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Entanglement in strongly fluctuating quantum many-body states
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Entanglement in strongly fluctuating quantum many-body states
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ENTANGLEMENT IN STRONGLY FLUCTUATING QUANTUM MANY-BODY STATES by Weifei Li 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 2007 Copyright 2007 Weifei Li Acknowledgments I recently heard about a study that showed the benefits of a large social network of family and friends. While the study did not specifically mention the importance of social support in the completion of a dissertation, I can personally attest to the immeasurable value of my family, friends, and mentors while tackling this endeavor. First, I would like to acknowledge the support, encouragement, and guidance of my advisor Dr. Stephan Haas. He has been an incredible mentor, assisting in my academic and professional development. I am indebted to Stephan for the opportunity to work on the topics of entanglement, which made this dissertation possible, and for his diligent and efficient reviews of my work. I wish to express my gratitude to my dissertation com- mittee member, Dr. Todd Brun, for his guidance and expertise in the quantum informa- tion and quantum computation arena. I would also like to extend my great appreciation to the other dissertation committee members, Dr. Gene Bickers, Dr. Werner Dappen and Dr. Clifford Johnson , for their experience and insight. Also thanks go out to Dr. Tu-nan Chang, Dr. John Nodvik, Dr. Hubert Saleur and Dr. Richard Thompson for their encouragement and support during my graduate school study. ii And I would like to pay my heartiest thanks to my wonderful colleagues for their greatest support and invaluable discussions throughout my graduate study. They are Rong Yu, Letian Ding, Dr. Tommaso Roscilde, Dr. Omid Nohadani, Dr. David Parker, Jason Thalken, Amy Cassidy, Wen Zhang and Dr. Pinaki Sengupta. Special thanks are given to my many marvelous friends who offered me mental breaks during my graduate school study at USC. I also want to acknowledge my solid base of friends from my years living in the West Coast who have encouraged me in my endeavors. A far from complete list includes Liuning Zhou, Manjiang Zhang, Bo Lei. Finally, I would not be here without the love, devotion, and encouragement of my family. My love goes out to my wife, Ling Wang, who have always encouraged me to aim high; my parents, Dejin Li and Suqin Huang, who have been blessing and support- ing me silently; my brothers, Min, Ling and Jie, who enforced my deadline, encouraged me along the way and challenged me to always work harder as we were growing up. Last but not least, I would like to thank those people who have contributed in some way to the success of my academic endeavors. Giving an even modest list is an impos- sible mission. Some of them are Betty Byers, Lisa Swanson, Gokhan Esirgen and Mary Beth. iii Table of Contents Acknowledgments ii List of Figures vi Abstract ix Chapter 1 Introduction 1 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Quantum Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Reviews on Entanglement in Strongly Fluctuating Quantum Many-Body States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 2 Scaling Behavior of Entanglement in Fermion Model 19 2.1 Model and Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 One-dimension Free-Fermion Model . . . . . . . . . . . . . . 21 2.1.2 High-dimensiond>1 Free-Fermion Model . . . . . . . . . . . 37 2.2 Phase Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Scaling Behavior of Entanglement . . . . . . . . . . . . . . . . . . . . 49 Table 2.1 Summary of the entanglement scaling properties, co-dimension, density of states and decay of correlations in the three phases of the model Eq.(2.48) ind=2;3. . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 3 Scaling Behavior of Entanglement in Boson Model 59 3.1 Model and Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Scaling Behavior of Entanglement . . . . . . . . . . . . . . . . . . . . 70 Chapter 4 Scaling Behavior of Entanglement in Spin Model 79 4.1 Resonating Valence Bond State . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Scaling Behavior of Entanglement . . . . . . . . . . . . . . . . . . . . 85 iv 4.2.1 One-dimensional Shortest Range Resonating Valence Bond States 85 4.2.2 General Short Range Resonating Valence Bond states . . . . . . 91 4.2.3 Long Range Resonating Valence Bond States . . . . . . . . . . 103 Chapter 5 Conclusions and Future Works 106 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 References 111 v List of Figures 1.1 Concurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Block entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 1D Ising QPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Block entropy for critical and non-critical quantum spin systems: Non- critical: Ising chain (empty squares), H XY (a = 1:1;° = 1); XXZ chain (filled squares), H XXZ (¢ = 2:5;¸ = 0). CriticalS L » log 2 (L) (stars) for the Ising chain, H XY (a = 1;° = 1); (triangles) for the critical XX chain with no magnetic field, H XY (a = 1;° = 0); in a finite XXX chain of N = 20 spins without magnetic field (diamonds), H XXZ (¢=1;¸=0) [VLRK03] . . . . . . . . . . . . . . . . . . . . . 14 2.1 Phase diagram of the model Eq.(2.8) for one dimension. The Ising model, ° = 1, has a critical point at ¸ = 1. The XX model, ° = 0, is critical in the interval¸2[0;1]. The whole line¸=1 is also critical. 28 2.2 Correlation function in parameter space:f(¸;°)2f0;1;2;3g N f0;1;2;3gg. The x-axis and y-axis of insets are distance and correlation function respectively. we can see algebra decay of the correlation function in f0· ¸ < 2;° = 0g andf0· ¸· 2;° > 0g and exponential decay in flambda =2;° =0g andf2·¸g . . . . . . . . . . . . . . . . . . . . 45 2.3 Phase diagram of the model Eq.(2.48) for the case d = 2. The roman numbers for the various phases are explained in the text. Representative contour plots of the dispersion relation ¤ k are also shown. There the black areas corresponds to¤ k = 0 and the white areas to the top of the band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Spectrum in momentum space for two dimension whenf¸=1;° =0g. The intersection circle of the two surfaces is the Fermi surface of the system which means ¹ d=1. . . . . . . . . . . . . . . . . . . . . . . . . 48 vi 2.5 Spectrum in momentum space for two dimension whenf¸=1;° =1g. The two surfaces only cross at two points which means ¹ d=2. . . . . . 49 2.6 Spectrum in momentum space for two dimension whenf¸=1;° =2g. The two surfaces only cross at one point which means ¹ d=2. . . . . . . 50 2.7 Spectrum in momentum space for two dimension whenf¸=1;° =3g. The two surfaces have no cross section which means ¹ d=2. . . . . . . . 51 2.8 Scaling of the block entropy S L in d = 2 for ° = 0 (left panel) and ¸ = 0 (right panel). The solid lines correspond to fits according to the formulaS L = C 3 Llog 2 (L)+BL+A. . . . . . . . . . . . . . . . . . . 52 2.9 ¸-dependence of theC coefficient in Eq.(1.4) ind = 2 andd = 3. The values extracted from fits to our numerical data are compared with the predictions of Ref. [GK06]. In d = 2, the exact form of C(¸) can be obtained, which is equal to 2 ¼ cos ¡1 (¸¡1) . . . . . . . . . . . . . . . 54 2.10 Scaling of the block entropy S L in d = 2 for ° = 1 (left panel) and ¸=1 (right panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1 The block entropy of one-dimension infinite bosonic chain for critical and non-critical cases. The critical point is:¸ = ¡1, h = 0:5 and ° = 0:5. The non critical point is:¸ =¡1:00001,h = 0:5 and° = 0:5. For critical point the block entropy can be fitted very well by 1 3 log(L). For the non-critical case, although only slightly away from critical case, the block entropy saturates very fast. . . . . . . . . . . . . . . . . . . . . . 72 3.2 The block entropy of two-dimension infinite bosonic lattice for critical and non-critical cases. The critical cases we choose are:¸ = ¡4, h = 0:5,° = 1:5 and ¸ = ¡4, h = 1,° = 1. The non critical point is:¸ = ¡4:2,h=1 and° =1. For all the cases the block entropy can be fitted very well byL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.1 Simplest example of singlet product state . . . . . . . . . . . . . . . . 81 4.2 Example of an arbitrary singlet product state . . . . . . . . . . . . . . . 82 4.3 R and L states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Block entropy of one dimension RVB state with shortest range . . . . . 87 4.5 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 Irreducible configuration vs reducible configuration . . . . . . . . . . . 92 vii 4.7 First three irreducible configuration B1,B2,B3 withR =3 . . . . . . . 93 4.8 Expansion ofÃ(N) around the length2L withR =3 . . . . . . . . . . 94 4.9 Only the bonds near the boundary is cut by tracing . . . . . . . . . . . 100 4.10 The block entropy of a long range RVB state: the superposition is equal weight. The curve can be fitted by 1 2 logL asymptotically. . . . . . . . . 104 viii Abstract In this thesis, the scaling behavior of entanglement is investigated in quantum sys- tems with strongly fluctuating ground states. We relate the reduced density matrices of quadratic fermionic and bosonic models to their Green’s function matrices in a uni- fied way, and calculate exactly the scaling of the entanglement entropy of finite systems in an infinite universe. In these systems, we observe quantum phase transition by tuning the parameters of the Hamiltonian. Our study shows that although in one dimension there is a unique relation between the quantum phase transition and the scaling behavior of entanglement, this is not necessarily true in higher dimensions. By exactly solving a spinless fermionic system in two and three dimensions, we investigate the scaling behavior of the block entropy in critical and non-critical phases. We find that the scaling behavior of the block entropy is not exclusively controlled by the decay of the correlation function, and critical phases exist which exhibit the area law, i.e., the entropy of entanglement scales with the area of a given subsystem. We identify two regimes of scaling. The scaling of the block entropy crucially depends on the nature of the excitation spectrum of the system and on the topology of the Fermi ix surface. Noticeably, in the critical phases the scaling violates the area law and acquires a logarithmic correction only when a well-defined Fermi surface exists in the system. A more stringent criterion for the violation of the area law is thus conjectured, based on the co-dimension which describes the topology of gapless excitation in momentum space. According to this conjecture, logarithmic corrections to the area law appear only in critical phases with codimension ¹ d = 1. For all other critical systems, as well as for the non-critical ones with finite ground-state entanglement, the area law holds. In free bosons systems with a generic quadratic Hamiltonian, we verify that the scal- ing behavior of the block entropy in higher dimensions always follows the area law, and we explain why logarithmic corrections to area law are impossible. Spin systems are harder to deal with. Few results can be obtained in higher dimen- sions. We study one candidate ground state, i.e. the resonant valence bond (RVB) state. In one dimension we find that: for short-range RVB states, the block entropy saturates to a constant, whereas the block entropy for long-range RVB states diverges logarithmi- cally. We also prove that the block entropy in higher dimensions follows the area law for short-range RVB states. This is consistent with the one-dimensional relation between the quantum phase transition and scaling behavior of entanglement. x Chapter 1 Introduction 1.1 Background and Motivation Richard Feynman was among the first to recognize the potential of quantum compu- tation [Fey82]. In 1982 he proposed an abstract model that showed how a quantum system could be used to perform computations. Since then, research in quantum com- putation and quantum information [NC00] has been blossoming. Quantum computation and information science has benefitted much from other fields, and vice versa the tools and ideas developed in this field have also given new insights to other areas such as quantum physics and strongly correlated systems[OAFF02, TN02]. This is because quantum computation and information focusses on quantum coherence and quantum correlations, or quantum entanglement. Any physical system described by the laws of quantum mechanics can be considered from the perspective of quantum information by 1 means of entanglement theory. The nature of many-body entanglement in various solid- state models has been the focus of recent interest. The motivation for this effort is two- fold. On one hand, these systems are of interest for the purpose of quantum information processing and quantum computation [Bur04]. On the other hand, at a fundamental level, the study of entanglement represents a purely quantum way of understanding and characterizing quantum phases and quantum phase transitions in many-body physics [OAFF02, TN02, VPC04, VLRK03]. The purpose of this work is to study entangle- ment properties in a many-body quantum system characterized by local interactions. This introduction is to give some elementary background about basic concepts of quan- tum computation and information, and elaborate on its relation with other fields of physics, such as quantum many-body physics. We begin by considering the defini- tion of entanglement and the von Neumann entropy. Then, we consider quantum phase transition. At the end of this introduction, we give an overview of the dissertation. 1.2 Entanglement Definitions of entanglement vary depending on whether we consider only pure states or a general set of mixed states. In this work, we mainly focus on pure states, which can 2 be represented with a single quantum state vector, and the states referred hereafter are pure states only by default. If a quantum system contains multiple subsystems, in general it is not possible for these subsystems to have a well-defined state. Instead the subsystems are entangled, and the state of the whole system is considered to be an entangled state. Mathematically, this can be expressed as:Ã(x;y)6=Ã(x)Ã(y). For example, the Bell states ( j"#>§j#"> p 2 , j"">§j##> p 2 ) are famous examples of entangled states. A striking feature of entangled statesjªi is that an accurate local description of such states is impossible, namely each subsystemA of the total systemU can have a finite entropy, quantified as the von-Neumann entropy S A = ¡Tr½ A log 2 ½ A of its reduced density matrix ½ A = Tr UnA jªihªj, whereas the total system clearly has zero entropy. Generally for a pure state of a bipartite system the entanglement can be measured by the entropy of the reduced density matrix for either system: E(ª AB )=S A =¡Tr(½ A log 2 ½ A )=S B =¡Tr(½ B log 2 ½ B ) (1.1) The entropy of entanglementS A of the subsystem is a reliable estimate of the entangle- ment between the subsystemA and the remainder of the system,UnA. There are other bipartite entanglement measures such as concurrence[Woo98]. 3 For multipartite systems, quantifying the entanglement in a general way is still an open question. Even for a three-particle system, we can not follow the above procedure. When doing partial tracing, different results will be obtained by tracing out different subsystems. For example, for a state such as:jà >= aj100 > +bj010 > +cj001 >, we have:S(½ A )6=S(½ B )6=S(½ B )6=S(½ AB )6=S(½ BC )6=S(½ AC ). Also for three particle case, there are two different classes of entangled states: W = j"##>+j#"#>+j##"> p 3 and GHZ = j""">+j###> p 2 . The states belonging to different classes can not be transformed from one to another by local transformation. To overcome this difficulty we apply bipartite entanglement to multipartite system. Typ- ically concurrence and block entropy are measures used to study the entanglement of such systems. For a system with the reduced density matrix ½ ij , one can define the concurrence by C =maxf0;¸ 1 ¡¸ 2 ¡¸ 3 ¡¸ 4 g (1.2) where ¸ i are eigenvalues of the matrix q ½ ij ¾ i y ¾ j y ½ ¤ ij ¾ i y ¾ j y , and ¸ 1 is the largest eigenvalue. Concurrence measures the pairwise entanglement between two particles. On the other hand, block entropy measures the entanglement between one block of the system and the remainder of the system. It has the same form as the definition of the entropy of entanglement, where A and B are partitions of the system. In Fig.1.1 and Fig.1.2, the relevant partitions for concurrence and block entropy are sketched. A 4 Figure 1.1: Concurrence Figure 1.2: Block entropy special case for the block entropy is when the size of the block under consideration is 1. In this case, the block entropy is named local entanglement[Zan02, ZW02] for it exhibits the correlations between a local state and the other parts of the system. Quantum entanglement was already pointed out to be a crucial element of quan- tum mechanics by Schr¨ odinger[Sch35]. In recent years, entanglement has become the fundamental physical resource for quantum computers and quantum com- munication technology. It is a key ingredient in quantum computation[Pre98], quantum teleportation[BBC + 93], quantum cryptography[Eke91], and superdense coding[BBM92]. On the other hand, entanglement also plays a central role in the study of strongly correlated quantum systems [Pre00, ON02a, ZW02], since a highly entangled ground state is at the heart of a large variety of collective quantum phenom- ena. Milestone examples are the ground states used to explain superconductivity and 5 the fractional quantum Hall effect, namely, the BCS ansatz [BCS57] and the Laughlin ansatz [Lau83]. Ground-state entanglement is, most promisingly, also a key factor to the understanding of quantum phase transitions, where it accounts for the appearance of long-range pure quantum correlations at zero temperature [TN02]. In the next section we will give a brief introduction of quantum phase transitions. 1.3 Quantum Critical Phenomena Quantum mechanical systems are known to undergo phase transitions at zero tempera- ture when a suitable control parameter in their Hamitonian is varied[Sac99, SGCS97]. At the critical point where the quantum phase transition occurs, the ground state of the system undergoes a qualitative change in some of its properties. The transition takes place at a quantum critical value of a tunable parameter such as pressure, composi- tion or magnetic field strength. A quantum phase transition occurs when collective ordering of the system disappears, but this loss of order is driven solely by the quan- tum fluctuations, i.e. not by temperature. Examples include transitions in quantum Hall systems[SGCS97], localization in Si-MOSFETs (metal oxide silicon field-effect transistors; [KKF + 94]) and the superconductor-insulator transition in two-dimensional systems[HLG89, vdZFE + 92]. 6 A simple model in which a quantum phase transition occurs is the one-dimensional Ising model: H =¡J X i (¾ z i ¾ z i+1 +g¾ x i ) (1.3) where J > 0 is an overall energy scale, and g > 0 is an external transverse field. Consider the ground states of H for small g and large g. For g = 0, the first term dominates and the spin all align themselves either along the +z or¡z directions. The system is in a ferromagnetic state. Wheng 6= 0, the external field introduces tunneling that flips spins. For very large g (g À 1), the second term dominates, and to leading order in 1 g , the ground state is simply¦ i j"> i +j#> i p 2 or¦ i j"> i ¡j#> i p 2 . Now the state is either one of these two states for each spin. So the system is in paramagnetic state. It is evident that ground state forgÀ 1 is very different from the ground state ofg = 0. There is a critical point between these two limits, g = g c , which can be calculated quantitatively: g c =1 for this simple model[Sac99]. We will detail the calculation in a later part. Figure 1.3: 1D Ising QPT We can study the quantum phase transition from another viewpoint: ground state energy and correlations. Consider a Hamiltonian,H(g), which is a function of a dimensionless 7 couplingg. For any reasonable g, the ground state energy will vary smoothly asg is var- ied. However, there may be special points, like g = g c , where there is non-analyticity in some property of the ground state: we identifyg c as the position of a quantum phase transition. We will focus on second order quantum phase transitions in this dissertation. These are defined as the transitions at which the characteristic energy scale of fluctua- tions above the ground state vanishes and long-range correlations emerge, in the sense of correlations that decay only slowly with distance, asg approachesg c . Let the energy ¢ represent a scale characterizing a significant spectral density of fluctuations at zero temperature forg 6= g c . Thus¢ could be the energy of the lowest excitation above the ground state, if this is non-zero, i.e., there is an energy gap¢, or if there are excitations at arbitrarily low energies in the infinite lattice limit, i.e. the energy spectrum is gap- less,¢ is the scale at which there is a qualitative change in the nature of the frequency spectrum from its lowest frequency to its higher frequency behavior. We find that as g approachesg c ,¢ vanishes as¢»jg¡g c j zº and» »jg¡g c j º , where» is the charac- teristic length scale on which correlation occur in the system. In particular we see that ¢ » » ¡z . We hence identify the quantum phase transition by a vanishing energy gap from the above definition. Both classical and quantum critical points are governed by a diverging correlation length, although quantum systems possess additional correlations that do not have a classical counterpart. These quantum correlations are described by entanglement. The 8 role of entanglement at a phase transition is not entirely captured by statistical mechan- ics. A complete classification of the critical many-body state requires the introduction of concepts from quantum information theory. Because the ground-state wave function undergoes qualitative changes at a quantum phase transition, it is important to under- stand how its genuine quantum aspects evolve throughout the transition. Will entangle- ment between distant subsystems be extended over macroscopic regions, as correlations are? Will it carry distinct features of the transition itself and show scaling behavior? Or in a more specific way, assuming that the system U corresponds to the whole universe ind dimensions, a fundamental question concerns the scaling behavior of the entropy of entanglementS L of an hypercubic subsystemL d (hereafter denoted as a block) with its sizeL. Indeed, such scaling probes directly the spatial range of entanglement: when the block size exceeds the characteristic length over which two sites are entangled, the block entropy should become subadditive, and scale at most as the area of the block bound- aries, following a so-called area law: S L »L d¡1 . A crucial question is then if and how the scaling of the block entropy changes when the nature of the quantum many-body state evolves in a critical way by passing through a quantum phase transition, and how the characteristic spatial extent of entanglement relates to the correlation length of the system. Answering these questions is important for a deeper understanding of quantum phase transitions, and also from the perspective of quantum information theory. Hence results that bridge these two areas of research are of great relevance. 9 An additional motivation for recent studies of block entropy in many-body physics is to view this as a route to better numerical algorithms for finding ground states: knowing that ground states of local Hamiltonians have much less entanglement, even at criticality, than generic quantum states both explains the success of the density-matrix renormaliza- tion group algorithm in one dimension [Whi92, OR95] and motivates recent proposals for analogous numerical methods in higher dimensions [VWPGC06]. 1.4 Reviews on Entanglement in Strongly Fluctuating Quantum Many-Body States This question has been extensively addressed in the case of one-dimensional spin sys- tems [VLRK03, LRV04, RM04, Laf05], in chains of harmonic oscillators [Skr05, CEPD06] and in related conformal field theories (CFT) [CC04, Kor04]. Here it was found unambiguosly that states with exponentially decaying (connected) correlators fol- low the one-dimensional area lawS L » L 0 , i.e. saturating to a finite value, whereas for critical states, displaying power-law decaying correlations, a logarithmic correction to the area law is always present: S L = [(c+ ¹ c)=6]log 2 L, where c is the central charge of the related CFT. The asymptotic value of the block entropy is found to diverge log- arithmically with the correlation length, S 1 » log 2 (»), which clearly establishes the 10 relationship between entanglement and correlations. The above picture holds true only in the presence of short-range interaction; On the contrary, in the presence of long-range interactions the divergence of the correlation length can still be accompanied by the area law [DHH + 05, CEPD06]. In higher spatial dimensions, fewer results are available, and the general relationship between the block entropy scaling and the correlation properties of the quantum state is still unclear, even for short-range interactions. In free-boson systems, it has been gen- erally proven that the area law is satisfied for non-critical systems [PEDC05, CEPD06]. For free-fermion systems, on the other hand, it has been proven [Wol06, GK06] that crit- ical systems with short-range hopping and a finite Fermi surface exhibit a logarithmic correction to the area law S L = C 3 (log 2 L)L d¡1 : (1.4) Next we will show basic ideas about these studies by discussing one-dimensional spin chains. Osterloh et al. [OAFF02] showed that in a specific class of one-dimensional mag- netic systems, the quantum phase transition (QPT) is associated with a change of entanglement, and that the entanglement exhibits scaling behavior in the vicinity of the transition point. This behavior was discussed in detail for the Heisenberg model [ON02b, VPM04, VPC04, LRV04, VLRK03] and for the Hubbard model [GDLL04]. 11 It is believed that the ground-state entanglement also plays a crucial role in other QPTs, such as the change of conductivity in the Mott insulator-superfluid transition [Geb97] and the quantum Hall effect [Lau83]. Many of the relevant features, such as the transi- tion from a simple product state to a strongly entangled state, occur over a wide range of parameters and persist for infinite systems as well as for systems with as few as two spins [ABV01, ZSGL03, Wan02, KS02]. These systems, especially the Heisenberg spin model, are central both to condensed-matter physics and to quantum information the- ory. In quantum information processing, the Heisenberg exchange interaction has been shown to provide a universal set of gates [IAB + 99, RB01], and in quantum communi- cation, information can be propagated through a Heisenberg spin chain [Bos03, Sm04]. Here we focus on the Heisenberg model. And because we are interested in the scaling behavior of entanglement, the measure of entanglement we use is the block entropy. In quantum spin chains with [RM04] or without disorder[VLRK03, LRV04] it has been shown that the block entropy, or the entanglement of a segment of N ¸ 1 spins with the remainder, is logarithmic inN at quantum critical point for systems. In [VLRK03, LRV04] the authors analyzed the entanglement in two spin- 1 2 chains, the so-called XXZ and XY models, with Hamiltonians: H XXZ = N¡1 X l=0 ¡ ¾ x l ¾ x l+1 +¾ y l ¾ y l+1 +¢¾ z l ¾ z l+1 ¡¸¾ z l ¢ (1.5) H XY =¡ N¡1 X l=0 ³ a 2 [(1+°)¾ x l ¾ x l+1 +(1¡°)¾ y l ¾ y l+1 ]+¾ z l ´ (1.6) 12 where ¾ are the Pauli matrices. These models consider a variety of spin-spin interac- tions between nearest neighbors as well as the effect of an external magnetic field along the z direction, and are used to describe a large range of one-dimensional quantum sys- tems [Sac99]. We recall that the H XXZ Hamiltonian for ¢ = 1 describes spins with antiferromagnetic isotropic Heisenberg interaction, whereas theH XY Hamiltonian cor- responds, for ° = 1, to the quantum Ising chain, and that both Hamiltonians coincide for¢=° =0, where they become theXX model. Osterloh et al. [OAFF02] and Osborne and Nielsen [18] have considered the entangle- ment in theXY spin model, Eq. 1.6, in the vicinity of a quantum phase transition. Their analysis focussed on single-spin entropies, or local entanglement, and on two-spin quan- tum correlations, or concurrences. It suggests that these one- and two-spin entanglement measures display a peak either near or at the critical point. To capture the large-scale behavior of quantum correlations at a critical regime, G. Vidal et al [VLRK03] studied the entanglement between a block of L contiguous spins and the remainder of the chain when the spin chain is in its ground statejª g > : S L =¡Tr(½ L log 2 ½ L ) (1.7) where ½ L = Tr N¡L jª g >< ª g j is the reduced density matrix for B L , a block of L spins. The XXZ model, Eq. 1.5, can be analyzed by using the Bethe ansatz [Bet31]. 13 A chain of up to 20 spins has been studied numerically. The XY model, Eq. 1.6, is an exactly solvable model (see for instance [Sac99]), and it is possible to evaluate an infinite chain, N ! 1. The results of XXZ and XY model are plotted in Fig1.4. The general picture emerging from the these numerical results is that there is a clear Figure 1.4: Block entropy for critical and non-critical quantum spin systems: Non- critical: Ising chain (empty squares), H XY (a = 1:1;° = 1); XXZ chain (filled squares), H XXZ (¢ = 2:5;¸ = 0). Critical S L » log 2 (L) (stars) for the Ising chain, H XY (a = 1;° = 1); (triangles) for the critical XX chain with no magnetic field, H XY (a = 1;° = 0); in a finite XXX chain of N = 20 spins without magnetic field (diamonds),H XXZ (¢=1;¸=0) [VLRK03] distinction between non-critical and critical entanglement, corresponding to two forms of structurally inequivalent ground states. 14 Non-critical regime: For all values (¢;¸) and (a;°) for which the spin chains (1.5) and (1.6) are non-critical [Sac99], the entropy of entanglement S L either vanishes for all L [e.g., when a sufficiently strong magnetic field aligns all spins into a product, unen- tangled state] or grows monotonically as a function of L until it reaches a saturation value for a certain block size L 0 . Thus non-critical ground-state entanglement corre- sponds to a weak, semi-local form of entanglement driven by the appearance of a length scale L 0 . For any L, the reduced density matrix ½ L is effectively supported on just a small, bounded subspace of the Hilbert space of L spins. Such entanglement is called semi-local because a good approximation to ½ L can be obtained by diagonalizing the Hamiltonian corresponding to the blockB L and only a few extra neighboring spins. Critical regime: In critical chains, on the contrary, the entropy S L diverges as log(L). Critical ground-state entanglement corresponds to a stronger form of entanglement, one that embraces the system at all length scales simultaneously. Hence the divergence behavior of the block entropyS L can be used as a signature of QPT in these cases. Also because of the strong entanglement between all parts of the chain, the approximation of the reduced density matrix is certain to fail by just local diagonalization. 15 1.5 Outline of the Dissertation In this dissertation, we address the scaling behavior of entanglement in fermionic, bosonic and spin systems. In Chapter 2, we investigate a general quadratic fermionic Hamiltonian both in d = 2 and d = 3. Upon tuning the Hamiltonian parameters, this model has distinct criti- cal phases with and without a finite Fermi surface, as well as non-critical phases. The scaling behavior of the block entropy is accurately obtained via exact numerical diago- nalization. In non-critical states we find that the area law indeed holds, and we confirm that logarithmic corrections to this law are present in critical states with a finite Fermi surface, as found in Refs. [Wol06, GK06]. The prefactorC of theL-dependence ofS L in Eq.(1.4) is found to be very accurately predicted by a formula based on the Widom conjecture [GK06]. On the other hand, for critical states with a Fermi surface of zero measure, we find that the corrections to the area law are either absent or sublogarith- mic. This means that the relationship between entanglement and correlations in higher dimensional systems is different than in d = 1, and that a crucial role is played by the geometry of the Fermi surface or, alternatively, by the density of states at the ground state energy. 16 Next, in Chapter 3 we consider bosonic systems with a general quadratic Hamiltonian. We show reasons why the scaling behavior in this system is different from the fermionic systems ind>1 dimensions. In these two chapters we focus on quadratic Hamiltonian models which are solvable analytically. The ground states of these models are Gaussian states. By definition, a Gaussian state is characterized completely by the first and second statistical moments of the field operators. Gaussian states play an important role in quantum information among all quantum states in infinite-dimensional systems[BvL05]. In quantum optics, for example, they are often encountered as states of field modes of light. From an experi- mental point of view, they can be created and manipulated relatively easily[KW87] with linear optical elements, and one can use them for quantum cryptography [GAW + 03] and quantum teleportation [FSB + 98, BTSL02]. On the theoretical side, separability [GKLC01, Sim00] and distillability[GDCZ02] criteria for bipartite systems have been fully developed. One key reason for these successes is the fact that Gaussian states are completely specified by their first and second moments so that questions concerning properties of Gaussian states can be translated into properties of (comparatively small) finite-dimensional matrices. Moreover, pure Gaussian states are intimately related to Heisenberg’s uncertainty relation since they minimize such a relation for position and momentum operators. 17 In Chapter 4, we study the scaling behavior of the block entropy in antiferromagnetic spin systems. The general Heisenberg model is very difficult to solve. Numerical cal- culations of block entropies for 1D and 2D systems are limited to small system size far away from the asymptotic scaling regime. So we propose an alternative way: we con- struct trial wave functions, which are known to be good candidate for the ground state of Heisenberg Hamiltonians, and in some special cases they are also known to be the exact ground state. We study the behavior of block entropy for these wave functions. In this section we will introduce such a class of trial wave functions: the resonating valence bond states(RVB). Our work is mainly on short-range RVBs. Long-range RVBs are dif- ficult to study. We examine a special case: the infinite-range RVB with equal weight of superposition. The general long-range RVB is left for future work. Finally in chapter 5 we summarize our results and give a outline of topics for future works. 18 Chapter 2 Scaling Behavior of Entanglement in Fermion Model The study of entanglement in many-body systems close to quantum phase transitions is a hot topic of investigation, given that entanglement captures the pure-quantum part of correlations and it is therefore expected to deepen our understanding of the quantum critical state, in which quantum effects show up at all length scales. At a practical level, the behavior of entanglement at quantum critical points is a fundamental ingredient to ascertain the efficiency of methods like the density matrix renormalization group, and, in general, it dictates how much information we need to faithfully represent the collective state of the system. A fundamental quantity to investigate is the so-called block entropy of entanglement, which gives the entanglement between a connected block and the rest of the system: such a quantity is generally believed to scale like its boundaries away from critical points (the so-called ”area law”), but quantum critical corrections to it are expected. Investigating directly the entanglement behavior in strongly correlated quantum systems undergoing a quantum phase transition is a formidable task, which has 19 been so far undertaken in one dimension only due to the availability of exact solutions and well established field theories. In dimensions two and higher no such solutions are available, and much less is known about the behavior of entanglement in general. In this work we adopt the strategy of investigating a model of spinless free fermions in two and three dimensions which has the advantage of being exactly solvable and, yet, of displaying correlation properties typical of quantum critical phases, as well as of non-critical ones. This model is a higher-dimensional generalization of the model of free fermions exactly mappable onto the XY model in a transverse field in 1d. In this respect, it can be thought of as the model for the effective fermionic degrees of freedom of a strongly correlated system undergoing a quantum phase transition. Making use of a numerically exact method, we investigate the scaling behavior of the block entropy over its whole phase diagram of the system. The key ingredient of our model is that it has two distinct critical phases with and without a finite Fermi surface, as well as a non-critical phase with a gap to particle-hole excitations. This helps us to identify the condition under which the scaling behavior of the block entropy will show a correction to the area law. In non-critical states we find that the area law indeed holds, and we find that logarithmic corrections to such law are present in critical states with a finite Fermi surface, as previously found analitically. On the other hand, for critical states with a Fermi surface of zero measure, we find that the corrections to the area law 20 are either absent or sublogarithmic. This means that the relationship between entan- glement and correlations in higher dimensional systems is different than in one dimen- sional, where logaritmic corrections are always observed at quantum critical points, and that in higher dimensions a crucial role is played by the geometry of the Fermi surface. The question remains open whether a zero-measure Fermi surface with a finite density of states at the Fermi energy or with a fractal dimension can give corrections to the area law. 2.1 Model and Formula 2.1.1 One-dimension Free-Fermion Model Mapping and diagonalization In one dimension, free-fermion model are equivalent to XY spin chain [LSM61] and hard-core bosons [Gir60]. We will start by reviewing the main features of the XY spin chain and by identifying some of the critical regions in the space of parameters that define the model. In course of reviewing, we show the equivalence between these three models. Then, we proceed to compute the ground state jª g > of the system, from 21 which we obtain the reduced density matrix½ L forL contiguous sites. The knowledge of the eigenvalues of ½ L allows us to compute its entropy S L and, therefore, have a quantification of entanglement in free-fermion model. The XY model describes a chain of N spins with nearest neighbor interactions and an external magnetic field, as given by the Hamiltonian H XY = X l £ (1+°)S x l S x l+1 +(1¡°)S y l S y l+1 +¸S z l ¤ (2.1) Here l labels the N spins, whereas parameter ¸ is the intensity of the magnetic field, applied in thez direction, and parameter° determines the degree of anisotropy of spin- spin interaction, which is restricted to thexy plane in spin space. Notice that the sign of the interaction can be changed by applying a 180 degree rotation along thez axis (in spin space) to every second spin. Since this is a local transformation, the ground state of the original and transformed XY Hamiltonians are related by local unitary operations. Therefore both ground states are equivalent as far as entanglement properties are concerned. In this sense entanglement depends on fewer details than other properties of the chain, such as the magnetization. 22 The XY model embraces two famous spin models: Ising model and XX model. When ° =1, that is the interaction is restricted to thex direction in spin space, thenH XY turns into the Ising Hamiltonian with transverse magnetic field, H Ising =¡ 1 2 X l ¡ ¾ x l ¾ x l+1 +¸¾ z l ¢ : (2.2) If, instead, we consider the interaction to be isotropic in the xy plane, ° = 0, then we recover the XX Hamiltonian with transverse magnetic field, H XX =¡ 1 2 X l µ 1 2 [¾ x l ¾ x l+1 +¾ y l ¾ y l+1 ]+¸¾ z l ¶ : (2.3) In one dimension the spin one-half operator can be made into exact fermion operators. This transformation was discovered by Jordan and Wigner (1928) [JW28]. The Jordan- Wigner transformation is: S x j = exp[¼i j¡1 X l=1 c + l c l ] c j +c + j 2 (2.4) S y j = exp[¼i j¡1 X l=1 c + l c l ] c j +c + j 2i (2.5) 23 Where c + l and c l are fermionic creation and annihilation operators. By the relation:c + l c l = 1 2 +S z l , the inverse transformation can be easily got as: c j = exp[¡¼i j¡1 X l=1 ( 1 2 +S z l )] S x j ¡iS y j 2 (2.6) c + j = exp[¡¼i j¡1 X l=1 ( 1 2 +S z l )] S x j +iS y j 2 (2.7) Applying the transformation to XY model, we get: H = 1 2 X (c + i c i+1 +°c + i c + i+1 +c + i+1 c i +°c i+1 c i+1 ¡2¸c + i c i ) (2.8) Now the Hamiltonian is a simple quadratic form in Fermi operators and can be exactly diagonalized. The particular simplicity of H depends on the fact that the spins can be arranged in a definite order, that interactions occur only between neighboring spins in this ordering, and that thez-component interactions of spin do not enter. If the interac- tions were to extend tonth nearest neighbors,H would involve a polynomial of order2n in Fermi operators. In two-dimensional models, it can be readily seen that any ordering and nontrivial scheme of interactions must lead to a Hamiltonian involving a polynomial roughly of order2N for a system ofN 2 spin. Thus in higher dimensions spin models can not be mapped into quadratic fermionic models in a general way. Indeed most approx- imate analyses assume that in two or three dimensions the spins behave approximately as bosons rather than as fermions [Mah90]. In one dimension we are making maximum use of the nearest neighbor character of the interactions. 24 The quadratic fermionic Hamiltonian can be mapped to hard-core bonson model by another Jordan-Wigner transformation. The transformation is: c + j = b + j exp[¡i¼ j¡1 X l=1 b + l b l ] (2.9) c j = exp[i¼ j¡1 X l=1 b + l b l ]b j (2.10) Whereb + j andb + are bosonic creation and annihilation operators. Next we will focus on fermionic models. The diagonalization of quadratic fermionic Hamiltonian takes two steps. The first step is fourier transformation: c + j = 1 p N X k c + k exp[¡i 2¼jk N ] c j = 1 p N X k c k exp[i 2¼jk N ] (2.11) The inverse fourier transformation can be get by the orthogonal relation 1 N P j exp[i 2¼j N (l¡l 0 )]=± l;l 0. It is: c + k = 1 p N X j c + j exp[¡i 2¼jk N ] (2.12) c k = 1 p N X j c j exp[i 2¼jk N ] (2.13) 25 Applying the fourier transformation, the Hamiltonian now has the form as: H = X k [(¡¸+cos( 2¼k N ))c + k c k + i° 2 sin( 2¼k N )(c + k c + ¡k +c ¡k c k )] (2.14) The last step is the so-called Bogoliubov transformation. It can be expressed as: f + k = cos(® k )c + k +isin(® k )c ¡k f k = cos(® k )c k ¡isin(® k )c + ¡k (2.15) where: tan(2® k ) = °sin( 2¼k N ) ¸¡cos( 2¼k N ) (2.16) cos(2® k ) = ¡ ¸¡cos( 2¼k N ) q (¸¡cos( 2¼k N )) 2 +° 2 sin 2 ( 2¼k N ) (2.17) Now Hamiltonian takes a diagonal form: H = X k ¤ k f + k f k (2.18) ¤ k = r (¸¡cos( 2¼k N )) 2 +° 2 sin 2 ( 2¼k N ) 26 The thermodynamical limit is obtained by definingÁ=2¼k=N and taking theN !1 limit: H = X Á ¤ Á f + Á f Á (2.19) ¤ Á = q (¸¡cosÁ) 2 +° 2 sin 2 Á (2.20) whereÁ2 [¡¼;¼] is the variable in momentum space. We can use the explicit expres- sion of the spectrum to discuss the critical region of our model in parameter space(°;¸). From spectrum we can see the system has two critical region:¸=1, which is critical XY model; and° =0;¸2[0;1], which is critical XX model or one-dimension metal. The detail analysis of the critical regions can be found in [BM71]. The phase diagram is plotted in Fig.2.1. In conformal field theory, these two critical regions have different central charge. For critical XX region, the central charge is 1, and for critical XY region, the central charge is 1 2 . It turns out that the central charge plays an important role in scaling behavior of entanglement entropy [HLW94, CC04, JK04a, IJK05]. 27 Critical XY 0 λ Insulator γ 1 1 Ising Critical Ising Metal or Critical XX Figure 2.1: Phase diagram of the model Eq.(2.8) for one dimension. The Ising model, ° = 1, has a critical point at ¸ = 1. The XX model, ° = 0, is critical in the interval ¸2[0;1]. The whole line¸=1 is also critical. Reduced Density Matrix and Entanglement Entropy When the Hamiltonian of a quantum system is diagonalized, it is quite simple to obtain the ground state of the system. The ground state ofH is annihilated by allf: f Á jª g >=0 (2.21) 28 It is the vacuum of the fermionic modes and can be written as: jª g >= Y Á (cos(® Á )j0> Á j0> ¡Á ¡isin(® Á )j1> Á j1> ¡Á ) (2.22) wherej0 > Á andj1 > Á are the vacuum and single excitation of the Á-th momentum mode,c Á , respectively. We get the following relations of expectation value for operators: <f Á >=0 <f Á f Á 0 >=0 <f + Á f Á 0 >=0 <f Á f + Á 0 >=± Á;Á 0 (2.23) Due to the fact that the ground state is the tensor product of the modes, Wick’s theorem is valid. Any expectation value for a product of operatorsf Á andf + Á can be expressed in terms of<f Á f Á 0 >,<f Á f + Á 0 > and their complex conjugates. This means thatjª g > is a Gaussian state which is characterized by the expectation values of the first and second moments. In [CP01, Pes04a, CH04, BMC06], they showed that the reduced density matrix corresponding to parts of the system can be expressed as the exponential of a quadratic pseudo-Hamiltonian of the subsystem. From this expression we can calculate entanglement of entropy from correlation matrix. Next we will briefly summarize those formulas and give a procedure on the calculations. 29 We know that the reduced density matrix of the subsystem of size L can be written as[CP01, Pes04a, CH04]: ½ L =Tr N¡L (½)=A 0 exp[¡H] (2.24) whereA 0 is the normalization constant andH is a quadratic Hermite operator: H = i=L;j=L X i=1;j=1 [c + i A i;j c j + 1 2 (c + i B i;j c + j ¡c i B i;j c j )] (2.25) The Hermiticity ofH requires thatA be a Heimitian matrix andB be an antisymmetric matrix: A + = A andB + =¡B. SinceH is not necessary translational invariant now, it can not be diagonalized by Fourier transformation and Bogoliubov transformation as above. Next we will use generalized Bogoliubov transformation to do the job. We can find a linear transformation of the form: d k = i=L X i=1 (g ki c i +h ki c + i ) d + k = i=L X i (g ki c + i +h ki c i ) (2.26) 30 with theg ki andh ki real and canonical (i.e. thed + k andd k should also be Fermi operators) and which gives forH the form: H = L X k=1 " k d + k d k +constant (2.27) If this is possible, then: [d k ;H]=" k d k (2.28) Substituting Eq.2.26 into above equation and setting the coefficients of each operator equal, we obtain a set of equations for theg ki andh ki : " k g ki = X j (g kj A ji ¡h kj B ji ) (2.29) " k h ki = X j (g kj B ji ¡h kj A ji ) (2.30) By introducing the linear combinations: Á ki =g ki +h ki (2.31) à ki =g ki ¡h ki (2.32) 31 We get a simplified matrix equation array: " k Á k =à k (A+B) (2.33) " k à k =Á k (A¡B) (2.34) Plugging one into another, we eliminateÁ k orà k from the equation array and get: " 2 k Á k =Á k (A¡B)(A+B) (2.35) " 2 k à k =à k (A+B)(A¡B) (2.36) Á and à are orthogonal matrix. The generalized Bogoliubov transformation in matrix form is: d= Á+à 2 c+ Á¡Ã 2 c + d + = Á+à 2 c + + Á¡Ã 2 c (2.37) whered,d + ,c andc + are column vectors ofd k ,d + k ,c i andc + i respectively; andÁ andà are matrices with elementsÁ ki andà ki respectively. Now the reduced density matrix has the form: ½ L =A 0 exp " ¡( L X k=1 " k d + k d k +constant) # =A 0 exp " ¡ L X k=1 " k d + k d k # (2.38) 32 ByTr(½ L )=1, we can getA 0 = Q L k=1 1 1+exp(" k ) . The final form of the reduced density matrix is: ½ L = L Y k=1 exp(¡" k d + k d k ) 1+exp(¡" k ) (2.39) Define:º k = 1¡exp(¡" k ) 1+exp(¡" k ) , we can get a decomposition of the reduced density matrix: ½ L = L Y k=1 1+º k 2 exp · ¡ln µ 1+º k 1¡º k ¶ d + k d k ¸ = L Y k=1 ( 1+º k 2 ¡º k d + k d k ) = L O k=1 ½ k (2.40) where½ k = 1+º k 2 ¡º k d + k d k and it can be written in matrix form as: ½ k = 0 B @ 1+º k 2 1¡º k 2 1 C A (2.41) 33 In above derivation, we use the operator equivalence:exp(d + k d k ln¸)´1+(¸¡1)d + k d k . From the decomposition we can calculate the block entropy: S L = ¡Tr(½ L log(½ L )) = L X k=1 · ln(1+exp(¡" k ))+ " k 1+exp(" k ) ¸ = ¡ L X k=1 µ 1+º k 2 ln 1+º k 2 + 1¡º k 2 ln 1¡º k 2 ¶ (2.42) The remaining task is to calculate" k orº k from which we can calculate block entropy. Next we will show how to calculate" k orº k from correlation matrix. We know that the reduced density matrix ½ L reproduces all expectation values in the subsystem, that is:< O >= Tr(½O) = Tr(½ L O) for any operatorO in the subsystem. The expectation value ofd + k d k andd k d + k can be calculated: <d + k d k >=Tr(½ L d + k d k )= exp(¡" k ) 1+exp(¡" k ) = 1¡º k 2 <d k d + k >=Tr(½ L d k d + k )= 1 1+exp(¡" k ) = 1+º k 2 (2.43) 34 In the other way, they can be calculated from correlation matrices by applying the generalized Bogoliubov transformation Eq2.37. We can get the equation connecting <d + k d k > to<c + i c j >: <d + d T ¡dd +T > = 2Á(<c + c T >¡<c + c +T >¡ I 2 )à T = 2Ã(<c + c T >+<c + c +T >¡ I 2 )Á T (2.44) where ¢¢¢ T means transpose of ¢¢¢ and I is identity matrix. Combine and simplify Eq2.44, we get: diag(º 2 1 ;º 2 2 ;¢¢¢ ;º 2 L )= 4Á(<c + c T >¡<c + c +T >¡ I 2 )(<c + c T >+<c + c +T >¡ I 2 )Á T (2.45) Define a new correlation matrix as:G ij = 2(< c + i c j > + < c + i c + j > ¡ ± ij 2 ) =< (c + i ¡ c i )(c + j +c j )>, the equation 2.45 can be written as: G T G=Á T diag(º 2 1 ;º 2 2 ;¢¢¢ ;º 2 L )Á (2.46) Note the sub-index ofG andG T are supposed to be kept inside the block, that is,(1;L) because the transformations from d + and d to c + and c are restricted in the block. In fact, if the range of sub-index is the whole system, we will have:G T G=I. 35 The correlation matrixG can be calculated by applying Eq 2.11 and Eq 2.15 and using the relations in Eq 2.23: G mn = 1 2¼ Z ¼ ¡¼ dÁ ¸¡cosÁ+i°sinÁ p (¸¡cosÁ) 2 +° 2 sin 2 Á exp(iÁ(n¡m)) (2.47) wherem;n2(1;L). So the whole procedure of calculation is clear now: first calculate correlation matrixG from Eq2.47 with the sub-index restricted inside the block; then diagonalize G T G and getºs; finally we can calculate block entropy fromºs by Eq2.42. In [VLRK03], Vidal investigated numerically the scaling behavior of block entropy and got it as a logarithm function of block size in critical XX and critical XY model. For the noncritical regimes of the XY model, the block entropy will either vanish or saturate to some value for a certain block size. The prefactor of the logarithm function was fitted very well by one third of central charge of the model. Our calculation also reproduce and verify this relation. Jin and Korepin [IJK05, JK04b] employed Toeplitz matrix proper- ties to diagonalize the correlation matrix and got the scaling behavior of block entropy analytically. Peschel[Pes04b] got the same results using corner transfer techniques to solve the problem. Keating and Mezzadri [KM04] applying random matrix theory also obtained the same conclusions. Calabrese and Cardy [CC04] using conformal field the- ory got the similar results and related the prefactor to cental charge analytically. It seems 36 that we can make a conclusion about the above results: the scaling behavior of block entropy is controlled by quantum phase transition and the logarithm divergence of it is a signature of critical regime. Next we will generalize the fermionic model 2.8 to higher dimension: d = 2 andd = 3, and study the behavior of entanglement entropy of them. In higher dimension we will see the conclusion that the logarithm divergence of block entropy is a signature of critical regime is not exactly right. 2.1.2 High-dimensiond> 1 Free-Fermion Model Formula We consider a bilinear spinless fermionic system on ad-dimensional hypercubic lattice with hopping and pairing between nearest-neighbor lattice sites H = X hiji h c y i c j +°(c y i c y j +c j c i ) i ¡ X i 2¸c y i c i (2.48) ¸ is the chemical potential, while ° is the pairing potential. The sum of P hiji extends over nearest-neighbor pairs. The above Hamiltonian is a d > 1 generalization of the 1d spinless fermionic Hamiltonian 2.8. Although in d > 1 the exact relationship to the XY model is lost, we can imagine the above Hamiltonian to represent the effective 37 fermionic degrees of freedom of an interacting system with quantum-critical phases. The formula in higher dimension are similar to one-dimension. We only give the outline of the derivation here and all the details can be got by one-dimension analog. A more insightful expression for the Hamiltonian of Eq. (2.48) is obtained upon Fourier transformation to momentum space: c i = Z ¼ ¡¼ Z ¼ ¡¼ dk 4¼ 2 e ik¢i c k (2.49) Hamiltonian in momentum space is: H = X k h ¡2t k c y k c k +i¢ k (c y k c y ¡k +c ¡k c k ) i t k =¸¡ d X ®=1 cosk ® ¢ k =° d X ®=1 sink ® (2.50) The pairing potential ink-space,¢ k , clearly reveals ap-wave symmetry. This Hamiltonian can be diagonalized exactly by a Bogoliubov transformation: f y k =cos(® k )c y k +isin(® k )c -k (2.51) 38 where: tan(2® k ) = ° P d ®=1 sink ® ¸¡ P d ®=1 cosk ® (2.52) cos(2® k ) = ¡ ¸¡ P d ®=1 cosk ® q (¸¡ P d ®=1 cosk ® ) 2 +° 2 ( P d ®=1 sink ® ) 2 (2.53) The Bogoliubov transformation gives: H = X k ¤ k f + k f k ¤ k =2 q t 2 k +¢ 2 k (2.54) The correlation functions in two dimension can be get as: <c + i c j > = 1 2 ± ij + 1 8¼ 2 Z Z ¼ ¡¼ dk x dk y (¸¡cos(k x )¡cos(k y ))exp ik¢(i¡j) p (¸¡cosk x ¡cosk y ) 2 +° 2 (sink x +sink y ) 2 <c + i c + j > = i 8¼ 2 Z Z ¼ ¡¼ dk x dk y °(sin(k x )+sin(k y ))exp ik¢(i¡j) p (¸¡cosk x ¡cosk y ) 2 +° 2 (sink x +sink y ) 2 (2.55) We then proceed to the evaluation of the block entropy of entanglement. The ground state of Eq.(2.48) is known to be a Gaussian state, whose density matrix can be expressed as the exponential of a quadratic fermion operator [Gau60, Pes04a]. To obtain the reduced density matrix of a L d subsystem, Grassman algebra is needed [CP01]. Using a Bogoliubov transformation, the reduced density matrix½ L can then be written as ½ L =A exp à ¡ L 2 X l=1 " l d + l d l ! ; (2.56) 39 whered + l ,d l are the new Fermi operators after the transformation, andA is a normaliza- tion constant to ensure Tr(½) = 1. The single-particle eigenvalues " l can be obtained fromhc + i c j i andhc + i c + j i by the following formula[Pes04a]: (C¡F ¡ I 2 )(C +F ¡ I 2 )= 1 4 P diag © tanh 2 ¡ " 1 2 ¢ ;tanh 2 ¡ " 2 2 ¢ ;:::;tanh 2 ¡ " L 2 ¢ª P ¡1 (2.57) whereC i;j =hc + i c j i andF i;j =hc + i c + j i;P is the orthogonal matrix that diagonalizes the left side of the above equation. The Block entropy can then be calculated in terms of" l as: S L = L 2 X l=1 ½ ln[1+exp(¡" l )]+ " l exp(" l )+1 ¾ (2.58) 2.2 Phase Analysis Depending on the parameters ° and ¸, this system has a rich phase diagram, includ- ing metallic, insulating and (p-wave) superconducting regimes, as shown in Fig. 2.3. The different phases are certainly distinguished by the different decay of the correlation function, which tells apart the critical from the non-critical phases. Now we will study the correlation function of two dimension model as a special case. The ground state 40 of the system can be conveniently thought in two different ways depending on whether °6=0 or° =0. METALLIC PHASE, ° = 0: in this simple case the c-fermions are free and have a cosine band as energy dispersion and a chemical potential of 2¸ setting the Fermi energy. The Fermi surface is therefore given by FS:¡cosk x ¡cosk y =¸ (2.59) and, for k x ;k y ! 0, it is a circle with radius k F = 2 p 1+¸=2 (here we assume the gauge transformation c i ! c i (¡i) i on one of the two sublattices, so that the hopping amplitude is negative and the cosine band is convex). The correlation function is given by hc y 0 c r i= 1 N X k e ik¢r hc y k c k i= 1 N X k e ik¢r µ(k F ¡k) (2.60) Passing from a sum to an integral, and taking the approximation of a circular Fermi surface (valid for¸&¡2, namely near the bottom of the band) 41 hc y 0 c r i= 1 (2¼) 2 Z ¼ ¡¼ dk x Z ¼ ¡¼ dk y e ik¢r µ(k F ¡k)¼ Z k F 0 dkk Z 2¼ 0 dÁe ikrcosÁ (2.61) We first integrate overk, getting hc y 0 c r i¼ 1 (2¼) 2 Z dÁ · e ik F rcosÁ ¡1 r 2 cos 2 Á ¡ ik F e ik F rcosÁ rcosÁ ¸ (2.62) Changing to the variablez =e iÁ integrated over the unit circle, each of the two terms in the integral has two poles on the integration contour atz =§i. Using the prescription of encircling one of the poles while excluding the other, e.g. z = §i¡² (² ! 0), we obtain that the first term in the integral vanishes while the second one gives hc y 0 c r i¼¡ k F 4¼r : (2.63) Hence we get a power-law decay with power 1 which coincides with the gapless excita- tion of the system. It is also meaningful to have the Fermi vector in the numerator: when we hit the bottom of the band withk = 0 the Fermi vector vanishes (in fact¸!¡2), and the ground state becomes empty of particles, so that the correlator clearly vanishes. In the opposite limit of a full band, ¸! 2, the approximation a circular Fermi surface 42 breaks down, there appear Fermi arcs instead, and we have to evaluate the integral over the Fermi volume more carefully. SUPERCONDUCTING PHASE ° 6= 0: We take for definiteness the positive solu- tion for the spectrum, ¤ k = 2 p (¸¡cosk x ¡cosk y ) 2 +° 2 (sink x +sink y ) 2 . In this case the ground state for the f-fermions is the vacuum, given that all energies are positive. The elementary quasiparticle excitations are gapless at the point nodes k x =arccos(¸=2),k y =¡k x . Inverting the Bogolyubov transformation, we obtain that hc y k c k i=hf y k f k i+ 1 2 (1¡cos(2® k )) (2.64) where the first term on the right hand side vanishes in the ground state. On the other side, for°!0 we obtain thatcos(2® k )!1 so thathc y k c k i!hf y k f k i. The fundamental integral to be evaluated is the following I(x;y)= Z ¼ ¡¼ dk x Z ¼ ¡¼ dk y jcosk x +cosk y ¡¸j p (cosk x +cosk y ¡¸) 2 +° 2 (sink x +sink y ) 2 cos(k x x+k y y) (2.65) 43 so that hc y 0 c r i= 1 2 (± r;0 ¡I(x;y)) (2.66) The above integral can only be calculated numerically, because it has non-integrable poles» 1=(k x(y) ¡k 0;x(y) ). Attacking it with the residue theorem and the substitution, e.g,z =e ik x and³ =e ik y does not make it very treatable. Numerical evaluation of the correlation function in different regions through integration over the first Brillouin zone (FBZ), <c + i c j >= Z FBZ d d k (2¼) d t k 2¤ k e ik¢(i¡j) ; (2.67) shows an expected power-law decay in the regions which have a gapless excitation and an exponential decay in the region which have a gapped excitation. Fig2.2 shows the numerical results for two dimension. We calculate correlation functions for the parame- ter space:f(¸;°)2f0;1;2;3g N f0;1;2;3gg. In graph we can see algebra decay of the correlation function inf0 · ¸ < 2;° = 0g andf0 · ¸ · 2;° > 0g and exponential decay inf¸ = 2;° = 0g andf2 · ¸g. Whenf¸ = 2;° = 0g the system is in insu- lator phase, and the ground state is a product state. This is the reason that the system has a gapless excitation whereas the correlation function decays exponentially. For the 44 Figure 2.2: Correlation function in parameter space:f(¸;°) 2 f0;1;2;3g N f0;1;2;3gg. The x-axis and y-axis of insets are distance and corre- lation function respectively. we can see algebra decay of the correlation function in f0 · ¸ < 2;° = 0g and f0 · ¸ · 2;° > 0g and exponential decay in flambda =2;° =0g andf2·¸g other cases, the gapless excitations coincide with the algebra decay and from this we can identify the critical and noncritical regions. Nonetheless, a classification which turns out to be relevant for the study of entanglement is based on the features of the gapless excitation manifold ¤ k = 0. Such manifold can 45 be characterized by the density of states at the ground-state energy g(0), and by the so-called co-dimension [V ol03, V ol05] ¹ d, defined as the dimension of k-space minus the dimension of the ¤ k = 0 manifold. We notice that the existence of a finite Fermi surface at zero energy implies that ¹ d = 1 and g(0) > 0, while in absence of a finite Fermi surface we have ¹ d ¸ 2 and g(0) > 0 or g(0) = 0 depending on the dispersion relation¤ k around its nodes. According to this classification, we can distinguish three phases: Phase I,f0 · ¸ < d;° = 0g, andf¸ = 0;° > 0g if d = 2. For ° = 0, Eq. (2.48) reduces to a simple tight-binding model, which is in a metallic state with a 2d-fold symmetric Fermi surface as far as¸·d. We plot the spectrum in momentum space for two dimension in Fig2.4. In d = 2, for ¸ = 0 the system is still a metal with a well defined Fermi surface, which is simply k x = k y §¼, and whose symmetry is lowered by the presence of the ° term in the Hamiltonian. In this phase, g(0) > 0, and ¹ d = 1 everywhere except at the pointf¸=d;° =0g where ¹ d=2. Phase II,f0 < ¸· d;° > 0g, and ifd = 3f¸ = 0;° > 0g. Away from the boundary lines of this phase, the system is in ap-wave superconducting state, with a finite pairing amplitude hc y k c y ¡k i 6= 0. Such pairing amplitude vanishes at the boundaries of this region. The dispersion relation ¤ k has point nodes in d = 2 and line nodes in d = 3. 46 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 λ=2,γ=0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 λ=1,γ=0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 λ=1,γ=1 I II III -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 λ=2,γ=1 λ=0,γ=1 λ=3,γ=0 0 γ λ 2 Figure 2.3: Phase diagram of the model Eq.(2.48) for the case d = 2. The roman numbers for the various phases are explained in the text. Representative contour plots of the dispersion relation¤ k are also shown. There the black areas corresponds to¤ k = 0 and the white areas to the top of the band. We plot the spectrum in momentum space ford = 2 in Fig2.5 forf¸ = 1;° = 1g and Fig2.6 forf¸=1;° =2g. Everywhere in this phaseg(0)=0 and ¹ d=2. 47 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Lambda(k x ,k y ) lambda = 1, gamma = 0: METAL k x k y Lambda(k x ,k y ) Figure 2.4: Spectrum in momentum space for two dimension whenf¸=1;° =0g. The intersection circle of the two surfaces is the Fermi surface of the system which means ¹ d=1. Phase III,f¸ > dg. In this phase the system is in an insulating state with a gap in the excitation spectrum. We plot the spectrum in momentum space ford = 2 in Fig2.7 for f¸=1;° =3g. Hereg(0)=0 and ¹ d=d. This shows that, in terms of the spectral properties, the above system has two distinct critical phases, (I andII), which are both gapless but have different gapless excitation topology, and a non-critical phase (III). 48 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Lambda(k x ,k y ) lambda = 1, gamma = 1: BAD METAL k x k y Lambda(k x ,k y ) Figure 2.5: Spectrum in momentum space for two dimension whenf¸ = 1;° = 1g. The two surfaces only cross at two points which means ¹ d=2. 2.3 Scaling Behavior of Entanglement Ind=1 our formulas reproduce the scaling of the block entropy as observed in the XY model in a transverse field [VLRK03, LRV04]. Ind=2 the phase diagram is richer, and we need to consider the various phases one by one. We begin with the critical metallic phase (I),f° = 0;0 · ¸ < dg. For this case a logarithmic correction to the area law, S L =(C(¸)=3)(log 2 L)L d¡1 is observed for all values of¸, as shown in Fig. 2.8. 49 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -4 -2 0 2 4 Lambda(k x ,k y ) lambda = 2, gamma = 1: BAD METAL k x k y Lambda(k x ,k y ) Figure 2.6: Spectrum in momentum space for two dimension whenf¸ = 1;° = 2g. The two surfaces only cross at one point which means ¹ d=2. This is in full agreement with the results of Refs.[Wol06, GK06], which predict this behavior in presence of a finite Fermi surface. More specifically, Ref. [GK06] also supplies us with an explicit prediction for the ¸ dependence of C(¸), based on the Widom’s conjecture [Wid82], in the form C(¸)= 1 4(2¼) d¡1 Z @ Z @¡(¸) jn x ¢n p jdS x dS p (2.68) 50 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -4 -2 0 2 4 Lambda(k x ,k y ) lambda = 3, gamma = 1: INSULATOR k x k y Lambda(k x ,k y ) Figure 2.7: Spectrum in momentum space for two dimension whenf¸ = 1;° = 3g. The two surfaces have no cross section which means ¹ d=2. where is the volume of the block normalized to one, ¡(¸) is the volume enclosed by the Fermi surface, and the integration is carried over the surface of both domains. For d=2 the calculation ofC(¸) is pretty simple and we can get: C(¸)= 2 ¼ cos ¡1 (¸¡1) (2.69) For three dimension the calculation ofC(¸) is done numerically after proper simplifica- tion. To calculateC(¸) in three dimension, we need to divide the parameter space[0;3] into two parts. 51 Figure 2.8: Scaling of the block entropy S L in d = 2 for ° = 0 (left panel) and ¸ = 0 (right panel). The solid lines correspond to fits according to the formula S L = C 3 Llog 2 (L)+BL+A. For0·¸·1: J =4f Z cos ¡1 (¡¸) 0 cos ¡1 (¸+cosx¡1)dx+cos ¡1 (¡¸)¤¼¡ Z ¼ cos ¡1 (¡¸) cos ¡1 (¸+cosx+1)dxg (2.70) 52 For1·¸·3: J =4f Z ¼ cos ¡1 (2¡¸) cos ¡1 (¸+cosx¡1)dxg (2.71) To calculate the coefficientC(¸), we have C(¸)= 2¤6¤J 4¤4¼ 2 = 3J 4¼ 2 (2.72) For example, if ¸ = 1, we have J = 4f R ¼ 0 xdxg = 2¼ 2 , so we get C(1) = 3¤2¼ 2 4¼ 2 = 3=2 = 1:5. And if¸ = 0, we haveS = 4¤2 R ¼ 2 0 cos ¡1 (cosx¡1)dx = 8¤3:11108 = 24:8882, so we can calculate® = 3¤24:8882 4¼ 2 =1:891. A numerical fit of the calculated asymptotic behavior of S L through the formula S L = C 3 L d¡1 log 2 (L)+BL d¡1 +AL d¡2 +::: provides us with the exact result for the C(¸) prefactor. In Fig 2.9 the prediction of Ref. [GK06], Eq(2.68), for the casef0 · ¸ · d;° =0g is compared to our numerical results both ford=2 andd=3. The agreement is clearly striking(the error is about 0.002). Moreover, forf¸ = 0;° > 0g in d = 2 the formula Eq(2.68) predicts C = 1, which is also accurately verified by our data as shown in Fig 2.8. This proves that the formula Eq(2.68) is essentially providing a complete analytic form for the leading behavior of the block-entropy scaling in arbitrary dimensions for systems with a finite Fermi surface. 53 Figure 2.9: ¸-dependence of the C coefficient in Eq.(1.4) in d = 2 and d = 3. The values extracted from fits to our numerical data are compared with the predictions of Ref. [GK06]. In d = 2, the exact form of C(¸) can be obtained, which is equal to 2 ¼ cos ¡1 (¸¡1) The model that was chosen has an extremely rich phenomenology. Related analyti- cal studies of Wolf [Wol06] and Gioev/Klich [GK06] do draw a rigorous connection between the existence of a regular Fermi surface and the breaking of the area law, but these approaches do not address the case of critical systems in the absence of a well- defined Fermi surface. We then turn to the other two phases, II and III. Two scans 54 Figure 2.10: Scaling of the block entropyS L ind = 2 for° = 1 (left panel) and¸ = 1 (right panel). through these phases, at fixed ° = 1 and at fixed ¸ = 1 are shown in Fig. 2.10. We observe that logarithmic corrections are absent in both, and only sublogarithmic correc- tions are possible. Forf¸¸d;° =0gS L =0 identically, and the state is not entangled. While the area law is expected to hold in the non-critical phase III, it is surprising to observe it enforced also in the critical phaseII, which has a diverging correlation length. 55 S L ¹ d g(0) hc y i c j i Phase I »(log 2 L)L d¡1 1 >0 power-law decay Phase II »L d¡1 2 0 power-law decay Phase III »L d¡1 d 0 exp. decay Table 2.1: Summary of the entanglement scaling properties, co-dimension, density of states and decay of correlations in the three phases of the model Eq.(2.48) ind=2;3. This clearly reveals that the connection between block-entropy scaling and correlation properties is not straightforward ind>1. Our results for the entanglement behavior, co-dimension, density of states and corre- lation properties are summarized in Table 2.1. A crucial difference between the two critical phases I and II is the co-dimension ¹ d, and the density of states at the ground state energy. We have ¹ d = 1 and g(0) > 0 in the phase I, which shows logarithmic corrections to the area law, whereas ¹ d = 2 and g(0) = 0 in the phase II, in which the area law is verified up to sublogarithmic corrections. The special casef¸ = 0;° > 0g makes this difference stand out as we move from d = 2 to d = 3. In this parameter region, for d = 2 we have ¹ d = 1 and g(0) > 0, while for d = 3 we have ¹ d = 2 and g(0) = 0. S(L)» Llog(L) is observed for the former case, while S(L)» L 2 for the latter, which does not show a logarithmic correction. It is therefore tempting to conjecture that a codimension ¹ d = 1 or, alternatively, a finite density of states at the ground state energyg(0) > 0 is a sufficient condition for critical phases in d > 1 to show violations of the area law. For the fermionic system under 56 consideration, ¹ d = 1 requires the existence of a finite Fermi surface, which is the basic assumption of the proof of area-law violation in Refs. [Wol06, GK06]. In absence of a finite Fermi surface, it remains to be proven whether a finite density of statesg(0) > 0 might lead alone to a violation of the area law (This assumption seems to be contradicted by the casef¸ = d;° = 0g of the model under investigation, where g(0) > 0 and yet the prefactor C vanishes. Nonetheless this special point corresponds to a non-entangled state with S L = 0 for every L.). Indeed, one can speculate that the density of states at the Fermi energy may play an important role, given that so far no system has been investigated which has no Fermi surface but a finite density of states - this will be a subject of future investigations. This conjecture would generalize the results ford=1, where the co-dimension can only take the value ¹ d = 1, and only critical phases withg(0) > 0 have been explored in the literature. Also this conjecture can explain why in higher dimension quadratic bosonic models area law will always hold. The reason is that in higher dimension bosonic models it is impossible to have ¹ d=1. We will detail this later. It is believed that the different behavior of systems with d > 1 with respect to d = 1 systems can be understood by mapping them onto d = 1 systems with long-range interactions. This is not true in this case. If this were in fact quantitatively true, d = 1 systems with long-range interactions should exhibit much more serious violations of the 57 area law than just logarithmic corrections, bur rather something like a ”volume law” to match the entropy scaling of the higher dimensional systems. However, this has never been reported in the literature. On the contrary, the examples on one- or higher- dimensional systems with long-range interactions available in the literature show that the area law is always enforced in bosonic systems [CEPD06], whereas for Ising spin systems the area law is observed for interactions decaying faster than the inverse of the distance [DHH + 05]. To conclude, in this study we verified the conjecture proposed in the work of Gioev and Klich on the quantitative relationship between the Fermi surface and the violation of the area law in critical fermionic systems in d > 1. And we find that fermionic criticality without a Fermi surface and with zero density of states at the Fermi energy does not lead to a violation of the area law. Further investigations in systems withd > 1 are clearly needed to confirm this picture, and to clarify whether more severe violations of the area law are possible in presence of infinitely degenerate ground states or in systems with a fractal Fermi surface [Wol06]. 58 Chapter 3 Scaling Behavior of Entanglement in Boson Model Quite recently a class of systems composed of bosonic modes of systems with contin- uous degrees of freedom have come under intensive investigations[JKP01, DGCZ00, Sim00, WW01, PHE04, WGK + 04]. There are a number of reasons for this interest. First, bosonic modes are the appropriate subsystems the entanglement of which must be calculated when dealing with systems of identical boson particles[vE03, GF02, Ved03]. Second, states of continuous systems are widely encountered in many branches of physics in which entanglement plays a role, i.e. namely in quantum optical setups, in atomic ensembles interacting with electromagnetic fields [JKP01], in the motion of ions in ion traps and in low excitations of bosonic field theories. Third, for a class of such states, namely Gaussian states, analytical measures of entanglement have been defined and calculated in closed form. Finally, a large class of interesting many body systems of this type, when written in terms of suitable coordinates, are in fact free systems. This 59 latter reason makes a fuller investigation of the above questions in such systems much easier than in spin systems. This chapter provides answers to the scaling behavior of the entanglement of regions with their exterior in a general setting of bosonic lattice systems with a quadratic Hamil- tonian. In one dimension, it was found the block entropy has a logarithmic dependence on block size when system is in critical region, and saturates to a constant value when system is away from critical region[Sre94, BR04] for bosons. In higher dimensions d > 1, it was proved analytically that the entanglement away from criticality is essen- tially proportional to the surface area of subsystem (area law)[PEDC05, MC06]. At criticality, the correlation length diverges and one may expect corrections to the area law as in fermionic systems[LDY + 06]. However, numerical evidences for area law dur- ing criticality were found[Sre94, BMC06, MC06] and Callan and Wiczek derived the area law for black hole entropy in approximative field theoretical calculations for all d>1 [CW94]. Next we will briefly derive the formulas for calculations and present the results and explanation of ours. 60 3.1 Model and Formula We begin with a linear harmonic chain of N local oscillators. It can be modeled by Hamiltonian with canonical variables(q i ;p i ) which denotes position and momentum of i-th oscillator. H = E 0 2 X (p 2 i +q 2 i ¡®q i q i+1 ) (3.1) where the dimensionless parameter® characterizes the strength of the coupling between adjacent neightbor sites. Note that such a Hamiltonian can be obtained from the standard Hamiltonian of a chain with springlike nearest-neighbor harmonic couplings: H = 1 2 X · ¼ 2 i M +M! 2 » 2 i +K(» i ¡» i¡1 ) ¸ (3.2) by means of the canonical variable rescaling: q i = s M! r 1+ 2K M! 2 » i p i = ¼ i r M! q 1+ 2K M! 2 (3.3) 61 and the identification E 0 = ! r 1+ 2K M! 2 ® = 2K M! 2 +2K (3.4) The last relation provides a restriction0 < ® < 1 to the possible values of the coupling constant. The limit of strong coupling between neighboring oscillators, 2K M! 2 ! 1, corresponds to®!1, and the weak coupling limit to®!0. The Hamiltonian 3.1 can be brought to a normal form by introducing a set of annihilation (creation) operators¯(µ k ) (¯ y (µ k ) ) satisfying the commutation relations [¯(µ k );¯ y (µ l )]=± kl ; (3.5) with the indexing angular variableµ k playing the role of a dimensionless wave number or pseudo-momentum and taking the values µ k = 2¼k N ; (k =0;1;:::;N¡1): (3.6) Defining the dispersion relation (in units ofE o ) º(µ k )´ p 1¡®cosµ k ; (3.7) 62 and expressingq n andp n in the form q n = 1 p N X k 1 p 2º(µ k ) £ ¯(µ k )e iµ k n +h:c: ¤ ; (3.8) p n = ¡i p N X k r º(µ k ) 2 £ ¯(µ k )e iµ k n ¡h:c: ¤ ; (3.9) the Hamiltonian 3.1 achieves the normal form H =E o X k º(µ k ) · ¯ y (µ k )¯(µ k )+ 1 2 ¸ ; (3.10) which is then diagonalized by the Fock states of the creation/annihilation operators. In particular, we will be interested in the ground statej0>, satisfying ¯(µ k )j0>=0; (3.11) for allµ k . This state can be solved in a more general framework. We can write 3.1 in a general form as: H = E 0 2 X (p 2 i +q i M ij q j ) (3.12) 63 where M ij is a Heimitian positive definite matrix. The Hamiltonian can be simplified by an orthogonal transformation:p 0 =Up andq 0 =Uq, wherep,p 0 ,q andq 0 are column vectors, andU is the unitary matrix that brings M into diagonal form: M =U + diag(! 2 1 ;! 2 2 ;:::;! 2 N )U (3.13) Now the Hamiltonian is: H = E 0 2 X (p 0 2 i +! 2 i q 0 2 i ) (3.14) because U is a unitary matrix, p 0 and q 0 are still canonical. The ground state of this Hamiltonian in the coordinate and momentum representations assume the Gaussian form as: à o (q 0 ) / Y exp(¡ ! i q 02 i 2 ) à o (p 0 ) / Y exp(¡ p 02 i 2! i ) (3.15) they can be written in matrix form as: à o (q 0 ) / Y exp(¡ 1 2 q 0T diag(! 1 ;! 2 ;:::;! N )q 0 ) à o (p 0 ) / Y exp(¡ 1 2 p 0T diag(! 1 ;! 2 ;:::;! N )p 0 ) (3.16) 64 By<q 0 i q 0 j >=± ij 1 2! i and<p 0 i p 0 j >=± ij ! i 2 , we can rewrite above equations as: à o (q 0 ) / Y exp(¡ 1 4 q 0T <q 0 q 0T > ¡1 q 0 ) à o (p 0 ) / Y exp(¡ 1 4 p 0T <p 0 p 0T > ¡1 p 0 ) (3.17) The canonical variables can be transformed back and we obtain the equations in original variables: à o (q) / Y exp(¡ 1 4 q T <qq T > ¡1 q) à o (p) / Y exp(¡ 1 4 p T <pp T > ¡1 p) (3.18) and we have the relations between<qq T >,<pp T > andM: <qq T > = 1 2 M ¡ 1 2 <pp T > = 1 2 M 1 2 (3.19) where the covariance matrices< qq T > and< pp T >, forq andp respectively, satisfy the generalized uncertainty relation< qq T >< pp T >= I 4 . Furthermore, since the state is Gaussian, higher moments of the oscillator coordinates or momenta are expressible in terms of the two-point correlation functions. Thus, the relevant physical information associated with the ground state is contained in the correlation functions. 65 Although we can follow the procedure in [Sre94] to calculate block entropy in Hamil- tonian 3.1 and 3.12, they can be generalized to a more comprehensive model by field operatorsb andb + . We can write Hamiltonian 3.1 in field operators and see what form it has. By the transformation:p j = i p 2 (b + j ¡b j ) andq j = 1 p 2 (b + j +b j ), we have: H = E 0 2 X h b + i b i +b i b + i ¡ ® 2 (b + i b i+1 +b i b + i+1 )¡ ® 2 (b + i b + i+1 +b i b i+1 ) i (3.20) This Hamiltonian has the same coefficient for the hopping terms and pairing terms. So we can generalize this Hamiltonian by allowing different coefficient: H = X £ ¡2¸b + i b i +h(b + i b i+1 +b i b + i+1 )+°(b + i b + i+1 +b i b i+1 ) ¤ (3.21) The diagonalization of the above quadratic bosonic Hamiltonian takes two steps similar to fermion case. The first step is fourier transformation: b + j = 1 p N X k b + k exp[¡ikr j ] b j = 1 p N X k b k exp[ikr j ] (3.22) where k = 2¼l N and l 2 (1;N). Applying the fourier transformation, the Hamiltonian now has the form as: H = X k £ (¡2¸+2hcos(k))b + k b k +°cos(k)(b + k b + ¡k +b ¡k b k ) ¤ (3.23) 66 The second step is the so-called Bogoliubov transformation. It can be expressed as: ¯ + k = cosh(® k )b + k ¡sinh(® k )b ¡k ¯ k = cosh(® k )b k ¡sinh(® k )b + ¡k (3.24) Apply the Bogoliubov transformation and make the coefficients of off-diagonal term equal zero, we can get diagonalized Hamiltonian. Now Hamiltonian takes a diagonal form: H = 2 X k ¤ k ¯ + k ¯ k (3.25) ¤ k = p (¸¡hcos(k)) 2 ¡° 2 cos 2 (k) (3.26) ® k can be solved from the following equations: cosh(2® k ) = ¡¸+hcos(k) ¤ k sinh(2® k ) = ¡ °cos(k) ¤ k (3.27) The ground state of this Hamiltonian is a Gaussian state. It is annihilated by all¯: ¯ k jª g >=0 (3.28) 67 It is the vacuum of the bosonic modes. We get the following relations of expectation value for operators: <¯ k > = 0 <¯ k ¯ k 0 > = 0 <¯ + k ¯ k 0 > = 0 <¯ k ¯ + k 0 > = ± Á;Á 0 (3.29) As in fermionic models, the reduced density matrix can be written as an exponential of a quadratic Hermite operator and the quadratic Hermite operator can be diagonalized. Finally we have: ½ L =A 0 exp " ¡ L X k=1 " k ´ + k ´ k # (3.30) ByTr(½ L ) = 1, we can getA 0 = Q L k=1 (1¡exp(¡" k )). The final form of the reduced density matrix is: ½ L = L Y k=1 (1¡exp(¡" k ))exp(¡" k ´ + k ´ k ) (3.31) 68 We can calculate<´ + k ´ k > and<´ k ´ + k > from½ L as: <´ + k ´ k >=Tr(½ L ´ + k ´ k )= exp(¡" k ) 1¡exp(¡" k ) <´ k ´ + k >=Tr(½ L ´ k ´ + k )= 1 1¡exp(¡" k ) (3.32) By these equations we can relate " k with < b + i b j > and < b + i b + j > as in chapter 2. DefineG AA =< (b + +b)(b + +b) T >= 1 2 < qq T >,G BB =< (b + ¡b)(b + ¡b) T >= ¡ 1 2 < pp T > and¹ k = 1+exp(¡" k ) 1¡exp(¡" k ) = coth( " k 2 ), whereb + andb are the column vectors of corresponding field operators, b + i and b i , with sub-index restricted inside the block (1;L). It turns out that they have the following relation: G AA G BB =P T diag(¡¹ 2 1 ;¡¹ 2 2 ;¢¢¢ ;¡¹ 2 L )P (3.33) Note the sub-index of G AA and G BB are supposed to be kept inside the block, that is, (1;L). In fact, if the range of sub-index is the whole system, we will have:G AA G BB =I by Eq3.19. The block entropy can be calculated as: S L = ¡Tr(½ L log(½ L )) = L X k=1 · ¡ln(1¡exp(¡" k ))+ " k exp(" k )¡1 ¸ = L X k=1 µ ¹ k +1 2 ln ¹ k +1 2 ¡ ¹ k ¡1 2 ln ¹ k ¡1 2 ¶ (3.34) 69 The correlation matrix G AA and G BB can be calculated by applying Eq 3.22 and Eq 3.24 and using the relations in Eq 3.29: G AA mn = <2b + m b n +2b + m b + n +± mn > = 1 2¼ Z ¼ ¡¼ dk ¡¸+hcosk¡°cosk ¤ k exp(ik(m¡n)) (3.35) G BB mn = <¡2b + m b n +2b + m b + n ¡± mn > = ¡ 1 2¼ Z ¼ ¡¼ dk ¡¸+hcosk+°cosk ¤ k exp(ik(m¡n)) (3.36) wherem;n2(1;L). Formula for higher dimensions can be built by analog to fermionic case in chapter 2. So the whole procedure of calculation is clear now: first calculate correlation matrix G AA andG AA from Eq3.35 and 3.36 with the sub-index restricted inside the block; then diagonalize G AA G BB and get ¹’s by Eq3.33; finally we can calculate block entropy from¹’s by Eq3.34. 3.2 Scaling Behavior of Entanglement The entanglement of a finite region for a one-dimensional bosonic fields has been pre- viously investigated in connection with the black hole entropy ”area law” [BKLS86, 70 Sre94]. By employing methods of conformal field theory it has been shown [CW94, HLW94] that in the massless case entanglement behaves like 1 3 ln L ² where ² plays the role of the UV cutoff. In [BR04], Botero and Reznik obtained the same result in strong coupling limit by modewise decomposition technique. In one dimension, the introduc- ing of UV cutoff ² is necessary. When the excitation of system is gapless, system is critical. We can easily get the critical region by the spectrum of Hamiltonian in Eq3.26 and the restriction in Eq3.27:¡¸=jh+°j or¡¸=jh¡°j. Suppose that¡¸=h+° andh+° > 0. The integrand in correlation matrixG AA ’s elements is: f AA mn = ¡¸+hcosk¡°cosk ¤ k exp(ik(m¡n)) = s ¡¸+hcosk¡°cosk ¡¸(1+cosk) exp(ik(m¡n)) (3.37) It has a pole at k = §¼ and the integral over the pole will diverge. The divergence comes from including modes of arbitrary small wavelength. The way to remove the divergence is introducing a fundamental length, the lattice spacing ², which act as a UV cutoff. This procedure is the regularization needed in calculation to remove the divergence: introduce cutoff first and then remove the cutoff from the final results. It was proven the final results about the scaling behavior of block entropy do not depend on the specific form of the ultraviolet cutoff which is used [Sre94]. We calculate the block entropy of one-dimension infinite bosonic chain for critical and non-critical cases. The critical point is:¸ =¡1,h = 0:5 and° = 0:5. The non critical 71 Figure 3.1: The block entropy of one-dimension infinite bosonic chain for critical and non-critical cases. The critical point is:¸ =¡1,h = 0:5 and° = 0:5. The non critical point is:¸=¡1:00001,h=0:5 and° =0:5. For critical point the block entropy can be fitted very well by 1 3 log(L). For the non-critical case, although only slightly away from critical case, the block entropy saturates very fast. point is:¸ = ¡1:00001, h = 0:5 and ° = 0:5. The results are plotted in Fig3.1. For critical point the block entropy can be fitted very well by 1 3 log(L). For the non-critical case, although only slightly away from critical case, the block entropy saturates very fast. In calculation of the critical case, we apply appropriate cutoff to make sure the convergence of the integral. 72 We apply the theory to a special system which has fixed particle numbers,W , and only hopping interaction. The Hamiltonian is: H = X £ ¡2¸b + i b i +h(b + i b i+1 +b i b + i+1 ) ¤ (3.38) The Hamiltonian can be diagonalized by fourier transformation Eq3.22: H = X ² k b + k b k (3.39) ² k = ¡2¸+2hcosk (3.40) The system is critical when¸ =¡1 andh =¡1 and the gapless excitation is atk = 0. We can calculate the correlation matrix in this case: <b + i b j > = 1 N X k exp[ik(r i ¡r j )]<b + k b k > = W N (3.41) Define the particle density as:n= W N . The correlation matrix can be get as: G AA = 0 B B B B B @ 1+n n ¢¢¢ n n 1+n ¢¢¢ n ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢ 1 C C C C C A (3.42) 73 The eigenvalue of this matrix is:¹ 1 = 1+nL and¹ 2 = ¹ 3 =¢¢¢ = ¹ L = 1. The block entropy is: S L = L X k=1 µ ¹ k +1 2 ln ¹ k +1 2 ¡ ¹ k ¡1 2 ln ¹ k ¡1 2 ¶ ¼ lnL+ 2 nL +Const: (3.43) The block entropy is still proportionallnL. However, it has different prefactor from 1 3 . In this system we have fixed particle number which is quite different from free bosons. Free bosonic system can have arbitrary number of particles and have higher symmetries. This is an example that the more symmetries, the less entanglement. We will see another example in chapter4. For higher dimension with fixed particle number and only hopping interaction, we have: S L /DlnL (3.44) The block entropy is logarithmic in all dimensions. This is because in higher dimension the system has the permutation invariant symmetry which does not show up in other free bosonic systems. The higher symmetry lowers the entanglement in the system. In higher dimensions d > 1, it was proved analytically that the entanglement away from criticality is essentially proportional to the surface area of subsystem (area law)[PEDC05, MC06]. At criticality, the correlation length diverges and one may expect 74 corrections to the area law as in fermionic systems[LDY + 06]. However, numerical evi- dences for area law during criticality were found[Sre94, BMC06, MC06] and Callan and Wiczek derived the area law for black hole entropy in approximative field theoreti- cal calculations for alld> 1 [CW94]. We calculate the block entropy of two-dimension infinite bosonic lattice for critical and non-critical cases. For two dimension, the diago- nalized Hamiltonian and spectrum are: H = 2 X k ¤ k ¯ + k ¯ k (3.45) ¤ k = q [¸¡hcos(k x )¡hcos(k y )] 2 ¡° 2 [cos(k x )+cos(k y )] 2 (3.46) The critical cases we choose are:¸ =¡4,h = 0:5,° = 1:5 and¸ =¡4,h = 1,° = 1. The non critical point is:¸ =¡4:2,h = 1 and° = 1. The results are plotted in Fig3.2. For all the cases the block entropy can be fitted very well by L. In calculation of the critical case, we do not apply any cutoff because the pole on the integral path is not a real pole: it can be removed by changing the variables into polar coordinate. So the integrals in Eq3.35 and 3.36 converge. The reason for no logarithmic correction to area law is that in bosonic system the codi- mension can not be1 except in one dimension, where the logarithmic correction shows up. All the gapless excitation in momentum space occurs only in several points for two- dimensional bosons according to Eq3.46. We conjectured that a codimension ¹ d = 1 is a sufficient condition for critical phases in d > 1 to show violations of the area law in 75 Figure 3.2: The block entropy of two-dimension infinite bosonic lattice for critical and non-critical cases. The critical cases we choose are:¸ = ¡4, h = 0:5,° = 1:5 and ¸ =¡4,h = 1,° = 1. The non critical point is:¸ =¡4:2,h = 1 and° = 1. For all the cases the block entropy can be fitted very well byL. chapter2. It is valid for fermions. And by the results in bosonic system, we believe it also holds for bosons. Although we can construct a bosonic system with ¹ d=1, this kind of systems is usually unstable. We will see this by an example. Suppose we want to adjust out Hamiltonian 76 to have a spectrum with ¹ d = 1 in two dimension. By Eq3.46, it is possible only when ¸=0. Now we have the spectrum as: ¤ k = q (h 2 ¡° 2 )[cos(k x )+cos(k y )] 2 = p h 2 ¡° 2 jcos(k x )+cos(k y )j (3.47) The spectrum requires thatjhj > j°j to be reasonable. Now the gapless excitation in momentum space occurs in a linecos(k x )+cos(k y )=0 and hence the codimension, ¹ d, is1. We need to check the original Hamiltonian that has this spectrum to make sure it is a realistic model. The Hamiltonian is: H = X <i; j> £ h(b + i b j +b i b + j )+°(b + i b + j +b i b j ) ¤ (3.48) Transform to canonical representation by: p j = i p 2 (b + j ¡b j ) andq j = 1 p 2 (b + j +b j ), we have: H = X <i; j> · h+° 2 (q i q j +q j q i )+ h¡° 2 (p i p j +p j p i ) ¸ = X <i; j> h+° 4 £ (q i +q j ) 2 ¡(q i ¡q j ) 2 ¤ + h¡° 4 £ (p i +p j ) 2 ¡(p i ¡p j ) 2 ¤ 77 Change the variable to center of mass frame by:q c i j = q i +q j 2 , q r i j = q i ¡q j 2 , p c i j = p i +p j 2 andp r i j = p i ¡p j 2 , we have: H = X <i; j> (h+°) £ q c2 i j ¡q r2 i j ¤ +(h¡°) £ p c2 i j ¡p r2 i j ¤ = X <i; j> h+° h¡° £ p c2 i j +! 2 q c2 i j ¤ ¡ h+° h¡° £ p r2 i j +! 2 q r2 i j ¤ = h+° h¡° X <i; j> ~!(b c+ i j b c i j ¡b r+ i j b r i j ) (3.49) In the diagonalized Hamiltonian there are two set of field operators,b c andb r . And the two set operators have opposite effects on the total energy. This means one field has negative energy and we could decrease the energy of system to an arbitrary value, even negative, by creation of the particles of this field. The system could not be in a stable state and the Hamiltonian is not realistic. In summary, we calculated the scaling behavior of entanglement in bosonic models for one and two dimensions. The explanation about the area law in critical bosonic system was given and we related it to the conjecture in chapter 2. Further investigations in systems with d > 1 and finite temperature are clearly needed. We will give a detail description on future topics in chapter 5. 78 Chapter 4 Scaling Behavior of Entanglement in Spin Model The theory of antiferromagnetism continues to be one of the challenging subjects of modern theoretical physics. Antiferromagnetic phenomena are important, and are observed in a wide variety of materials. The equations are difficult to solve. Even for the simplest model of antiferromagnetism, the Hensenberg model and XY model, no full consistent treatment is available in general. In one dimension, Heisenberg model of spin- 1 2 and XXZ model can be exactly solved by Bethe’s ansatz[Bet31] numerically. Also from chapter 2, we know XY model can be mapped to free spinless fermions in one dimension and hence exactly solvable analytically. In high dimensions, analytical solu- tions are not available for Heisenberg model and XY model, only numerical solutions are possible. The scaling behavior of entanglement in spin models is harder to get. One difficulty is from the intractability of the models. The other one is from the intractability of the calculation of the block entropy even when the numerical results of ground states are in hand. There is no Monte Carlo way found to calculate block entropy. 79 In one dimension, the scaling behavior of entanglement in XY model and XXZ model of spin- 1 2 was thoroughly studied by a lot of researchers[LRV04, VLRK03, IJK05]. J. Zhao et al numerically studied the block entropy of an XXZ model with impurity [ZPW06]. Another random spin system with random singlet state as ground state was studied ana- lytically [RM04] and numerically [Laf05]. The logarithmic divergence was found to be the signature of the criticality in all these cases. For models with spin»1, spin opera- tors can be mapped to boson operators by Holstein-Primakoff transformation[HP40]. In this case the scaling behavior of entanglement is the same as boson and the theory and results of it are in chapter [HP40]. In higher dimensions, fewer results are available. An reasonable guess from analog to bosons is that area law still holds in higher dimension spin model. Because general Heisenberg model is very difficult to solve. Numerical calculations of block entropies for one and two dimension systems are limited to small size system which is far away from the asymptotic scaling regime. So we take an alternative way: we construct trial wave functions, which are known to be good candidate for the ground state of Heisenberg Hamiltonian, and in some special cases they are also known to be the exact ground state. And then study the behavior of block entropy for these wave functions. In the following section we will introduce such class of trial wave functions: the Resonating Valence Bond States(RVB). 80 4.1 Resonating Valence Bond State For a quantum spin system on a lattice, we consider two statesI andII which are two equivalent valence bond states (or singlet coverings ) described by the wavefunctions ª I and ª II . A general wavefunction can be constructed as the superposition of such states: ª=aª I +bª II (4.1) We can say that the system is “resonant” between states I and II. Fig.4.1 gives an exam- ple for the simplest singlet product state. In general, if there are possible equivalent Figure 4.1: Simplest example of singlet product state structures I, II, III, IV ,..., described by the wavefunctions ª I , ª II , ª III , ª IV ,..., then the linear combination of these wavefunctions is a possible resonant state of the system. ª=aª I +bª II +cª III +dª IV ¢¢¢ (4.2) This kind of state is named resonating valence bond (RVB) state. 81 It is believed that the low-energy physics of some Heisenberg antiferromagnets is of the resonating valence bond(RVB) type[And73]. RVB states are proved to be the ground state of some special Heisenberg models[FKK + 89, BG97, BM91, SS81, Kum02]. It is known that for any finite bipartite lattice the ground state of the antiferromagnetic Heisenberg Hamiltonian is a singlet of total spin(S total = 0)[Mar55]. And in the s z representation the ground state wave function is real and satisfies a sign rule[Mar55]. Moreover, it is possible to represent any singlet states by a linear superposition of singlet product state, i.e. RVB state, corresponding to all possible pairings of sites into singlets. So the ground state of antiferromagnetic spin system can be in general represented by RVB states. A general RVB state is of form[LDA88]: ª= X i l ;j l Y l f i l ;j l (i l ;j l ) (4.3) Where (i;j) = (j " i # j > ¡j # i " j >)= p 2 and f i l ;j l stands for any positive function to satisfy Marshall’s sign rule and can be interpreted as a weight factor for a singlet as a function of its length. l runs through all the particles such that every spin forms a singlet with another spin. And the sum over i l ;j l takes all the combination of singlets into consideration. A general component of the superposition is sketched in Fig.4.2. Figure 4.2: Example of an arbitrary singlet product state 82 RVB states usually are classified into two type: short range and long range. A short range RVB wave function is the one with f i l ;j l = f(i l ¡j l ) decaying exponentially or faster in distance i l ¡ j l , while a long range RVB typically have a power-law or less decay. The state with the shortest possible range is the dimer RVB state, for which f(i l ¡ j l ) = 1 for i and j nearest neighbors, and is zero otherwise. It was shown that short range RVB states, which represent quantum-disordered states, describe sys- tem with gapped excitation, short range spin-spin correlation, and away from criticality [LDA88]. While a long range RVB state may not exist in nature, it can serve as a limit (or a paradigm) for related states that do exist. The second is that some artificial states such as RVB can indeed be synthesized, e.g. by using quantum dots and gates. Some parts of nano-scale experimental physics have progressed to the point that they can realize microscopic Hamiltonians at will. In the context of quantum computing this synthetic approach has already been partially successful. The long-range RVB states are relevant to represent two types of states, the quantum critical states with power-law decaying correlations, and the classically ordered states with correlations decaying to a finite value. The distinction between quantum critical and classically ordered depends crucially on the power P of algebraic decay of the amplitude for the long-range singlets, schematically jª>= X singlet coverings Y ij A ji¡jj P (i;j) (4.4) 83 where (i,j) is a singlet between the i-th and the j-th site and the sum runs over all possible singlet coverings of the lattice. In two dimensions it is known that, forP < 5, the RVB ground state has long-range Neel order. This means that, for amplitudes decaying fast enough, using RVB states we can represent a quantum critical state without long-range order [LDA88, San05]. So, the motivation to study long-range RVB states is to capture the main features of quantum critical states in any dimensions without diagonalizing exactly any Hamiltonian at a quantum critical point. Once we have a good control on RVB states we can try and use them as variational states for quantum critical Hamilto- nians, maybe with more variational parameters. We can start from RVB states of the form in (4.3) to calculate block entropy. But it is still difficult to calculate block entropy of such a general RVB state and some restrictions or simplifications on f(i l ¡j l ) are needed. Next we start from the simplest RVB state and then extend to a general short range RVB state. Finally we study a long range RVB state, where the superposition is equal weight. 84 4.2 Scaling Behavior of Entanglement 4.2.1 One-dimensional Shortest Range Resonating Valence Bond States The simplest case of RVB state is 1D and f(i¡j) = ± i;j§1 . This means only nearest- neighboring particles can form singlet with each other. Suppose we are considering an infinite chain and choosing one site as a reference point denoted as0. Then the system has only two equivalent valence bond states: jR>=¢¢¢(0;1)(2;3)(4;5)¢¢¢ (4.5) jL>=¢¢¢(1;2)(3;4)(5;6)¢¢¢ (4.6) The following graph illustrates these two states. jL > andjR > is proved to be the Figure 4.3: R and L states 85 ground state of Majumdar-Gosh Hamiltonian [SS81, Kum02]: H MG =J N X i=1 (2S i ¢S i+1 +S i ¢S i+2 ) (4.7) The general wavefunction is the superposition of these two states. First we consider a symmetric superposition: ª= jR>+jL> p 2 (4.8) The density matrix of this state is: ½= jR ><Rj+jR ><Lj+jL><Rj+jL><Lj 2 (4.9) We choose a block ofl sites and we trace out the remaining part of the chain. The off- diagonal term will vanish by the tracing operation. Let us denotej(1;2) >< (1;2)j ¢¢¢j(l¡1;l)><(l¡1;l)j asjÁ 1;l ><Á 1;l j. Whenl is odd the reduced density matrix is: ½ l = 1 4 [(j" 1 ><" 1 j+j# 1 ><# 1 j)jÁ 2;l ><Á 2;l j +jÁ 1;l¡1 ><Á 1;l¡1 j(j" l ><" l j+j# l ><# l j)] = 1 4 2 6 6 4 0 B @ 1 0 0 1 1 C A 1 jÁ 2;l ><Á 2;l j+jÁ 1;l¡1 ><Á 1;l¡1 j 0 B @ 1 0 0 1 1 C A l 3 7 7 5 86 For evenl the reduced density matrix is: ½ l = 1 2 [ 1 2 (j" 1 ><" 1 j+j# 1 ><# 1 j)jÁ 2;l¡1 ><Á 2;l¡1 j 1 2 (j" l ><" l j+j# l ><# l j) +jÁ 1;l ><Á 1;l j] = 1 2 2 6 6 4 1 2 0 B @ 1 0 0 1 1 C A 1 jÁ 2;l ><Á 2;l j 1 2 0 B @ 1 0 0 1 1 C A l +jÁ 1;l ><Á 1;l j 3 7 7 5 We now calculate the entropy of this density matrix numerically. The result is plotted in Fig.4.4. From the graph we can see the block entropy grows monotonically as a Figure 4.4: Block entropy of one dimension RVB state with shortest range function ofl until it reaches a saturation value 2 for a block sizel 0 » 8. This suggests 87 the system we are studying is away from the quantum critical point according previous discussion. And this means for this system we can approximate the density matrix ½ l by diagonalizing the Hamiltonian corresponding to the block B l and only 8 extra neighboring spins. Also we can obtain an asymptotic form of the block entropy analytically by considering the limitingl!1. In this limit the vectors in½ l are orthogonal to each other because their inner product or overlap decreases exponentially as 2 ¡ l 2 . So in the odd-l case the block entropy is: S(1) = ¡ 4 X 1 4 log 2 1 4 = 2 (4.10) And in the evenl case the block entropy is: S(1) = ¡( 4 X 1 8 log 2 1 8 + 1 2 log 1 2 ) = 2 (4.11) This gives the same result as the numerical calculation and thus verifies the method we are using. Also, because the overlap of the states decreases according 2 ¡ l 2 , the block entropy approach 2 exponentially. We can determine the analytical form of the block 88 entropy by diagonalizing the inner product matrix. The block entropy for odd block size l andl¸3 is; S(l) = 2¡ 1 2 log 2 (1¡C 2 )¡ C 2 log 2 1+C 1¡C (4.12) WhereC =( 1 p 2 ) l¡1 . For evenl andl¸4 the block entropy is: S(l) = 19 8 ¡ 5 16 log 2 (1¡D 2 )¡ p 9+32D 2 16 log 2 5+ p 9+32D 2 5¡ p 9+32D 2 (4.13) WhereD =( 1 p 2 ) l . Whenl is sufficiently large, the above equations simplify to: S(l) = 2¡ 2 ¡l ln2 (4.14) This analytically verifies that the block entropy saturates at some value (in this case 2) exponentially. Now let us consider the more general case: ª=®jR >+¯jL> (4.15) 89 where® and¯ are positive and satisfy the relation: ® 2 +¯ 2 = 1. By the same way we can get the limit value of the block entropy for this general state. The block entropy for oddl!1 is: S(1)=1¡® 2 log 2 ® 2 ¡¯ 2 log 2 ¯ 2 (4.16) But for even l ! 1 the block entropy has two possible value depending on the block chosen: S(1)= 8 > < > : 2® 2 ¡® 2 log 2 ® 2 ¡¯ 2 log 2 ¯ 2 2¯ 2 ¡® 2 log 2 ® 2 ¡¯ 2 log 2 ¯ 2 (4.17) We can see that the results for even l reflects the oscillating strength of the singlets along the chain. The strength of the singlet is distributed evenly along the chain only Figure 4.5: General Case when® = ¯ and the asymptotic value of block entropy attains maximum value. When 90 ® 6= ¯ the state loses the translational symmetry. We see again the link between the entanglement behavior and the symmetry of the system. 4.2.2 General Short Range Resonating Valence Bond states A general RVB state of2N spin chain can be written as jà N >= X k f k j(a k 1 ;b k 1 );(a k 2 ;b k 2 );(a k 3 ;b k 3 );¢¢¢ ;(a k N ;b k N )> (4.18) Here,(a k s ;b k s ) means that two spins form a singlet as (a k s ;b k s ) = j"a s # b s >¡j" b s #a s > p 2 , k labels all the possible configuration of pairing in the spin chain andf k is the weight of a particular configurationk. We can keep anyf k positive by switching the indices of a singlet. In this part, we will study the RVB states with open boundary and the range of our valence bond pairing is finite. Let us set the maximum range of valence bond state asR. It means that thef k has the form¦ s £(ja k s ¡b k s j¡R), where£(x) is the step function. 91 B) 6 5 4 3 2 1 A) 1 2 3 4 5 6 Figure 4.6: Irreducible configuration vs reducible configuration An Example: Short Range Resonating Valence Bond States with Bond Length R=3 We will use the bond lengthR = 3 as an example to illustrate the idea to calculate the block entropy. In next section we will generalize this idea to an arbitrary bond length. Let us define B n as irreducible configurations for a spin chain with a length 2n. The meaning of ’irreducible’ here means that we can not divide B n as separated pieces of configuration. For example, the configuration shown in Fig4.6A can not be divide to separated pieces and hence it is irreducible. As a contrast, Fig4.6B can be divided to separated pieces and therefore it is not a irreducible. Note that the number of irreducible configurations in a block of 2n spins depends on the range of bond length, R. With the increasing of R, the irreducible configurations in a given block of2n spins also will increase. Fig.4.7 shows the three irreducible B1;B2;B3 when R = 3. Notice in this figure, we have already taken the convention that all a k s is taken from sublattice A and all b k s is taken from sublattice B. Then One can always keep the f k as positive. In the remaining of the chapter, this convention is always used. 92 1 2 1 B1 4 3 2 1 B2 B3 6 5 4 3 2 Figure 4.7: First three irreducible configuration B1,B2,B3 withR =3 Our crucial observation is that for a spin chain of open boundary with R = 3, we can expand theà N as jÃ(N)>=jÃ(N¡1)B1>+jÃ(N¡2)B2>+jÃ(N¡3)B3> (4.19) Using this equation, we can easily calculate the number of total configurations. Let us denote the number of configurations of pairing in a stateÃ(N) as®(Ã(N)), we can use the above equation to find®(Ã(N)) as the terms in the right hand side of Eqn.4.19 are independent configurations to each others. We get: ®(Ã(N))=®(Ã(N¡1))®(B1)+®(Ã(N¡2))®(B2)+®(Ã(N¡3))®(B3) (4.20) In principle one can calculate the ®(Ã(k)) for 1 · k · R and ®(Bk) for 1 · k · 3 hand by hand as R is finite using this recursion relation. 93 3) 2) 1) (L) (N−L−1) (L−2) (N−L−2) (L−1) (N−L−1) (L−1) (N−L) ψ ψ ψ ψ ψ ψ ψ ψ 2N−2L 2L 4) Figure 4.8: Expansion ofÃ(N) around the length2L withR =3 For calculating the block entropy, We will exploit this kind of recursion idea further by expandÃ(N) even in the middle of the spin chain as jÃ(N)>=jÃ(L)Ã(N¡L)>+jÃ(L¡1)B2Ã(N¡L¡1)>+ jÃ(L¡1)B3Ã(N¡L¡2)>+jÃ(L¡2)B3Ã(N¡L¡1)> (4.21) Fig4.8 gives direct picture for the above equation. It is very easy to see this expansion also gives finite terms. For R=3, we only have four terms as shown in Fig.4.8. In this meaning, we can use this expansion to analyze the block entropyS 2L (Notice our definition ofS 2L is the block entropy in the regime[0;2L] as our total chain is[0;2N]). As it is seen, we concentrate our block entropy to even value of block size. However it is straightforward to generalize our method to odd value of block entropy. We will come to this point at the end of this session. 94 Let us rewriteBs with0·s·N in the spin up-down basis: jÃ(N)>= jÃ(L)Ã(N¡L)>+jÃ(L¡1) j#">j#"> p 2 Ã(N¡L¡1)> +jÃ(L¡1) j"#>j"#> p 2 Ã(N¡L¡1)> ¡jÃ(L¡1) j"">j##> p 2 Ã(N¡L¡1)> ¡jÃ(L¡1) j##>j""> p 2 Ã(N¡L¡1)>+¢¢¢ (4.22) With this expansion, it is possible to calculate the block entropy ofS 2L on the basis of: jÃ(L) >;jÃ(L¡1)#">;jÃ(L¡1)"#>;jÃ(L¡1)"">;jÃ(L¡1)##>;¢¢¢ , when we trace out the other 2N ¡ 2L sites. This basis is named valence bond(VB) basis. The number of vectors in the basis should be finite. The reason is that our basis will be complete after we count all the states likejÃ(L¡R+1)"#¢¢¢#> since our expansion stops atjÃ(L¡R+1)BRÃ(N¡L¡1) >. We get that the number of vectors,T 3 (3 means whenR = 3), in this basis is equal to2 0 +2 2 +2 4 . To see this, we can run over all division(even spins+even spins) for even-valued spin number from 0 to2R, keep in mind that ourBs represent the chain with the length2s. For simplicity, let us labelNull,j"">,j"#>,j##,j#">,#"#">,¢¢¢ asC 0 ,C 1 ,C 2 ,C 3 ,C 4 ,C 5 ,¢¢¢ , we can write every state inside the blockjA L i >=jà L¡f C i i C i >, wheref is the total number of sites inC i divided by two. For example, C 1 has two sites and thenf C 1 = 1 and C 5 has four sites and hence f C 5 = 2. In the same way, it is easy to conclude that f BR =R. 95 Generally, when we study the reduced density matrice of a wave-function whose form is jª>= § i jA i >jB i > a , wherea is a normalized vector defined asa 2 =<ªjª> andjB > is the part needed to be traced out, we can calculate the inner product ofjB i > andjB j >, B i;j =<B i jB j >, with anyi andj. According the definition of reduced density matrix, we can get the reduced density matrix as ½ L = Tr A (jª><ªj) = X ij <B i jB j > a 2 jA i ><A j j (4.23) Hence the matrix element is B i;j in the basis ofjA i >< A j j. However, jA i > is not an orthogonal basis, we need to calculate the inner product matrix whose element is written as A i;j =< A i jA j > and transfer the basis to an orthogonal representation by diagonalizing A with a unitary matrixP asP + AP = ¸ A (¸ A is a diagonalized matrix) before meaningful calculation. After some straightforward algebra, we can express the reduced density matrix as: ½ L = 1 a 2 p ¸ A P + A BP A p ¸ A (4.24) on the normalized orthogonal basis. When applying the above procedure to our former valence bond(VB) basis, we need to work out first the matrix A. If we denote the VB basis in a length2n asjA n i > and the 96 inner product matrix of the basis is A n . So the inner product matrix of our finite basis for a 2L length block can be written as A L . Now we can try to work out the matrix element hand by hand. Let us calculate the matrix element A L 0;j when 0 · j · T R . Our crucial observation is that A L 0;j can be written as P R p=1 P p s p A L¡1 p as 0 · p · T and s is a constant vector that can be calculated by the following kind of procedure. First, let us expand <A L 0 j=<à L C 0 j as: <à L C 0 j=<à L¡1 B1C 0 j+<à L¡2 B2C 0 j+<à L¡3 i B3C 0 j (4.25) So, we can write <à L C 0 jA L¡f C i C i >=<à L¡1 B1C 0 jA L¡f C i C i >+ <à L¡2 B2C 0 jA L¡f C i C i >+<à L¡3 B3C 0 jA L¡f C i C i > (4.26) It is natural to rewriteBm with0·m·3 in basis of2 ¡ m 2 § k;s C k C s with the constraint f C s =f C i andf Bm =f C k +f C s as we had noted our basisC is a complete description to 97 the states for any spin chain with the length from0 to2R. For example,B1 = (C 1 ¡C 2 ) p 2 andB2= (C 1 C 1 +C 2 C 2 ¡C 4 C 3 ¡C 3 C 4 ) 2 p 2 . Then we can revise Eq4.26 as <à L C 0 jA L¡f C i C i > = X m X s X k 2 ¡ m 2 <à f¡C m k ¡C m s C m k jA L¡f C i ><C s jC i > = X m X k 2 ¡ m 2 <à f¡C m k ¡C m i C m k jA L¡f C i > (4.27) For the last equation, we use the orthogonal relation betweenC s andC i iff Cs =f C i . The above equation arrays are difference equations. Because the coefficients of the equa- tions are inner product, we can always write the difference equations into a symmetric form. Hence the solution of the above equation array is in general asc 1 ® L 1 +c 2 ® L 2 +¢¢¢ . WhenL is large enough the solution of the difference equation is asymptotically of the form:c® L , where® is the largest root. To calculate the remaining matrix element A L m;j withm;n6=0, we only need to calculate the case of m > n for symmetric reason. Because we can expand C t with 0 · t · R in basis of C k C s with the constraint f C t = f C k +f C s , we can always reduce A L m;j to A L¡f C j k;0 in a recursion way: <Ã(L¡1)#"jÃ(L¡1)"#> = 0 <Ã(L¡2)"#"#jÃ(L¡1)"#> = <Ã(L¡2)#"jÃ(L¡1)> <Ã(L¡2)"###jÃ(L¡1)##> = <Ã(L¡2)"#jÃ(L¡1)> (4.28) 98 So we can also write A L m;j =o m;j ® L for large L as we have absorb the factor® ¡f C j into o m;j . The algebra is thus closed and self-contained. With this above argument, we can always write the matrix A L i;j = ® L A 0 with A 0 is a constant matrix which is independent with L for large enough L. The same argument can be extent to the matrix B N¡L i;j . One can also directly work out the relation between A N¡L i;j and B N¡L i;j as B N¡L i;j is some linear combination of A N¡L i;j . So B N¡L i;j can also be written as® N¡L B 0 , where B 0 is a constant matrix, ifN¡L>> 1. Finally, our Eq4.24 becomes ½= 1 a 2 p ¸ A P + A BP A p ¸ A = ® N a 2 p ¸ A 0P + A 0 B 0 P A 0 p ¸ A 0 (4.29) Notice the overall normalized constant a 2 is equal to < à N jà N >= A N 0;0 and then a 2 =a 0 ® N , wherea 0 is a constant. So we get: ½= 1 a 2 0 p ¸ A 0P + A 0 B 0 P A 0 p ¸ A 0 (4.30) Thus for largeL;N¡L>>1, our conclusion is that the reduce matrix in fact comes to a constant matrix independent to the size of block. Obviously the block entropy should be also a constant. 99 L R R Figure 4.9: Only the bonds near the boundary is cut by tracing General Short Range Resonating Valence Bond States The above example shows that for R = 3 the block entropy of RVB state approach a constant. Now we can generalize this result to a general short range RVB state with bond length R and give the upper limit of block entropy. A general short range RVB state is composed of any possible covering: jª>= X c f c jc> (4.31) We want to calculate entropy of a blockL which is fair large(>> 2R). To get reduced density matrix we need to do trace operation on density matrix. The trace operation only cut off at most2R bonds for each covering, see Fig4.9. 100 We can group the covering according to the cutting and label each group as a new basis. The short range RVB state can be rewritten as: jª>= K X j=1 f j jj > (4.32) ½= K X j=1;j 0 =1 f j;j 0jj ><j 0 j (4.33) wherejj > is the summation of the coverings which have the bonds linking the same inside sites with any outside sites and the inside sites are decoupled from the outside sites asj"> or#> by tracing. The total number of the groups can be calculated as: K = C 0 2R +2C 1 2R +¢¢¢+2 2R C 2R 2R = 3 2R (4.34) Letji> denote the states ofjj > after tracing. The reduced density matrix is: ½ L = K X i=1;i 0 =1 f i;i 0ji><i 0 j (4.35) Now the reduced density matrix is written in a finite dimension matrix and the dimension is independent of L when L is large enough, L À R. By diagonalizing the matrix we have: ½ L = K X l=1 ¸ l jl ><lj (4.36) 101 And the block entropy is:S L = ¡ P K l=1 ¸ l log¸ l with the constraint that:¸ l ¸ 0 and P ¸ l = 1. The function f(x) = ¡xlogx is concave in x 2 [0;1]. So we have: P K l=1 f(¸ l ) K ·f( P K l=1 ¸ l K )= 1 K logK. Finally we have: S L ·logK =2Rlog3 (4.37) Our results show that for arbitrary long (finite) range RVB states, the block entropy in one dimension will be less than or saturate to a constant. The constant turns out to be proportional to the range of bond length which makes the upper bound of block entropy a meaning of a quantifier of correlation. The above procedure can be applied to two dimension case. In two dimension, the total number of groups is upper bounded bya bRL , wherea andb are constants not related to L whenL is large enough. We thus get S L ·logK»RL (4.38) This is the area law in higher dimensions. So our results show that for arbitrary long (finite) range RVB states, the block entropy in one dimension will be less than or saturate to a constant and follows area law in higher 102 dimensions. This result is in agreement with those in other cases because short range RVB states essentially represent non-critical cases. 4.2.3 Long Range Resonating Valence Bond States For arbitrary long range RVB state, if the superposition is equal weight, the wavefunc- tion has permutation invariant symmetry. For this kind of wave functions the dimension of the basis can be reduced tod = L+1 by re-grouping all2 L vectors according to the symmetry, and this leads to the upper limit of block entropy islogL. To calculate RVB states of infinite range with equal weight in 1D case, we need to calculate reduced density matrix. We can follow the similar procedure in calculations of short range RVB states. When expanding the reduced density matrix, we can use theL+1 vectors with permutation invariant symmetry as basis and this will drastically reduce the vector space from exponentially to linearly increasing. We calculate the block entropy of the half of a 2L spin chain. By dividing the system into two equal- length part, we can use another symmetry: thez component of spins for two parts will cancel each other to make the whole system a singlet. Numerically we can calculate the system of size up to 800. Our result shows the block entropy reaches the upper bound and followslogL behavior but the cenctral charge calculated by the fitting is around3=2 103 Figure 4.10: The block entropy of a long range RVB state: the superposition is equal weight. The curve can be fitted by 1 2 logL asymptotically. which seems to be different from the other models. The reasons for this may be that the symmetry in this system is different and what we calculated is the entanglement between two equal parts of a system. Our result is in agreement with exact diagonlization when the length of chain is small. This verifies our method and result directly. This result is essentially consistent with the argument that logarithmic behavior is a signature of criticality since long range RVB states represent quantum critical states. 104 We have thoroughly studied short-range and a special long range RVB states. The next step is to study a general long-range RVB states. We parameterize thef i;j , coefficients of the RVB wavefunction as: f i;j = e ¡ ji¡jj » ji¡jj ® (4.39) where » describe the characteristic length of singlets. We will investigate in particu- lar the relation between S 1 and » which can tell us the dependence of the asymptotic behavior of the block entropy on the length of the singlets. And for a given » and ®, we will investigate the scaling behavior ofS l . And this will tell us the scaling behavior of block entropy when » is fixed. These two relations give a good description of the entanglement behavior in RVB states. For the general one dimensional long-range RVB state the block entropy may have a different behavior fromlog(L). 105 Chapter 5 Conclusions and Future Works 5.1 Conclusions In this thesis, we studied the scaling behavior of entanglement in quantum systems which have strongly fluctuating ground states. These systems have rich phase diagrams. Making use of numerically exact methods, we investigated the scaling behavior of the block entropy over the entire phase diagram. Focussing on two- and three-dimensional free-fermion system with a generic quadratic Hamiltonian, we adopted the strategy of investigating a model which has the advantage of being exactly solvable and, yet, of displaying correlation properties typical of quan- tum critical phases, as well as non-critical phases. This model is a higher-dimensional generalization of the model of free fermions exactly mappable onto the XY model in a transverse field in 1d. The key ingredient of this model is that it has two distinct critical phases with and without a finite Fermi surface, as well as a non-critical phase with a 106 gap to particle-hole excitations. This helps us to identify conditions under which the scaling behavior of the block entropy will show corrections to the area law. In non- critical states, we found that the area law indeed holds, and we found that logarithmic corrections to this law are present in critical states with a finite Fermi surface, as pre- viously shown analytically. On the other hand, for critical states with a Fermi surface of zero measure, we found that the corrections to the area law are either absent or sub- logarithmic. This means that the relationship between entanglement and correlations in higher dimensional systems is different than in one dimension, where logarithmic cor- rections are always observed at quantum critical points, and that in higher dimensions a crucial role is played by the geometry of the Fermi surface. In bosonic lattice systems with a generic quadratic Hamiltonian, we explored the spec- trum of the Hamiltonian. Having studied the phase diagram and the topology of gapless excitations of the system, we gave an explanation on why logarithmic corrections to the area law do not exist. The general Heisenberg model is very difficult to solve. Numerical calculations of block entropies for one and two dimension systems are limited to small size systems, far away from the asymptotic scaling regime. So we chose an alternative way: we constructed trial wave functions, which are known to be good candidate for the ground state of Heisenberg Hamiltonian, and in some special cases they are also known to be the exact ground state. We investigated the scaling of the block entropy for these wave functions. 107 The candidate ground states we chose were RVB states. In one dimension the results are consistent with the fermion and boson systems: for short-range RVB states, the block entropy saturates to a constant; while for long-range RVB states the block entropy diverges logarithmically. In higher dimensions, the area law of block entropy was proven for short range RVB states. 5.2 Future Works The theory of scaling behavior of entanglement in condensed matter is far from com- plete. More work needs to be done to arrive at a unified picture for it. There are several directions to extend this research. The first extension is to consider more general Hamiltonians. Strongly correlated sys- tems have rich physics that mean field theory cannot catch. So it is meaningful to study the scaling behavior of entanglement in systems such as the Hubburd model, Luttinger liquids and spin chains with frustration. Also systems with randomness or impurities are quite different. For example, in one-dimension Luttinger liquids, the effective strength of a defect depends on the sign of the interaction [EA92, KF92] and goes to zero for attraction while it diverges for repulsion, as the system size increase. So the topics 108 about how the introduction of randomness and impurity changes the entanglement in the system and its relation with quantum phase transition are quite interesting. Another extension is to study the finite temperature case. Entanglement is often con- sidered necessarily a low-temperature phenomenon that becomes less important as the temperature is increased. The suppression of the entanglement by decohering actions such as temperature, noise, etc., is one of the central problems in quantum computation and quantum information theory. Therefore the concept of entanglement for finite tem- perature or mixed state is of primary relevance. In finite temperature, the block entropy contains two part: quantum correlation (or quantum entropy) and thermal entropy. The quantum correlation can be used for quantum information process. But how to identify quantum entropy out of thermal entropy is still an open question. The third extension is to study the sub-leading term of block entropy for some system. In higher dimension conformal field theory is not applicable. In most cases area law holds for block entropy. Fradkin and Moore[FM06] found the sub-leading term can be related to some conformal symmetry and be universal with a central chargec of the associated two dimensional conformal field theory. Combined with recent results on sub-leading corrections to entanglement entropy in massive topological phases [KP06, LW06], it now appears that the physics of entanglement entropy in d > 1 is considerable richer than the area law might suggest, even when the area law is applicable. 109 Another line of investigation will involve the relation between symmetry and entangle- ment. In chapter 3 and 4 we have seen that the scaling behavior of entanglement is affected by the symmetry of the state. Whether there are universal relations in other states and systems is yet to be done. 110 References [ABV01] M. C. Arnesen, S. Bose, and V . Vedral. Natural thermal and mag- netic entanglement in the 1d heisenberg model. Physical Review Letters, 87:017901, 2001. [And73] P. W. Anderson. Mater. Res. Bull., 8:153, 1973. [BBC + 93] C. H. Bennett, Gilles Brassard, Claude Crpeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Physical Review Letters, 70:1895, 1993. [BBM92] C. H. Bennett, G. Brassard, and N. D. Mermin. Quantum cryptography without bells theorem. Physical Review Letters, 68:557, 1992. [BCS57] J. Bardeen, L. N. Cooper, , and J. R. Schrieffer. Theory of superconduc- tivity. Physical Review, 108:1175, 1957. [Bet31] H. A. Bethe. Z. Phys., 71:205, 1931. [BG97] I. Bose and Asimkumar Ghosh. Exact ground and excited states of frustrated antiferromagnets on the cav4o9 lattice. Physical Review B, 56:3149, 1997. [BKLS86] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin. Quantum source of entropy for black holes. Phys. Rev. D, 34:373, 1986. [BM71] E. Barouch and B. McCoy. Statistical mechanics of the x y model. ii. spin-correlation functions. Phys. Rev. A, 3:786, 1971. 111 [BM91] Indrani Bose and Partha Mitra. Frustrated spin-1/2 model in two dimen- sions with a known ground state. Physical Review B, 44:443, 1991. [BMC06] T. Barthel and U. Schollwoeck M.C. Chung. eprint, cond-mat:0602077, 2006. [Bos03] S. Bose. Quantum communication through an unmodulated spin chain. Physical Review Letters, 91:207901, 2003. [BR04] A. Botero and B. Reznik. Spatial structures and localization of vacuum entanglement in the linear harmonic chain alonso botero. Phys. Rev. A, 70:052329, 2004. [BTSL02] W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam. Experimental demonstration of continuous variable polarization entanglement. Phys- ical Review Letters, 89:253601, 2002. [Bur04] G. Burkard. Theory of solid state quantum information processing. eprint, cond-mat:0409626, 2004. [BvL05] S. L. Braunstein and P. van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77:513, 2005. [CC04] P. Calabrese and J. Cardy. Entanglement entropy and quantum field the- ory. J. Stat. Mech., page P06002, 2004. [CEPD06] M. Cramer, J. Eisert, M. B. Plenio, and J. Drei¯ig. Entanglement-area law for general bosonic harmonic lattice systems. Physical Review A, 73:012309, 2006. [CH04] Siew-Ann Cheong and Christopher L. Henley. Many-body density matri- ces for free fermions. Phys. Rev. B, 69:075111, 2004. [CP01] M.C. Chung and I. Peschel. Density-matrix spectra of solvable fermionic systems. Physical Review B, 64:064412, 2001. [CW94] C. Callan and F. Wilczek. On geometric entropy. Physics Letters B, 333:55, 1994. [DGCZ00] L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller. Inseparability criterion for continuous variable systems. Physical Review Letters, 84:2722, 2000. [DHH + 05] W. D¨ ur, L. Hartmann, M. Hein, M. Lewenstein, and H.-J. Briegel. Entan- glement in spin chains and lattices with long-range ising-type interac- tions. Physical Review Letters, 94:097203, 2005. 112 [EA92] S. Eggert and I. Affleck. Magnetic impurities in half-integer-spin heisen- berg antiferromagnetic chains. Phys. Rev. B, 46:10866, 1992. [Eke91] A. K. Ekert. Quantum cryptography based on bells theorem. Physical Review Letters, 67:661, 1991. [Fey82] R. P. Feynman. Simulating physics with computers. Int. J. Theor. Phys., 21:467, 1982. [FKK + 89] F. Figueirido, A. Karlhede, S. Kivelson, S. Sondhi, M. Rocek, and D. S. Rokhsar. Exact diagonalization of finite frustrated spin-(1/2 heisenberg models. Physical Review B, 41:4619, 1989. [FM06] Eduardo Fradkin and Joel E. Moore. Entanglement entropy of 2d con- formal quantum critical points: Hearing the shape of a quantum drum. Physical Review Letters, 97:050404, 2006. [FSB + 98] A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik. Unconditional quantum teleportation. Science, 282:706, 1998. [Gau60] M. Gaudin. Nucl. Phys, 15:89, 1960. [GAW + 03] F. Grosshans, G. V . Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grang- ier. Quantum key distribution using gaussian-modulated coherent states. Nature, 421:238, 2003. [GDCZ02] G. Giedke, L.-M. Duan, J. I. Cirac, and P. Zoller. Quant. Inf. Comput., 1:79, 2002. [GDLL04] S. J. Gu, S. S. Deng, Y . Q. Li, and H. Q. Lin. Entanglement and quantum phase transition in the extended hubbard model. Physical Review Letters, 93:086402, 2004. [Geb97] F. Gebbhard. The Mott Metal-Insulator Transition: Models and Methods. Springer-Verlag, New York, 1997. [GF02] J. R. Gittings and A. J. Fisher. Describing mixed spin-space entanglement of pure states of indistinguishable particles using an occupation-number basis. Phys. Rev. A, 66:032305, 2002. [Gir60] M. Girardeau. Relationship between systems of impenetrable bosons and fermions in one dimension. Journal of Mathematical Physics, 1:516, 1960. 113 [GK06] D. Gioev and I. Klich. Entanglement entropy of fermions in any dimen- sion and the widom conjecture. Physical Review Letters, 96:100503, 2006. [GKLC01] G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac. Entanglement crite- ria for all bipartite gaussian states. Physical Review Letters, 87:167904, 2001. [HLG89] D. B. Haviland, Y . Liu, and A. M. Goldman. Onset of superconductivity in the two-dimensional limit. Physical Review Letters, 62:2180, 1989. [HLW94] Ch. Holzhey, F. Larsen, and F. Wilczek. Nucl. Phys. B, 424:443, 1994. [HP40] T. Holstein and H. Primakoff. Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev., 58:1098, 1940. [IAB + 99] A. Imamoglu, D. D. Awschalom, G. Burkard, F. P. Di-Vincenzo, D. Loss, M. Sherwin, and A. Small. Quantum information processing using quan- tum dot spins and cavity qed. Physical Review Letters, 83:4204, 1999. [IJK05] A. R. Its, B.-Q. Jin, and V . E. Korepin. Entanglement in the xy spin chain. J. Phys. A: Math. Gen., 38:2975, 2005. [JK04a] B.-Q. Jin and V . E. Korepin. J. Stat. Phys., 116:79, 2004. [JK04b] B.-Q. Jin and V . E. Korepin. Quantum spin chain, toeplitz determinants and fisher-hartwig conjecture. J. Stat. Phys., 116:79, 2004. [JKP01] B. Julsgaard, A. Kozhekin, and E. S. Polzik. Experimental long-lived entanglement of two macroscopic objects. Nature, 413:400, 2001. [JW28] P. Jordon and E. Wigner. Z. Phys., 47:631, 1928. [KF92] C. L. Kane and M. P. A. Fisher. Transmission through barriers and reso- nant tunneling in an interacting one-dimensional electron gas. Phys. Rev. B, 46:15233, 1992. [KKF + 94] S. V . Kravchenko, G. V . Kravchenko, J. E. Furneaux, V . M. Pudalv, and M. D’Iorio. Possible metal-insulator transition at b=0 in two dimensions. Physical Review B, 50:8036, 1994. [KM04] J. P. Keating and F. Mezzadri. Random matrix theory and entanglement in quantum spin chains. Commun. Math. Phys., 252:543, 2004. [Kor04] V . Korepin. Universality of entropy scaling in one dimensional gapless models. Physical Review Letters, 92:096402, 2004. 114 [KP06] Alexei Kitaev and John Preskill. Topological entanglement entropy. Phys- ical Review Letters, 96:110404, 2006. [KS02] G. L. Kamta and A. F. Starace. Anisotropy and magnetic field effects on the entanglement of a two qubit heisenberg xy chain. Physical Review Letters, 88:107901, 2002. [Kum02] B. Kumar. Quantum spin models with exact dimer ground states. Physical Review B, 66:024406, 2002. [KW87] H. J. Kimble and D. F. Walls. J. Opt. Soc. Am. B, 4:10, 1987. [Laf05] N. Laflorencie. Scaling of entanglement entropy in the random singlet phase. Physical Review B, 72:140408, 2005. [Lau83] R. B. Laughlin. Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Let- ters, 50:1395, 1983. [LDA88] S. Liang, B. Doucot, and P. W. Anderson. Some new variational resonating-valence-bond-type wave functions for the spin-? antiferro- magnetic heisenberg model on a square lattice. Physical Review Letters, 61:365, 1988. [LDY + 06] Weifei Li, Letian Ding, Rong Yu, Tommaso Roscilde, and Stephan Haas. Scaling behavior of entanglement in two- and three-dimensional free- fermion systems. Phys. Rev. B, 74:073103, 2006. [LRV04] J. I. Latorre, E. Rico, and G. Vidal. Ground state entanglement in quantum spin chains. Quant. Inf. and Comp., 4:48, 2004. [LSM61] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Ann. Phys., 16:407, 1961. [LW06] Michael Levin and Xiao-Gang Wen. Detecting topological order in a ground state wave function. Physical Review Letters, 96:110405, 2006. [Mah90] Gerald D. Mahan. Many-Particle Physics. Plenum Press, New York, 1990. [Mar55] W. Marshall. Proc. Roy. Soc. (London), A232:48, 1955. [MC06] M.B. Plenio J. Dreißig M. Cramer, J. Eisert. An entanglement-area law for general bosonic harmonic lattice systems. Physical Review A, 73:012309, 2006. 115 [NC00] M. A. Nielsen and I. Chuang. Quantum Computation and Quantum Infor- mation. Cambridge University Press, Cambridge, 2000. [OAFF02] A. Osterloh, L. Amico, G. Falci, and R. Fazio. Scaling of entanglement close to a quantum phase transition. Nature(London), 416:608, 2002. [ON02a] T. J. Osborne and M. A. Nielsen. Quantum Inf. Process., 1:45, 2002. [ON02b] T. J. Osborne and M. A. Nielsen. Entanglement in a simple quantum phase transition. Physical Review A, 66:032110, 2002. [OR95] Stellan ¨ Ostlund and Stefan Rommer. Thermodynamic limit of density matrix renormalization. Physical Review Letters, 75:3537, 1995. [PEDC05] M. B. Plenio, J. Eisert, J. Dreißig, and M. Cramer. Entropy, entanglement, and area: Analytical results for harmonic lattice systems. Physical Review Letters, 94:060503, 2005. [Pes04a] I. Peschel. Calculation of reduced density matrices from correlation func- tions. J. Phys. A: Math. Gen., 36:L205, 2004. [Pes04b] I. Peschel. On the entanglement entropy for a xy spin chain. J. Stat. Mech., page P12005, 2004. [PHE04] M. B. Plenio, J. Hartley, and J. Eisert. New J. Phys., 6:36, 2004. [Pre98] J. Preskill. Proc. Roy. Soc. A: Math., Phys. and Eng., 454:469, 1998. [Pre00] J. Preskill. J. Mod. Opt., 47:127, 2000. [RB01] R. Raussendorf and H. J. Briegel. A one-way quantum computer. Physi- cal Review Letters, 86:5188, 2001. [RM04] G. Refael and J. E. Moore. Entanglement entropy of random quantum critical points in one dimension. Physical Review Letters, 93:260602, 2004. [Sac99] S. Sachdev. Quantum Phase Transition. Cambridge Univsity Press, Cam- bridge, 1999. [San05] Anders W. Sandvik. Ground state projection of quantum spin systems in the valence-bond basis. Physical Review Letters, 95:207203, 2005. [Sch35] E. SchrÄ odinger. Discussion of probability relations between seperated systems. Proceedings of the Cambrige Philosophical Society, 31:555, 1935. 116 [SGCS97] S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar. Continuous quan- tum phase transitions. Rev. Mod. Phys., 69:315, 1997. [Sim00] R. Simon. Peres-horodecki separability criterion for continuous variable systems. Physical Review Letters, 84:2726, 2000. [Skr05] S. O. Skrøvseth. Entanglement in bosonic systems. Physical Review A, 72:062305, 2005. [Sm04] V . Subrah-manyam. Entanglement dynamics and quantum-state transport in spin chains. Physical Review A, 69:034304, 2004. [Sre94] M. Srednicki. Entropy and area. Physical Review Letters, 71:666, 1994. [SS81] B. S. Shastry and B. Sutherland. Physica B, 108:1069, 1981. [TN02] T.J.Osborne and M. A. Nielsen. Entanglement in a simple quantum phase transition. Physical Review A, 66:032110, 2002. [vdZFE + 92] H. S. J. van der Zant, F. C. Fritschy, W. E. Elion, L. J. Geerligs, and J. E. Mooij. Field-induced superconductor-to-insulator transition in josephson junction arrays. Physical Review Letters, 69:2971, 1992. [vE03] S. J. van Enk. Entanglement of electromagnetic fields. Phys. Rev. A, 67:022303, 2003. [Ved03] V . Vedral. Central Eur. J. Phys., 1:289, 2003. [VLRK03] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev. Entanglement in quantum critical phenomena. Physical Review Letters, 90:227902, 2003. [V ol03] G.E. V olovik. The Universe in a Helium Droplet. Clarendon Press, Oxford, 2003. [V ol05] G.E. V olovik. eprint, cond-mat:0505089, 2005. [VPC04] F. Verstraete, M. Popp, and J. I. Cirac. Entanglement versus correlations in spin systems. Physical Review Letters, 92:027901, 2004. [VPM04] J. Vidal, G. Palacios, and R. Mosseri. ibid., 69:022107, 2004. [VWPGC06] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac. Criticality, the area law, and the computational power of projected entangled pair states. Physical Review Letters, 96:220601, 2006. [Wan02] X. Wang. ibid, 66:034302, 2002. 117 [WGK + 04] M. M. Wolf, G. Giedke, O. KrÄ uger, R. F. Werner, and J. I. Cirac. Gaussian entanglement of formation. Phys. Rev. A, 69:052320, 2004. [Whi92] Steven R. White. Density matrix formulation for quantum renormaliza- tion groups. Physical Review Letters, 69:2863, 1992. [Wid82] H. Widom. Toeplitz centennial(tel aviv, 1981). Operator Theory: Adv. Appl., 4:477, 1982. [Wol06] M. M. Wolf. Violation of the entropic area law for fermions. Physical Review Letters, 96:010404, 2006. [Woo98] W. K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Physical Review Letters, 80:2245, 1998. [WW01] R. F. Werner and M. M. Wolf. Bound entangled gaussian states. Physical Review Letters, 86:3658, 2001. [Zan02] P. Zanardi. Quantum entanglement in fermionic lattices. Physical Review A, 65:042101, 2002. [ZPW06] Jize Zhao, Ingo Peschel, and Xiaoqun Wang. Critical entanglement of xxz heisenberg chains with defects. Physical Review B, 73:024417, 2006. [ZSGL03] L. Zhou, H. S. Song, Y . Q. Guo, and C. Li. Enhanced thermal entan- glement in an anisotropic heisenberg xyz chain. Physical Review A, 68:024301, 2003. [ZW02] P. Zanardi and X. Wang. J. Phys. A.:Math. Gen., 35:7947, 2002. 118
Abstract (if available)
Abstract
In this thesis, the scaling behavior of entanglement is investigated in quantum systems with strongly fluctuating ground states. We relate the reduced density matrices of quadratic fermionic and bosonic models to their Green's function matrices in a unified way, and calculate exactly the scaling of the entanglement entropy of finite systems in an infinite universe. In these systems, we observe quantum phase transition by tuning the parameters of the Hamiltonian. Our study shows that although in one dimension there is a unique relation between the quantum phase transition and the scaling behavior of entanglement, this is not necessarily true in higher dimensions.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Li, Weifei
(author)
Core Title
Entanglement in strongly fluctuating quantum many-body states
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
03/16/2007
Defense Date
12/08/2006
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
entanglement,Fermi surface,OAI-PMH Harvest,quantum critical region
Language
English
Advisor
Haas, Stephan W. (
committee chair
), Bickers, Nelson Eugene, Jr. (
committee member
), Brun, Todd A. (
committee member
), Dappen, Werner (
committee member
), Johnson, Clifford (
committee member
)
Creator Email
weifeili@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m314
Unique identifier
UC1135633
Identifier
etd-Li-20070316 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-324291 (legacy record id),usctheses-m314 (legacy record id)
Legacy Identifier
etd-Li-20070316.pdf
Dmrecord
324291
Document Type
Dissertation
Rights
Li, Weifei
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
entanglement
Fermi surface
quantum critical region