the AFM channel, followed by AFM
0
,
SDW ,CDW andEI
0
.
0.2 0.4 0.6 0.8 1.0 1.2 1.4
5
10
15
SC
CDW
EI'
EI
SDW
AFM'
AFM
G
j
‘h
0
t
Figure 3.4: Flow of the susceptibilities: Here, we set h
0
= u
0
= v
0
, h
1
= u
1
= v
1
=
0:8h
0
,h
2
= u
2
= v
2
= 0:1h
0
, andg
0
= g
1
= g
2
= 0 (h
0
> 0) . TheFM instabilities
occupy a large region in parameter space, and fine-tuning is not necessary.
The instabilitiesAFM,AFM
0
, andSDW indicate broken spin symmetry, thus lead-
ing to magnetically ordered ground states. Since the pairing in (3.27) is a pairing of dif-
ferent bandsc

,f

, it does not have an obvious connection with the spin density operator.
72
However, it can be related to local magnetization in a more sophisticated manner. To
illustrate this, we follow Ref. [9], and define a local spin operator by
S(r) =
X
ss
0
a
y
s
(r)
ss
0a
s
(r);
where is (
x
;
y
;
z
), and a
y
s
(r) (a
s
(r)) represents local field creation (annihilation)
operators with spins. An explicit expression for these operators is given in Eq. (3.6). The
local magnetization can then be expressed in terms of the spin operator,
M(r) =
g
B
V
X
ss
0
hS(r)i (3.32)
where the average is taken with respect to the dominant ground state obtained from the
RG. Here,g is theg-factor,
B
is the Bohr magneton, andV is the volume of the system.
If we expand the local field operator into Bloch waves (3.6), we obtain
M(r) =
g
B
V
X
ss
0
c;f
X
A;B
X
K
2
K
1
u

A;K
2
(r)u
B;K
1
(r)e
i(K
2
K
1
)r
k
hA
y
K
2
;s

ss
0B
K
1
;s
0i; (3.33)
where the Bloch wave function is'
A;K
(r) =e
iKr
k
u
A;K
(r),r = r
?
+r
k
,r
?
is the out-
of-plane vector, r
k
is the in-plane vector, and u
A;K
(r) is a periodic function with r
k
!
r
k
+ma
1
+na
2
, wherem,n are integers.
Let us also introduce aSDW gap function (we will discuss this result in the section
about Mean field analysis),

sdw
ss
0
g
1
+u
1
=
X

0
;k
S
z
hc
y
s;k

x

0

z
ss
0f

0
s;k
i: (3.34)
73
From the gap equation, only when A
y
pairing with B in different bands has non-zero
contribution. Therefore, we observe that
Z
unit cell
d
3
rM(r) = 0 (3.35)
This equation follow because of Bloch’s wave function in different bands are orthogonal
to each others. Therefore, this implies that the spin density is modulating within a prim-
itive unit cell. Furthermore, in a unit cell, which must contains different (opposite) net
spin direction. This ordering pattern are replicated exactly for each unit cell inAFM and
AFM
0
, and which lead to the anti-ferromagnatism. ForSDW , it is replicated in stagger-
ing manner.
Using this formulation, pairing in theSDW channel (c
y
Ks
(k)
z
ss
0f
K
0
s
0(k)) can be easily
identified by this observable with ordering vectorQ. Similary, forCDW , we introduce
the local charge density operator,
(r) =
1
V
X

ha
y

(r)a

(r)i: (3.36)
From the mean field Hamiltonian, the trial ground state solutions forEI andEI
0
are
equivalent to excitonic insulator given in Ref. [38]. In these insulating states, the electrons
from the conduction band and the holes from the valence band form bound states.
Furthermore, AFM
0
and EI
0
break valley symmetry, i.e. time reversal symmetry,
because in these ground states, the symmetry exchanging K and K
0
is absent. This can
lead to non-trivial insulating states[41, 78, 79].
74
3.4.2 Case II:g
0
,g
1
,g
2
6= 0
For non-vanishing values ofg
0;1;2
, much of the discussion is similar to the previous section.
However, sinceg
0;1;2
connect different channels in the flow rate equations, they do not give
simple analytical results that show how the
j
evolve. Instead, in this more general case
the flow rate equations in (3.20-3.23) need to be solved numerically.
Due to the large parameter space spanned by the possible sets of bare couplings, it is
impossible to explore the entire phase diagram. In this section, we only select a region to
scan, illustrating how non-zero values ofg
0;1;2
affect the results from the previous section.
Using the bare values from Fig. 3.4, we scang
0
andg
1
. Wheng
0;1:2
is small, we obtain
results very similar to theg
0;1;2
= 0 case, withAFM occupying large regions of the phase
diagram. However, wheng
1
becomes large, we instead obtain the more complicated phase
diagram shown in Fig. 3.5.
To form an excitonic insulator,EI andEI
0
order intricately compete with other insta-
bilities. Without the scattering processes g
0;1;2
(case I), fine-tuning bare couplings to
enhance EI and EI
0
instabilities and suppress the others is inevitable. However, intro-
ducing nonzerog
0;1;2
, the flow ofg
0;1;2
significantly affects this result, which can automat-
ically enhance or suppress the orders. For instance, wheng
1
is small, theg
0
starts flowing
towards more positive values (see Fig. 3.6). Consequently, theAFM ordering tendency is
enhanced (which enhances the divergence of
AFM
). On the other hand, in Fig. 3.6, when
g
1
becomes large, g
0
flows towards increasingly negative values, this suppressingAFM
order. Because of this suppression,EI,EI
0
andSDW emerge in the largeg
1
region as
shown in Fig. 3.5.
75
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æ æææææ
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æææ
æ
æ
àà
à
ààà à
ààààà
àààà àà
àààààààààà
àààààààààà
àààààààààà àà
ààààààààààààààà
àààààààààààààà
àààààààààààààà àà
ààààààààààààààà
àààààààààààààààà à
ààààààààààààààààààààà
àààààààààààààààààààà
ààààààààààààààààààààà
ààààààààààààààààààààààà
àààà ààààààààààààààààààààà
ààààààààààààààààààààààààààààààà
ààà ààààààààààààààààààààààà
àààààààààààààààààààààààà àà
ààààààààààààààààààààààààààà à
àààààààààààààààààààà à à à
à àààààààààààààààààààààààààà àààà à
àààààà ààààààààààààààààààààààààà à
àààààààààààààààààààààààààààààà à à à
àà àààààààààà àààààààààààààààààààà
ààààà àààààààààààààààààààà à
ààààààààààààààààààààààààààà àààà
ì
ì
ì
ì
ì
ì
ì
ìì ì ì ì
ì ì
ì ì ì
ì ì ì ì
ììì ì
ìì ìì
ì ì ìì ì
ì ì
ìì ì ìì
ì ì ì ì ì ì
ì ì ì ìì ìì ì ì
ì ìì ì ìì ìì ì ì ì
ì ì ì ì ìì ì ìì ì
ì ì ì ìì ìì ì ì ì
ì ì ì ìì ì ìì ìì ììì
ì ìì ì ì ì ì ì ì ì
ì ìì ì ì ì ì ìì ì ì ì ì ì ì ìì
ìì ì ì ì ììììì ì ì ì ì
ò
ò
ò
ò ò
òò
òò òò
òò ò ò
ò ò
ò ò òòò
òò ò
òò ò
òò òò òò
òò ò òò ò ò
òò ò ò ò
0.2 0.4 0.6 0.8 1.0
g
0
� h
0
0.2
0.4
0.6
0.8
1.0
g
1
� h
0
ò EI'
ì EI
à SDW
æ AFM
Figure 3.5: Phase Diagram for a representative choice of bare couplingsh
0
=u
0
=v
0
=
1, h
1
= v
1
= u
1
= 0:8h
0
, and h
2
= u
2
= v
2
= g
2
= 0:1h
0
. The phase diagram is
determined by monitoring which channels divergence first during the RG flow.
Furthermore, charge density wave order is very unlikely to dominate, since

CDW
(0) =
SDW
(0) 2(v
2
+g
2
) [see Table 3.2], and g
2
always grows into the pos-
itive regime, as long as only repulsive interactions are considered. We observe that the
divergence of spin density wave order is always stronger than charge density wave order.
For the same reason,
AFM
(0) =
AFM
0(0) 2(u
2
+g
2
), and thus FM order is more
favorable thanAFM
0
.
Similarly, superconducting order is not expected to dominate for small bare valuesh
2
andg
2
. In order to produce dominant BCS instabilities one needs thath
2
+g
2
>h
1
+g
1
or h
2
+g
1
> h
1
+g
2
, such that
SC
(0) or
SC
0(0) is positive. However, as we argue
before, imposin such condition on the parameters we need the large momentum transferred
processes stronger than the small momentum one. This may not intuitive to establish.
76
0.2 0.4 0.6 0.8
-4
-2
0
2
4
6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
g
1
� h
0
HbareL
g
0
� h
0
t
Figure 3.6: The flow ofg
0
as the bare value ofg
1
varies. We setg
0
=g
2
= 0:1h
0
, and the
remaining bare couplings are same as Fig. 3.4
3.5 Mean Field Analysis of the Ground States
In this section, we summarize the mean field analysis for the ground states FM, FM
0
,
SDW , EI, EI
0
, and CDW . The idea of the mean field approximation[97] is to guess
a trial ground state which can minimize the total energy of the many-body system. With
a given trial ground state, the original Hamiltonian can be approximated by an effective
quadratic mean field Hamiltonian which can be solved by self-consistent diagonalizing.
The procedure will be briefly shown in the following. First, let the mean field Hamil-
tonian be
H
MF
=H
0
+H
pair
H
0
is the free Hamiltonian given in Eq. (3.4), and
H
pair
=
X

0
;kk
0

0
;
0(k;k
0
)c
y
;k
f

0

0
;k
0 +h:c:
77
For simplicity, since the coupling constants are independent of k and k
0
, we are able to
assume

0(k;k
0
)'
(0)

0
;
0

2
(kk
0
). Therefore, the pairing gap function
(0)

0
;
0
(order parameter) is given by

afm

0
;ss
0
G
afm
=
X
k
hc
y
s;k

0

z
ss
0f

0
s
0
;k
i;

afm
0

0
;ss
0
G
afm
0
=
X
k
hc
y
s;k

z

0

z
ss
0f

0
s
0
;k
i;

sdw

0
;ss
0
G
sdw
=
X
k
hc
y
s;k

x

0

z
ss
0f

0
s
0
;k
i;

ei

0
;ss
0
G
ei
=
X
k
hc
y
s;k

0

ss
0f

0
s
0
;k
i;

ei
0

0
;ss
0
G
ei
0
=
X
k
hc
y
s;k

z

0

ss
0f

0
s
0
;k
i;

cdw

0
;ss
0
G
cdw
=
X
k
hc
y
s;k

x

0

ss
0f

0
s
0
;k
i:
whereG
j
=
j
(0) is a linear combination of the bare coupling constants listed in Table
3.2. To find the trial ground state, a new quasi-particle (mixture ofc;v bands) is introduced
which diagonalizes the mean field Hamiltonian,

;k
=v

0
;
0
k
c
;k
u

0
;
0
k
f

0
;k

;k
=u

0
;
0
k
c
;k
+v

0
;
0
k
f

0
;k
Whereju(k)j
2
+jv(k)j
2
= 1.
f

(k);
y

0

0
(k
0
)g =

0

0
2
(kk
0
)
f

(k);
y

0

0
(k
0
)g =

0

0
2
(kk
0
)
The other commutation relation are zero. Then, the diagonalized Hamiltonian can be
written as
H
MF
=
X
;k
E(k)[
y

(k)

(k)
y

(k)

(k) ]
78
where,
E(k) =
q
j(k)j
2
+j
(0)
j
2
;
and
u

0
;
0
k
=
1
p
2
(

0

0)

1 +
(k)
E(k)
1
2
;
v

0
;
0
k
=
1
p
2

(0)

0
;
0
j
(0)

0
;
0
j
!

1
(k)
E(k)
1
2
;
(k) =j

(k)j is given by Eq. (3.2)
Therefore, using a Hartree-Fock state as the trial ground state for the quasi-particles,
we have
Q
;k

y

(k)j0i =j

(0)i, wherej0i is the state with no particles (vacuum). To
calculate
(0)
, one can applyj

(0)i to calculate the order parameter and obtain the gap
equations,

(0)

0
;
0
=
X
k
G
(0)

0
;
0
q
j(k)j
2
+j
(0)

0
;
0
j
2
which are then solved self-consistently. Note that the mean field method yields only qual-
itative results for low-dimensional systems.
3.6 Discussion
Projecting out the orbital field operatorsa
1K
anda
2K
in (3.3) and taking the spatial con-
tinuum limit, the non-interacting low energy effective model of BLG can be approximated
by a massive chiral fermion model[62]. By symmetry arguments, all possible two-body
interactions can be obtained.[50, 92, 101] Under this limit, this model exhibit a very rich
79
and exotic low energy phenomenology because of the newly emerging valley and pseu-
dospin degree of freedom.
In contrast to the emerging degree of freedom approach, we utilize the band represen-
tation point of view which are not necessary to impose continuum limit. Projecting out
the gapped bands, BLG is effectively viewed as a conventional two-band model and all its
interactions can be immediately obtained according to the band index. In this approach,
the instabilities of BLG is clearly interpreted as the peculiar nature of the FS and the low
energy physics exhibit rich excitonic orders.
The ground states in this paper have been classified according to their band index, and
the pseudospin index is implicitly contained in (3.3). Therefore, the pseudospin symmetry
breaking is not made explicit in our model. This leads to a different physical interpreta-
tion of the ordered states. Because of this reason, not all the of the possible competing
ground states that are found in Ref. [20, 49, 50, 91, 92, 101] can be obtained by our
study, especially the gapless nematic state which emerges naturally in some of the previ-
ous studies.[20, 49, 50, 91, 92, 101]
Furthermore, in Refs. [20, 49] it is discussed that only nine independent coupling
constants are allowed by symmetry. However, within our model twelve independent cou-
pling constants are obtained, because the projection procedure is different in our approach.
Hence, it do not preserve all of the symmetry in pseudospin space, and the interaction
terms in our treatment allow more interaction processes.
In addition, the ‘which-layer’ [68] or pseudospin[64] symmetry breaking is not obvi-
ous in the present approach. As discussed in section 3.4, to extract layer order, it would be
necessary to know the appropriate Bloch wave function or its Wannier representation. The
80
more complicated representation of pairing gap functions in real space are a disadvantage
of our approach.
A recent functional renormalization group (fRG) study[80] has demonstrated the
advantage of retaining all the lattice structure, and integrating out energy modes without
ambiguities. Furthermore, their approach takes into account the complication of angu-
lar dependence in the interactions. Their study has shown an interesting “three-sublattice
CDW instability”. This instability is quickly disappears as the on-site interaction becomes
dominant. Since their model Hamiltonian (extended Hubbard model) is different from the
one studied in this paper, a direct comparison with their results is not straightforward.
The results in this thesis are valid only of the one-loop level. Typically, higher-loop
contributions can be neglected by invoking 1=N arguments.[81] However, this type of
argument is not very strong for this model, because the Fermi surface contains just two
points, resulting inN = 4 only. Also, the results presented here only apply for the weak
coupling limit. Any strong enough coupling to break down the perturbative expansion
will invalidate the preceding discussion. In the strong coupling limit, results from the tree
level analysis cannot be trusted, and using the same effective action as in (3.18) will not
guarantee correct results. When the order of the tree and loop diagrams is comparable,
new effective models and non-perturbative approaches may be needed.
Furthermore, the instabilities in this thesis are driven by perfect particle-hole nesting.
The presence of disorder can destroy this symmetry and thus change the phase diagram.
Moreover, doping away from the charge neutrality point, the FS becomes a line rather
than a few points. As shown in a recent study of doped monolayer graphene,[44] and also
in Ref. [80], functional RG is a promising alternative method to study the BLG doping
problem. In the doped case, functional RG may be superior to our approach, since the
81
shape of FS evolves non-trivially upon doping. In addition, the effects of phonons are
not considered in the presence work. Their inclusion may modify the coupling constants
significantly, possibily turning some interactions from repulsive into attractive.
Another issue which cannot be easily resolved within our framework is due to the
lack of knowledge of the precise values of bare coupling constants at the energy scale

1
=2. Because of this, the assumption of being in the weak coupling regime needs further
justification. As pointed out in Ref. [49], an intuitive argument can be given as follows.
From recent optical experiments,[36, 51, 74, 102] the high-energy regime of BLG have
been found to be well described by a two-band model. This suggests that the interactions
in BLG are not strong enough to break down entirely the quasi-particle picture, with good
momentum quantum numbers. Therefore, the coupling is very likely to be weak. To
compute precise values that include the effects of screening, ab initio calculations would
be required. However, this is beyond the scope of this paper. Since the flow rate equations
are sensitive to the bare values of the coupling constants, working with unknown twelve
coupling constants, pinning down the most stable ground state is a difficult task.
In parameter space, the free part of the action is non-analytic at the point with non-
zero trigonal warping. Due to this reason, perturbative RG may not work properly at that
given point. In particular, the scaling rule of the fermonic field cannot be defined, and one
does not know whether that point is a Gaussian fixed-point or not. Therefore, we enforce
using
3
= 0 as the fixed-point to define the scaling rule, and always treat the trigonal
warping term as a quadratic perturbation. Another recent treatment has included the effect
of trigonal warping.[20]
The results of our model are consistent with recent current transport spectroscopic
experiments.[31, 93] Specifically, a magnetic field dependent gap is expected in a ground
82
state with excitonic order. [25] Therefore, magnetically order ground states are not a
necessary condition to exhibit this property. To make connections with experiments, the
physical properties of the ground states discussed in this thesis need to be analyzed, in
particular how these ground states respond to external perturbations, especially to currents.
Also in the experiments, considering the effects of disorder and boundaries is important.
83
Chapter 4
Iron pnictide and electronic nematicity
4.1 Introdution
A new unconventional superconducting state has recently been discovered in doped iron
pnictide (FeAs) materials .[39, 40, 86] Although FeAs is similar to the cuprates in some
respects, i.e. it is a layered and exhibits magnetic ordering in its parent compound, some
of its electronic properties are fundamentally different. In contrast to cuprates, the elec-
tronic structure of FeAs is effectively described by a multi-band model.[12, 34, 48, 84]
In addition, FeAs is believed to be less correlated than the cuprates. These observations
suggest that their mechanism of superconductivity may be different. In FeAs, the Fermi
surface (FS) breaks down into several pockets. Due to this feature, scattering processes at
low energy can give rise to non-trivial many-body physics. This may explain the origin of
superconductivity in FeAs with only repulsive interactions.[17]
In addition to the superconducting state, another intriguing property in FeAs has
recently been observed. The electronic properties exhibit a directional preference,[14,
15, 16, 22, 43, 66, 88] regardless of the system’s intrinsic 4-fold crystal rotational symme-
try (C
4
symmetry). This anisotropy may be related to nematicity due to the interactions
between electrons.[30] The new phase is metallic and associates with magnetic order. This
observation has led to speculations about the relationship between magnetic fluctuations
and nematic ordering. Some theoretical studies have shown that magnetic fluctuations can
84
play an important role in giving rise to nematic order.[24, 27, 83, 98] The functional renor-
malization group (FRG) point of view[89, 100] provides another interesting perspective to
understand the stability of the metallic phase in FeAs. Because of the intricate geometrical
structure of the FS, the coupling parameters in the forward scattering channels are relevant
under RG transformation instead of just being marginal. This suggests that some non-
trivial scattering processes may occur in the direct channels which can potentially break
the stability of the FS.
Based on these observations, the central goal of this study is to investigate how mag-
netic fluctuations can affect the direct channels in FeAs, and yield instabilities. This chap-
ter is organized as the following. In Sec. 4.2, we will discuss the effective model that
will be used in the study. Sec. 4.3 will show how the magnetic fluctuations affect the
density fluctuations and give rise to instabilities in the isotropic metallic phase. Sec. 4.4,
the nematic order and its magnetic fluctuations will be discussed at the mean field level.
In Sec. 4.5, we will briefly discuss these results in the context of experiments and other
theoretical studies.
4.2 Model Hamiltonian
We extend the model of Ref. [27] by including the forward scattering (density-density)
interactions within intra- and inter- FS pockets. The free part of the Hamiltonian is given
by
H
0
=
X
;k

k
c
y
;ks
c
;ks
; (4.1)
where s is the spin, and represents the location of the FS pockets. = represents
the hole FS pocket with its center located at (0; 0), and = X;Y represent the electron
85
FS pockets with centers located at Q
1
= (; 0) and Q
2
= (0;) respectively (see Fig.
4.1). The energy dispersions are

k
=
0

k
2
2m
,
X
k+Q
1
=
0
+
k
2
x
2mx
+
k
2
y
2my
, and

Y
k+Q
2
=
0
+
k
2
x
2my
+
k
2
x
2my
, wherem,m
x
, andm
y
are the band masses,
0
is the offset
energy, and is the chemical potential. For simplicity, we shift the momentum of theX
(Y ) pocket toQ
1
(Q
2
) i.e.
X
k+Q
1
!
X
k
(
Y
k+Q
2
!
Y
k
). Furthermore, summing over all
spins is implicitly assumed throughout the paper.
Y
X
Q
2
=(0, Π)
Q
1
=(Π, 0)
G
Figure 4.1: A schematic diagram showing the FS structure of a typical FeAS in the
unfolded Brillioun Zone: The bold curves represent the FS of the model in the isotropic
metallic phase. the dotted curves represent the unequally renormalized FS pockets in the
nematic phase withh

i> 0.
In the low temperature limit, the dominant scattering processes only take place in the
vicinity of the FS. Moreover, because of momentum conservation, all scattering processes
are strictly constrained to the available phase space for two-body scattering processes. We
86
are interested in the processes with small momentum transfer (or up to a nesting vector in
exchange channels), as these processes are likely to develop collective modes.[1] There-
fore, keeping only direct, exchange, and BCS channels in the model is sufficient, since
these dominate over the others in phase space.
Although most of the momentum channels are being discarded, there is still ambiguity
which of these three special channels are relevant to the problem. We can further elim-
inate some of them by using the following intuitive arguments. First, for electron-hole
(e-h) interactions, because of the FS pocket nesting spin-density-wave (SDW) fluctuations
are the most important collective mode, especially as the system temperature approaches
the magnetic ordering phase transition temperature (T
N
). Therefore, the exchange channel
in the e-h interactions is kept. Normally, the exchange channel of this interaction breaks
into spin triplet and singlet channels, which correspond to SDW and charge-density-wave
(CDW) respectively. However, the CDW fluctuations are insignificant and can be dis-
carded, because their characteristic energy scale is much lower than for the SDW. By
using similar arguments, all the direct and BCS channels in the e-h interactions can be
discarded. For the electron-electron (e-e) interactions, since we wish to investigate the
instabilities of the isotropic paramagnetic phase (metallic), the direct channels should be
kept. Following the same reasoning for the e-h interactions case, the exchange and BCS
channels are irrelevant in e-e interactions. Therefore, the effective interacting Hamiltonian
becomes
H
int
=
u
spin
2
X
=X;Y
X
q
s
;q
s
;q
+
u
(1)
4
2
X
=X;Y
X
q

;q

;q
+u
6
X
q

X;q

Y;q
;
(4.2)
87
where s
;q
=
P
k
c
y
;k+q;s

ss
0c
;ks
0, is the Pauli matrix vector, and
;q
=
P
k
c
y
;k+q;s
c
;ks
. The first term of Eq. (4.2) combined withH
0
is the original model
given in Ref. [27]. u
spin
is the coupling constant for electron and hole pocket interactions
in the triplet channels. The last two terms are newly introduced, describing the density-
density interactions between theX andY FS pockets.u
(1)
4
is the coupling constant for the
intra-electron pocket interactions, andu
6
is the coupling constant for inter-electron pocket
interactions. The notation of the coupling constants in (4.2) are the same as in Ref. [56],
and angular dependence has been ignored. Because of theC
4
symmetry in the system, the
electron intra pocket interactions [second term in (4.2)] have the same coupling constants.
The above interaction terms only capture dominant scattering processes of the electrons
in the vicinity of the FS. The full interaction terms include arbitrary momentum transfer
contributions as well.[56]
4.3 Instabilities of the isotropic phase
To investigate whether the model in Sec. 4.2 can develop any instability, one can use
rigorous diagrammatic methods. However, we use a simple alternative approach to tackle
the problem. This approach is based on the functional integration formalism, introducing
auxiliary bosonic fields to decouple the fermonic quartic terms (Hubbard-Stratonovich
transformation). By integrating out the fermonic degrees of freedom, one can obtain the
action of these bosonic fields which describes the collective modes of the system at low
energy. In the following, we demonstrate this procedure and investigate the stabilities of
the isotropic metallic phase.
88
4.3.1 Hubbard-Stratonovich Transformation
In order to perform the Hubbard-Stratonovich transformation, it is important to check
whetheru
(1)
4
u
6
is positive or negative. Moreover, we will see later that this is also the
key parameter determining whether instabilities can emerge in the direct channels. For
now, we assumeu
(1)
4
u
6
is negative and setu
6
+u
(1)
4
=u
0
n
andu
6
u
(1)
4
=u
n
(positive).
Then we introduce a set of auxiliary bosonic fields
q
= (
+
q
;

q
;M
X
q
;M
Y
q
), where
q = (i
n
;q) and
n
is the bosonic Matsubara frequency.
By inserting the ‘fat’ unity (4.3) into the partition function integrand (Z =
R
D[

; ]1e
S
),
1 =
Z
D[] exp
"

1
2
X
;q
2
u
spin
M

q
M

q

1
2
X
q

2
u
0
n

+
q

+
q
+
2
u
n

q

q

#
: (4.3)
Here,u
n
must be positive to ensure the Gaussian functional integral in (4.3) is convergent,
and the1 (‘fat’ unity) represents the normalization constant.
The quartic fermonic interaction terms in (4.3) can be generated by shifting the auxil-
iary fields by
M

q
!M

q
+
u
s
4
s

q
;
+
q
!
+
q
+i
u
0
n
4
[
X
q
+
Y
q
];

q
!

q
+
u
n
4
[
X
q

Y
q
]; (4.4)
where s

q
=
P
k

k+q;s

ss
0

ks
0,

q
=
P
k

k+q;s

ks
, and k = (i!
n
;k). !
n
is the
Fermionic Matsubara frequency. Since u
0
n
> 0, the shift in
+
includes an extra i fac-
tor. This is ensures that the generated quartic fermonic interaction terms carry opposite
signs and cancel with those in (4.2).
89
Therefore, the fermonic quartic terms in the action are canceled out by inserting
‘1’ with the shifted auxiliary fields (4.4). By introducing the Nambu spinor

k
=
(

X
k"
;

X
k#
;

Y
k"
;

Y
k#
;

k"
;

k#
), the new action can be compactly expressed as
S =
X
kk
0

k
G
1
kk
0

k
0 +
1
u
spin
X
;q
M

q
M

q
+
X
q

2
u
0
n

+
q

+
q
+
2
u
n

q

q

;
(4.5)
where
G
1
kk
0
=G
1
0;kk
0
+V
kk
0: (4.6)
G
1
0;kk
0
is given by
0
B
B
B
@
G
1
X;k
0 0
0 G
1
Y;k
0
0 0 G
1
;k
1
C
C
C
A

kk
0; (4.7)
andV
kk
0 is given by
0
B
B
B
@
i
+
kk
0
+

kk
0
0 M
X
kk
0
0 i
+
kk
0

kk
0
M
Y
kk
0
M
X
kk
0 M
Y
kk
0 0
1
C
C
C
A
; (4.8)
whereG
1
;k
=i!
n
+

k
, and =X;Y; .

must change sign under exchanging x and y axis to preserve the original C
4
symmetry in (4.5) because the boson-fermion coupling terms. Intuitively, this can be
realized, once the connection with mean field theory has been established. With this
picture, one may identify the bosonic fields as, M

q
/
P
k
h

k+q;s

ss
0

ks
0i and

q
/
P
k
h

X
k+q;

X
k

Y
k+q;

Y
k
i.
90
Integrating out the fermionic fields, we obtain the action for the bosonic fields,
S[] =Tr lnG
1
+
1
2
X
;q
1
u
spin
M

q
M

q
+
1
2
X
q

1
u
0
n

+
q

+
q
+
1
u
n

q

q

:
(4.9)
Note thatM

=M

n, wheren is some arbitrary direction of the spin. Since

change
sign underC
4
rotation, any non-zero ground state expectation value of

implies spon-
taneousC
4
rotational symmetry breaking. Therefore, we can identify

as the nematic
order parameter, and Eq. (4.9) describes how the magnetic and nematic collective modes
interact with each other.
4.3.2 Isotropic metallic phase
The extremum of the action is given by
S[]

q

q =hqi
= 0: (4.10)
Solving this equation involves the inversion ofG
1
. However, without knowing the
functional form of
q
, this is a formidable task, sinceG
1
is generally not diagonal in
momentum-frequency space. Therefore, in order to proceed, we need to guess a solu-
tion which simultaneously satisfies
P
q
G
kq
[G
1
]
qk
0 =I
33

kk
0 and Eq. (4.10). Since we
want to investigate the stability of the metallic phase, one obvious and desirable choice is
hM
X
q
i =hM
Y
q
i = 0 (paramagnetic), andh

q
i = 0 (isotropic). In generalh
+
q
i can be
a nonzero constant. For instance, [100] the FS pockets can shrink or expand according to
its non-zero expectation values. However, the hole and electron FS pockets must be renor-
malized, such that no net charge is introduced into the system (Luttinger theorem).[53]
91
Modifying the FS pockets in this manner does not influence our result qualitatively, since
angular dependence in the coupling constants is ignored, particularly, which do not break
C
4
rotation. Therefore, we assumeh
+
q
i = 0.
Using the saddle point approximation, we expand Eq. (4.9) around the extremum with
h
q
i = 0 and include fluctuations beyond the mean field level. Thus, we obtain
tr lnG
1
= tr ln[G
1
0
(1 +G
0
V)] = tr lnG
1
0

X
n
1
2n
tr(G
0
V)
2n
;
(4.11)
and
V
kk
0 =
0
B
B
B
@

kk
0
0 M
X
kk
0
0

kk
0
M
Y
kk
0
M
X
kk
0 M
Y
kk
0 0
1
C
C
C
A
; (4.12)
whereM
X;Y
kk
0
and

kk
0
are the fluctuations of the collective modes nearh
q
i = 0. This
yields the effective action
S
eff
'
1
2
X
q

q
(
2
u
6
u
(1)
4
+ 2
q
)

q
+
1
2
X
;q
M

q

1
;q
M

q
+
X
x;
(

x
)
2
(M

x
)
2
;
(4.13)
where
q
=
P
k
G
;k
G
;k+q
,
1
;q
=
2
u
spin
+ 2
P
k
G
;k
G
;k+q
, and =
P
k
G
;k
(G
;k
)
3
,
andx = (;x) represent the imaginary time and real space coordinate respectively. Note
that the frequency and momentum dependence in can be discarded because of power
counting[63] as only local interactions are considered.[37] Moreover, the other terms
beyond Gaussian only yield higher order corrections for the action and are not impor-
tant to the problem (before the magnetic order sets in), except in the last term of (4.13),
92
which accounts for the interactions between magnetic and nematic order fluctuations. The
physical meaning of this term is that magnetic fluctuations associated with the fluctua-
tions of the X and Y FS pockets unequally can lower the total energy, since this term
is negative ( is negative). If the magnetic fluctuations are small, these contributions are
negligible. When the magnetic fluctuations become large in the vicinity of the magnetic
ordering transition point, the stability of theh

q
i = 0 isotropic phase can break down.
4.3.3 Instabilities due to magnetic fluctuations
To find the instabilities, we can integrate outM

. For smallq,
1
;q
=r
0
+
j
n
j +f
;q
,
wherer
0
=
1
;q=0
,
is the Landau damping coefficient, andf
;q
= q
2
x
(1) +q
2
y
(1
) +
z
q
2
z
is a generalized anisotropic function with1 < < 1, and the upper (lower)
sign refers to =X ( =Y ).[27] In this paper, we are not interested in the out-of-plane
anisotropy. Therefore, we set
z
to zero. We thus can obtain the effective action for

,
S
eff
=
1
2
X
q

q
(
2
u
n
+ 2
q
)

q
+ tr ln[r
0
+
@

+f
;^ q
+(

x
)
2
];
(4.14)
where ^ q are momentum operators. The trace in the above equation can be evaluated in
momentum-frequency space, yielding
P
;q
ln[r
0
+
j
n
j + f
;q
+

q

q
]: Now,
expanding the logarithmic function with respect to (

)
2
, we can further simplify it
by keeping only the leading term,
S
eff
'
1
2
X
q

q
"
2
u
6
u
(1)
4
+ 2
q
+
X

r
0
+
j
n
j +f
;q
#

q
: (4.15)
93
As the temperature approaches T
N
from above, near q = 0, the first two terms in the
coefficient of this Gaussian term remain finite and positive, but the last term is negative and
diverges. Therefore, the coefficient approaches zero and eventually turns into the negative
regime. This signals a new instability (in the long-wavelength fluctuations,q' 0), setting
in prior to the divergence of the magnetic susceptibility. This instability occurs in the direct
channels and it is similar to Pomeranchuk instabilities.[77] This implies a deformation of
the FS in the ground state.[30] Also note that since
q
> 0, this instability cannot arise
without the aid of magnetic fluctuations, and nesting between electron and hole pockets is
essential in this scenario.
4.4 Nematic ordered phase
So far, we have shown that the isotropic metallic phase can be unstable near T
N
due to
magnetic fluctuations. This suggests that a new phase occurs between the isotropic metal-
lic and the magnetic phase. In this section, we will identify this as nematic ordering. The
effective action for the nematic order parameter will be derived within mean field theory.
4.4.1 Mean field theory
To obtain the Ginzburg-Landau action, one cannot simply replace

by its order param-
eter in (4.13). This method works only when the order parameter is small and all the terms
beyond the Gaussian approximation are positive. Although the order parameter is small
in our case, the quartic term (magnetic-nematic fluctuation coupling) is negative. Any
truncation of the series expansion beyond the quartic terms in (4.11) will potentially lead
to an ill defined (diverging) functional integral. Therefore, instead of expanding around
94
h
q
i = 0 in the-field manifold, we only expand
+
,M
X
andM
Y
around zero and keep

in an arbitrary functional form. Thus,G
1
0
is replaced byG
0
1
0
,
G
0
1
0;kk
0 =G
1
0;kk
0
+
0
B
B
B
@

kk
0
0 0
0

kk
0
0
0 0 0
1
C
C
C
A
; (4.16)
andV is replaced byV
0
,
V
0
kk
0 =
0
B
B
B
@
0 0 M
X
kk
0
0 0 M
Y
kk
0
M
X
kk
0 M
Y
kk
0 0
1
C
C
C
A
: (4.17)
Again, the fluctuations of
+
are ignored. Thus,
S
eff
=
1
2
X
q
"
2
u
n

q

q
+
2
u
spin
X

M

q
M

q
#
trlnG
0
1
0
trln[1 +G
0
0
V
0
]: (4.18)
So far, the above equations have not assumed any functional form for

q
. In order
to proceed, we assume that the stable ground state energetically favors homogeneous con-
figurations in space (preserving translational symmetry) and time (static). Then, we can
invertG
0
1
0
exactly and expand (4.18) aroundh
+
q
i =hM

q
i = 0. Namely,
S
eff
'S
nem
[

] +
1
2
X
;q
M

q
~
1
;q
M

q
; (4.19)
S
nem
[

] =
1
2
2
u
n
(

)
2

X
k
ln[G
1
X;k
+

][G
1
Y;k

];
(4.20)
95
and ~
1
;q
= 2=u
spin
+ 2
P
k
1=[G
1
;k
(G
1
;k+q

)], where the upper (lower) sign refers
to = X ( = Y ). The effective action in (4.19) is our main result that describes
the low-energy magnetic collective modes in the nematic phase. S
nem
is the Ginzburg-
Landau action of the nematic order parameter, and the second term describes the magnetic
fluctuations. The original magnetic and nematic fluctuations coupling term in (4.13) is
implicitly contained in the second term of Eq. (4.19). One can quickly check that in
this phase the magnetic fluctuations are unequal in the X and Y direction. Namely, the
susceptibility (the correlation of magnetic fluctuations) ishM

k
M

k
0i ~
;kk
0, which
is

dependent.
Furthermore, at this point the system still possesses a SDW instability, since ~
1
;q
approaches zero at a finite temperatureT
N
. ButT
N
may be different from the prediction
in isotropic phase, because non-zero values of

modify the nesting condition between
the electron and hole FS pocket. This modification lifts the degeneracies between stripe
orders.
Note that the magnetic fluctuations peak aroundQ
1
andQ
2
orq = 0 at low energies.
Therefore, it is natural to impose a cutoff frequency!
c
forM

q
. This frequency is also
the characteristic frequency that describes the typical wavelength of the magnetic fluctua-
tions. To make a connection with the phenomenology of magnetic susceptibilities, we can
constrain ourself to the fluctuating modes below this cutoff. Hence again, by expanding
~
1
;q
aroundq = 0, we obtain ~
1
;q
= ~
1
;q=0
+
j
n
j +f
;q
, but
andf
;q
can be different
in the isotropic phase. If the FS renormalization is not too drastic in nematic phase, we
can further expand ~
1
;q=0
around the isotropic phase. Thus, we have ~
1
;q=0
=r
0
+

,
where is some constant that depends on the microscopic details of the material. This
result coincides with Ref. [26].
96
The mean field equation of the nematic order parameter can be obtained by varying
S
nem
with respect to

,
2

u
n
=
X
k
"
1
G
1
X;k
+

1
G
1
Y;k

#
: (4.21)
This equation can be straightforwardly be solved numerically, giving the location of
the minimum ofS
nem
. One can also quickly check that

= 0 satisfies the mean field
equation trivially.
4.4.2 Nematic ordering phase transition
To search for the non-zero of

, instead of evaluating the mean field equation, we plot
the value of S
nem
[

] versus

at a given temperature numerically. This demonstrate
howS
nem
can develop a minimum at non-zero

, as temperature is varied. In order to do
this, we evaluate the Matsubara sum in (4.20),
S
nem
[

] =
1
2
2
u
n
(

)
2

X
k
ln[1 +e
(
X
k
+

)
][1 +e
(
Y
k

)
]:
(4.22)
As shown in Sec. 4.3, the isotropic state is unstable only when the magnetic fluctuations
become important. Therefore, it is natural to take!
c
as the cutoff energy for this effective
model. Then,
P
k
!
R
!c
0
m
(2)
2
d
R
2
0
d.
The input parameters are
2
' 0:2
0
, = 0:05
0
,[26] where
0
' 0:2 eV,[34]
and the magnetic fluctuations cutoff energy !
c
= 3:4 meV.[28] In addition, we set
a = (1=u
n
)(2)
2
(
0
=m), which is a dimensionless parameter controlling the transition
temperature in the mean field approximation. For demonstration purposes, we setu
n
= 0:1
97
and use the hole Fermi pocket momentum from Ref. [34]. Thus we havea' 1:5. Using
these input parameters, we obtain the plot shownthe plot is in Fig. 4.2.
æ æ æ æ æ æ æ æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
à à à à
à
à
à à à à
à
à
à
à
à
à
à
à
à
à
à
ì ì
ì
ì
ì
ì
ì
ì
ì
ì ì ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ò ò
ò
ò
ò
ò
ò
ò
ò
ò
ò ò ò
ò
ò
ò
ò
ò
ò
ò
ò
ô ô
ô
ô
ô
ô
ô
ô
ô
ô
ô
ô
ô ô ô
ô
ô
ô
ô
ô
ô
ç ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç ç ç
ç
ç
ç
ç
ç
0.2 0.4 0.6 0.8 1.0
-0.8
-0.6
-0.4
-0.2
0.2
ç 20
ô 18
ò 16
ì 14
à 12
æ 10
ΒΕ
0
S
_
nem
D
-
Ε
0
Figure 4.2: The nematic order phase transition occurs at some finite temperature where

S
nem
=S
nem
(2)
2
=(m
2
0
).
According to Fig. 4.2, the transition temperature from the mean field approximation
is
0
' 12 or T ' 193 K and

' 0:4
0
. These values are clearly overestimates
compared to experiments. This is because mean field theory does not taking account the
correlations between fluctuations. This deviation is even more pronounced because in the
low effective dimensionality. However, the mean field result at least estimates the correct
order of magnitude.
So far, magnetism does not appear to play any direct role in the nematic phase. How-
ever, the order parameter depends on the cutoff!
c
(this is similar to the mean field analysis
in BCS theory). This determines the position of the minimum, where by a smaller cutoff
implies smaller

or lower transition temperature and vice versa. To further investigate
98
the relationship between magnetic fluctuations and nematic order calls for more sophisti-
cated methods beyond mean field theory.
4.5 Discussion
4.5.1 The control parameter
For the nematic order considered in this paper,u
n
=u
6
u
(1)
4
is the key parameter. It not
only determines whether the system can sustain nematic ordering, but also determines the
transition temperature to the nematic order from the isotropic phase. These coupling con-
stants depend on microscopic details, and are difficult to obtain in general. Nevertheless,
this could explain the fact that nematic ordering is not found in every FeAs material.
Why u
n
plays such a crucial role in this scenario can be understood by the follow-
ing intuitive picture. u
6
andu
(1)
4
are related to the coupling strength between two quasi-
particles, depending on whether both reside in different electron FS pockets or the same
electron FS pocket respectively. Ifu
6
>u
(1)
4
means that two quasi-particles tend to occupy
the same electron pocket rather than different pockets because this yields a smaller poten-
tial energy. Therefore, instabilities of the original isotropic ground state may arise, when-
ever this scattering processes surpass the other channels.
4.5.2 Comparison with other work
The nematic order considered in this study also yields an anisotropic magnetic fluctuation,
which can explain results from recent neutron scattering experiment.[21] It also implies
anisotropic resistivity in the nematic phase.[14, 26, 88]
99
In earlier theoretical work, it was shown explicitly that nematic order can be developed
solely from the effect of magnetic fluctuations, and yields an anisotropic magnetic fluctu-
ating state. In the phenomenological approach based on aJ
1
J
2
model,[24, 83, 98] or
a preemptive nematic order from a microscopic itinerant model,[27] nematicity originates
from the competition between two different stripe orders. Another alternative explanation
was given by orbital ordering. [8, 6, 10, 11, 46, 47, 52, 54, 55, 71, 72, 85, 99] In this
scenario, the interactions between orbitals drive local ordering in orbital occupancy, and
resulting in rotational symmetry breaking. However, all of these consequences are similar
to our approach. It is the FS pocket in X and Y that are renormalized unequally in the
ground state.
In our approach, the nematic order is interpreted using a quasi-particle picture.
Because of the instability in the direct channel, the FS fluctuations (the excitations of
the quasi-particles) can yield lower energies than its original isotropic FS configuration.
Therefore the system tends to deform an isotropic FS to lower the ground state energy.
In addition, only the Gaussian terms of the magnetic fluctuation were kept in our study.
Hence, the competition with stripe orders does not play an essential role in this scenario.
However, it would be interesting to study the interplay between stripes and nematic order
(

) by including fluctuations beyond the Gaussian approximation.
Furthermore, although the results in this paper are based on an itinerant picture, we
can still observe that non-zero

implies orbital ordering. As in Ref. [34], the itinerant
band electrons are linear combinations of the Fed-orbitals. Particularly, near the FS, the
main orbital components in the X (Y) pocket can bed
yz
(d
xz
) ord
xz
. If the ground state
occupation number in thed
xy
orbital is zero, one may approximately treat

as the orbital
polarization. Even if thed
xy
orbital occupation number is not zero, this will not change
100
the result qualitatively, sinced
xy
orbital is invariant underC
4
rotation. To explore further
the relationship between

and orbital polarization, a different model would be needed
that retain orbitals information. We leave this to a future study.
4.5.3 Concluding remarks
In summary, we have discussed possible instabilities of the isotropic metallic phase within
an itinerant model. By using the Hubbard-Stratonovich transformation, several auxiliary
bosonic fields that describe the magnetic and nematic order collective modes were intro-
duced. By perturbing the action of these bosonic fields in the isotropic phase, if the inter
electron pocket interaction (u
6
) is greater than the intra electron pocket interaction (u
(1)
4
),
the magnetic fluctuations can drive an instability in the direct channels. From mean field
analysis, we argue that nematic order is a favored ground state which spontaneously breaks
C
4
symmetry but preserves translational symmetry. In the nematic ordered state, the size
of the FS pockets inX andY becomes unequal. For

> 0 (

< 0), the X (Y) pocket
is larger than the Y (X) pocket. Because of this distortion, the magnetic fluctuations also
exhibit anisotropy in X and Y directions. A similar nematic order can possibly occur in
FS hole pocket, if angular depend in the coupling constants is considered.
101
Chapter 5
Conclusion
In this thesis, we have studied the electronic correlation effects in bilayer graphene and iron
pnictides. These systems exhibit intriguing physical properties which are tightly related to
the interactions within different bands electrons. One of the remarkable consequences is
that, many instabilities of the Fermi surface are found under these effects and lead to the
discovery of novel quantum phases in experiments and theories.
In our theoretical studies, bands touching in bilayer graphene lead to critical behaviors
beyond the Fermi-liquid paradigm. In contrast to monolayer graphene, electronic corre-
lation effects play a crucial role in determining the system ground states properties. Our
results is able to explain the observations qualitatively and offer several insights why the
bilayer graphene Fermi surface is unstable. In addition, the study suggests excitonic order
with anti-ferromagnetism in the ground state, which can naturally explain the magnetic
field dependent insulating gap which has been observed in the bilayer graphene.
The method in this thesis can be extended for few layer graphene system. At least in
the tri-layer ABC stacking graphene, based on the thesis aspect, the Fermi surface is highly
unstable against electron-electron interactions because of its cubic energy dispersion. In
addition, the coupling constants with momentum dependent may be relevant. This could
lead to many exotic collective modes and ordering in the ground. This feature is not found
in others conventional materials, because such scattering processes are highly irrelevant.
102
In the study of iron pnictide, we have shown explicitly from microscopic theory that
the magnetic fluctuations can produce an instability of the isotropic metallic phase. Fur-
thermore, the study suggests that the instability occurs in the forward scattering processes,
and this eventually gives rise to a Pomeranchuk instability. These results have succeeded
to identify nematic order, which reproduces those important results from the other theo-
retical candidate models, such as preemptive nematic order and orbital ordering, although
the mechanism are different from this thesis.
Pomeranchuk instabilities in Fermi liquid have been sought for a long time after the
theoretical realization. In the system with single connected Fermi surface, this can only
occur, if some peculiar interactions between electrons is introduced. However, from this
thesis study, due to the multi-bands electronic structure of iron pnictide, this instability
naturally emerges. Furthermore, the theoretical method has been developed in this thesis
can be generalized to study other part in the iron-pnictide phase diagram. It is particularly
useful to study the coexistence of magnetic and superconductivity region. Finally, the
discovery of iron-pnictide has opened a new page for highT
C
materials. There still remain
many unsolved puzzles in iron-pnictides. Its many intriguing phenomonena challenge
many conventional notions in condensed matter physics.
In conclusion, when the band degrees of freedoms play a non-negligible role in the low
energy limit, many exotic collective motions of electrons can emerge and affect the physics
profoundly. In addition to those new materials that are mentioned in this thesis, there are
still remaining many unknown multi-band systems which could exhibit fascinating physics
yet to be discovered. The theoretical study of graphene and iron-pnictide in this thesis not
only yields new insight and understanding for multi-band system, but also paves the way
for discovering in new fascinating materials.
103
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112

Abstract (if available)

Abstract The recent dominant trends in condensed matter physics research can be roughly summarized into three newly discovered materials: topological insulators, graphene, and iron‐based superconductors. All these materials exhibit many intriguing properties which are fundamentally related to their electronic band structure. Therefore, this lead to many intense investigations on multi‐band electronic system to explore new physics. ❧ The physics of multi‐band electronic structure is fascinating in several aspects. Without many‐body effects, because of the gauge freedom of Bloch states, topological insulators can give rise a robust metallic behavior at its boundaries. In graphene, the touching between conduction and valence band at Fermi level yields a new criticality class which exhibit many unconventional electronic properties, especially its quasi‐relativistic behavior. Turning to the many‐body effects, for instance, the iron‐based superconductors can sustain an superconducting ground state despite of no attractive interactions in the system. Therefore, a deeper understanding for the conventional notions in condensed matter physics has put forward by many of these experimental observations. ❧ In this thesis, the many‐body effects in multi‐band systems are the main focus, especially the study of graphene and iron‐based superconductors which can be compared to experiments. These theoretical studies intend to understand how the underlying electronic bands degree of freedom can give rise to Fermi‐liquid instabilities, and how these effects can be related to intriguing physical properties. ❧ We first study the electrons correlation effects in bilayer graphene by a renormalization group technique. In this study, we build a microscopic model of bilayer graphene from a tight‐binding approach. In our finding, the peculiar Fermi surface configuration leads to critical behavior which is beyond the Fermi‐liquid paradigm. Furthermore, due to the electron‐electron interactions between different bands, excitonic instabilities are found in many different scattering channels. This analysis suggest a collection of competing orders in the system ground states. This result is consistent with the experimental observation that bilayer graphene is an insulator. ❧ Next, we study nematic order in the metallic phase of iron pnictides. In contrast to graphene, the density of states is finite at the Fermi surface. By careful investigating the scattering processes near these Fermi surface, and then identifying the most relevant collective modes from these processes, we find that a Pomeranchuk instability can be driven by magnetic fluctuations. This instability eventually leads to the break down of the isotropic metallic phase which electronic system exhibit broken crystalline rotational symmetry but preserve translation symmetry. As the experiment suggests, this can be a candidate for nematic order in the metallic phase.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Song, Kok Wee (author)

Core Title Electronic correlation effects in multi-band systems

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 04/14/2014

Defense Date 02/26/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag excitonic instabilties,Fermi liquid,graphene,iron pnictides,nematic order,OAI-PMH Harvest,superconductivity

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haas, Stephan W. (committee chair), Bickers, Gene (committee member), Daeppen, Werner (committee member), Däppen, Werner (committee member), Haselwandter, Christoph (committee member), Jonckheere, Edmond A. (committee member)

Creator Email kokweeso@usc.edu,songkok25@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-376102

Unique identifier UC11295835

Identifier etd-SongKokWee-2348.pdf (filename),usctheses-c3-376102 (legacy record id)

Legacy Identifier etd-SongKokWee-2348.pdf

Dmrecord 376102

Document Type Dissertation

Format application/pdf (imt)

Rights Song, Kok Wee

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA