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University of Southern California Dissertations and Theses
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Finite size effect and Friedel oscillations for a Friedel-Anderson impurity by FAIR method
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Finite size effect and Friedel oscillations for a Friedel-Anderson impurity by FAIR method
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FINITE SIZE EFFECT AND FRIEDEL OSCILLATIONS FOR A FRIEDEL-ANDERSON IMPURITY BY FAIR METHOD by Yaqi Tao A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) Dec 2012 Copyright 2012 Yaqi Tao To my family ii Acknowledgements First I would like to express my gratitude to my PhD advisor, Professor Gerd Bergmann, for his guidance on my research. He is a great mentor who has given me tremendous support and encouragement during my PhD study. It is he who has lead me into the area of Kondo problem and who has brought me a lot of insightful ideas and advice. I would like to thank Dr. Liye Zhang, Dr. Go Tateishi and all my colleagues and friends for many invaluable discussions. I express my love and gratitude to my grandpa, Shoupeng Chen, grandma, Aixiang Shen, parents, Yan Tao and Zhenghua Chen, for their love and support throughout my life. Specially, I express my love and gratitude to my husband, Wen Zhang, for his continuous encouragement and countless suggestions. Finally I would like to thank Professors Richard Thompson, Stephan Haas, Jianfeng Zhang and Werner Dappen for serving on my PhD advisory committee. iii Table of Contents Dedication ii Acknowledgements iii List of Figures vi Abstract ix Chapter 1: Introduction 1 Chapter 2: Model and Method 4 2.1 Wilson’s states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 FAIR method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Friedel Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3 Friedel-Anderson Hamiltonian . . . . . . . . . . . . . . . . 10 2.2.4 Kondo Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Appendix:Matrix elements for Friedel-Anderson and Kondo Hamil- tonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Friedel Anderson Model . . . . . . . . . . . . . . . . . . . 17 2.3.2 Kondo Model . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 3: Kondo Cloud 26 3.1 Continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Finite Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 FAIR calculation for finite size sample . . . . . . . . . . . . 32 3.2.2 Wave function of Wilson states in finite size sample . . . . . 33 3.2.3 Singlet state of the Friedel-Anderson impurity . . . . . . . . 35 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 iv Chapter 4: Friedel Oscillations about a Friedel-Anderson Impurity 50 4.1 Friedel Oscillations of a symmetric Friedel-Anderson impurity . . . 51 4.1.1 Derivation of the formulas . . . . . . . . . . . . . . . . . . 53 4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Friedel Oscillations of a simple Friedel impurity . . . . . . . . . . . 69 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Chapter 5: Linear response to the external magnetic field 75 5.1 General theory of linear response to an external perturbation . . . . 75 5.2 Linear response to the external magnetic field for a Kondo Hamilto- nian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3 Linear response calculation for a Friedel-Anderson impurity . . . . 79 5.3.1 Two numerical examples . . . . . . . . . . . . . . . . . . . 82 Chapter 6: Conclusions 86 Bibliography 88 v List of Figures 2.1 Wilson band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The ground state energies of the mean-field solution, the magnetic and the singlet FAIR solution . . . . . . . . . . . . . . . . . . . . . 12 2.3 The magnetic moment as a function of the Coulomb exchange energy U, using the mean-field solution and the magnetic FAIR solution . . 13 2.4 The energy difference between the singlet and triplet states. . . . . . 17 3.1 The net integrated density of the s electron within a distance r from the impurity for spin up and spin down, as well as total density and polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 The net integrated density of one magnetic component of the FA singlet state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Comparison of Dr.Bergmann’s result and my result . . . . . . . . . 37 3.4 Envelope and average of the oscillations of 2p(). p() is the polarization density of conduction electrons . . . . . . . . . . . . . 38 3.5 The polarization density for Wilson band with = 2 . . . . . . . . 39 3.6 The polarization density for Wilson band with = p 2 . . . . . . . 40 3.7 The polarization density forjV sd j 2 = 0:05 . . . . . . . . . . . . . . 40 3.8 Integrated Kondo cloud in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:1 . . . . . . . . . . . . . . . . . . 42 3.9 Integrated Kondo cloud in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:05 . . . . . . . . . . . . . . . . . . 42 3.10 Integrated Kondo cloud in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:02 . . . . . . . . . . . . . . . . . . 43 vi 3.11 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:1 . . . . . . . . . . . . . . . . . . 44 3.12 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:05 . . . . . . . . . . . . . . . . . . 45 3.13 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:04 . . . . . . . . . . . . . . . . . . 45 3.14 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:02 . . . . . . . . . . . . . . . . . . 46 3.15 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:1 . . . . . . . . . . . . . . . . . . 48 3.16 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:05 . . . . . . . . . . . . . . . . . . 48 3.17 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:04 . . . . . . . . . . . . . . . . . . 49 3.18 The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:02 . . . . . . . . . . . . . . . . . . 49 4.1 The amplitudeA(l) of the Friedel oscillations for a symmetric FA impurity withjV sd j 2 = 0:03 and different values of U with E d = U=2. The abscissa is the distancel = log 2 () from the impurity where = 2r= F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 The amplitude A(l) of the charge oscillations of the spin-up and spin-down electrons in the magnetic component of the singlet state. 67 4.3 The phase relations of the charge oscillations of the magnetic half of the FA impurity with U = 1, E d = 0:5 andjV sd j 2 = 0:03 as a function of the distancel = log 2 (). The full and open trian- gles show the positions of the minima and maxima of the Friedel oscillations for the spin-up and spin-down electron (up and down triangles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4 The amplitude of the polarization oscillation of the magnetic com- ponent for the U = 1 FA impurity as a function of the distance l =log 2 (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 vii 4.5 The phase relations of the charge oscillations of the magnetic half of the FA impurity withU = 0:1, E d =0:05 andjV sd j 2 = 0:03 as a function of the distancel = log 2 (). The full and open trian- gles show the positions of the minima and maxima of the Friedel oscillations for the spin-up and spin-down electron (up and down triangles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.6 The amplitudes of the Friedel oscillations for different Friedel reso- nances withE d = 0 andjV sd j 2 = 10 j withj = 1; 2; 3; 4; 5. . . . . . 71 4.7 The logarithm of the half-width F for the Friedel impurity (full circles) and the logarithm of the Kondo energy for the FA impurity are plotted versus the correspondingl 1=2 = log 2 ( 1=2 ) values. The latter are obtained from Fig.(4.6) and Fig.(4.1) as the position where A(l) takes the value 1. The two full lines are straight lines with the slope -1. Their vertical separation is 1.25, corresponding to a ratio of 2 1 :25 2:4 between Kondo energy and Kondo resonance half width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 viii Abstract A compact solution consisting of 4-8 Slater states (FAIR solution) is introduced to treat the Friedel Anderson and Kondo impurity problem. The ground state energy is obtained with impressively high accuracy. Net integrated polarization density is calculated and it confirms the existence of Kondo cloud. Finite size effect in the impurity problem is studied using FAIR method. It is shown that the formation of a Kondo ground state requires a minimum sample size and is accompanied by the presence of Kondo cloud. The Friedel Oscillations in the vicinity of a Friedel-Anderson impurity are investi- gated by FAIR method. The development of Friedel oscillation with a phase shift of=2 outside the Kondo radius is confirmed. And the amplitudeA() of the Friedel oscilla- tions show a very similar behavior to that of a simple non-interacting Friedel impurity with a narrow resonance at the Fermi level. This similarity supports the concept of a ”Kondo” resonance. And the Kondo resonance half width FA is suggested to be FA E =2:4, whereE is the Kondo energy calculated from susceptibility. ix Chapter 1 Introduction When magnetic atoms, such as cobalt, are added into some metals like copper and gold, an anomalous increase in the resistance of metals at low temperature has been observed since 1930s. Not until 1964 did Jun Kondo give out a satisfactory explana- tion for this phenomenon, which is therefore called Kondo effect. Since 1960s, the occurrence of localized magnetic moments on iron-group ions has also been discov- ered. All the observations led to discussions on the interaction between spins of the localized impurity and conduction electrons. And the properties of magnetic impu- rities in a metal became one of the most intensively studied problems in solid state physics.[25][32][22][12][34][1][30] [13][27][26][31][29] The work of Friedel[16][17][18][19] and Anderson[3] laid the foundation to under- stand why some transition-metal impurities form a local magnetic moment while others do not. Kondo[24] showed that multiple scattering of conduction electrons by a mag- netic impurity yields a divergent contribution to the resistance in perturbation theory. Kondo’s paper stimulated a large amount of theoretical and experimental work which changes our understanding of d and f impurities completely. After decades of effort, a number of methods for Kondo and Friedel-Anderson problem have been developed, ranging from scaling, renormalization, Fermi-liquid theory, slave bosons to large-spin limit. It was shown that at zero temperature, the Friedel-Anderson impurity is in a singlet state. And with the help of Bethe-ansatz, the exact solutions of the Kondo and Friedel- Anderson impurities were derived. But they all use very sophisticated and demanding 1 formalisms. Therefore even comparing the results from different techniques can be dif- ficult and those theoretical papers often represent a huge challenge to experimentalists. What’s more, those methods use simplified band structures and reduce the five-fold orbital degeneracy of d-impurity to a single orbit, thus they are not describing a real system. The Friedel Artificially Inserted Resonance (FAIR) method is a new numerical method developed by my advisor Dr. Gerd Bergmann. It introduces a compact solu- tion which consists of just 4-8 Slater states. This method was first applied to Friedel- Anderson impurity problem. After the rotation of the basis in the Hilbert space and the optimization of the coefficients of the FAIR states, an impressively good result of the ground state energy was calculated. The result is more accurate than the Mean Field Theory approximation and shows the same resolution as the calculation by Gunnars- son and Schonhammer, who included 10 7 basis states into their calculation. Then this method was introduced into the Kondo problem, and some well known properties of the Kondo problem were derived in a simpler way. FAIR method can be applied to systems with an arbitrary density of states and take the orbital degeneracy into account, which approaches the real system. Moreover, the explicit form of the FAIR solution makes the physical interpretation much easier than some of the sophisticated solutions and therefore is well suited to solve problems which are difficult for other methods, such as the Kondo cloud problem, the Friedel-Anderson or Kondo state in finite size structures and so on. In this paper, two investigations by FAIR method are presented. First, we studied the finite size effect in magnetic impurity problem. We mainly investigated the rela- tionship between sample size and the presence of Kondo cloud. Second, we studied the Friedel oscillations around a Friedel-Anderson impurity. The comparison between the 2 Friedel oscillation of a FA impurity and a simple Friedel impurity provides us with very important information about the ”Kondo resonance” at the Fermi level. The paper is organized as follows. In Chapter 2, the Hamiltonian used by Friedel- Anderson and Kondo to treat magnetic impurity problems will be discussed. And the FAIR method in combination with the use of Wilson states will be introduced. The main results obtained by FAIR method will also be reported in this chapter. In Chap- ter 3, the calculation of Kondo cloud will be described in detail. It is first calculated for infinitely large samples in magnetic and singlet states. Then the calculation for finite size sample is performed and a confusion about the oscillations in the envelope of the polarization density curve is clarified. In Chapter 4.2, the Friedel oscillations about a Friedel-Anderson impurity will be analyzed. And then a comparison with the Friedel oscillation of a simple Friedel impurity will be conducted. Finally, in Chapter 6, the conclusions are reached and the possible future work is discussed. Some relevant derivations are shown in the Appendix of each chapter if available. 3 Chapter 2 Model and Method 2.1 Wilson’s states The description of metal electrons by plane wave is very successful in many cases. But if one wants to manipulate the whole electron basis, one runs into troubles. Because in a macroscopic metal sample, the number of electrons is of the order ofN = 10 23 . A rotation of the whole basis in Hilbert space requiresN(N 1)=2 Euler angles, which is a huge number and is beyond the computational ability of computers. Fortunately, Wilson introduced a very handy discrete energy band to replace the real continuous band. Although it’s somewhat artificial, it seizes the essentials of the problem. He considers an energy band ranging from -1 to +1 with constant density of states and divide it into N cells in such a way that the boundaries of those cells have values: 1; 1 ; 1 2 ; 1 3 ;:::; 1 N=21 ; 0; 1 N=21 ;:::; 1 3 ; 1 2 ; 1 ; 1 where is a parameter larger than 1, which is usually set to be 2. In other words, the division is done on a exponential scale. Theth cell is confined by [ 1 ; 1 +1 ] Figure 2.1: Wilson band 4 for = 0; 1;:::;N=2 2 and [ 1 N ; 1 N1 ] for =N=2 + 1;N=2 + 2;:::;N. The two cells nearest to the center are special. They are confined by [ 1 N=21 ; 0] and [0; 1 N=21 ] respectively(Fig.2.1). Denote the boundaries ofth cell to be [ ; +1 ]. The logarithm scale is used because a very small energy spacingE near the fermi energy is especially important for obtaining the Kondo ground state. Suppose that the volume of the host metal isV h and the atomic volume isV A , then the total number of atoms isZ =V h =V A = P N1 =0 Z The density of states in atomic volumeV A (per spin) is = 1 2 , because there is only one state in each atomic volume (per spin) and it is evenly distributed on a energy band of width 2. The separation of energy levels is constant: E = 2V A =V h which is of the order of 10 8 . As a result, there are still a large number of states in each energy cell. Wilson further simplified this situation. He did such a trasformation of all the states in each cell, that only one resulting state interact with the impurity and all the other states in the cell have no interaction with the impurity. Suppose inth cell, there areZ electron states which we denote as' , ( = 0; 1; ;Z 1). They all interact with the impurity via a constant s-d matrix elementV sd . The transformation matrix is defined as A, which has the simple formA = exp(i2=Z )= p Z . And the new states are formed as (r) = 1 p Z Z1 X =0 ' (r) exp(i2=Z ) ( = 0; 1; 2;:::;Z v 1) (2.1) (0) = 8 < : 1 p Z P Z1 =0 ' (0) 0 = 0 = 1; 2;:::;Z v 1 Only the states for = 0 has a finite interaction with the impurity. For all other states with 6= 0, the s-d interaction is zero. As a result these states can be ignored 5 in the calculation. We end up with only one state in each energy cell. This state is the superposition of the original states in its cell, which absorbs all the interaction with the impurity in that cell. Those states are called Wilson’s states or delegate states, denoted as (r) = 1 p Z Z1 X =0 ' (r) (2.2) And the energy of the state is simply E v = 1 Z Z1 X =0 " = +1 + 2 (2.3) 2.2 FAIR method 2.2.1 Overview Professor Bergmann recently introduced an approach called FAIR method to treat mag- netic impurity problem[5] [6]. Consider the Hamiltonian of a band with a finite number N of free electron states H 0 = N1 X =0 " c y c (2.4) wherec y andc are the creation and annihilation operators of the states. From these band states a new statea y 0 is composed with random but normalized coefficients a y 0 = N1 X =0 0 c y (2.5) 6 And then N-1 statesa y i (i = 1; 2;:::;N 1) are constructed in the manner that they are orthogonal toa y 0 and to each other. In this basis, the Hamiltonian matrix is not diag- onal any more. We proceed to sub-diagonalize it, namely we discard the first column and first row and diagonalize the remaining (N-1)*(N-1) matrix. In this way we trans- form the basis once more and get the new basis n a y 0 ;a y 1 ;a y 2 ;:::;a y N1 o , which is uniquely determined bya y 0 . In the new basis, the s-band Hamiltonian has the form H 0 = N1 X i=1 E i a y i a i +E 0 a y 0 a 0 + N1 X i=1 V fr (i) a y 0 a i +a y i a 0 (2.6) This Hamiltonian is very similar to a Friedel Hamiltonian (we will discuss this in the section 2.2.2). Herea y 0 representing an artificial Friedel resonance is analogous to thed y resonance state. Therefore this state is called an Friedel Artificially Inserted Res- onance state (FAIR) state. Since we can constructa y 0 in an arbitrary way and afterwards manipulate and optimize it, it gives the FAIR method the adaptability. 2.2.2 Friedel Hamiltonian The simplest model that describes the magnetic impurity in metal problem is the Friedel model. The Hamiltonian has the form: H 0 = X " c y c +E d d y d + X v sd; [d y c +c y d] (2.7) wherec y andc are the creation and annihilation operators of s-band electron states andd y andd represent the the creation and annihilation operators of d-resonance states. This Hamiltonian describes a spinless metal host with an s-band in which a transition metal atom is dissolved. The ten-fold degeneracy of a real d-impurity is simplified to a single d-state without spin. The s-electrons interact with the d-impurity via the hopping 7 matrix elementv sd; . If the s-d interaction can be expressed by a-like potential, then v sd; is a constant. v sd; = Z ' d (r)v sd (r)' (r)d 3 r = Z ' d (r)v sd (r) r 1 V h e ikr d 3 r (2.8) Let V 0 sd = Z ' d (r)v sd (r) r 1 V A e ikr d 3 r (2.9) then v sd; = r V A V h V 0 sd = V 0 sd p Z (2.10) This Hamiltonian is a single particle Hamiltonian, which in principle can be diago- nalized. But it can’t be done using the original energy band due to the large number. So we apply the Wilson’s band here, keeping only the N delegate states. H 0 = N1 X =0 " c y c +E d d y d + N1 X =0 V sd ()[d y c +c y d] (2.11) The corresponding coupling to the impurityV sd () can be derived as follows: V sd () = Z ' d (r)v sd (r) (r)d 3 r = Z ' d (r)v sd (r) 1 p Z Z X =1 ' (r)d 3 r (2.12) = 1 p Z Z X =1 Z ' d (r)v sd (r)' (r)d 3 r = 1 p Z Z X =1 v sd; = p Z v sd; = r Z Z V 0 sd =V 0 sd r +1 2 This yields the sum rule 8 N1 X =0 jV sd ()j 2 = N1 X =0 V 0 sd 2 ( +1 ) 2 = V 0 sd 2 (2.13) Professor Bergmann observed numerically that the (n+1)-electron ground state can be written in terms of FAIR states as = h Aa y 0 +Bd y i n Y i=1 a y i 0 (2.14) where 0 is the vacuum state. In the FAIR basis, the total Friedel Hamiltonian takes the form H 0 = N1 X i=1 E i a y i a i +E 0 a y 0 a 0 + N1 X i=1 V fr (i) a y 0 a i +a y i a 0 (2.15) +E d d y d +V 0 sd (0) h d y a 0 +a y 0 d i + N1 X i=1 V 0 sd (i)[d y a i +a y i d] with E i = P N1 j=0 j i " j i (i = 0; 1;:::;N 1),V 0 sd (i) = P N1 j=0 V sd (j) j i . And the energy expectation value of ground state is E gs =A 2 E 0 +B 2 E d + 2ABV 0 sd (0) + n X i=1 E i (2.16) Then we rotate the statea y 0 = P N1 =0 0 c y in the Hilbert space until the minimum expectation energy is obtained. Meanwhile, we can also get the ground state energy by diagonalizing the total Hamiltonian matrix. It turns out that the energy expectation 9 value found by FAIR approach agrees with the ground energy from the diagonaliza- tion approach up to 10 14 , which is almost the internal accuracy of the computer! Dr. Bergmann proved that Eq. (2.14) is the exact ground state of the Friedel impurity.[4] The success in this simple problem shows the effectiveness of FAIR method. It can be applied to more complicated Hamiltonians, such as Friedel-Anderson and Kondo Hamiltonians. 2.2.3 Friedel-Anderson Hamiltonian If we take spin into account, the simplified single channel d-state will have a two-fold degeneracy for spin up and spin down. If both states are occupied, they repel each other due to the Coulomb interaction. This is described by the Friedel-Anderson (FA) Hamiltonian[3] : H 0 = ( N1 X =0 " c y " c " +E d d y " d " + N1 X =0 V sd ()[d y " c " +c y " d " ] ) (2.17) + ( N1 X =0 " c y # c # +E d d y # d # + N1 X =0 V sd ()[d y # c # +c y # d # ] ) +Ud y " d " d y # d # For FA Hamiltonian, the spin-flip process is a two-step process. If a s-electron comes in, it first has to hop to an empty d-state with the same spin and then the other occupied d-state with opposite spin hops to an s-state. Those two steps combined together give a spin-flip process of both the s and d-electron. 10 magnetic state It is a many particle problem which can not be solved by diagonalization. If we forget about the coulomb interaction part for a short while, the FA Hamiltonian just consists of two Friedel Hamiltonian with different spins. So the potential magnetic solution is the product of two states of the form of Eq.(2.14). [7]Therefore, the state can be written as MS = h A 0 a y 0" +B 0 d y " ih A 00 b y 0# +B 00 d y # i n Y i=1 a y i" n Y i=1 b y i# 0 (2.18) = h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i n Y i=1 a y i" n Y i=1 b y i# 0 wherea y 0 andb y 0 are two FAIR states. This state consists of an electron background Q n i=1 a y i" Q n i=1 b y i# 0 multiplied with the sum of four two-electron states, each of which hasS z = 0. And the coefficients should fulfill the normalization condition: A 2 +B 2 + C 2 +D 2 = 1. The FA Hamiltonian can be expressed in those two sets of FAIR bases in a similar way to Eq.(2.15). Although Eq. (2.14) is only the exact ground state for mean field FA Hamiltonian(i.e. approximateUd y " d " d y # d # byUhn d" ihn d# i), we can use this as an approximate ground-state for FA Hamiltonian. The expectation value of the FA Hamiltonian in the state Eqn.(2.18) is E gs = A 2 (E a 0 +E b 0 ) +B 2 (E a 0 +E d ) +C 2 (E b 0 +E d ) +D 2 (2E d +U) +2(AC +BD)V a sd (0) + 2(AB +CD)V b sd (0) + n X i=1 (E a i +E b i ) We minimize the ground-state energy by optimizing the two FAIR statesa y 0 andb y 0 , as well as the four coefficients A,B,C,D under the constraintA 2 +B 2 +C 2 +D 2 = 1. 11 Figure 2.2: The ground state energies of the mean-field solution, the magnetic and the singlet FAIR solution The resulting ground-state energy lies considerably below the mean-field energy, as shown in Fig. 2.2. Moreover, the new solution suppresses the magnetic moment up to a considerably larger value ofU' 0:46, which is almost twice the value of the mean-field theory(Fig.2.3). This FAIR state (Eq. (2.18)) is in a symmetric state for small U, but becomes asym- metric between spin up and spin down for large U, similar as the mean-field state. As a result, the FAIR state in Eq.(2.18) is called magnetic state. singlet state We discussed the magnetic state in the previous section, however, it is well known that the ground state of the Friedel-Anderson impurity is a singlet state, i.e. it is symmetric in spin-up and spin-down electrons. From MS , a mirror state can be constructed by reversing all the spins. Combining the two states yields the approximate singlet state.[8] 12 Figure 2.3: The magnetic moment as a function of the Coulomb exchange energy U, using the mean-field solution and the magnetic FAIR solution SS = h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i n Y i=1 a y i" n Y i=1 b y i# 0 (2.19) + h Ab y 0" a y 0# +Cb y 0" d y # +Bd y " a y 0# +Dd y " d y # i n Y i=1 b y i" n Y i=1 a y i# 0 TheB term is not orthogonal to theC term and theB term is not orthogonal to the C. This introduces some additional terms in the expectation energy. (see section 2.3 for reference). The same as before, we variate the two FAIR statesa y 0 andb y 0 , as well as the coefficients under the constrainth SS j SS i = 1, until the minimum expectation energy is obtained. The expectation energy of the singlet state as a function of Coulomb interaction U is shown in Fig.(2.2). Its energy clearly lies below that of that of the magnetic state, confirming that singlet state is the ground state. 13 2.2.4 Kondo Hamiltonian For the treatment of a magnetic impurity, Kondo used a different Hamiltonian than Friedel-Anderson, [24]shown as follows: H 0 = ( N1 X =0 " c y " c " +E d d y " d " ) + ( N1 X =0 " c y # c # +E d d y # d # ) +H sd (2.20) whereH sd is the exchange Hamiltonian with exchange interactionJ(r) H sd = Z 2J(r)s(r)Sd 3 r (2.21) Generally,J(r) is approximated by a(r) function,J(r) =v a J(r), wherev a is the atomic volume. SoH sd has the form H sd = 2v a Js(0)S (2.22) = 2v a J s z (0)S z + 1 2 [s + (0)S +s (0)S + ] where " (0) and # (0) are field operators of the s-electron, andS z ,S ,S + are the spin operators of the impurity. s z (0) = 1 2 [ " (0) " (0) # (0) # (0)] s + (0) = " (0) # (0) s (0) = # (0) " (0) So 14 H sd = v a J h S + y # (0) " (0) +S y " (0) # (0) i +v a JS z h y " (0) " (0) y # (0) # (0) i The spin-flip process in Kondo model is similar to the one described by FA model, but lack of detailed sketch of the interaction. The incoming s-electron is scattered by the d-impurity, and the d-electron and out coming s-electron reverse their spin orientations. It has been shown by Schrieffer and Wolff [33]that there is a close relation between FA and Kondo impurity. The FA Hamiltonian can be transformed into an exchange Hamiltonian used by Kondo with a positive antiferromagnetic exchange interactionJ > 0. In particular, for large values of U and large negative values ofE d , the effectiveJ has the form J'jV sd j 2 U (U +E d )E d > 0 The Kondo impurity is in some respect the limiting case of the FA impurity. For a Kondo impurity, the d-state is always singly occupied, either with spin-up or spin-down. By increasing the Coulomb interaction U, the FA impurity approaches the properties of a Kondo impurity. This can be seen from the effect that the coefficientA;A 0 ;D;D 0 approach zero as U increases. Therefore, for the Kondo Hamiltonian, we assume the following approximate ground state by removing the zero and double d-occupancy terms from the FA ground state SS (Eq.(5.20)).[11] 15 K = h Ba y 0" d y # +Cd y " b y 0# i n Y i=1 a y i" n Y i=1 b y i# 0 + h Cb y 0" d y # +Bd y " a y 0# i n Y i=1 b y i" n Y i=1 a y i# 0 The expectation energy of Kondo Hamiltonian in this state involves many terms. We write a computer programm to calculate them. Again, we variate the two FAIR states and the coefficientsB;C;B;C under the constrainth K j K i = 1, until the minimum energy is reached. The resulting energy is defined as Kondo ground state energy. In this process, we also get a byproduct: the first excited state. To minimize the expectation energy under the constraint, we use the method of Lagrange multiplier and form an eigenvalue equation of the coefficientsB;C;B;C. The solution gives four eigenvalues. The lowest one corresponds to the ground energy and the second one corresponds to the first excited energy. The total spin square S 2 = ( P i s i ) 2 of those two states has been calculated. For the value of J=0.1, we find S 2 = 0:04 in the ground state and S 2 = 1:99 in the first excited state. It indicates that the ground state is close to a singlet state and the first excited state is close to a triplet state, as we expected. The structure of the coefficient vector B;C;B;C contains interesting information of the states. In singlet state, they are pairwise equal(B = B;C = C) , while in triplet state mirror states have opposite signs(B =B;C =C). However, this triplet state is not necessarily the triplet state with lowest energy. To obtain the lowest triplet state, we impose the conditions B =B;C =C during minimization, which automatically force the triplet state. We denote the resulting triplet state the relaxed triplet state and the previous triplet state the unrelaxed one. The energy difference between relaxed triplet and singlet states is defined as Kondo energy. The Kondo energy is plotted vs. 1=(2J 0 ) in Fig.( 2.4) 16 Figure 2.4: The energy difference between the singlet and triplet states. Our Kondo energy lies between two theoretical curves: (i)E st = D exp [1=(2J 0 )] (ii)E st = p 2J 0 D exp [1=(2J 0 )]. Both expressions are used in literatures as approximate values of Kondo energy k B T . It confirms that the FAIR method represents the physics of the Kondo impurity accurately. 2.3 Appendix:Matrix elements for Friedel-Anderson and Kondo Hamiltonian 2.3.1 Friedel Anderson Model The Friedel-Anderson Hamiltonian is given again by the following 17 H 0 = N1 X i=0 E i a y i" a i" + N1 X i=1 V fr (i) a y 0" a i" +a y i" a 0" +E d d y " d " + N1 X i=0 V 0 sd (i)[d y " a i" +a y i" d " ] + N1 X i=0 E i a y i# a i# + N1 X i=1 V fr (i) a y 0# a i# +a y i# a 0# +E d d y # d # + N1 X i=0 V 0 sd (i)[d y # a i# +a y i# d # ] +Ud y " d " d y # d # (2.23) Denote K " = N1 X i=0 E i a y i" a i" + N1 X i=1 V fr (i) a y 0" a i" +a y i" a 0" +E d d y " d " K # = N1 X i=0 E i a y i# a i# + N1 X i=1 V fr (i) a y 0# a i# +a y i# a 0# +E d d y # d # V " = N1 X i=0 V 0 sd (i)[d y " a i" +a y i" d " ] V # = N1 X i=0 V 0 sd (i)[d y # a i# +a y i# d # ] Then H 0 =K " +K # +V " +V # +Ud y " d " d y # d # (2.24) The approximate ground state is 18 SS = h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i n Y i=1 a y i" n Y i=1 b y i# 0 (2.25) + h Ab y 0" a y 0# +Cb y 0" d y # +Bd y " a y 0# +Dd y " d y # i n Y i=1 b y i" n Y i=1 a y i# 0 or abbreviated as the following: using # fora 0 SS = A#1#2 +B#1d2 +Cd1#2 +Dd1d2 +A 0 #2#1 +C 0 #2d1 +B 0 d2#1 +D 0 d2d1 We order the operators in such a way that the first operator is always associated with spin up and the second is associated with spin down. After the expansion, we can see that only the following terms present. Scalar Product Part S < ##; 11>=<a 1y 0 n1 Y i=1 a 1y i ja 1y 0 n1 Y i=1 a 1y i >= 1 (2.26) S < ##; 21>=<a 2y 0 n1 Y i=1 a 2y i ja 1y 0 n1 Y i=1 a 1y i > (2.27) = <a 2y 0 ja 1y 0 > <a 2y 0 ja 1y n1 > <a 2y n1 ja 1y 0 > <a 2y n1 ja 1y n1 > (2.28) 19 S <dd; 11>=<d y n1 Y i=1 a 1y i jKjd y n1 Y i=1 a 1y i >= 1 (2.29) S <dd; 21>=<d y n1 Y i=1 a 2y i jKjd y n1 Y i=1 a 1y i > (2.30) = <a 2y 1 ja 1y 1 > <a 2y 1 ja 1y n1 > <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > (2.31) Kinetic Energy Part K = N1 X i=0 E (i)a y i a i + N1 X i=1 V fr (i) h a y 0 a i +a y i a 0 i +E d d y d 20 K < ##; 21> (2.32) = <a 2y 0 n1 Y i=1 a 2y i jKja 1y 0 n1 Y i=1 a 1y i > (2.33) = <a 2y 0 n1 Y i=1 a 2y i N1 X i=0 E 1 a (i)a 1y i a 1 i + N1 X i=1 V 1 fr (i)a 1y i a 1 0 a 1y 0 n1 Y i=1 a 1y i > (2.34) = N1 X i=0 E 1 a <a 2y 0 ja 1y 0 > <a 2y 0 ja 1y n1 > <a 2y n1 ja 1y 0 > <a 2y n1 ja 1y n1 > (2.35) + N1 P i=n Vfr (i)<a 2y 0 ja 1y i > <a 2y 0 ja 1y 1 > <a 2y 0 ja 1y n1 > N1 P i=n Vfr (i)<a 2y 1 ja 1y i > <a 2y 1 ja 1y 1 > <a 2y 1 ja 1y n1 > N1 P i=n Vfr (i)<a 2y n1 ja 1y i > <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > (2.36) K <dd; 11> (2.37) = <d y n1 Y i=1 a 1y i jKjd y n1 Y i=1 a 1y i > (2.38) = <d y n1 Y i=1 a 1y i N1 X i=0 E 1 a (i)a 1y i a 1 i +E d d y d d y n1 Y i=1 a 1y i > (2.39) = n1 X i=1 E 1 a (i) +Ed (2.40) 21 K <dd; 21> (2.41) = <d y n1 Y i=1 a 2y i jKjd y n1 Y i=1 a 1y i > (2.42) = * d y n1 Y i=1 a 2y i N1 X i=0 E 1 a (i)a 1y i a 1 i + N1 X i=1 V 1 fr (i)a 1y 0 a 1 i +E d d y d d y n1 Y i=1 a 1y i + (2.43) = ( n1 X i=0 E 1 a (i) +E d ) <a 2y 1 ja 1y 1 > <a 2y 1 ja 1y n1 > <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > (2.44) 0 V 1 fr (1) V 1 fr (n 1) <a 2y 1 ja 1y 0 > <a 2y 1 ja 1y 1 > <a 2y 1 ja 1y n1 > <a 2y n1 ja 1y 0 > <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > (2.45) = K <dd; 11>S <dd; 21> (2.46) 0 V 1 fr (1) V 1 fr (n 1) <a 2y 1 ja 1y 0 > <a 2y 1 ja 1y 1 > <a 2y 1 ja 1y n1 > <a 2y n1 ja 1y 0 > <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > (2.47) s-d coupling energy part V = N1 X i=0 V sd (i) h d y a i +a y i d i (2.48) V <d#; 11> (2.49) = <d y n1 Y i=1 a 1y i N1 X i=0 V 1 sd (i)d y a 1 i a 1y 0 n1 Y i=1 a 1y i >=V 1 sd (0) (2.50) 22 V <d#; 21> (2.51) = <d y n1 Y i=1 a 2y i N1 X i=0 V 1 sd (i)d y a 1 i a 1y 0 n1 Y i=1 a 1y i > (2.52) = V 1 sd (0) V 1 sd (1) V 1 fr (n 1) <a 2y 1 ja 1y 0 > <a 2y 1 ja 1y 1 > <a 2y 1 ja 1y n1 > <a 2y n1 ja 1y 0 > <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > (2.53) The coulomb interaction energy part D SS Ud y " d " d y # d # SS E = D 2 D d1d2 Ud y " d " d y # d # d1d2 E +D 02 D d3d4 Ud y " d " d y # d # d3d4 E +DD 0 D d1d2 Ud y " d " d y # d # d3d4 E +D 0 D D d3d4 Ud y " d " d y # d # d1d2 E = UD 2 +UD 02 +UDD 0 S dd13S dd24 +UD 0 DS dd31S dd42 2.3.2 Kondo Model Scalar Product and Kinetic Energy Parts are the same as Friedel Anderson Model except that E d should be set to 0 in the Kondo model. The coulomb interaction term is not relevant here. And the s-d coupling part is different from the FA model, which will be discussed below. 23 s-d coupling energy part H sd = J xy h S + y # (0) " (0) +S y " (0) # (0) i +J z S z h y " (0) " (0) y # (0) # (0) i S + =d y " d # (2.54) S =d y # d " (2.55) B =B#1d2 +Cd1#2 +C 0 #2d1 +B 0 d2#1 Let H jz =J z S z h y " (0) " (0) y # (0) # (0) i (2.56) H jxy =J xy h S + y # (0) " (0) +S y " (0) # (0) i (2.57) where b " (x) = P i ' a i (x)a i" and' a i (x) = P i c (x). c (x) is the wavefunc- tion of the th Wilson state Eq.(3.2). And let A i =' a i (0) = X i c (0) = N1 X k=0 i r +1 2 (2.58) Then H jz < ##; 11>=J z n1 X i=0 A 1 i (0) 2 (2.59) H jz <dd; 11>=J z n1 X i=1 A 1 i (0) 2 (2.60) 24 H jz < ##; 21> = 0 B B B B B B B @ S < ##; 21> 1 A 1 0 (0) A 1 n1 (0) A 2 0 (0) <a 2y 0 ja 1y 0 > <a 2y 0 ja 1y n1 > A 2 n1 (0) <a 2y n1 ja 1y 0 > <a 2y n1 ja 1y n1 > 1 C C C C C C C A J z H jz <dd; 21> = 0 B B B B B B B @ S <dd; 21> 1 A 1 0 (0) A 1 n1 (0) A 2 0 (0) <a 2y 1 ja 1y 1 > <a 2y 0 ja 1y n1 > A 2 n1 (0) <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > 1 C C C C C C C A J z H jxy <d#; 11>=A 1 0 (0) p J xy (2.61) H jxy <d#; 21> = A 1 0 (0) A 1 1 (0) A 1 n1 (0) <a 2y 1 ja 1y 0 > <a 2y 1 ja 1y 1 > <a 2y 0 ja 1y n1 > <a 2y n1 ja 1y 0 > <a 2y n1 ja 1y 1 > <a 2y n1 ja 1y n1 > p J xy The superscript ofA indicates the number of basis. 25 Chapter 3 Kondo Cloud One of the most controversial aspects of Kondo ground state is the so-called Kondo cloud within the radius of Kondo length K = ~v F k B T K , wherek B T K is the Kondo energy andv F is the Fermi velocity of the s electrons. Simply speaking, Kondo cloud is a cloud composed of s-spins which screens the impurity d-spin. The proponents of the Kondo cloud argue that if one divide the Kondo ground state into two parts with opposite d spins, then in each component there is an s electron within the Kondo sphere which compensates the d spin. The s-electron forms a singlet state with the d-spin and this bond is broken above the Kondo temperature, namely, the screening cloud disappears. Several experimental investigations were stimulated in the 1970s, and most of them used nuclear magnetic resonance (NMR). In a Cu sample with dilute Fe-Kondo impurities, the experiment showed that the spin polarization at shells of Cu atoms around impurity did not change at all when the temperature crosses the Kondo temperature, i.e., the Kondo cloud was not detected. The conclusion that Kondo cloud does not exist was made at that time. However, in recent theoretical papers, the argument was made that NMR experiments cannot possibly detect the Kondo cloud because of the low density of the cloud. Because of the large volume of the Kondo sphere, the Kondo cloud spreads so thinly that the change of polarization felt by an individual Cu atom cannot be detected in an NMR experiment. A number of theoretical papers have been published since 2000, proposing to observe the Kondo cloud in sub-micron structures while the experimental side has not improved so far. 26 Since the FAIR solution consists of only several slater determinants, it is very suit- able for calculating Kondo cloud.[9] 3.1 Continuous case Since the Wilson energy band is given in units of Fermi energy, the momentum will be measured in units of the Fermi wave number and the coordinate x gives the position in units of half Ferimi wave length f =2. Assume a linear dispersion relation " = k 1, (for 06 k 6 2). Then the wavefunction of s-electron states before the Wilson transformation are ' k (x) = r 2 L cos(kx) and the boundary condition yields k = 2n+1 L and n = 0;:::;N 1. This wave function is not the full wavefunction, because we ignore the sin part which does not contribute at x=0. We call this reduced wavefunction. We then sum over all the wavefunctions in one energy cell to get the wavefunc- tion of Wilson state. The boundaries ofth cell is [ ; +1 ], indicating the k range is [ + 1; +1 + 1]. Then the wavefunction of the Wilson state representingth cell is v (x) = 1 p ( +1 )L X +1<k< +1 +1 r 2 L cos(kx) (3.1) The normalization coefficient comes from the fact that the th cell hasZ v = ( +1 )L states. In infinitely large sample, k is almost continuous, the summation above can be treated as integral, which yields 27 v (x) = 2 p 2 p ( +1 ) sin x( +1 ) 2 x cos x(2 + +1 + ) 2 (3.2) The density of the wavefunction v (x) is given by j v (x)j 2 = 8 ( +1 ) sin 2 x( +1 ) 2 (x) 2 cos 2 x(2 + +1 + ) 2 The cosine part is a fast oscillation which yields Friedel oscillations if integrated. We are only interested in density on a much larger scale, so we average this part and sub- stitute cos 2 x(2+ +1 + ) 2 by 1 2 . The wavefunctions have very different spatial ranges, with the largest to be 2 26 in the case of N=50(number of Wilson states). So it is not very useful to calculate the density as a function of x. As a result, we calculate the integrated density instead, from 0 tor = 2 l v (r) = Z r 0 j v (x)j 2 dx = 2 v+3 Z r 0 sin 2 x 2 v+2 (x) 2 dx = 2 Z 2 l =2 v+2 0 sin 2 (u) (u) 2 du = I(lv 2) whereI(s) = 2 R 2 s 0 sin 2 (u) (u) 2 du and s is an integer. It can be seen that the single ruler integral I(s) yields the integrated density for almost all wavefunctions v (x), which dramatically reduce the computation time. Our ground state is expressed in terms of new statesa y i = P N1 =0 i c y , wherec y is the Wilson states. Namely, the density of the wavefunction ofa y i is 28 i (2 l ) = Z 2 l 0 N1 X =0 i v (x) 2 dx Besidesj v (x)j 2 terms, interference terms v (x) v+ (x) are also present. Fortu- nately, interference terms can also be expressed by some ruler intergralsI 0 (v;) I 0 (v;) = 2 p 2 Z 2 s 0 sin (u) sin u 2 (u) 2 cos 3u 1 1 2 du In the numerical evaluation, we calculate (i) the s-electron density with and without d impurity, (ii) integrate both densities from x=0 to r, where r is increased on an expo- nential scale, r = 2 l , (iii) form the net integrated density as the difference and plot it versusl = log 2 (r). FA Magnetic state For FA magnetic state MS , the net integrated density of spin up and spin down, as well as the total density and spin polarization are calculated and shown in Fig.(3.1) There is a sudden drop beyond 2 20 , and it corresponds to some rim or surface of the sample and therefore is of no interest for the density distribution around the impurity. Below r = 2 20 , the curve is almost flat, indicating no polarization cloud around the impurity. It is consistent with the result that the magnetic state has a moment of = 0:93 B . FA singlet state For FA singlet state, the net integrated density of spin up and spin down, as well as the total density and spin polarization are calculated for first half of the singlet state Eq.(5.20) MS = h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i n Y i=1 a y i" n Y i=1 b y i# 0 29 Figure 3.1: The net integrated density of the s electron within a distance r from the impurity for spin up and spin down, as well as total density and polarization This part is of the same form as the magnetic state and the second half is just revers- ing all the spin directions of the first part. Therefore the magnetic state is the building block of the singlet state. So the spin polarization of one of the magnetic components is of interest here. The result is shown in Fig.(3.2) The sudden drop of density at 2 20 is again the surface effect. The density of spin down increases, while density of spin up decreases, resulting in a spin polarization of -0.46 at the distance r = 2 11:6 . The reason why the density of spin up is negative is that we didn’t count the density of d-electron. The net d-spin up of this magnetic component is C 2 B 2 = 0:46. Therefore, the d-spin up is completely canceled by the spin polarization of the s-electron. The same situation happens to the other half of the state. So the whole state Eq.(5.20) does not possess magnetic moment, which is consistent with the fact that it is a singlet state. 30 Figure 3.2: The net integrated density of one magnetic component of the FA singlet state It’s worth to mention that the range of the Kondo cloud (i.e. Kondo length) depends on the strength of s-d interaction. WhenjV 0 sd j 2 = 0:1 is used instead of 0.04, the polar- ization cloud extends only over a distance ofr = 2 6:4 . It shows that the stronger the s-d interaction, the more compact the Kondo cloud is. The cloud for Kondo impurity is in principle the same as FA singlet results. It is discussed in detail in Prof. Bergmann’s paper.[9] 3.2 Finite Size Effect Finite size effect is always of interest in Physics study, because finite size sample is what we deal with in real life. From the continuous case, we see that the Kondo cloud exist for singlet state, but not for magnetic state. As a result, the presence of Kondo cloud can be treated as an evidence of existence of Kondo effect. I investigated the relationship between sample size and the presence of Kondo cloud. 31 3.2.1 FAIR calculation for finite size sample Suppose an impurity is put at the center (x = 0) in a one dimensional sample of length L. The wavefunction of s-electron states before the Wilson transformation are' k (x) = q 2 L cos(kx). This is the same as the continuous case, except we have not applied the unit of half wave length f =2 to coordinatex yet. The boundary condition k ( L 2 ) = 0 gives k = (2n+1) L = (2n+1) Na , where a is the lattice constant, and n = 0;:::;N 1. Again this is the reduced wave function because we ignore the Sine part which does not contribute atx = 0. The Fermi (smallest) wave length in the sample is f = 2a, consequently, the Fermi wave number is k f = 2= f = =a. Using the unit of half Fermi wave length and Fermi wave number, the wavefunction becomes ' k (x) = r 2 N cos(kx) , k = 2n + 1 N (n = 0;:::;N 1) (3.3) L becomes N because L=( f =2) = L=a = N. We assume the linear dispersion relation" =k 1, and scale the energy levels into the range of -1 to +1. Construction of Wilson frame is the same as described in section 2.1. In most cases, 40 or 50 is chosen. Put the scaled energy levels into those cells according to their value. And the Wilson states are the superposition of all the states in each cell. Then the wavefunction of the Wilson state representingth cell is v (x) = 1 p Z v X k12cellv r 2 N cos(kx) And its energy isE v = 1 Z P Z1 =0 " , namely, the average energy of those states in cellv. Unless we have a large number of atoms, several energy cells near 0 won’t contain any energy levels, since the spacing of energy levels are large. We simply remove those cells. IfN = 2 11 = 2048 is used for the total number of atoms, the actual number of Wilson states is 22, instead of 40 or 50. The corresponding s-d interactionV sd can also 32 be calculated easily by Eqn.(2.13)V sd () = q Z Z V 0 sd . Then we use the FAIR method described in section 2.2 to solve it, getting the ground state energy and the coefficients. 3.2.2 Wave function of Wilson states in finite size sample In finite sample, each Wilson state involves a finite number of original states, e.g., 0 (x) is a sum of 256 terms in the case N = 2048. However, unlike in continuous case, the summation in the wavefunction of Wilson states can’t be treated as an integral, for the reason that the wave numbers are discrete. If we simply leave the summation to the computer, the densityj 0 (x)j 2 would involve 256 2 = 65 536 terms. Even if only j 0 (x)j 2 is integrated, it would be a disaster that for each integration step the computer need evaluating 65 536 terms, not to say this is only one of 22 2 = 484 combinations to be integrated. So the wavefunctions have to be simplified. v (x) = 1 p Z v X k12cellv r 2 N cos(kx) = 1 p Z v r 2 N X k2cellv 1 2 (e ixk +e ixk ) P k2cellv e ixk is a geometric series with common ratioq = e ix 2 N , since the differ- ence between two successive k’s is 2 N . 3.3 X k2cellv e ixk = e ixk 0 v (1q Zv ) 1q where k 0 v is the smallest k in cell v, and Z v is the total number of states in cell v (Z v = P k2cellv I). v (x) = 1 p Z v r 2 N 1 2 e ixk 0 v (1e ixZv 2 N ) 1e ix 2 N + e ixk 0 v (1e ixZv 2 N ) 1e ix 2 N ! 33 e ixk 0 v (1e ixZv 2 N ) 1e ix 2 N + e ixk 0 v (1e ixZv 2 N ) 1e ix 2 N = e ixk 0 v (1e ixZv 2 N )e ix 1 N 1e ix 2 N e ix 1 N + e ixk 0 v (1e ixZv 2 N )e ix 1 N 1e ix 2 N e ix 1 N = e ixk 0 v 1e ixZv 2 N e ix 1 N e ixk 0 v 1e ixZv 2 N e ix 1 N e ix 1 N e ix 1 N = e ixk 0 v 1e ixZv 2 N e ix 1 N e ixk 0 v 1e ixZv 2 N e ix 1 N 2i sin( x N ) = e i(xk 0 v x N ) e i(xk 0 v + 2xZv N x N ) e i(xk 0 v x N ) +e i(xk 0 v + 2xZv N x N ) 2i sin( x N ) = 2i sin(xk 0 v x N ) + 2i sin(xk 0 v + 2xZv N x N ) 2i sin( x N ) = sin(xk 0 v + 2xZv N x N ) sin(xk 0 v x N ) sin( x N ) = sin(xk 0 v + 2xZv N x N ) sin(xk 0 v x N ) sin( x N ) = 2 cos((xk 0 v + 2xZv N x N +xk 0 v x N )=2) sin( x N ) sin((xk 0 v + 2xZ v N x N (xk 0 v x N ))=2) = 2 csc x N cos(xk 0 v + xZ v N x N ) sin xZ v N = 2 csc x N cos(x k 0 v + Z v 1 N ) sin xZ v N And we finally reach a succinct expression as follows: v (x) = r 2 Z v N csc x N cos x k 0 v + Z v 1 N sin xZ v N This expression converges to the continuous Wilson wavefunction in Eq.(3.2), when N!1, confirming the validity of the expression. 34 The Wilson states v (x) orc y v represent the free electron states. With the impurity, we express the ground state in terms of new statesa y i = P N1 v=0 v i c y v as discussed before. This means the wave function ofa y i is a i (x) = P N1 v=0 v i v (x) and the its density is a i (x) a i (x). 3.2.3 Singlet state of the Friedel-Anderson impurity To study the real Kondo cloud, we use the Friedel-Anderson Hamiltonian. As before, we separate the singlet states into two magnetic components ss = MS ("#) + MS (#"). When calculating the integrated density, we only consider one of the component. MS ("#) = [A s;s a y 0+" a y 0# +A s;d a y 0+" d y # +A d;s d y " a y 0# +A d;d d y " d y # ] n1 Y i=1 a y i+" n1 Y i=1 a y i# 0 The density of spin-up is " = A s;s 2 n1 X i=0 a i+ (x) a i+ (x) + A s;d 2 n1 X i=0 a i+ (x) a i+ (x) + A d;s 2 n1 X i=1 a i+ (x) a i+ (x) + A d;d 2 n1 X i=1 a i+ (x) a i+ (x) = [ A s;s 2 + A s;d 2 ] n1 X i=0 a i+ (x) a i+ (x) +[ A d;s 2 + A d;d 2 ] n1 X i=1 a i+ (x) a i+ (x) where a i+ (x) = P N1 v=0 v i (+) v (x) ( v i (+) is from the coefficient matrix of spin- up base). 35 The density of spin-down is # = A s;s 2 n1 X i=0 a i (x) a i (x) + A s;d 2 n1 X i=1 a i (x) a i (x) + A d;s 2 n1 X i=0 a i (x) a i (x) + A d;d 2 n1 X i=1 a i (x) a i (x) = [ A s;s 2 + A d;s 2 ] n1 X i=0 a i (x) a i (x) +[ A s;d 2 + A d;d 2 ] n1 X i=1 a i (x) a i (x) where a i (x) = P N1 v=0 v i () v (x) ( v i () is from the coefficient matrix of spin-down base). To get net density, we subtract the background density back = P n1 i=0 v (x) v (x) from both P n1 i=0 a i+ (x) a i+ (x) and P n1 i=1 a i (x) a i (x). And polarization is defined asp = # " . We use two ways to present polariza- tion:1) integrated net polarization. 2) 2p():(p() is the polarization density) My result in infinitely large sample To confirm the validity of my program, I first did the calculation for infinite sample and compared my result with Dr. Bergmann’s. It’s shown in Fig.(3.3). 1) Integrated net polarization They are basically the same, except that my curve has a bump near the surface of the theoretical sample. The reason could be: a) I run the energy-minimization program up to a greater precision. b) I used the exact expression for the two wavefunctions N=21 (x) and N=2 (x). In the continuous case, three ruler integrals yield both the integrated density of Wil- son states and the interaction terms. However, in finite size case, such ruler integrals 36 Figure 3.3: Comparison of Dr.Bergmann’s result and my result do not exist. We have to integrate them in a brute force way. Without using the ruler integral, the calculation of integrated density requires huge computational efforts. I did that with the help of parallel computing. For the calculation in Fig.(3.3), 64 processors are used and each processor takes 21 hours to finish the task. 2) 2polarization density Since integration takes so much time, we try to present the results without integration. The most informative quantity is the polarization density. However, since there is fast oscillations, It would be a mess if we show the complete plot of polarization density vs. log(). And actually we are only interested in the envelope of the polarization density function, so we search for the maximums and minimums of the function around = 2 i (i is a positive integer).(It has been found that the minimums always occur at integer 0 s and maximums occur at half integer before the theoretical boundary of the sample.) Then we plot the maximums and minimums 37 Figure 3.4: Envelope and average of the oscillations of 2p().p() is the polarization density of conduction electrons of 2p() vs log()(Fig.(3.4)). The reason of multiplying 2 is that the polariza- tion density is found to decay as 1=2. The curve in the middle indicates the average of maximum and minimum curve. For small distances, the average is close to zero. The absolute value of average increases as increases and reaches its maximum around = 2 8 , and then gradually fades away. This means that Kondo cloud is developed before the boundary of the theoretical sample. But in this case, we don’t get the exact range of cloud, i.e. the Kondo length. (Note: In my program, I used folded wavefunction, so to get correct density, result should be divided by two.) A interesting phenomenon appears if we search maximums and minimums more frequently, e.g. around = 2 i=10 (i is positive integer). We get similar curves as above but with small oscillations. The period of those oscillations is around 2 (Fig.(3.5)). We suspect that this oscillation is artificial due to the choice of Wilson state with = 2. 38 Figure 3.5: The polarization density for Wilson band with = 2 So I calculated the polarization density for N=80 and = p 2 with all other parameters remaining the same. The results are shown in Fig.(3.6). It’s shown that the period of oscillations becomes smaller, and is around the value of p 2, which confirms our suspicion. So this oscillation is of no physics interest and searching maxima and minima around integer values ofl is sufficient. We also find that the strength of s-d interaction Vsd affects the position where mini- mum of 2p() occurs. In Fig.(12),jV sd j 2 = 0:04, and minimum is atl = 8. But if changingjV sd j 2 = 0:05, the minimum is atl = 6 instead (Fig.(3.7)). My result in finite size sample Wilson found in his renormalization approach that the formation of a Kondo ground state requires that the sample size is much larger than the Kondo length K = ~v F k B T K . The physical origin of this critical size lies in the fact that the Kondo effect creates a resonance at the Fermi energy whose width is of the order of Kondo energy = k B T K . The electrons can only feel the resonance when their 39 Figure 3.6: The polarization density for Wilson band with = p 2 Figure 3.7: The polarization density forjV sd j 2 = 0:05 40 energy spacing is sufficiently small so that a wave packet with an energy width much less thank B T K can be constructed. This condition can be translated to a critical sample size, because the larger the sample size, the smaller the energy spacing, with the limiting case of continuous energy levels in an infinitely large sample. If Kondo cloud is an indication of the existence of Kondo effect, we expect that when the number of atoms decreases, Kondo cloud would gradually disappear. To study this, we can calculate the integrated kondo cloud for samples with different number of atoms. But alternatively, we can also reduce the width of resonance to reach the similar situation. According to Friedel and Anderson, the width of Kondo resonance is proportional to the strength of s-d interaction, specifically, = hv 2 sd ig 0 . So we can fix the number of atoms and reducev 2 sd to get the same effect. Since varyingv 2 sd is easier to handle than changing the number of atoms, we choose the former way. In our calculation, we fix the number of atoms to be 2048. 1) Integrated net polarization The integrated Kondo cloud from the d-spin up component has been calculated for a finite size sample using the FA Hamiltonian with variousjV 0 sd j 2 values. Three typical figures are shown here. From Fig.(3.8) to Fig.(3.10), thejV 0 sd j 2 is reduced from 0.1 to 0.02. It can be seen that the polarization curve is pushed further and further to the surface of sample asjV 0 sd j 2 is reduced. In the case withjV 0 sd j 2 = 0:02, the spin-down, total density and polarization curve collapse almost to a single curve. And they don’t change much up to 2 6 and then significantly go down. The spin up curve is almost flat, indicating that the spin up density doesn’t change much after the impurity is introduced. Obviously, in this case the polarization cloud doesn’t exist in the sample, instead it is on the edge of the sample, which means that the Kondo effect fades away. 41 Figure 3.8: Integrated Kondo cloud in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:1 Figure 3.9: Integrated Kondo cloud in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:05 42 Figure 3.10: Integrated Kondo cloud in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:02 Now we check the Kondo energy (width of Kondo resonance) and energy spacing. The Kondo energy for N=40 is shown in Table (3.1) and Kondo energy for N=50 is shown in Table (3.2). E s is the ground energy for singlet state andE t is the energy for relaxed triplet state. Table 3.1: Kondo energy for different s-d coupling strength and N=40 jV 0 sd j 2 E s E t KondoEnergy 0:1 3:63797491 3:628112742 0:009862168 0:05 3:559651547 3:55882949 0:000822057 0:02 3:522459321 3:522458973 3:4779E 07 And the energy spacing around fermi level is 0.00048. We can see that when jV sd j 2 = 0:02, Kondo energy is significantly smaller than the energy spacing. In this case, the infinitely strongly bounded singlet state can not be formed, namely, there is no 43 Table 3.2: Kondo energy for different s-d coupling strength and N=50 jV 0 sd j 2 E s E t Kondoenergy 0:1 3:637978672 3:628118743 0:009859929 0:05 3:559655383 3:558834308 0:000821074 0:04 3:546451776 3:546327655 0:000124121 0:02 3:522463655 3:522463638 1:666E 08 Figure 3.11: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:1 Kondo effect occurring. It is consistent with the disappearance of the Kondo cloud as we expected. 2) 2polarization density In polarization density figures (Fig.(3.11)- Fig.(3.14)), we can also see the difference. Kondo cloud is pushed further and further to the surface of the sample whenjV sd j 2 decreases from 0:1 to 0:02. 44 Figure 3.12: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:05 Figure 3.13: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:04 45 Figure 3.14: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:02 3.3 Conclusions The real space properties are calculated from the FAIR solution. It is found that for a FA magnetic state, there is no spin polarization in the vicinity of impurity. For the magnetic component of a singlet state, there is spin polarization within the Kondo length K , which completely screens the magnetic moment of d spin. The extension of Kondo cloud is found to depend on the strength of s-d interaction. The stronger the interaction, the smaller the range of Kondo cloud is. Finite size effect in impurity problem is investigated in the frame of FAIR method. The formalism for finite size sample is derived. The spin polarization is calculated for several differentjV 0 sd j 2 values. AsjV 0 sd j 2 decreases, the Kondo cloud is pushed further and further to the edge of the sample. At the critical value ofjV 0 sd j 2 = 0:02, Kondo cloud doesn’t exist in the sample any more. Meanwhile, the Kondo resonance width 46 is found to be smaller than the energy spacing around fermi energy, indicating that the Kondo ground state can’t be formed. All those observations lead to the conclusion that a minimum sample size is required to obtain the Kondo effect and Kondo cloud disappears as Kondo effect fade away. The choice of, the parameter of Wilson band, can affect the period of oscillations of the polarization density. 3.4 Appendix Kondo length r K = d" dk =" K " ="=" f energy in unit of fermi energy k =k=k f wave number in unit of fermi wave number d" dk = " f k f d" dk = " f k f since" =k 1 So r K = " f k f " K = 1 (" K =" f ) (2= f ) Expressr K in unit of f =2 K = r K f =2 = 1 " K where" K is the Kondo energy in unit of" f The original figure for Fig.(3.11)-Fig.(3.14) Those figures search for maxima and minima around = 2 i=10 (where i takes positive integer values). 47 Figure 3.15: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:1 Figure 3.16: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:05 48 Figure 3.17: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:04 Figure 3.18: The polarization density in finite size sample with the number of atoms to be 2048 andjV 0 sd j 2 = 0:02 49 Chapter 4 Friedel Oscillations about a Friedel-Anderson Impurity Affleck, Borda and Saleur [2] (ABS) studied the formation of Friedel oscillations in the vicinity of the Kondo impurity. It has the form Fr (r) 0 = C D r D [F ( r r K ) cos(2k F rD 2 + 2 p ) cos(2k F rD 2 )] (4.1) where D is the dimension of the system,F ( r r K ) is a universal scaling function that is the same for all D, P is the phase shift at the Fermi surface produced by the potential scattering, r K = ~ F =k B T K is the Kondo length andC D has the valueC 1 = 1=(2), C 2 = 1=(2 2 ),C 3 = 1=(4 2 ). ( P = 0 when the potential scattering is ignored.) The universal functionF ( r r K ) is found to approach 1 whenr r K , therefore the standard formula for Friedel Oscillation produced by potential scattering is recovered. However whenr r K ,F ( r r K ) approaches -1, and again the potential scattering result is recov- ered, except that the phase shift picks up an additional 2 . This extra phase shift is due to the Kondo scattering. ABS proved the universality of F ( r r K ) by both numerical renormalization-group (NRG) calculation and analytical derivation. Analytical expressions forF ( r r K ) are also derived in both limitsrr K andrr K . 50 F ( r r K ) ! 1 3 2 =[8 ln 2 (r K =r)] rr K (4.2) F ( r r K ) !1 +wr K =4r 3(w) 2 r 2 K =(32r 2 ) rr K (4.3) In this chapter, we investigate the Friedel oscillation of a set of symmetric Friedel- Anderson impurities. We use FAIR method to calculate the Friedel oscillation and con- firm thatF ( r r K ) is universal and approaches 1 asrr K and approaches -1 asrr K . And the behavior of the amplitude of the Friedel oscillation is compared to that of a simple Friedel impurity with a very narrow resonance at the Fermi level. The similarity between the two confirms the existence of the ”Kondo” resonance at the fermi level. 4.1 Friedel Oscillations of a symmetric Friedel- Anderson impurity When calculating the Friedel oscillation of the Friedel-Anderson impurity, the Wilson states with energy ratio of = 2 are too far apart to yield accurate results, as Affleck, Borda and Saleur (ABS) has already noticed. For example, the Wilson state with highest energy 3 4 represents all the states in the energy range of [ 1 2 ; 1]. It means that this state has an energy uncertainty of 1 4 . So the ratio of the energy uncertainty and the energy is about 1 3 in the case of = 2. One of the advantage of FAIR method is that we can easily sub-divide the Wilson states geometrically, i.e. using = p 2 or even = 4 p 2. However, compared to usingN = 60 and = 2, taking = p 2 results in doubling the Wilson states, which meansN = 120 now. And with = 4 p 2, the number of states become 240. If we optimize the whole basis from scratch, it takes forever. To reduce the computational time, we use an interpolation procedure. 51 The interpolation procedure works as follows: The FAIR state is expressed as a linear combination of the Wilson states with normalized coefficients. a y 0 = N1 X =0 0 c y (4.4) where c y is the creation operator of the Wilson state c . Taking N = 60 as an example, 60 coefficients are required to determine the FAIR state. The coefficients squaredj 0 j 2 gives the contribution of the state c toa y 0 . Since the Wilson state c is constructed from the original electron state ' K , the coefficients squared divided by the corresponding Wilson band width gives the average contribution of the original state ' K toa y 0 . We denote this quantity as the probability density of the FAIR statep() = j 0 j 2 =( +1 ). Denotep() =p() when < < +1 , then the integral over the energy is normalized. Z 1 1 p()d = N1 X =0 Z +1 j 0 j 2 =( +1 )d = N1 X =0 j 0 j 2 = 1 (4.5) When N is large enough, the probability density curve p() is smooth. Therefore from a finite number of points on the curve, one can get those points in between by interpolation. As the width decreases logarithmically towards the center of the band, the probability density increases dramatically. To gain better accuracy for interpolation, we take the log of the probability density. The Wilson band is subdivided by setting = p 2. Each cell is therefore divided into two cells and now 120 coefficients are needed to determine a FAIR state. We obtain the approximate log probability density of the new FAIR state by parabolic interpolation of the three nearest points on the old log probability density curve. The last step is to transform the log probability density back into the coefficients and normalize them. This procedure yields an approximate FAIR state. Optimizing the state starting from this 52 point requires much less computational effort. From the optimized FAIR state withN = 120, we subdivide the Wilson band again, and repeat the same parabolic interpolation procedure to produce the FAIR state withN = 240. In the case ofN = 240, the energy uncertainty for the largest energy level takes the value of about 0.09. All the results in this chapter are based on a number of Wilson statesN = 240. 4.1.1 Derivation of the formulas Different from the Kondo cloud calculation in chapter 3, we do not only calculate the electron density of half the solution, i.e. the magnetic component of the FA singlet state. Instead, we calculate the electron density of the whole state Eq.(5.20) SS = h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i n Y i=1 a y i" n Y i=1 b y i# 0 + h Ab y 0" a y 0# +Cb y 0" d y # +Bd y " a y 0# +Dd y " d y # i n Y i=1 b y i" n Y i=1 a y i# 0 Density of spin up To get the spin up density, apply the number density operator: b y " (x) c " (x) to the FA singlet state Eq.(5.20), where b " (x) = P i ' a i (x)a i" and' a i = P i c (x). c (x) is the wavefunction of the th Wilson state Eq.(3.2). v (x) = 2 p 2 p ( +1 ) sin x( +1 ) 2 x cos x(2 + +1 + ) 2 53 D SS b y " (x) b " (x) SS E = D A b y " (x) b " (x) A E + D B b y " (x) b " (x) B E + D C b y " (x) b " (x) C E + D D b y " (x) b " (x) D E + D A b y " (x) b " (x) A E + D B b y " (x) b " (x) B E + D C b y " (x) b " (x) C E + D D b y " (x) b " (x) D E +2 0 B @ D A b y " (x) b " (x) A E + D B b y " (x) b " (x) C E + D C b y " (x) b " (x) B E + D D b y " (x) b " (x) D E 1 C A Each term in the above expression is calculated as follows: D A b y " (x) b " (x) A E =A 2 n1 X i=0 ' a i (x)' a i (x) D B b y " (x) b " (x) B E =B 2 n1 X i=0 ' a i (x)' a i (x) D C b y " (x) b " (x) C E =C 2 n1 X i=1 ' a i (x)' a i (x) +' d (x)' d (x) ! D D b y " (x) b " (x) D E =D 2 n1 X i=1 ' a i (x)' a i (x) +' d (x)' d (x) ! D A b y " (x) b " (x) A E =A 2 n1 X i=0 ' b i (x)' b i (x) D B b y " (x) b " (x) B E =B 2 n1 X i=1 ' b i (x)' b i (x) +' d (x)' d (x) ! 54 D C b y " (x) b " (x) C E =C 2 n1 X i=0 ' b i (x)' b i (x) D D b y " (x) b " (x) D E =D 2 n1 X i=1 ' b i (x)' b i (x) +' d (x)' d (x) ! D A b y " (x) b " (x) A E = * Aa y 0" b y 0# n Y i=1 a y i" n Y i=1 b y i# b y " (x) b " (x) Ab y 0" a y 0# n Y i=1 b y i" n Y i=1 a y i# + = * Aa y 0" n Y i=1 a y i" b y " (x) b " (x) Ab y 0" n Y i=1 b y i" +* b y 0# n Y i=1 b y i# a y 0# n Y i=1 a y i# + = AA * b " (x)a y 0" n Y i=1 a y i" b " (x)b y 0" n Y i=1 b y i" +* b y 0# n Y i=1 b y i# a y 0# n Y i=1 a y i# + 55 * b " (x)a y 0" n Y i=1 a y i" b " (x)b y 0" n Y i=1 b y i" + = * X j ' a j (x)a j" a y 0" n Y i=1 a y i" X k ' b k (x)b k" b y 0" n Y i=1 b y i" + = X j X k ' a j (x)' b k (x) * a j" a y 0" n Y i=1 a y i" b k" b y 0" n Y i=1 b y i" + = X j X k ' a j (x)' b k (x)(1) j+k * a j" a y 0" n Y i=1 a y i" b k" b y 0" n Y i=1 b y i" + = X j X k ' a j (x)' b k (x)(1) j+k D a y 0" a y j1" a y j+1" a y n" b y 0" b y k1" b y k+1" b y n" E = ' a 0 ' b 0 ' a 0 ' b 1 ' a 0 ' bn D a y 1 jb y 0 E D a y 1 jb y 1 E D a y 1 jb y n E D a y 2 jb y 0 E D a y 2 jb y 1 E D a y 2 jb y n E . . . . . . . . . . . . D a y n jb y 0 E D a y n jb y 1 E a y n jb y n D a y 0 jb y 0 E D a y 0 jb y 1 E D a y 0 jb y n E ' a 1 ' b 0 ' a 1 ' b 1 ' a 1 ' bn D a y 2 jb y 0 E D a y 2 jb y 1 E D a y 2 jb y n E . . . . . . . . . . . . D a y n jb y 0 E D a y n jb y 1 E a y n jb y n + D a y 0 jb y 0 E D a y 0 jb y 1 E D a y 0 jb y n E D a y 1 jb y 0 E D a y 1 jb y 1 E D a y 1 jb y n E ' a 2 ' b 0 ' a 2 ' b 1 ' a 2 ' bn . . . . . . . . . . . . D a y n jb y 0 E D a y n jb y 1 E a y n jb y n + + (1) n D a y 0 jb y 0 E D a y 0 jb y 1 E D a y 0 jb y n E D a y 1 jb y 0 E D a y 1 jb y 1 E D a y 1 jb y n E . . . . . . . . . . . . D a y n1 jb y 0 E D a y n1 jb y 1 E D a y n1 jb y n E ' an ' b 0 ' an ' b 1 ' an ' bn 56 = n X k=0 (1) k D a y 0 jb y 0 E D a y 0 jb y 1 E D a y 0 jb y n E D a y 1 jb y 0 E D a y 1 jb y 1 E D a y 1 jb y n E . . . . . . . . . . . . D a y k1 jb y 0 E D a y k1 jb y 1 E D a y k1 jb y n E ' a k ' b 0 ' a k ' b 1 ' a k ' bn D a y k+1 jb y 0 E D a y k+1 jb y 1 E D a y k+1 jb y n E . . . . . . . . . . . . D a y n jb y 0 E D a y n jb y 1 E a y n jb y n D B b y " (x) b " (x) C E = * Ba y 0" d y # n Y i=1 a y i" n Y i=1 b y i# b y " (x) b " (x) Cb y 0" d y # n Y i=1 b y i" n Y i=1 a y i# + = * Ba y 0" n Y i=1 a y i" b y " (x) b " (x) Cb y 0" n Y i=1 b y i" +* d y # n Y i=1 b y i# d y # n Y i=1 a y i# + = BC * b " (x)a y 0" n Y i=1 a y i" b " (x)b y 0" n Y i=1 b y i" +* d y # n Y i=1 b y i# d y # n Y i=1 a y i# + D C b y " (x) b " (x) B E = * Cd y " b y 0# n Y i=1 a y i" n Y i=1 b y i# b y " (x) b " (x) Bd y " a y 0# n Y i=1 b y i" n Y i=1 a y i# + = CB * b " (x)d y " n Y i=1 a y i" b " (x)d y " n Y i=1 b y i" +* b y 0# n Y i=1 b y i# a y 0# n Y i=1 a y i# + 57 D D b y " (x) b " (x) D E = * Dd y " d y # n Y i=1 a y i" n Y i=1 b y i# b y " (x) b " (x) Dd y " d y # n Y i=1 b y i" n Y i=1 a y i# + = * Dd y " n Y i=1 a y i" b y " (x) b " (x) Dd y " n Y i=1 b y i" +* d y # n Y i=1 b y i# d y # n Y i=1 a y i# + = DD * b " (x)d y " n Y i=1 a y i" b " (x)d y " n Y i=1 b y i" +* d y # n Y i=1 b y i# d y # n Y i=1 a y i# + * b " (x)d y " n Y i=1 a y i" b " (x)d y " n Y i=1 b y i" + = n X j=0 n X k=0 ' a j " (x)' b k " (x) * a j" d y " n Y i=1 a y i" b k" d y " n Y i=1 b y i" + = n X j=1 n X k=1 ' a j " (x)' b k " (x) * a j" d y " n Y i=1 a y i" b k" d y " n Y i=1 b y i" + = n X j=1 n X k=1 ' a j " (x)' b k " (x) * a j" n Y i=1 a y i" b k" n Y i=1 b y i" + = n X k=1 (1) k1 D a y 1 jb y 1 E D a y 1 jb y 2 E D a y 1 jb y n E D a y 2 jb y 0 E D a y 2 jb y 1 E D a y 2 jb y n E . . . . . . . . . . . . D a y k1 jb y 0 E D a y k1 jb y 1 E D a y k1 jb y n E ' a k ' b 0 ' a k ' b 1 ' a k ' bn D a y k+1 jb y 0 E D a y k+1 jb y 1 E D a y k+1 jb y n E . . . . . . . . . . . . D a y n jb y 1 E D a y n jb y 2 E a y n jb y n 58 Density of spin down To get the spin down density, apply the number density operator: b y # (x) c # (x) to the FA singlet state Eqn.(5.20), where b # (x) = P i ' a i (x)a i# and' a i = P i c (x). c (x) is the wavefunction of the th Wilson state Eqn.(3.2). SS = h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i n Y i=1 a y i" n Y i=1 b y i# 0 (4.6) + h Ab y 0" a y 0# +Cb y 0" d y # +Bd y " a y 0# +Dd y " d y # i n Y i=1 b y i" n Y i=1 a y i# 0 D SS b y # (x) b # (x) SS E = D A b y # (x) b # (x) A E + D B b y # (x) b # (x) B E + D C b y # (x) b # (x) C E + D D b y # (x) b # (x) D E + D A b y # (x) b # (x) A E + D B b y # (x) b # (x) B E + D C b y # (x) b # (x) C E + D D b y # (x) b # (x) D E +2 0 B @ D A b y # (x) b # (x) A E + D B b y # (x) b # (x) C E + D C b y # (x) b # (x) B E + D D b y # (x) b # (x) D E 1 C A Each term in the above expression is calculated as follows: D A b y # (x) b # (x) A E =A 2 n1 X i=0 ' b i (x)' b i (x) D B b y # (x) b # (x) B E =B 2 n1 X i=1 ' b i (x)' a i (x) +' d (x)' d (x) ! 59 D C b y # (x) b # (x) C E =C 2 n1 X i=0 ' b i (x)' b i (x) D D b y # (x) b # (x) D E =D 2 n1 X i=1 ' b i (x)' b i (x) +' d (x)' d (x) ! D A b y # (x) b # (x) A E =A 2 n1 X i=0 ' a i (x)' a i (x) D B b y # (x) b # (x) B E =B 2 n1 X i=0 ' a i (x)' a i (x) D C b y # (x) b # (x) C E =C 2 n1 X i=1 ' a i (x)' a i (x) +' d (x)' d (x) ! D D b y # (x) b # (x) D E =D 2 n1 X i=1 ' a i (x)' a i (x) +' d (x)' d (x) ! D A b y # (x) b # (x) A E = * Aa y 0" b y 0# n Y i=1 a y i" n Y i=1 b y i# b y # (x) b # (x) Ab y 0" a y 0# n Y i=1 b y i" n Y i=1 a y i# + = * Ab y 0# n Y i=1 b y i# b y # (x) b # (x) Aa y 0# n Y i=1 a y i# +* a y 0" n Y i=1 a y i" b y 0" n Y i=1 b y i" + = AA * b # (x)b y 0# n Y i=1 b y i# b # (x)a y 0# n Y i=1 a y i# +* a y 0" n Y i=1 a y i" b y 0" n Y i=1 b y i" + 60 * b # (x)b y 0# n Y i=1 b y i# b # (x)a y 0# n Y i=1 a y i# + = X j X k ' b j (x)' a k (x) * b j# b y 0# n Y i=1 b y i# a k# a y 0# n Y i=1 a y i# + = n X k=0 (1) k D b y 0 ja y 0 E D b y 0 ja y 1 E D b y 0 ja y n E D b y 1 ja y 0 E D b y 1 ja y 1 E D b y 1 ja y n E . . . . . . . . . . . . D b y k1 ja y 0 E D b y k1 ja y 1 E D b y k1 ja y n E ' b k ' a 0 ' b k ' a 1 ' b k ' an D b y k+1 ja y 0 E D b y k+1 ja y 1 E D b y k+1 ja y n E . . . . . . . . . . . . D b y n ja y 0 E D b y n ja y 1 E b y n ja y n D B b y # (x) b # (x) C E = * Ba y 0" d y # n Y i=1 a y i" n Y i=1 b y i# b y # (x) b # (x) Cb y 0" d y # n Y i=1 b y i" n Y i=1 a y i# + = * Bd y # n Y i=1 b y i# b y # (x) b # (x) Cd y # n Y i=1 a y i# +* a y 0" n Y i=1 a y i" b y 0" n Y i=1 b y i" + = BC * b # (x)d y # n Y i=1 b y i# b # (x)d y # n Y i=1 a y i# +* a y 0" n Y i=1 a y i" b y 0" n Y i=1 b y i" + 61 D C b y # (x) b # (x) B E = * Cd y " b y 0# n Y i=1 a y i" n Y i=1 b y i# b y # (x) b # (x) Bd y " a y 0# n Y i=1 b y i" n Y i=1 a y i# + = CB * b # (x)b y 0# n Y i=1 b y i# b # (x)a y 0# n Y i=1 a y i# +* d y " n Y i=1 a y i" d y " n Y i=1 b y i" + D D b y # (x) b # (x) D E = * Dd y " d y # n Y i=1 a y i" n Y i=1 b y i# b y # (x) b # (x) Dd y " d y # n Y i=1 b y i" n Y i=1 a y i# + = * Dd y # n Y i=1 b y i# b y # (x) b # (x) Dd y # n Y i=1 a y i# +* d y " n Y i=1 a y i" d y " n Y i=1 b y i" + = DD * b # (x)d y # n Y i=1 b y i# b # (x)d y # n Y i=1 a y i# +* d y " n Y i=1 a y i" d y " n Y i=1 b y i" + 62 * b # (x)d y # n Y i=1 b y i# b # (x)d y # n Y i=1 a y i# + = n X j=1 n X k=1 ' a j (x)' b k (x) * a j# n Y i=1 a y i# b k# n Y i=1 b y i# + = n X k=1 (1) k1 D b y 1 ja y 1 E D b y 1 ja y 2 E D b y 1 ja y n E D b y 2 ja y 0 E D b y 2 ja y 1 E D b y 2 ja y n E . . . . . . . . . . . . D b y k1 ja y 0 E D b y k1 ja y 1 E D b y k1 ja y n E ' b k ' a 0 ' b k ' a 1 ' b k ' an D b y k+1 ja y 0 E D b y k+1 ja y 1 E D b y k+1 ja y n E . . . . . . . . . . . . D b y n ja y 1 E D b y n ja y 2 E b y n ja y n The total density is calculated as the sum of spin-up and spin-down densities. To obtain the Friedel oscillation, the total background density should be subtracted. As shown above, introducing both magnetic components not only doubles the computa- tional time but also the interference between the two components has to be included, which has a very complicated form, although its contribution to the total density is little. A large number of multi-electron scalar products in the form of determinants has to be calculated in order to get the interference. But fortunately, this has to be done only once and can be used for all distances. 63 4.1.2 Results We restrict our calculation to one-dimensional(1D) case and zero potential scattering (i.e. P = 0). In fact, the Hamiltonian in any dimension can be mapped into a one- dimensional model by expanding it in spherical harmonics and using the fact that only the s-wave harmonic interacts with the impurity in the case of a function interac- tion. Therefore, the treatments of the two- and three-dimensional cases are essentially the same as that of the 1D case. For zero potential scattering, the form of the Friedel oscillation Eqn.(4.1) reduces to Fr (r) 0 = 1 2r [1F ( r r K )] cos(2k F r 2 ) (4.7) DenoteA(r) = 1F ( r r K ), thenA(r) is the amplitude of the Friedel oscillation and is a universal scaling function as well. The same as in Chapter 3, the momentum is measured in units of the Fermi wave number and the position is measured in units of half wave length f =2, i.e. =r=( f =2). In these units, the Friedel oscillations have the period ”one”. We multi- ply the total density by 2 and extract the maxima and minima, take the differences and divide by 2 to get the amplitudeA(). Such a procedure is performed for a set of differ- ent parameters. The resulting amplitudesA() should be valid for all dimensions.A() is shown as the function of the logarithm of the distancel = log 2 () in the following figure (Fig.(4.1)). The right curve (stars) in Fig.(4.1) has the largestU = 1 and behaves very similar to a Kondo impurity that is discussed in Ref.[2][10]. For large distances the amplitude A() reaches the value 2; and for short distances the amplitude essentially vanishes. The transition between these two regions occurs at thel1 2 point whereA(l) = 1. For 64 Figure 4.1: The amplitudeA(l) of the Friedel oscillations for a symmetric FA impurity withjV sd j 2 = 0:03 and different values of U with E d =U=2. The abscissa is the distancel =log 2 () from the impurity where = 2r= F . U = 0:8 one obtains a similar curve as for U = 1:0, only shifted to the left. With decreasing U thel1 2 point for the amplitude moves to shorter distances. ForU = 1 andU = 0:8 the positions of thel1 2 points are at 13.1 and 9.4. If one shifts one of the curves by 3.7 then one obtains a single curve. It represents the universal curve for the FA impurity with a full magnetic moment. IfU is further decreased then the curves for A(l) are further shifted to the left and for U 0:4, they don’t reach the horizontal axis anymore. Essentially the whole environment of the impurity is filled with Friedel oscillations. For the smallest value ofU = 0:1, it is in the perturbative region while forU = 1:0 it is in the local moment region (see for example Krishna-murtha et al. [? ]). Generally the properties of the symmetric FA impurity are characterized by the parameterU=(), where = jV sd j 2 . In Table (4.1), the values of U=() are collected for different 65 examples. Our curves for the Friedel oscillations approach the local moment behavior somewhere aboveU=() = 4. Table 4.1: The effective coupling strengthU= for the different parameter of Fig.(4.1). r andS are discussed in the following text. U jV sd j 2 U= r S E 0:1 0:03 0:66 0:51 0:303 0:0532 0:2 0:03 1:33 0:67 0:354 0:0304 0:4 0:03 2:67 0:83 0:434 0:01016 0:6 0:03 4:05 0:90 0:479 0:00273 0:8 0:03 5:33 0:93 0:497 3:283 10 4 1:0 0:03 6:67 0:95 0:500 1:78 10 5 The result can be discussed further by looking at the two magnetic components of the singlet state separately. The two magnetic components have a similar form to the magnetic state, however, the FAIR states a y 0 and b y 0 are largely different. For each magnetic component, and for large effective coupling strength U=(), the sum S = (A 2 + B 2 + C 2 + D 2 ) is close to 0.5, i.e. almost normalized to 1=2, so the interference between the two components approaches zero. However, with decreasing U=(), the interference becomes more and more important, thus the sumS is consid- erably smaller than 1=2. We definer = (B 2 C 2 )=(A 2 +B 2 +C 2 +D 2 ), and as long asS 0:5, the magnetic moment takes the value= B r. The values ofr andS are also collected in Table (4.1). From Table (4.1), we may see that for U = 1, E d = 0:5 andjV sd j 2 = 0:03, each magnetic component of the singlet state is very well normalized to the value of S = 0:5. Therefore it is interesting to study their properties. In Fig.(4.2) the half of the ground state with the net d-spin down is investigated. Both the spin-up and the spin- down bands of the conduction electron gas show charge oscillations with the period one. Their amplitudes are identical within the accuracy of the calculation. At large distances they reach the valueA = 0:5. This means that the total charge oscillation of both spins 66 for both magnetic components yields the value two which agrees with the results of Fig.(4.2). However, for short distances one observes for each spin band a value of 0.05. This value appears to contradict the vanishing Friedel oscillations in Fig.(4.1). Figure 4.2: The amplitudeA(l) of the charge oscillations of the spin-up and spin-down electrons in the magnetic component of the singlet state. The reason for this apparent contradiction lies in the phase differences of the spin-up and spin-down oscillations. Taking the nearest integer values as the reference points, we calculate the deviation of the positions of the maxima and minima from the reference point , and those are the phase shifts in units of 2. Those phase shifts of the charge oscillations are plotted versus the distance for both the spin-up part and the spin- down part of the conduction electrons in Fig.(4.3). The full triangles show the phase shifts of the minima of the charge oscillations and the open triangles represent the phase shifts of the maxima. In addition the up-triangles represent the spin-up electrons and the down-triangles represent the spin-down electrons. 67 Figure 4.3: The phase relations of the charge oscillations of the magnetic half of the FA impurity withU = 1, E d =0:5 andjV sd j 2 = 0:03 as a function of the distance l =log 2 (). The full and open triangles show the positions of the minima and maxima of the Friedel oscillations for the spin-up and spin-down electron (up and down triangles) For large distances the maxima positions for both spin orientations lie at 0.75 and the minima at 0.25. Therefore the charge oscillations add constructively in this region. However for distances which are smaller than 1=2 , the maxima and minima of spin-up and down electrons are shifted by0:25 yielding a relative difference of = 0:5, corresponding to a phase difference of. Therefore the charge oscillations of spin-up and down electrons add destructively and cancel in this region. While the total charge oscillation in the magnetic half of the singlet solution cancels essentially to zero in the region of l l 1=2 , one obtains a polarization oscillation in this region. But for large distances, the the polarization oscillation cancels to zero. In Fig.(4.4) the amplitude of this polarization is plotted as a function ofl =log 2 (). We perform the same analysis for the FA impurity with U = 0:1, E d = 0:05 and jV sd j 2 = 0:03, and observe essentially two important differences. Fig.(4.5) shows that 68 Figure 4.4: The amplitude of the polarization oscillation of the magnetic component for theU = 1 FA impurity as a function of the distancel =log 2 (). (i) the phase difference between spin-up and down electrons at short distances is much less developed than in the local moment example in Fig.(4.3), and (ii) the transitional region where the phases split is shifted to much shorter distances. The fact that the phase differences at short distances is much less than is the reason that the two charge oscil- lations of spin-up and down electrons do not cancel. The Friedel oscillations maintain a relatively large amplitude even at short distances. 4.2 Friedel Oscillations of a simple Friedel impurity It had been observed earlier that the Friedel phase shifts and sum rules that were derived for non-interacting electron systems are also valid for interacting electron systems[14]. Those kind of similarities are of great help in understanding the interacting systems, many properties of which are very hard to investigate. One of such properties is the 69 Figure 4.5: The phase relations of the charge oscillations of the magnetic half of the FA impurity withU = 0:1,E d =0:05 andjV sd j 2 = 0:03 as a function of the distance l =log 2 (). The full and open triangles show the positions of the minima and maxima of the Friedel oscillations for the spin-up and spin-down electron (up and down triangles). Kondo resonance[23][21][28][20]. Regarding the opinions on the Kondo resonance, researchers are still locked in huge disagreements. Some say ”there is really no Kondo resonance”, and some others say ”yes, there is Kondo resonance, but its form is very complicated and no body can write it down”. Therefore, it is worthwhile to find out if the Friedel Oscillation of the Friedel-Anderson impurity performs similarly with the Friedel Oscillation of a simple non-interacting Friedel impurity. Fig.(4.6) shows the Friedel oscillations of a simple Friedel impurity with spin-up and down sub-bands. The resonance d-state lies at the Fermi level, i.e. E d = 0. The s-d-hopping matrix elementjV sd j 2 is varied by four orders of magnitude between 10 1 and 10 5 . There is a remarkable similarity between the Friedel oscillations of the FA impurity in Fig.(4.1) and of the Friedel impurity in Fig.(4.6). If we consider first the Friedel 70 Figure 4.6: The amplitudes of the Friedel oscillations for different Friedel resonances withE d = 0 andjV sd j 2 = 10 j withj = 1; 2; 3; 4; 5. impurity withjV sd j 2 = 10 5 then the overall shape of the A(l) curve is very similar to the A(l)-curve for the FA impurity with U = 1. Both curves approach the value two at large distances and zero at short distances. We also observed a similar A(l)- curve shifting behavior between the Friedel impurity as we increasedjV sd j 2 and the FA impurity when we reduced the value of U. For the Friedel impurity an increase in jV sd j 2 by a factor of 10 shifts the curve by log 2 (10) to the left. The two left curves in Fig.(4.6) no longer reach the value zero. This must be due to the fact that is no longer sufficiently small compared to the band width of one. The similarity with the Friedel impurity suggests that the reason here is also due to the finite band width and not due to the fact that the FA impurity withU=()< 4 is no longer in the local moment limit. For the Friedel impurity withjV sd j 2 = 10 5 the position of thel 1=2 point is atl 1=2 12:6, corresponding to a length of 1=2 = 2 12:6 6208. The product between 1=2 and 71 the resonance half-width = jV sd j 2 =2 = 1:57 10 5 is 1=2 0:1 and is universal (as long as is small compared with band width of the system). Figure 4.7: The logarithm of the half-width F for the Friedel impurity (full circles) and the logarithm of the Kondo energy for the FA impurity are plotted versus the corre- spondingl 1=2 = log 2 ( 1=2 ) values. The latter are obtained from Fig.(4.6) and Fig.(4.1) as the position whereA(l) takes the value 1. The two full lines are straight lines with the slope -1. Their vertical separation is 1.25, corresponding to a ratio of 2 1 :25 2:4 between Kondo energy and Kondo resonance half width. It should be emphasized that the amplitude A(l) saturates at the value of two for large distances only when the Friedel resonance lies exactly at the Fermi level. Varying E d either to a positive or negative value reduces the saturation amplitude. A shift by 2 reduces the saturation amplitude roughly by a factor of two. If we interpret the Friedel oscillation with a saturation amplitude of two as an indi- cator of a resonance at the Fermi level, then the similarity between the Friedel curves in Fig.(4.6) and the FA curves in Fig.(4.1) supports the interpretation that the FA impu- rity has a resonance at the Fermi energy. Here the reasoning can go as this: since the real space property Friedel oscillations are similar, then the k space property should 72 be similar as well and because we use the same dispersion relation for both cases, the distribution on the energy scale should also be similar. However, its half-width is not well known. We want to quantify this similarity and fit the half-width of the FA impurity. For this purpose we determine for each curve in Fig.(4.1) and Fig.(4.6) the distancel 1=2 = log 2 ( 1=2 ) whereA(l) takes the value 1. For the Friedel impurity we plot log 2 ( F ) as a function of l 1=2 = log 2 ( 1=2 ). That yields the full points in Fig.(4.7). And we assume that the half-width of the FA impurity is proportional to the Kondo energy (which is differently defined in different theoretical approaches). We choose for the FA impurity in Fig.(4.7) the definition of the Kondo energy by means of the susceptibility. We obtain the susceptibility through a linear response calculation described in Chapter 5, which yields E K . In Fig.(4.7) we plot the logarithm of this Kondo energy log 2 (E K ) versus l 1=2 = log 2 ( 1=2 ) for each curve in Fig.(4.1)(stars). Both sets of points for the FA and the Friedel impurity (circles and stars) lie on two straight lines with the slope of -1 and are separated by a vertical distance of 1.25 which corresponds to a factor of 2.4. If we assign to the FA impurity a resonance half-width of FA E K =2:4 then the values for F and FA fall on the same straight line with the slope of -1. Our calculations of the Friedel oscillations not only show that there is a Kondo resonance at the Fermi level but also suggest that even for relatively smallU=() the quasi-particle behavior is similar to that of a non-interacting Friedel impurity. 4.3 Conclusions In this chapter we investigate the Friedel oscillations of a set of symmetric Friedel- Anderson impurities. In the local moment limit (largeU=()) the normalized ampli- tude A(l) essentially vanishes for short distances and assumes the value 2 for large 73 distances. In this range of U=() the A(l)-curves show universal behavior because curves for differentU=() can be shifted into a perfect overlap. The physics behind this behavior of the amplitude of the Friedel oscillation is illuminated by the study of the simple Friedel impurity with a very narrow resonance at the Fermi level. For these non-interacting Friedel impurities one obtains real space Friedel oscillations which are very similar to those of the interacting FA impurities. The similarity between the Friedel oscillations of the FA and the Friedel impurity support the concept of a ”Kondo” reso- nance at the Fermi level. The position of the l 1=2 -point suggests that the Kondo reso- nance width of the FA impurity is 1/2.4 of the Kondo energy FA E =2:4. For the Friedel impurity the product of 1=2 = 2 l 1=2 and is 1=2 0:1. Our calculations of the Friedel oscillations not only show that there is a Kondo reso- nance at the Fermi level but also suggest that even for relatively small values ofU=() the quasi-particle behavior is similar to that of a Friedel impurity. It should be emphasized that the spatial behavior of the wave function and its charge and spin oscillations provides quantitative information about Kondo energy, Kondo length and resonance width of a FA impurity without the need of a magnetic field or excited states. 74 Chapter 5 Linear response to the external magnetic field 5.1 General theory of linear response to an external per- turbation In the Schroedinger picture, consider an interacting electron system, with Hamil- tonianH and wave-functionj S (t)i. The following Schroedinger equation should be satified. i~ @j S (t)i @t = b Hj S (t)i =>j S (t)i =e i b Ht=~ j S (0)i (5.1) Att =t 0 perturbation b H ext (t) is turned on, resulting in the new statej 0 S (t)i in the Schroedinger picture. i~ @ prime S (t) @t = h b H + b H ext (t) i j 0 S (t)i (5.2) Suppose j 0 S (t)i =e i b Ht=~ A (t)j S (0)i (5.3) whereA (t) =1 fortt 0 and should satisfy the following: i~ @A (t) @t =e i b Ht=~ b H ext (t)e i b Ht=~ A (t))i~ @A (t) @t = b H ext H (t)A (t) (5.4) 75 where b H ext H (t) is the external perturbation Hamiltonian in the Heisenberg picture. The solution to the Eqn.(5.4) is A (t) =1 i ~ Z t t 0 b H ext H (t 0 )dt 0 +:: (5.5) Plug Equ.(5.5) into Equ.(5.3), it yields j 0 S (t)i =e i b Ht=~ (1 i ~ Z t t 0 b H ext H (t 0 )dt 0 )j S (0)i (5.6) So the change of the state due to the external perturbation takes the form j 0 S (t)i = i ~ e i b Ht=~ Z t t 0 b H ext H (t 0 )dt 0 )j S (0)i (5.7) Given an operator b O S (t) in the Schroedinger picture. Its matrix element then has the form D b O (t) E ex = D 0 S (t) b O S (t) S (t) E = h 0 S (0)j 1 + i ~ Z t t 0 b H ext H (t 0 )dt 0 e i b Ht=~ b O S (t)e i b Ht=~ (5.8) 1 i ~ Z t t 0 b H ext H (t 0 )dt 0 j S (0)i = D 0 H (0) b O H (t) H (0) E (5.9) + i ~ 0 H (0) Z t t 0 b H ext H (t 0 )dt 0 ; b O H (t) H (0) (5.10) since H (0) = S (0) ( H (0) is the state vector in Heisenberg picture). If both states 0 S (0); S (0) are the ground-state 0 then the change in the matrix element due to the perturbation is 76 D b O (t) E = D b O (t) E ex D b O (t) E = i ~ Z t t 0 dt 0 D 0 h b H ext H (t 0 ); b O H (t) i 0 E (5.11) 5.2 Linear response to the external magnetic field for a Kondo Hamiltonian The Kondo Hamiltonian is again given by the following H 0 = ( N1 X =0 " c y " c " +E d d y " d " ) + ( N1 X =0 " c y # c # +E d d y # d # ) (5.12) +v a J h S + y # (0) " (0) +S y " (0) # (0) i (5.13) +v a JS z h y " (0) " (0) y # (0) # (0) i (5.14) And the Kondo ground state K = h Ba y 0" d y # +Cd y " b y 0# i n Y i=1 a y i" n Y i=1 b y i# 0 + h Cb y 0" d y # +Bd y " a y 0# i n Y i=1 b y i" n Y i=1 a y i# 0 If one applies a magnetic field in z-direction then the energy of electrons with their moment pointing upwards is reduced by E B = B B while those with their moment pointing down is increased by E B . Since the calculations for a Kondo impurity in a magnetic field in NRG (numerical renormalization group) generally assume that the 77 magnetic field couples only to the impurity spin, we follow this convention in our cal- culations. So the perturbation Hamiltonian due to the magnetic field has the form H ex =E B (b n d# b n d" ) (5.15) Using Eqn.(5.7), the change of the state due to the external magnetic field is j K i =E B i ~ e i b Ht=~ Z t t 0 dt 0 e iHt 0 =~ h (b n d" b n d# )e iHt 0 =~ Ba y 0" d y # +Cd y " b y 0# n Y i=1 a y i" n Y i=1 b y i# 0 + Cb y 0" d y # +Bd y " a y 0# n Y i=1 b y i" n Y i=1 a y i# 0 i The term in the square brackets yields h Ba y 0" d y # Cd y " b y 0# n Y i=1 a y i" n Y i=1 b y i# 0 + Cb y 0" d y # Bd y " a y 0# n Y i=1 b y i" n Y i=1 a y i# 0 i e iE 0 t 0 =~ which is perpendicular toj K i and can be expanded in terms of the other three solutions of the 4 4 secular matrix = 3 X =1 j ie iE 0 t 0 =~ wherej i has energyE . Introducing a factore t 0 with = 0 + for convergence reasons and movingt 0 to1 yields 78 j K i =E B 3 X =1 1 E ;0 j ie iE 0 t=~ (5.16) where (E E 0 )> 0. And the magnetic moment in units of B can be calculated as follows: the operator in the Schroedinger picture is (b n d" b n d# ). The expectation value of the momenthi is hi = 2Reh K (t)j (b n d" b n d# )j K (t)i (5.17) which yields hi = 2E B 3 X =1 1 E ;0 j j 2 (5.18) 5.3 Linear response calculation for a Friedel-Anderson impurity The Friedel-Anderson Hamiltonian is given again by the following H 0 = N1 X i=0 E i a y i" a i" + N1 X i=1 V fr (i) a y 0" a i" +a y i" a 0" +E d d y " d " + N1 X i=0 V 0 sd (i)[d y " a i" +a y i" d " ] + N1 X i=0 E i a y i# a i# + N1 X i=1 V fr (i) a y 0# a i# +a y i# a 0# +E d d y # d # + N1 X i=0 V 0 sd (i)[d y # a i# +a y i# d # ] +Ud y " d " d y # d # (5.19) 79 The approximate ground state is SS = h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i n Y i=1 a y i" n Y i=1 b y i# 0 (5.20) + h Ab y 0" a y 0# +Cb y 0" d y # +Bd y " a y 0# +Dd y " d y # i n Y i=1 b y i" n Y i=1 a y i# 0 Apply the linear response solution Eqn.(5.7), we find the change of the singlet ground state due to the external magnetic fieldH ex = (b n d" b n d# ) is j k i = i ~ e iHt=~ E B Z t t 0 dt 0 e iHt 0 =~ (b n d# b n d" )e iHt 0 =~ h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i j0 a" 0 b# i + h Ab y 0" a y 0# +Cb y 0" d y # +Bd y " a y 0# +Dd y " d y # i j0 b" 0 a# i = i ~ e iHt=~ E B Z t t 0 dt 0 e iHt 0 =~ (b n d# b n d" ) h Aa y 0" b y 0# +Ba y 0" d y # +Cd y " b y 0# +Dd y " d y # i j0 a" 0 b# i + h Ab y 0" a y 0# +Cb y 0" d y # +Bd y " a y 0# +Dd y " d y # i j0 b" 0 a# i e iE 0 t 0 =~ = i ~ e iHt=~ E B Z t t 0 dt 0 e iHt 0 =~ h Ba y 0" d y # +Dd y " d y # Cd y " b y 0# Dd y " d y # i j0 a" 0 b# i + h Cb y 0" d y # +Dd y " d y # Bd y " a y 0# Dd y " d y # i j0 b" 0 a# i e iE 0 t 0 =~ = i ~ e iHt=~ E B Z t t 0 dt 0 e iHt 0 =~ h Ba y 0" d y # Cd y " b y 0# i j0 a" 0 b# i + h Cb y 0" d y # Bd y " a y 0# i j0 b" 0 a# i e iE 0 t 0 =~ 80 h Ba y 0" d y # Cd y " b y 0# i j0 a" 0 b# i + h Cb y 0" d y # Bd y " a y 0# i j0 b" 0 a# i is perpendicular to the ground state and can be expanded in terms of the other seven solutions of the 8 8 secular matrix. (= P 7 =1 j i) To determine , calculate the scalar product between j i and h Ba y 0" d y # Cd y " b y 0# i j0 a" 0 b# i + h Cb y 0" d y # Bd y " a y 0# i j0 b" 0 a# i The coefficients of each term for both the states are shown in Table 5.1. Table 5.1: The coefficients of each term for both the original state and the change of state due to the external magnetic field Term j i h Ba y 0" d y # Cd y " b y 0# i j0 a" 0 b# i + h Cb y 0" d y # Bd y " a y 0# i j0 b" 0 a# i a y 0" b y 0# c 0 0 a y 0" d y # c 1 B d y " b y 0# c 2 C d y " d y # c 3 0 b y 0" a y 0# c 4 0 b y 0" d y # c 5 C d y " a y 0# c 6 B d y " d y # c 7 0 = h j h Ba y 0" d y # Cd y " b y 0# i j0 a" 0 b# i + h Cb y 0" d y # Bd y " a y 0# i j0 b" 0 a# i = c 1 Bc 2 C +c 5 Cc 6 B + c 5 Bc 6 C +c 1 Cc 2 B S < ##; 21>S <dd; 21> whereS < ##; 21 > andS < dd; 21 > are the multi-scalar product, the form of which is shown in section 2.3.1. So onlyc 1 c 2 c 5 c 6 are relevant here. 81 5.3.1 Two numerical examples With parametersjV 0 sd j 2 = 0:1;U = 1;E d =0:5 Table 5.2: The coefficients of the 7 other solutions to the 8 8 secular matrix for jV 0 sd j 2 = 0:1 Term = 0 = 1 = 2 = 3 = 4 a y 0" d y # c 1 0:75267 0:62915 0:14143 1:0204e 06 d y " b y 0# c 2 0:19985 0:7002 0:72332 4:1515e 06 b y 0" d y # c 5 0:19985 0:7002 0:72332 4:1515e 06 d y " a y 0# c 6 0:75267 0:62915 0:14143 1:0204e 06 = 5 = 6 = 7 3:1181e 06 0:036728 0:36098 B = 0:56536 3:4232e 06 0:4626 0:39202 C = 0:11195 3:4232e 06 0:4626 0:39202 C = 0:11195 3:1181e 06 0:036728 0:36098 B = 0:56536 Since onlyc 1 c 2 c 5 c 6 are relevant, the coefficients of the other irrelevant terms are not listed in Table 5.2. The corresponding energies ofj i are as follows (for = 0; 1;:::; 7) E 0 =3:637974655 E 1 =3:584844851 E 2 =3:105711941 E 3 =3:024669835 E 4 =2:788981432 E 5 =2:577041915 E 6 =2:411318184 E 7 =2:333550576 And the multi-scalar products are the following: S < ##; 21>=0:7416013694 82 S <dd; 21>=0:7416005859 After some algebra, one get 1 =0:774 72 2 = 0:0 3 = 0:434 43 4 = 0:0 5 =1: 014 4 10 6 6 = 0:0 7 =0:121 06 j k (t)i =E B 7 X =1 1 E ;0 j ie iE 0 t 0 =~ hi (5.21) = 2E B 7 X =1 1 E ;0 j j 2 (5.22) = E B 2 ( 2 1 E 1 E 0 + 2 3 E 3 E 0 + 2 5 E 5 E 0 + 2 6 E 6 E 0 + 2 7 E 7 E 0 )(5.23) = 23:231E B (5.24) Define the Kondo energy as k B T K = E B 4hi = 1 4 23:231 = 1: 076 1 10 2 This value is very close to the singlet-triplet energy difference 0:986 10 2 , which is our definition of Kondo energy used before. 83 With parametersjV 0 sd j 2 = 0:05;U = 1;E d =0:5 Table 5.3: The coefficients of the 7 other solutions to the 8 8 secular matrix for jV 0 sd j 2 = 0:05 Term coefficient = 1 = 2 = 3 = 4 a y 0" d y # c 1 0:69013 0:26233 0:043435 2:4195 10 7 d y " b y 0# c 2 0:07637 0:61415 0:63809 1:7592 10 6 b y 0" d y # c 5 0:07637 0:61415 0:63809 1:7592 10 6 d y " a y 0# c 6 0:69013 0:26233 0:043435 2:4194 10 7 = 5 = 6 = 7 9:4075 10 7 0:085711 0:18971 B =0:66182 1:7737 10 6 0:36393 0:31808 C =0:06736 1:7737 10 6 0:36393 0:31808 C =0:06736 9:4076 10 7 0:085711 0:18971 B =0:66182 The multi-scalar products are the following: S < ##; 21>= 0:40739036444 S <dd; 21>= 0:407390355017 And the corresponding energies ofj i are as follows (for = 0; 1;:::; 7) E 1 =3:553282091 E 2 =3:065069972 E 3 =3:033544938 E 4 =2:751985596 E 5 =2:678056735 E 6 =2:557306399 E 7 =2:521272037 84 hi (5.25) = 2E B 7 X =1 1 E ;0 j j 2 (5.26) = E B 2 ( 2 1 E 1 E 0 + 2 3 E 3 E 0 + 2 5 E 5 E 0 + 2 6 E 6 E 0 + 2 7 E 7 E 0 )(5.27) = 255:73E B (5.28) The Kondo energy is k B T K = E B 4hi = 1 4 255:73 = 9: 78 10 4 This value is very close to the singlet-triplet energy difference 8:22 10 4 , which is our definition of Kondo energy used before. 85 Chapter 6 Conclusions A new numerical method is introduced to solve the Friedel-Anderson and Kondo problems. Wilson states, which captures all the interactions with the impurity, are used, reducing countless free electron states down to several dozen. Based on Wilson states, two FAIR states are constructed to express the ground state. The whole orthoganal basis are uniquely determined by the FAIR states. By optimizing the FAIR states, as well as the coefficients, ground state energy are found. For Friedel-Anderson Hamiltonian, FAIR method yields much better results than the mean field approximation. For Kondo Hamiltonian, the groud state and first excited state energy are calculated. Their energy difference yields a good approximation for Kondo energy, which lies between two the- oretical curves: (i)E st =D exp [1=(2J 0 )] (ii)E st = p 2J 0 D exp [1=(2J 0 )]. The real space properties are calculated from the FAIR solution. It is found that for a FA magnetic state, there is no spin polarization in the vicinity of impurity. For the magnetic component of a singlet state, there is spin polarization within the Kondo length K , which completely screens the magnetic moment ofd spin. The extention of Kondo cloud is found to depend on the strength of s-d interaction. The stronger the interaction, the smaller the range of Kondo cloud is. Finite size effect in impurity problem is investigated in the frame of FAIR method. The formalism for finite size sample is derived. The spin polarization is calculated for several differentjV 0 sd j 2 values. AsjV 0 sd j 2 decreases, the Kondo cloud is pushed further and further to the edge of the sample. At the value ofjV 0 sd j 2 = 0:02, Kondo cloud doesn’t exist in the sample any more. Meanwhile, the Kondo resonance width is found to be 86 smaller than the energy spacing around fermi energy, indicating that the Kondo ground state can not be formed. All those observations lead to the conclusion that a minimum sample size is required to obtain the Kondo effect and Kondo cloud disappears as Kondo effect fade away. The choice of , the parameter of Wilson band, can add an artificial period of oscillations to the polarization density. In Kondo problem, the density of states has a resonance at the fermi level. Kondo resonance can explain many properties of the Kondo system and therefore worth study- ing. Kondo resonance was studied before. However, the mean field theory decouples the spin-up d-electron from the spin-down d-electron, therefore, the actual resonance is sug- gested to be broader than what the mean field suggests. And it is actually the Hubbard resonance not the Kondo resonance. NRG method can only calculate discrete energy levels, and they did some artificial broadening to obtain the density of states[34] [15]. We use FAIR method to calculate the Friedel oscillation of a Friedel-Anderson impu- rity with the parametersjV sd j 2 = 0:03, different values ofU = 0:1; 0:2; 0:4; 0:6; 0:8; 1:0 andE d =U=2. A doubling technique is used to construct the FAIR statea 0 andb 0 based on a Wilson band with N = 240, so that the energy levels are close enough to yield accurate results. The result confirms thatF ( r r K ) is universal and approaches 1 as rr K and approaches -1 asrr K . The results are derived for a 1D system, however, the treatments of the two- and three-dimensional cases are essentially the same as that of the 1D case. The behavior of the amplitude of the Friedel oscillation is compared to that of the simple Friedel impurity with a very narrow resonance at the Fermi level. The similarity between the two confirms the existence of the ”Kondo” resonance at the fermi level. The Kondo energy vsl 1=2 curve of a FA impurity is found to be linear and parallel to the resonance width vsl 1=2 curve of a simple Fiedel impurity. And it suggests that the Kondo resonance width is 1=2:4 of the Kondo energy obtained from linear response calculationE , FA E =2:4. 87 Bibliography [1] A.C.Hewson.TheKondoproblemtoheavyFermions. Cambridge University Press, 1993. [2] I. Affleck, L. Borda, and H. Saleur. Phys.Rev.B, 77:180404, 2008. [3] P. W. Anderson. Phys.Rev., 124:41, 1961. [4] G. Bergmann. Eur.Phys.J.B, 2:233, 1998. [5] Gerd Bergmann. Z.Phys.B, 102:381–383, 1997. [6] Gerd Bergmann. Eur.Phys.J.B, 2:233–235, 1998. [7] Gerd Bergmann. Phys.Rev.B, 73:092418, 2006. [8] Gerd Bergmann. Phys.Rev.B, 74:144420, 2006. [9] Gerd Bergmann. Phys.Rev.B, 77:104401, 2008. [10] Gerd Bergmann. Phys.Rev.B, 78:195124, 2008. [11] Gerd Bergmann and Liye Zhang. Phys.Rev.B, 76:064401, 2007. [12] P. Coleman. J.Magn.Magn.Mater., 47:323, 1985. [13] N.Read D.M.Newns. Adv.Phys., 36:799, 1987. [14] F.B.Anders, N.Grewe, and A.Lorek. A.Phys.B,Condens.Matter, 83:75, 1991. [15] Axel Freyn and Serge Florens. Phys.Rev.B, 79:121102, 2009. [16] J. Friedel. Adv.Phys., 3:446, 1954. 88 [17] J. Friedel. Can.J.Phys., 34:1190, 1956. [18] J. Friedel. Suppl.NuovoCimento, 7:287, 1958. [19] J. Friedel. J.Phys.Rad., 19:573, 1958. [20] G.A.Fiete, J.S.Hersch, E.J.Heller, H.C.Manoharan, C.P.Lutz, and D.M.Eigler. Phys.Rev.Lett., 86:2392, 2001. [21] G.Gruener and A.Zawadowski. Rep.Prog.Phys., 37:1497, 1974. [22] G.Gruener and A.Zawadowski. Prog.LowTemp.Phys.B, 7:591, 1978. [23] H.C.Manoharan, C.P.Lutx, and D.M.Eigler. Nature, 403:512, 2000. [24] Jun Kondo. ProgressofTheoreticalPhysics, 32:37, 1964. [25] M.D.Daybell and W.A.Steyert. Rev.Mod.Phys., 40:380, 1968. [26] J.H.Lowenstein N.Andrei, K.Furuya. Rev.Mod.Phys., 55:331, 1983. [27] N.E.Bickers. Rev.Mod.Phys., 59:845, 1987. [28] O.Ujsaghy, J.Kroha, L.Szunyogh, and A.Zawadowski. Phys. Rev. Lett., 85:2557, 2000. [29] P.Nozieres. J.Phys., 41:193, 1980. [30] P.Nozieres. Ann.Phys., 10:19, 1985. [31] P.Schlottmann. Phys.Rep., 181:1, 1989. [32] P.W.Anderson. Rev.Mod.Phys., 50:191, 1978. [33] J. R. Schrieffer and P. A. Wolff. Phys.Rev., 149:491, 1966. [34] K. G. Wilson. Rev.Mod.Phys., 47:773, 1975. 89
Abstract (if available)
Abstract
A compact solution consisting of 4-8 Slater states (FAIR solution) is introduced to treat the Friedel Anderson and Kondo impurity problem. The ground state energy is obtained with impressively high accuracy. Net integrated polarization density is calculated and it confirms the existence of Kondo cloud. ❧ Finite size effect in the impurity problem is studied using FAIR method. It is shown that the formation of a Kondo ground state requires a minimum sample size and is accompanied by the presence of Kondo cloud. ❧ The Friedel Oscillations in the vicinity of a Friedel-Anderson impurity are investigated by FAIR method. The development of Friedel oscillation with a phase shift of Pi/2 outside the Kondo radius is confirmed. And the amplitude of the Friedel oscillations show a very similar behavior to that of a simple non-interacting Friedel impurity with a narrow resonance at the Fermi level. This similarity supports the concept of a ”Kondo” resonance. And the Kondo resonance half width is suggested to be E/2.4, where E¬is the Kondo energy calculated from susceptibility.
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Tao, Yaqi
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Finite size effect and Friedel oscillations for a Friedel-Anderson impurity by FAIR method
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