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Spin gaps in two models of strongly correlated electron systems Sikkema, Arnold Eric 1997

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SPIN GAPS IN TWO  MODELS  OF  STRONGLY CORRELATED ELECTRON SYSTEMS By Arnold Eric Sikkema B . S c , University of Waterloo, 1991  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y  in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS AND A S T R O N O M Y  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA  January 1997 © Arnold Eric Sikkema, 1997  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Department of Physics and Astronomy The University of British Columbia Vancouver, B C Canada V 6 T 1Z1  Date:  Abstract  Two aspects of strongly correlated electron systems are studied in this thesis. Hubbard-type models are thought to be at the basis of high-critical-temperature superconductivity. One interaction which has not had much study is the nearest-neighbour hopping of on-site singlet pairs. We refine earlier renormalization group arguments and, using the density-matrix renormalization group method, numerically confirm their prediction that a spin gap opens at infinitesimal pair-hopping amplitude V > 0 in the one-dimensional tight-binding model. We also find a phase separation transition at a finite V > 0 as well as a spin-gap transition at a finite V < 0. The exotic magnetic behaviour of heavy-fermion materials involves an interplay between the screening of local moments from each other and the formation of a magnetic state of long range order. While the single-impurity Kondo Hamiltonian is thought to model some aspects of this behaviour, the properties of the Kondo lattice model away from half-filling are largely unknown. We determine the presence of a spin-gap region in the phase diagram of the one-dimensional Heisenberg-Kondo lattice model and make predictions about certain concealed dimerization order parameters.  ii  Table of C o n t e n t s  Abstract  ii  L i s t of Figures  vi  Acknowledgments  ix  Dedication  xi  1  2  3  Introduction  1  1.1  Why Study One-Dimensional Systems  2  1.2  Chains, Ladders, and Quasi-One-Dimensional Systems  3  S t r o n g l y C o r r e l a t e d E l e c t r o n Systems  8  2.1  Introduction  8  2.2  Symmetries of Strongly Correlated Electron Models  11  2.3  Bosonization: Theoretical Background  14  The Density-Matrix Renormalization Group M e t h o d  19  3.1  Introduction  19  3.2  The D M R G Procedure  21  3.2.1  Subdividing Sites To Improve Accuracy and Speed  22  3.2.2  The Infinite System Method  23  3.2.3  The Finite System Method  28  3.3  Recent Advances  29  iii  4  5  3.3.1  Low-Level Numerical Computation  31  3.3.2  Thermodynamic Limit Fixed Point Approximation  32  3.3.3  General Implementation  35  T h e One-Dimensional P a i r - H o p p i n g M o d e l  38  4.1  Introduction  38  4.2  Spin Gap and Phase Separation from the Strong Coupling Limit  41  4.2.1  Spin Gap  42  4.2.2  Phase Separation  .  44  4.3  Analytical Renormalization Group Studies of the Phase Diagram  46  4.4  Density-Matrix Renormalization Group Results  52  4.4.1  D M R G Details  52  4.4.2  Spin Gap for V > 0  52  4.4.3  Phase Transition at V - V « -1.5*  59  4.4.4  Phase Separation at V = V  59  c  cl  « 3.5t  T h e One-Dimensional Heisenberg-Kondo L a t t i c e M o d e l  61  5.1  Introduction  61  5.2  The Hamiltonian of the Heisenberg-Kondo Lattice Model  65  5.3  Relation to Experiment  67  5.4  Relation to Theories of High Temperature Superconductivity  69  5.5  Connection With the t-J Model and Ferromagnetism  70  5.6  Bosonization Analysis  80  5.6.1  The Heisenberg Ladders  81  5.6.2  The Heisenberg-Kondo Lattice Model  86  5.6.3  Concealed Dimerization Order Parameters  88  5.6.4  A Strongly Anisotropic Limit of the Kondo Lattice Model iv  . . . .  95  5.7  6  Density-Matrix Renormalization Group Results  96  5.7.1  Summary: Prediction of the Phase Diagram  96  5.7.2  Overview of Calculation Details  100  5.7.3  Determination of the Ferromagnetic State  100  5.7.4  Boundary Between Spin-Gap and Luttinger Liquid Phases . . . .  103  5.7.5  Characterization of the Spin-Gap Phase  103  5.7.6  Spin Gap Behaviour for Small Kondo Coupling  105  Conclusions  110  6.1  The One-Dimensional Pair-Hopping Model  110  6.2  The One-Dimensional Heisenberg-Kondo Lattice Model  110  Appendices  112  A F e r r o m a g n e t i s m i n the strong-coupling K o n d o lattice m o d e l  112  A.l  Zero Hopping Case  112  A.2  Small Hopping Limit  113  Bibliography  118  v  L i s t of Figures  3.1  Labeling of sites in D M R G approach  24  3.2  In the infinite system method of D M R G , we add site (or sub-site) i to the previous left block Bf to construct a new left (system) block Bf  +1  and  construct an appropriate right (environment) block 3.3  26  Starting the finite system method of D M R G upon completion of the infinite system method to form the first non-trivial new right (system) block and an appropriate left (environment) block  30  3.4  Sweeping the system-environment interface to the left in the general case.  31  3.5  Sweeping the system-environment interface to the right in the general case. 31  4.1  Third-order renormalization group flow diagrams, ignoring g±  50  4.2  Summary of numerical results for the pair-hopping model  53  4.3  Fitting the DMRG-computed spin gaps to the prediction from renormalization group  4.4  flows  55  Expectation values (S (i)) for for different values of V/t, for one electron z  added relative to half-filling. The unpaired electron delocalizes into the chain near V/t — 3.5. (The L = 60 chain is symmetric about its central link.) 4.5  56  Spin at a chain end (open squares) and net spin in the centre half of the chain (filled squares) as a function of V/t for a single added electron.  4.6  . .  57  Spin at a chain end (open squares) and net spin in the centre half of the chain (filled squares) as a function of V/t for two added electrons  vi  58  5.1  The lattice and coupling constants for the one-dimensional HeisenbergKondo lattice model  5.2  65  A typical configuration of spins and conduction electrons in a general state of the Heisenberg-Kondo lattice model. Double (single) arrows represent / (conduction-electron) spins  5.3  66  The phase diagram of the one-dimensional t-J model, determined by projecting the true ground state from a wave function determined by a variational Monte Carlo method (adapted from Ref. [58]; original figure courtesy of C. Stephen Hellberg)  5.4  75  A second version of the one-dimensional t-J model's phase diagram, based on a variational Monte Carlo method (adapted from Ref. [93]; original figure courtesy of Kenji Kobayashi). Note that the axes are interchanged from those of Fig. 5.3  76  5.5  The two-leg Heisenberg spin-| ladder  81  5.6  The two-leg Heisenberg zig-zag spin-| ladder  82  5.7  Renormalization group flow of the marginal interchain coupling Ji in the zig-zag spin ladder  83  5.8  The dimerization found in the two-leg Heisenberg spin-| zig-zag ladder. .  84  5.9  Phase diagram of the one-dimensional Heisenberg-Kondo lattice model at a particular filling which has a spin gap for J  K  —> oo  97  5.10 Reducing the filling factor by a small amount will adjust the phase boundaries from the solid lines to the dashed lines  98  5.11 Phases in the H - K L M for a filling which does not give a spin gap in the limit J K —• oo  99  5.12 The spin gap in the Heisenberg-Kondo lattice model along J function of J H shows a transition near J vii  H  = O.Qt  K  = 2t as a 104  5.13 Quadratic fit of ground state energies for a 32-site chain, at (JK,JE) — (32, 1.652), as a function of charge.  Residuals, together with numerical  uncertainties, are plotted on the greatly-expanded right-hand scale.  . . . 106  5.14 Quadratic fit of ground state energies for a 64-site chain, at (JK,</H) = (32,1.652), as a function of charge.  Residuals (together with numerical  uncertainties) are plotted on the right-hand scale  107  5.15 Testing the s m a l l - J bosonization prediction for two values of J H by plotK  ting ln(A /2) versus t/J s  A.l  109  K  Spectrum of the Kondo term for a single site. Here double (single) arrows represent / (conduction) electrons; for example | -frfl) = /{c|cj|0}.  viii  . . . 113  Acknowledgments  I would like to thank my graduate advisor, Ian Affleck for his dedication, support, and guidance for the past five years. I thank Steven White of the University of California at Irvine for extensive guidance in preparing a general advanced D M R G implementation. Stephen Hellberg, a postdoctoral fellow first at Florida State University and later at the Naval Research Laboratory, shared a multitude of insights with me from his study of the t-J  model; he was always  willing to address my questions, and did so with unbelievable promptness. I have benefited from discussions at U B C with, among others, Erik S0rensen, Jacob Sagi, Sebastian Eggert, Junwu Gan, Masaki Oshikawa, Alexandre Zagoskin, Mehrdad Sharifzadeh, Victor Barzykin, and Jeff Sonier. Bill Minor, a Ph.D. candidate at McMaster University, and I sharpened one another with our experiences dealing with correlated electron systems and D M R G . Samuel Moukouri, a postdoctoral fellow at l'Universite de Sherbrooke, kindly sent me some data to allow a verification of the correctness of my D M R G code on the Heisenberg-Kondo lattice model. Ron Parachoniak and Mary Ann Potts were always willing to help me in various ways with computer support. Several people during the course of my education stand out in my mind for contributing to my love of science. Thank you, Henry Plantinga of Emmanuel Christian High School, for wonderful science classes pointing out the beauty of God's creation, as a result of which I planned to become a botanical illustrator. Thank you, Hans VanDooren, also of Emmanuel, for inspiring me to study physics and entertaining my quest to prove that objects with zero total force cannot possibly move. Thank you, Dune Lamond of Centre Wellington District High School, for deepening my love of physics by  ix  showing yours. Many University of Waterloo professors—among them Robb Mann, A . D . Nagi, Ken Woolner—I thank for many hours of instruction and discussion during my undergraduate years. During the course of my studies and research, I have been blessed with many and various brothers and sisters who have encouraged and edified me in my academic endeavours. I thank God for interactions with John & Margaret Helder, Tony Jelsma, Jitse van der Meer, James Wanliss, Henry Sikkema, Wes Bredenhof, Bert Moes, Willie ten Haaf, Rienk Koat, Jeff Watts, Peter & Sarah Vandergugten, Rick Baartman, Dave DeWitt, Ken Kiers, and many other members of the Reformed community in Canada, especially in Langley. I thank my parents, Klaas and Ina Sikkema, for encouraging me to use every opportunity for post-secondary education, and for supporting me throughout. I thank my wife, Valerie, for everything she has meant to me increasingly; without her loving ministry of support, encouragement, understanding, and patience, life and the completion of this work would have been more difficult. I also thank our two wonderful children, William and Kristin, for their patience as well as their impatience as they faced me with the reality of what is of fundamental importance in this life. I acknowledge with gratitude the financial support of N S E R C (NSERC 1967 Science and Engineering Scholarship, 1991-95 plus NSERC-funded research assistantships, 199596) and the Izaak Walton Killam Memorial Foundation (Killam Predoctoral Fellowship, 1995-96). Finally, I thank God, whom I know as creator, sustainer, and redeemer of the cosmos, for transcending the range from neutrinos to galactic clusters to give hope to those who trust in Him.  Soli Deo Gloria  xi  Chapter 1  Introduction  In this thesis, we study two models of strongly correlated electron systems. These systems are of a general class for which a simple treatment in terms of nearly free electrons must be replaced by theoretical analyses which account for the observed variety of phases exhibiting striking charge and/or magnetic properties. The models studied herein are the one-dimensional pair-hopping and Heisenberg-Kondo lattice models. Both of these models are related to some degree to the question of high temperature superconductivity on which there has been an enormous effort in the past decade; the latter is also relevant to heavy-fermion systems and their exotic magnetic and superconducting behaviours. This overall introduction sets the stage by briefly discussing the importance of the theoretical study of one-dimensional systems and particularly spin gaps in the context of the quasi-one-dimensionality of weakly-interacting chains. Each of the next four chapters of this thesis has a more specific introduction as well. Strongly correlated electron systems are outlined in Chapter 2 to establish the theoretical background for later chapters. The numerical approach used in this thesis, namely the density-matrix renormalization group method, is explained in detail in Chapter 3. The following two chapters, namely Chapter 4 on the pair-hopping model and Chapter 5 on the Heisenberg-Kondo lattice model introduce the Hamiltonians of the models studied in this thesis, placing them in the relevant experimental and theoretical contexts, and describe in detail our analytical and numerical calculations.  1  Chapter 1.  1.1  2  Introduction  W h y S t u d y One-Dimensional Systems  It may seem to someone outside of the field of one- or two-dimensional electronic systems that working in one dimension is completely irrelevant to three-dimensional reality. However, there are in fact many reasons for working in lower-dimensional physics, in both condensed matter theory as well as other fields. First, restricting study to one dimension continues to demonstrate itself as an efficient and effective laboratory for the development and consideration of new theoretical ideas intended for application to real higher-dimensional systems. Both thermal and quantum fluctuation effects are larger in lower dimensions. Based on an inequality due to Bogoliubov [1], the Mermin-Wagner-Hohenberg theorem [2, 3] (which holds for classical and quantum systems) shows that there is no finite-temperature phase transition in either the one- or two-dimensional Heisenberg model with short-range interactions. Later, Coleman [4] showed that for quantum systems continuous symmetries in one dimension cannot be spontaneously broken even at zero temperature. No long range order is possible: ordered states are destabilized by quantum fluctuations due to the presence of infra-red divergences connected with Goldstone bosons. Thus the one-dimensional case provides an arena in which the physics is as non-classical as possible. Secondly, there are in fact real systems which demonstrate a high degree of lowdimensionality, at least in condensed matter physics. For example, Ni(C2H N2)2N021  8  (C10 ), known as N E N P , is well described as a Heisenberg spin-1 chain with interchain 4  spin-exchange coupling on the order of 10~ times its intrachain coupling [6]. In addition 4  to organic spin chains such as these, there is a well-studied class of quasi-one-dimensional organic conductors and superconductors [7]. Recent technological advancements in areas such as molecular beam epitaxy and lithography have also allowed the fabrication of 1  Lower-dimensional cosmology [5], for example, does not have a similar luxury accessible.  Chapter 1.  3  Introduction  one-dimensional "quantum wires" [8, 9]. In these cases, excitations involving two perpendicular directions are effectively suppressed quantum mechanically at sufficiently low temperatures by geometrically, compositionally, or electrostatically imposed restrictive potentials [8, 9]. One can then legitimately concentrate on those in the single spatial dimension of the chain. Thirdly, one-dimensional systems share the possibility with higher-dimensional systems of undergoing "quantum phase transitions" as some parameter is varied while holding the temperature fixed at T = 0. In experimental situations, this parameter could be pressure or doping, for example, such as in the case of high temperature superconductors where, upon the variation of doping, the ground state ranges from antiferromagnetic to superconducting. While there could always be new experimental realizations of a model to expand the accessible range of parameters, theoretical physicists enjoy the luxury of being able to vary at will any coupling constants, filling factors, doping levels, and any number of other variables. This is precisely what we do in this thesis: after selecting Hamiltonians to model certain types of behaviour, we consider possible quantum phase transitions as the model parameters are varied.  1.2  C h a i n s , Ladders, a n d Quasi-One-Dimensional Systems  Recently, a class of materials closely related to the high temperature superconducting cuprates has been observed to contain ladder structures, as recently reviewed by Dagotto and Rice [10]. A n w-leg ladder consists of a set of n infinite chains coupled together, usually by perpendicular rungs. For example, (VO)2P20 (vanadyl pyrophosphate), SrCu203, and L a 2 C u 0 all con7  2  5  tain 2-leg Heisenberg spin-| ladders [11, 12, 13]. In fact, strontium-copper-oxide has a  Chapter 1.  4  Introduction  rich stoichiometry-dependent structure: Sr„Cu i0 ra+i has n-leg ladders. The copperw+  2  oxide compounds mentioned here have rungs composed of C u — O — C u  O—Cu,  while in the 2-leg ladder vanadyl pyrophosphate, the vanadium (V) atoms are connected by two neighbouring paths through one oxygen each. Modelling these as Heisenberg spin\ ladders, when considering them as half-filled t-J ladders, one can dope the systems to produce t-J ladders and possibly Hubbard ladders.  2  If one's goal is to study a two-dimensional lattice, it turns out that one cannot do this by a simple one-leg-at-a-time procedure [10]. In fact, any ladder with a finite number M of legs, each of whose sites has d degrees of freedom, is exactly identical to a single chain having a Hilbert space of dimension d  M  at each site. That is, any finite ladder is just a  single chain with a likely-more-complicated Hamiltonian. For example, the ground state of the Heisenberg spin-| n-leg ladder is probably in the same phase as that of the spin-n/2 Heisenberg chain, which, as Haldane conjectured [14], is gapless only for n odd. (This has been demonstrated numerically for n = 2 taking the interchain coupling to range through all ferromagnetic and antiferromagnetic values while the intrachain coupling remains antiferromagnetic [15], and is also an exact statement for all n > 1 with an infinite ferromagnetic interchain coupling. In this latter case, each rung simply forms a fully symmetric (ferromagnetic) ground state with maximal spin n/2 and the intrachain couplings simply reproduce the single-chain Heisenberg couplings for that spin.) That having been said, the true two-dimensional system can be considered to be the limit of an n-leg ladder in cases where such a limit is defined appropriately. In the case of the Heisenberg ladders, while the ladder alternates between being gapped and gapless, the gaps tend to zero for large n, so that the two-dimensional Heisenberg lattice is in fact gapless and can be seen as the limiting case of n-leg ladders. T h e widely-studied Heisenberg, Hubbard, and t-J models mentioned in this paragraph are defined in Eqs. (5.35), (5.70), and (5.23) respectively. 2  Chapter 1.  5  Introduction  Thinking of ladder systems and chain systems interchangeably proves to be particularly useful when studying the one-dimensional Heisenberg-Kondo lattice model. Here we can think of the localized spins as constituting one chain and the conduction electrons as constituting another. These two chains are then coupled via Kondo-interaction rungs. Details of this analysis, particularly in terms of bosonization, are given in Chapter 5. The other model Hamiltonian studied in this thesis, namely the one-dimensional pairhopping model, can also be thought of as two chains of identical free spinless fermions with correlated hopping; however, this idea is not developed further in this thesis. There is, however, an important point to note about ladder systems involving fermions which have interchain as well as intrachain hopping. This is that statistics becomes more important, since in ladders particles can interchange positions without having to pass through each other [16]. In our cases of the pair-hopping and Heisenberg-Kondo models, this does not apply because there are no fermion hopping interaction terms on the ladder rungs. It should be noted, though, that statistics are not completely unimportant in even one-dimensional systems, since in condensed matter systems, the electrons which exist in the presence of a chain of ions embedded in a three-dimensional space certainly "know" they are fermions and can in fact interchange positions without difficulty [17]. Now if we had completely one-dimensional systems one could never observe phase transitions since they would always occur at zero temperature; however, due to the always-present interchain interactions, these are often raised to a finite critical temperature. A phase transition would be signaled by the divergence of a susceptibility, which can be computed by a mean-field-theoretic treatment of the interchain couplings to be  ^ - i ^ b f j -  ( L 1 )  where g is an interchain coupling constant and XID{T) is the temperature-dependent (and possibly exact) susceptibility under consideration for the one-dimensional chain. For this  Chapter 1.  6  Introduction  to diverge at some critical temperature T , we must have  3  c  T  (1.2)  lim S X I D ( T ) = 1,  where the interchain interaction is written in the general form (1.3) with Oi being an operator on chain i — 1,2. This must be true for the particular value of interchain coupling found in the system under study; however, as we are considering quasi-one-dimensional systems, it would be beneficial for it to apply for all g —• 0. Then one can consider the purely one-dimensional system to exhibit a finite-temperature phase transition in this limiting sense. For the above mean-field criterion to apply for all g —> 0, we require that the susceptibility diverge as T —> 0. For example, if xw ~ T  p  p < 0, then T ~ c  where  That is, for any non-zero g, there is a finite critical temperature  in the quasi-one-dimensional system. While the superconductivity of the cuprates continues to evade simple explanation, it is widely believed that an important aspect of the mechanism is that it involves weaklycoupled planes.  Similarly, to determine if superconductivity is favoured in the quasi-  one-dimensional case of weakly-coupled chains, while it certainly does not occur in true one-dimensional systems, one must compute the one-dimensional pair susceptibility which can be obtained by a Fourier transform of  4  (1.4)  XlD ,pair Let d be the scaling dimension of the pair operator 5  V'LV'R-  AS  the Fourier transform  introduces two more dimensions, the pair susceptibility becomes XiD, air(7 ) ~ T where 1  p  P  F o r an example of the application of this mean-field criterion to a spin system, see Ref. [18]. T h e operators ipL,R are for left- and right-moving electrons and will be introduced in Chapter 2; at this stage, the precise meaning is unimportant to this dimensional argument. A n operator O(x) is said to have scaling dimension d if its correlation functions decay like -2d {&(x)0{y))~\x-y\ 3  4  5  Chapter 1.  Introduction  7  p — 2d — 2 so that to satisfy the mean-field criterion of Eq. (1.1) in the weak-coupling limit, we require d < 1 [7, 19, 20]. To compute the scaling dimension d of the pair operator [21], we note that as a consequence of spin-charge separation any operator O can be expressed as a product O = OO s  (1.5)  c  of a spin operator and a charge operator (as indicated by the subscripts s and c). Thus the scaling dimension can be written as d = d + d. s  (1.6)  c  If the spin sector is gapless, the spin dimension would be d & \ [22]. On the other hand, s  if there is a gap in the spin sector, the spin operator may have a non-zero expectation value in the ground state, (O ) ^ 0, so that the spin dimension contribution to Eq. (1.6) s  may be taken to be d = 0. Thus having a spin gap in the one-dimensional system reduces s  the net scaling dimension of the pair operator d, greatly increasing the possibility that the mean field criterion d < 1 will be met, and resulting in the favouring of a superconducting instability. It is fitting at the close of this introduction to briefly mention a recently proposed mechanism of high temperature superconductivity put forward by Emery, Kivelson, and Zachar [23]. The formation of spin gaps in weakly-interacting self-organized onedimensional chains constitutes the basic ingredient of this model, which will certainly attract much attention on many fronts. It is certainly too early to determine the applicability or viability of this controversial model; however, we mention it here since it is undoubtedly a very interesting and rich model with several connections to those studied in this thesis. chapters.  These points of contact and comparison will be discussed in following  Chapter 2  Strongly C o r r e l a t e d E l e c t r o n Systems  Any survey of conferences or publications in the past five years will demonstrate that one of the most active area of research in condensed matter physics is that of correlated electron systems. This interest is in large part due to the decade-old discovery of highcritical-temperature superconductors and their continued frustrating denial of adequate theoretical understanding. While this class of materials and their various phenomena and propositional explanations are not directly subjects of this thesis, an important class of lattice models is, namely those of the tight-binding Hubbard type. In this thesis I will be studying two models of correlated electron systems, namely the pair-hopping model and the Heisenberg-Kondo lattice model. As these and most other strongly correlated electron models have in common their treatment of conduction electrons by a tight-binding model, it is important to clarify some points of notation and terminology upon the introduction of the physics of such systems. This chapter will also serve to introduce some of the theoretical background for following chapters' analyses of these models.  2.1  Introduction  Many condensed materials lend themselves, in the study of their electronic properties, to a classification as conductors, semi-conductors, or insulators. This categorization is often accurately modeled according to a one-electron band theory. Upon the computation of the one-electron dispersion relation, one arrives at continuum bands of one-electron energies 8  Chapter 2. Strongly Correlated Electron Systems  9  separated by band gaps. The electronic ground state is then considered dependent upon the number N of electrons participating in the highest occupied band (as determined by the valency of the lattice atoms) only insofar as it is a Slater determinant in which the lowest N one-electron states are occupied. A material modeled in this way is, then, a conductor if this band is only partly full since charge excitations are possible with infinitesimal energy, or an insulator if this band is completely full since charge excitations are suppressed by the presence of a gap to the next available band. Semiconductors are a special case of insulators in which the band gap is small enough to be only one to two orders of magnitude greater than room temperature.  Hole-doped (or electron-doped)  semiconductors have an almost-full (or almost-empty) band and are thus special cases of conductors with low conductivity. As is always the case, creation presents us with many exceptions which defy simple explanation. In this case, some materials which are expected to be metals are actually insulators, and others have surprising magnetic properties.  One-electron band theory  breaks down and gives way to correlated, or strongly correlated, electronic models: the interactions between the electrons themselves, which are by construction neglected in band theory, play the crucial role in the determination of the ground state. One can no longer speak of a band of electrons since the addition or removal of one electron will completely alter the ground state of the system. Despite this drastic revision necessary to describe strongly correlated electron systems, one aspect of the one-electron terminology continues to survive in common usage, namely the concept of band-filling. Since there is no longer necessarily a band in the sense of charge and spin excitations defining a continuum of energy eigenstates, the band filling fraction is meant only in the sense that the number of electrons under consideration corresponds to that filling in a one-electron, non-interacting, picture. That is, it becomes simply a shorthand to indicate that the number of conduction electrons is a certain  10  Chapter 2. Strongly Correlated Electron Systems  fraction of 2L, where L is the number of unit cells each capable of contributing or housing, in principle, up to two conduction electrons of opposite spin because of the Pauli exclusion principle. The tight-binding model used in the description of the kinetic contribution of the conduction electrons in strongly correlated electron systems to the full Hamiltonian is H = - i £ clc <*i)ot  + h.c,  j(T  where c\ (c ) is a creation (annihilation) operator for an electron of spin a =f>l a  (2.1) m  a  ia  Wannier orbital at lattice site i and (ij) denotes nearest-neighbour pairs. In the sum, bonds (ij) are summed over only once each. Here t is the matrix element for tunneling from one lattice site to a neighbouring one, i.e. the overlap of nearest-neighbour electron wavefunctions. The letters "h.c." denote the Hermitian conjugate of the immediately preceding term. The fermionic operators obey the anticommutation relations {CiaAf*} = * <#  (-)  {c ,c,7,} = 0.  (2-3)  6  6  2  2  and te  From the basic creation and annihilation operators c, it is helpful to construct operators representing spin and charge. The charge (or number) operator at site i is ni = Y^ i*>  (2- )  n  4  where  is the number operator for electrons of spin o at site i. The conduction electron spin operator at site i is £ = E L^a/3C a/3 c  i / 3  ,  (2.6)  11  Chapter 2. Strongly Correlated Electron Systems  where 1  0  a—  (2.7)  are the Pauli matrices.  2.2  S y m m e t r i e s of Strongly C o r r e l a t e d E l e c t r o n M o d e l s  In the thermodynamic limit of a crystal, there is by definition symmetry for translations which are integral linear combinations of the primitive translation vectors which define the unit cell of the lattice. While in band theory one uses translational symmetry immediately by considering Bloch waves as a basis for one-electron wave functions, this type of assumption may actually hinder the calculation of ground states of strongly correlated electron systems [24]. Often the important local details of the model, considered using basic operators like c\ , such as the relative number of electrons on neighbouring sites, are a  more fundamental to the system's behaviour. Having said that, it is still often helpful to consider translationally-symmetric states even in the finite lattice case, and this is done by assuming the system to be periodic in each of its dimensions. While many numerical studies of strongly correlated systems (such as the exact diagonalization and quantum Monte Carlo techniques [25]) work well in the periodic case, we use open (free end) boundary conditions since the numerical method of the density-matrix renormalization group employed in this thesis works best in this case. Many strongly correlated electron models, including those studied in this thesis, exhibit conservation of total spin quantum number, total spin in the z-direction, and total charge, i.e. [H,S ] = [H,S*] = [H,N} = 0, 2  (2.8)  12  Chapter 2. Strongly Correlated Electron Systems  where S  =  (2.9) i  N  =  (- ) 2  10  In addition, these operators also commute with one another, i.e. [S ,S } = [N,S } = [N,&}=0, z  2  (2.11)  z  so that the eigenvalues of H, N, S , and S are simultaneous good quantum numbers, z  2  which we will denote simply by E, N, S , and S(S + 1) respectively. (The Heisenbergz  1  Kondo lattice model which we study in Chapter 5 involves localized / electrons, represented by spin-1 operators Si, interacting with the conduction electrons; in this case instead of Eq. (2.9) we have S — J2i{$i + Si).) In the numerical treatment of strongly correlated Hamiltonians, we consider eigenstates which simultaneously diagonalize H, JV, and S . We do this by choosing to work in a representation in which the charge N and z  spin S are always diagonal, selecting a sector of Hilbert space with particular eigenvalues z  of N and S , and diagonalizing the Hamiltonian in this particular sector. The total spin z  quantum number <S is much more difficult to specify, and also quite difficult to measure, 2  since in terms of the fundamental operators c^ it is extremely non-local: <^  =  ,  ~  \o.\ aP ip \' \ '^6 j6':  C  Cr  C  C  0  1  C  (2.12) (2.13)  aijaPfS  where i and j both run throughout the entire lattice, i.e. they do not just represent nearest-neighbour terms. Besides these conservation laws, strongly correlated electron models share with many other physical models some basic transformations under the symmetries of space such as lt will be clear from the context whether by S or N we mean the operator or its eigenvalue.  l  z  13  Chapter 2. Strongly Correlated Electron Systems  reflection and rotation. In cases in which choosing the direction of quantization in the z-axis is arbitrary, i.e. in which there is full rotational symmetry, the eigenvalues of H, N, and S  2  will be independent of the eigenvalue of S ,  which may take on any value  z  ranging from —S to S. This can be quite useful in determining the total spin quantum number S of the ground state if it is only possible to specify the projection S  z  in the  method of investigation. Let us denote by E(S ) Z  projection S . z  the ground state energy in the sector specified by the spin  If E(S) < E{S + 1) and E{S) = £(<S - 1) = • • • = E{S ), min  |,  where  S integer  (2.14)  0, S half-odd-integer, then the absolute ground state (i.e. that in the Hilbert space unrestricted by the specification of S ) contains a state of spin S and no states of any higher spin. z  Strongly correlated electron models constructed from the starting point of the tightbinding model also often exhibit a "particle-hole" symmetry. This symmetry relates the creation of an electron to its destruction in the following way. Consider the transformation PH : c  J(T  -  (2.15)  Under this transformation, the nearest-neighbour hopping terms transform according to PH : 4 c  i + 1 )  , + h.c.  -c c\+i,a + h.c. = cj ia  +1>(r  c  J(T  + h.c. = c U i , < , + h-c, i+  (2.16)  i.e. they remain unchanged under this transformation. However, the number operators transform according to PH : n  ia  = c\ c  c c\ = 1 - c\ c  PH:ni  2-Hi  (2.18)  PH : N  2L-N,  (2.19)  a  ia  irr  a  a  i<7  = 1- n  icr  (2.17)  14  Chapter 2. Strongly Correlated Electron Systems  and similarly the conduction electron spin operator transforms according to PH:£ PH : 5 =  y  = (s?,s?,sn  S ,S\-S )  {S ,S ,S ) x  y  (2.20)  ii X  z  Z  = RS  (2.21) (2.22)  PH : S  Here R represents a rotation by TT about the y-axis. If this particle-hole symmetry applies to the full Hamiltonian, i.e. not only to its non-interacting kinetic part as shown here but also its interaction terms, then there will be a one-to-one correspondence between eigenstates with quantum numbers (E, S, S ,N) Z  and (E, <S, — S , 2L — N). In particular, z  if we wish to determine the properties of a system away from half-filling, we need only do this below half-filling and the physics above half-filling will be identical.  2.3  Bosonization: T h e o r e t i c a l B a c k g r o u n d  A very powerful tool in the study of one-dimensional systems such as spin chains and strongly correlated electron systems is bosonization. This is a procedure in which we rewrite operators in fermionic systems in terms of bosons to allow a simpler understanding of the low-energy physics involved. In one system we study (the pair-hopping model of Chapter 4) we may proceed directly to the bosonization, but since the other (the Heisenberg-Kondo lattice model of Chapter 5) involves what may be thought of as the coupling of a spin-| chain and a conduction-electron chain, our starting point will be the fermionization of the spin chain [26]. Since each site of a spin-| chain has only two degrees of freedom, it may be represented by a spinless fermion, the presence of which may be taken to be spin up, and absence representing spin down. To ensure that the commutation relations i6ij ^  .abc gc  '  e  c  (2.23)  Chapter 2. Strongly Correlated Electron Systems  15  are properly obtained, the Jordan-Wigner transformation S?  = 1>Hi~\  S~  =  (2-24)  ( - l ^ e x p ^ ^ M - j  must be employed. The generally-anisotropic Heisenberg Hamiltonian #  = £ J StS?  = Y, [-\J{StS-  a  H  +1  +1  ia  (2.25) 2  + h.c.) + J'SfS^]  (2.26)  - 1)^1 i ^ + i - §)] •  ( -27)  i  may then be written as H  = E  [~\J(4A I  + h.c.) + J {4^i Z  +  2  i  The J  z  = 0 case is simply free spinless fermions on a lattice, which have a dispersion  relation e = -Jcosk,  (2.28)  t  (we take the lattice spacing to be a = 1) so that the ground state is a Fermi sea with fc = 7r/2; all states with k < \k \ are occupied. We now proceed to the continuum limit. F  F  Considering only low-energy, long-wavelength excitations, we may write ipi* « e ^ ' ^ x ) + e - ^ ^ M z ) ,  3  (2.29)  where V^R are slowly varying, and indicate left- and right-moving components respectively; we have also generalized to the case of fermions with spin indexed by a, since we will be considering such systems which also have a Fermi sea as their non-interacting ground state. The expression Eq. (2.29) is valid for any kp, not only 7r/2; in particular we T h e sign of the X Y term has been reversed in the second equality for convenience by making a ir rotation about the 2-axis for every second site. W e employ throughout the convention of using variables names like x and y for the continuum representation of lattice positions i and j. 2  3  16  Chapter 2. Strongly Correlated Electron Systems  will use it for the situation away from half-filling in the Heisenberg-Kondo lattice model. The continuum limit has the non-interacting Hamiltonian H — ivp^2  _d_  dx  (2.30)  dx  dx  The relation between Bose and Fermi fields in one dimension is [26, 22, 27] ip (x)  oc exp[-iV4TT(p „(x)}  (2.31)  ipKvix)  oc exp[+iv 47T0R (a:)],  (2.32)  LtT  L  /  <T  where the constant of proportionality is cutoff-dependent.  The left- and right-moving  boson fields are written in terms of a boson field <f> and its dual 4> according to  4>Ux) = \[<j>„{x) + Mx)] <PR*(X)  = WM^I-Mx)].  (2.33) (2.34)  The dual field is Mx) where n  a  =J  X  dx U^x'),  (2.35)  1  is the field conjugate to 4><r obeying [<b {x\ Jl^x')] = i8 ,6{x - x ). 1  0  a(T  (2.36)  The free fermion Hamiltonian of Eq. (2.30) may now be expressed as a free boson Hamiltonian ^ y E / ^ f t ) ,  (2.37)  where  H t\x) {  EE Ul(x)  + (J:M*))  (2-38)  is the general dimensionless free boson Hamiltonian density. Including interactions between the fermions, such as we have in the case of J ^ 0 in the fermionized Heisenberg z  17  Chapter 2. Strongly Correlated Electron Systems  spin-^ chain, turn out to simply renormalize the boson fields according to *  ^ " / T ^  4>  4>-V^R,  ( 2  -  3 9 )  (2.40)  leaving a free boson Hamiltonian. The values of R and of v depend on interactions; in F  the Heisenberg spin-| chain case, the Bethe ansatz gives  = JS-S*  R  (2  '  41)  where cos# = J ^ / J . We will have occasion to use the values R = l/y/2-K and w = 7rJ/2 F  which apply for the isotropic antiferromagnetic (J = J > 0) Heisenberg spin-^ chain. z  In the case of a model involving spin-| fermions, one may define spin and charge bosons via *  =  ^ T T  ( 2  4  3  )  in which case the Hamiltonian becomes the sum of a charge and a spin Hamiltonian  H= fjdx V  «<£>(*) + I Jdx  (2.45)  In one dimension, the spin and charge degrees of freedom can always be separated in this way in the full interacting Hamiltonian. Two basic symmetries are respected by the continuum free fermion theory. There is a U ( l ) charge symmetry VU*) -  ^  ( a : )  VUz)  (2.46)  and an SU(2) spin symmetry iPUx)^E J( )^Ax), G  x  (2.47)  18  Chapter 2. Strongly Correlated Electron Systems  where G is an SU(2) matrix. These symmetries are chiral in that these transformation may be made independently on the left- and right-moving fermions. currents corresponding to these symmetries are a charge current J (x)=  4  £:^(z)VUz):  L  The conserved  (2-48)  cr  and a spin current MX)  ^1°(X)IOJTP (X),  =  LT  (2.49)  CT and similarly for L—>R. We will often be able to express operators for the Hamiltonian of interest in terms of these currents. As can be seen from the fermion-boson transformations, all physical operators are invariant under shifting <p -> cp + 2irR 4> -»• (p+l/R.  (2.50) (2.51)  This implies that the boson field 0 is to be thought of as a periodic variable, measuring arc-length on a circle of "boson compactification radius" R.  The charge and spin currents can also be thought of as charge and spin densities.  Chapter 3  The Density-Matrix Renormalization G r o u p M e t h o d  3.1  Introduction  In theoretical condensed matter physics and particularly in the study of strongly correlated electron systems, an important problem is to understand the nature of the ground state for a given Hamiltonian. The system could, for example, be insulating, superconducting, ferromagnetic, or antiferromagnetic, have only short-distance correlations, or exhibit some combination of these characteristics. In addition, it is important to understand something about the low-lying excitations to determine what its low-temperature properties are. The system could have some combination of charge gap, spin gap, singleparticle excitation gap, or superconducting gap. A l l of these questions can be answered by exactly diagonalizing the Hamiltonian, except that this can only be done for relatively short chains rather than the thermodynamic limit. Finite size effects may be quite important, and exact diagonalization on short chains may not provide accurate information. The "density-matrix renormalization group" ( D M R G ) method is a method of obtaining highly-accurate information on much longer chains, and perhaps in higher-dimensional systems as well. The D M R G method was invented by Steve White of U C Irvine in 1992 [28]. Kenneth Wilson had developed a renormalization group idea successfully in the solution of the Kondo problem [29]. Subsequent attempts at numerically solving a wider range of large-scale quantum lattice models remained largely unsuccessful for reasons not fully 1  A brief review of some of these early real-space renormalization attempts for quantum systems,  1  19  Chapter 3. The Density-Matrix Renormalization Group Method  20  understood until White correctly represented the physics occurring at the boundaries in terms of system/environment density matrices. In this formulation, the full system of interest is divided arbitrarily into what are called "system" and "environment" blocks; the interface between these two may be anywhere within the system. In fact, White demonstrated that the density-matrix prescription is the optimal one for the numerical application of the real-space renormalization group procedure [28, 30]. The success of the D M R G approach was quickly evident in its first few applications. Some basic ideas of the original approach for computing ground and low-lying states remained unchanged in White's formulation. One iteratively computes eigenstates of a Hamiltonian on a given section (or block) of the system and forms a new Hilbert space consisting of tensor products of states on this and a small additional contiguous section of the system (forming a new block), hoping to eventually arrive at a fixed point representing the thermodynamic limit. Computational limitations require that, at each iteration, a number of states in the block must be discarded. The discovery of White was that the traditional approach of keeping the lowest-energy states must be replaced by one of discarding states which correspond to the lowest eigenvalues of the density matrix of one block of the system with the other block being regarded as the environment. Although Wilson's real space renormalization group method worked well for the Kondo problem, it performs poorly when applied to other problems. This is because it completely neglects the quantum fluctuations at the boundaries of the blocks; there is not a significant improvement by increasing the number of states kept. The D M R G formulation provides a systematic method of choosing the states to keep for the new block, fully taking into account the quantum fluctuations at the boundaries. The D M R G method can be used to calculate the expectation value of any operator in the ground and low-lying excited states, such as energies and correlation functions, which ranged from 1979 to 1992, is contained in Section H I of Ref. [30].  Chapter 3. The Density-Matrix Renormalization Group Method  21  It is capable of studying much longer chains than exact diagonalization. It also avoids the problem of (quantum) Monte Carlo calculations where very low temperatures are difficult to attain, and in which fermionic Hamiltonians suffer from the sign problem;  2  D M R G works at zero temperature to find the ground state (and optionally some lowlying excited states), and has no sign problem at all. Although it is difficult to extend its application beyond one dimension or to implement long-range interactions, some success has in fact been achieved in this area. Two-chain problems have been quite successfully studied, such as the Hubbard [11, 13, 12], Heisenberg [15], and t-J [33] ladders, and very recently a study of four-leg t-J ladder has been reported[34]. Two-dimensional systems have also been studied by D M R G , for example in a very preliminary demonstration for the Heisenberg model on a square lattice [35] and more substantially in a studies of the Heisenberg model on the 1/5-depleted square lattice of C a V 0 9 in Ref. [36] and of the 4  7 x 10 [37], 8 x 8, and 4 x 18 [38] t-J lattices. In this chapter, I will describe in detail the D M R G method used for the numerical work reported in this thesis including some recent advances in the implementation of this powerful technique.  3.2  The D M R G Procedure  There are two closely-related implementations of the D M R G method. The first is the "infinite system method" in which the lattice is systematically increased in size with the intention of reaching a fixed point, beyond which certain quantities, such as the energy per site, no longer change. The second is the "finite system method" in which after the infinite system method is utilized in the "warmup" step to attain a desired system size, the accuracy is increased for that particular system size by repeating the D M R G 2  [32].  F o r reviews of the quantum Monte Carlo method and its fermionic sign problem, see Refs. [31] and  Chapter 3. The Density-Matrix Renormalization Group Method  22  procedure with the system/environment interface sweeping back and forth through the lattice. It has been observed [39] that in the finite system method, the accuracy of the results obtained is quite independent of the number of states kept in the warmup step or in initial sweeps in the finite system. As successively better representations of the entire lattice are computed and utilized, the finite system is clearly the most reliable especially in the computation of correlation functions. While periodic boundary conditions may be implemented in the D M R G method, the extra fluctuations at the added independent boundaries between blocks make the calculations much less accurate than in an open boundary condition implementation [39], which therefore is the method of choice in this work.  3.2.1  S u b d i v i d i n g Sites To Improve A c c u r a c y a n d Speed  At each iteration the fraction of the constituent block states kept is of necessity the reciprocal of the number of degrees of freedom in the added block, thus keeping the dimension of the Hilbert space constant and numerically manageable. Thus when having sites involving a number of different quantum numbers, such as those associated with localized / spins as well as conduction electrons, the accuracy lost in a large truncation as well as the time spent in adding the degrees of freedom could become important. This can be countered by regarding each site as consisting of a set of sub-sites, effectively absorbing only a few of the degrees of freedom at each step. In studying models involving / spins and conduction electrons, one could consider each site as being composed of three subsites: a spin site (either up or down), a conduction-electron spin-up site (either occupied or unoccupied), and a conduction-electron spin-down site. Thus instead of adding in the eight degrees of freedom at each iteration, three successive iterations involving only two degrees of freedom each will achieve the addition of a site with higher accuracy and speed. To avoid the potential introduction of systematic error, the sweep direction in  23  Chapter 3. The Density-Matrix Renormalization Group Method  which the sub-sites are added must be alternated. This method may be implemented together with the utilization of different sub-site representations. To build up to the desired finite size (in the "warmup" step) the full site representation may be used, as accuracy at this stage is not a serious concern. Once the target length is achieved, the sites may then be represented as |spin—|) <g> |conduction electrons)  (3.1)  in which the eight degrees of freedom are broken down into 2 x 4 .  Ramping up the  accuracy, the sites are finally represented as |spin-D <8> |spin-up electron) <8> |spin-down electron)  (3.2)  in which case we have 8 = 2 x 2 x 2 . A clear advantage of such an approach, apart from its intended one of minimizing truncation error, is that once it is fully implemented it becomes relatively simple to numerically study systems with impurities. For example, to model the absence of an / spin, one need only neglect the spin-| sub-site at the site of interest. This type of study is not undertaken in this thesis, but an important goal of this work is to ensure the continued viability and applicability of the code for future research programmes.  3.2.2  T h e Infinite S y s t e m M e t h o d  The left block for the first iteration will generally be the sum of a number (usually two) of individual site blocks, and for subsequent iterations it will be the sum of the possibly-truncated block from the previous iteration and one more added site or sub-site. In the sum of two blocks (call them blocks 1 and 2), the sum's states | i )  s u m  are those  formed from an outer product of the summands' states |z)i,2{lOsunJIT  2  = {L?>i ® |fc>2 I  l<j  <m l<k<m } u  2  (3.3)  24  Chapter 3. The Density-Matrix Renormalization Group Method  Here m; is the dimension of the Hilbert space of block i. The Hamiltonian of the sum is the sum of the individual Hamiltonians plus the interactions between the summands as determined by the lattice Hamiltonian. That is, # s u m = {Hi < g > I ) + ( / l ® H ) + # (l,2), 2  2  int  (3.4)  where Hi and U are the Hamiltonian and identity respectively on block i and i^i t(i,2) n  denotes the interactions between blocks 1 and 2. In the blocking scheme of the D M R G method, one always thinks of the blocks in terms of the original lattice in that each block represents a specific set of sites or sub-sites. This allows the determination of which operators are involved in the inter-block Hamiltonian as well as allowing direct measurements of operators on specific sites. We now consider the infinite system method as we build up the system to a specific finite size, with the number of sites or sub-sites being n . These sites are labeled in a s  specific ordering as indicated in Fig. 3.1.  1 2 3 4  n -l s  (a) In this implementation, each labeled site corresponds to a simple physical lattice site.  2 3 6 7 1 4 5 8  n -l s  (b) A n example in which each physical lattice site is subdivided into two labeled sub-sites, one representing / spins and the other representing conduction electrons. Figure 3.1: Labeling of sites in D M R G approach. The left block Bf , constructed by the addition of a site or sub-site to an existing left +l  block Bf, will be considered the "system", for which we must construct an "environment" block. Here B\ denotes the block spanning the % left-most sites or sub-sites of the lattice,  Chapter 3. The Density-Matrix Renormalization Group Method  25  and the bar indicates that the block has been truncated in a previous iteration if necessary. In cases where reflection is a symmetry of the model, we may use a reflection of the left block (as shown in Fig. 3.2a) plus additional sites (as shown in Fig. 3.2b), to construct 3  this right block. If reflection symmetry is not applicable or utilized, we must build the right block from the addition of a number of individual site (or sub-site) blocks including the interactions between them (as shown in Fig. 3.2c). These left blocks are approximations to the exact left blocks, with the accuracy ranging from exact for the first few values of i in which no truncation is necessary (more precisely, B\ is exact if U^ di < m where di is the dimension of the Hilbert space of =1  site or sub-site i, and m is the maximum number of states kept in system blocks) to deviations indicated by cumulative discarded weights. The left and right blocks contain states with a range of quantum numbers Q L and <2R (Q here denotes a set of additive quantum numbers, and Q the corresponding set of operators, for example total number iV of electrons and ^-component S of the total Z  spin), which range will in general have been truncated in previous iterations. That is,  Q\i)L  =  Q\J)R  =  where | i ) (1 < i < ra ) and \j) L  L  R  QLWIOL QR(J)\J)R,  (3.5) (3.6)  (1 < j < m^) are states in the left and right  blocks respectively. We now consider, instead of the full Hilbert space determined by the addition of the left and right blocks, specific combinations of states from the left and right blocks which sum to give the target quantum numbers Q of interest for the intermediate system size represented by the sum. These intermediate target quantum M y current implementation, in which the sub-sites represented by a block are assumed to be contiguous in the chosen ordering sense of the sub-sites, permits any block (including those containing incomplete lattice sites) to be reflected in any link of the lattice with the results still representing a block of contiguously labeled sub-sites. 3  Chapter 3. The Density-Matrix Renormalization Group Method  26  (a) The case in which reflection symmetry applies and is being used.  R(^ i) +  (b) A n example in which sub-sites are being used.  (c) The case in which reflection symmetry does not apply, or is not being used. Figure 3.2: In the infinite system method of D M R G , we add site (or sub-site) i to the previous left block Bf to construct a new left (system) block Bf and construct an appropriate right (environment) block. +1  27  Chapter 3. The Density-Matrix Renormalization Group Method  numbers are determined in order to most closely represent the current system size as a fraction of the target system size together with its quantum numbers of interest. While Moukouri et a/. [40] consider a weighted mixed state in which two pure states' quantum numbers bracket those required to represent the exact proportion based on the ratio of the current system length to the target system length, this approach is not necessary for the finite system method. The accuracy in the infinite system method step building up to the finite system method could undoubtedly be improved in this way, but, as noted earlier, the accuracy of this step is relatively unimportant in the overall finite system method. Thus the states of the intermediate-size system are {|^->  =  ®\J)R I  QL(1) +  QRU)  = Q}  •  (3-7)  We then determine and diagonalize the Hamiltonian for the system in that sector using Davidson's block diagonalization method [41] to find the ground state which may be written as m  L  TOR  l*)=££^(K>L®b'>R). i=lj=l  (3.8)  We now form the reduced density matrix of the system (the left block) by tracing over the degrees of freedom of the environment (the right block):  k  In Eq. (3.8) I have written the sums deliberately over the full product Hilbert space instead of just those product states of Eq. (3.7) contributing to its sector of interest because some states in small blocks turn out to be important in further iterations even if they do not make non-zero contributions to the density matrix. We diagonalize this density matrix by a unitary matrix U of eigenvectors: UpW = diag[u ,u , 1  2  • • • ,co ], t  (3.10)  Chapter 3. The Density-Matrix Renormalization Group Method  28  with 1 > ui > u > • • • > ui > 0. The eigenvalues u>i of the density matrix give the weight 2  of the corresponding eigenstates. We truncate the basis by keeping a predetermined number, ra, of the states with the largest weights (i.e. drop the rows of U corresponding to all but the highest ra eigenvalues of p). Using these eigenstates we construct a new basis for the left block Bf  +1  by keeping a maximum of m of the total number / of states. (The  resulting block is denoted Bf .) +1  That is, we transform and truncate to the new basis  the operators O needed to compute the system Hamiltonian or desired measurements according to O'lmxm]  =  {mxl]0[l l}Ul .  U  x  lxm]  (3.11)  The subscripts here are just used to indicate the dimensions of the matrices. The next block Bf  +1  forms the starting point for the next iteration as we again add a site or sub-site  to it until we have attained the target system size or possibly a renormalization group fixed point.  3.2.3  The Finite System Method  The infinite system method is useful in the computation of ground state energies and their dependence on system size, and can allow the limited computation of correlation functions; however, to obtain the greatest possible precision and to reliably determine correlation functions, one must employ the "finite system method", which improves upon the first-approximation results obtained from the infinite system method once it has reached a desired system size. This method sweeps the interface between the system and environment blocks back and forth a number of times, alternately considering the left block and the right block as the system and using previously computed system blocks as better representations of the environment. During the course of building up to the desired finite size L in the infinite system  Chapter 3. The Density-Matrix Renormalization Group Method  29  method, representations Bf are obtained and stored (on disk) for left segments of the lattice.  These left blocks, which are truncations of system blocks Bf in the infinite  system method, are now used to construct environment blocks as the system-environment interface is moved from right to left. The first right block, Bf,  will be simply that  corresponding to the right-most site in the case in which reflection symmetry is not employed (here n = n ; see Fig. 3.3a), or the reflection of the left block Bf_ s  (here  3  n = n /2 + 2; see Figures 3.3b,c). Bf denotes the block spanning the right-most lattice s  sites or sub-sites numbered from i to n , and, as before, a bar indicates that the block s  has been truncated in a previous iteration if necessary. To this block the next site (or sub-site), namely the (n — l )  t h  one, is added to form the new right (system) block  the left (environment) block will be the sum of Bf_  Bf^;  and the (n — 2 ) site or sub-site. We th  3  now diagonalize the Hamiltonian and compute the density matrix considering Bf_  x  system block, forming the new truncated block Bf_ , x  the  and proceed to move the system-  environment interface to the left, constructing better approximations to the right blocks Bf  for i = n-  l,n-2,...,3.  When the interface has been swept all the way left (the general step of this sweep is depicted in Figure 3.5) to between sites 2 and 3, we now reverse the sweep direction (as depicted in Figure 3.4), now using the newly obtained right blocks Bf as environment blocks and computing better approximations to the left blocks Bf.  This process is  repeated a number of times until the discarded weights are minimized and we have attained a fixed point for the given maximum number m of states kept per system block.  3.3  Recent Advances  While computer memory, especially random access ( R A M ) , and speed always provide limitations to numerical work attempting to understand systems in the thermodynamic  Chapter 3. The Density-Matrix Renormalization Group Method  O  30  O  R  B n-l  n-3  B  (a) The case in which reflection symmetry does not apply, or is not being used. R(^-3)  n-3  B  B  n-\  (b) The case in which reflection symmetry applies and is being used, in the simple case of having no sub-sites.  R  ( «-3) 5  (c) A depiction of how this works in a case in which reflection symmetry applies and is being used, and each site is represented by two sub-sites. Figure 3.3: Starting the finite system method of D M R G upon completion of the infinite system method to form the first non-trivial new right (system) block and an appropriate left (environment) block.  Chapter 3. The Density-Matrix Renormalization Group Method  31  R B i-l  ^ 3  Figure 3.4: Sweeping the system-environment interface to the left in the general case.  o  I  L B i+l  > 7? i+3 R  B  Figure 3.5: Sweeping the system-environment interface to the right in the general case. limit, it is necessary to employ advanced techniques to harness the available power. Some of these which I have implemented are discussed in this section.  3.3.1  Low-Level N u m e r i c a l C o m p u t a t i o n  The bulk of the purely-numerical aspects of the D M R G method involves matrix and vector algebra: diagonalization of large and small real symmetric matrices, basis transformations, etc. For this reason, a set of Fortran routines linked closely to the system hardware, known as B L A S (Basic Linear Algebra Subprograms) are utilized for all linear algebra computations. This results in significant processing time savings, even over the intrinsic matrix multiplication now available in Fortran 90. This does require a carefully constructed interface between Fortran and Fortran 90 due to the different matrix representations in the two languages and related bugs in the currently available I B M A I X Fortran compiler x l f ; a public-domain set of general interfaces, known as L A P A C K 9 0 ,  Chapter 3. The Density-Matrix Renormalization Group Method  32  originally scheduled to be (but not) released on the NetLib software repository in the summer of 1996 is unfortunately too late for my use.  3.3.2  Thermodynamic Limit Fixed Point Approximation  In early 1995, Ostlund and Rommer [42] considered the D M R G method in the case in which the renormalization converges to a fixed point, allowing the study of the thermodynamic fixed point. Following White [36], I implemented their ansatz as an improved starting point in each step of the finite system method. These advances are detailed in this section. The basic step of any real space renormalization group technique is the reduction of the Hilbert space of a block of sites. Generalizing the approach of Ostlund and Rommer, we write this reduction as  \a) =Y, f»)n®\P)n-u A  (3.12)  n  where {|a) } are the m states of the system block representing n sites, {|s) } are the w  n  n  states of the added site number n, and {A„[s]} are m  n  x m -i matrices effecting the n  transformation at this stage of the calculation; the D M R G method provides an optimal determination of these matrices. In cases in which we have translational symmetry (in the approximate sense of neglecting the open boundary conditions namely that each added site is identical in terms of its local contributions to the lattice Hamiltonian), as we approach the thermodynamic limit of large system size, it is reasonable to assume that A [s] —> A[s}. While at each iteration, one must use the D M R G method to compute the n  matrices A J s ] , it is helpful to approximate A [s] ~ -A _i[s] and to subsequently refine n  ra  this starting point. In my implementation of the D M R G method, I more generally add only a sub-site at each iteration. When we have translational symmetry in the above sense but modified  Chapter 3. The Density-Matrix Renormalization Group Method  33  by the n different sub-sites and their alternating order of introduction into the system ss  block, i.e. each sub-site is logically identical to its corresponding sub-site in every second lattice site, we expect the above limiting behaviour to be replaced by a periodic one: A [s] —> A ^ [ s ] , where 0 < n < 2n n  ss  labels the n  ss  different types of sub-sites (n =  n mod 2n ). In this case, we may approximate A [s] f» An_2nss[s] instead. ss  n  This generalization of the approach by Ostlund and Rommer is useful in the infinite system method for "translationally-symmetric" lattices as we increase the system size. Its applicability also extends to the finite system method regardless of any translationaly symmetry, in which case we can obtain a very good initial guess for the wavefunction on a superblock based on that determined at the previous iteration, after we perform the necessary change of basis. Consider the D M R G step in which a site is added onto a left block in the construction of a new left block. Let |a) be the states of left block Bf, where I is the rightmost site L  /  of the block, and let \s)i denote the states of site I. Then the basis states for the new left block are given by H+i =  E A\+A*U\P)\ ® l><+i-  (3-13)  5  s/3  Similarly, we had earlier obtained the states of the right block Bf^ : 3  K + 3 = £4+3[«WI*>l+3 ® |/?}? . +4  (3.14)  A basis state for the superblock formed by adding a left block, two sites, and a right block, is written as \ass'(5)f = \a)f ®  ® |a')/+2 ®  \P)f+3,  (3-15)  and a wavefunction is M i = E M^s'(3)\ass'p)f . B  ass P 1  (3.16)  34  Chapter 3. The Density-Matrix Renormalization Group Method  The next D M R G step will have as its basis states of the form \ass'P)f+  v  is truncation in going from \a)f <g>  Since there  to |a)/+i> the necessary transformation is not  exact. But because of the optimal nature of the density-matrix renormalization group method, the states |a)/  represent \tp)i as well as possible so in the transformation of  L +1  the wavefunction one may approximate[36] E N m - r + i H - i ,  (3.i7)  a  and insert this into Eq. (3.16) to obtain, using Eq. (3.15), |V)  m  ~  (3.18)  =  £  ipi(ass'p) • 1 • \ass'p)f  (3.19)  B  ass'fl  «  £  V/(as5'/3)|7>i  L +1  • ^(TKIa)!- ® 1  (8>  |3'>  /+2  < g > |^>^. . 3  (3.20)  We may now use the transformation matrices defined in Eqs. (3.13) and (3.14) to write this as \rp)i  +1  «  ^(aM'M+i[*]7«l7>j+i®k')i+2  £ cryss's"f}8  ®Af {s"U\ ") ®\6)f +3  = ' £  S  l+3  +4  ip, i(is's"8)\ s's"6)™, +  (3.21) (3.22)  1  -ys's"6  where 1>i+i(l8'8 S) = £ Af [ ] Mass'P)Af [s'% H  +1  S  ia  +3  s  (3.23)  as/3  are the coefficients of the new approximate wavefunction in terms of the old. This approximation significantly speeds up the D M R G calculation when several finite system sweeps are done[36].  Chapter 3. The Density-Matrix Renormalization Group Method  3.3.3  35  General Implementation  A major feature of my numerical implementation of the D M R G method is the harnessing of the new object-oriented Fortran language, Fortran 90, to enable the high-level specification of the lattice type, system Hamiltonian, and measurement operators, and blocks. The D M R G procedure as formulated here belies the size and complexity of the computer code required for its powerful general implementation. A few of the technical details are provided below to demonstrate the features which I have implemented. Blocks are systematically constructed by the concatenation of individual sites or subsites, together with the construction of operators on the resulting product space. Each of these sites or sub-sites may have an independently specified Hilbert space representing either a physically different site (such as an impurity or vacancy) or a different representation of the same physical lattice site. The Hamiltonian is specified in a very formal way. For example, the hopping term - *E  c U , . , + h.c.  (3.24)  (ij)<r  present in Hubbard-type models is added to an operator op in my Fortran 90 code simply by writing do j=2,L i=j-l;  idn=GPosDn(i); iup=GPosUp(i); jdn=GPosDn(j); jup=GPosUp(j)  c a l l Increment(op, (  (-t) *  (*( + (*(  enddo  + (*( + (*(  Chapter 3. The Density-Matrix Renormalization Group Method  3 6  The numerical values of the elements of the matrices corresponding to the formal operators are put in "by hand" for the simplest sub-site representations, for example  (t\s \ttt = 1 +  (i\c\\0)  ( 3 . 2 5 )  = 1  ( 3 . 2 6 )  from which the matrix elements for full physical sites are computed. Operators to be "measured", i.e. whose expectation values in the eigenstate of interest are to be computed, are also specified in this way. Having done this, the code must intelligently determine which operators or parts thereof are to be kept up-to-date, i.e. truncated and transformed to the new basis, based on what is required for the lattice Hamiltonian and desired measurement operators. To prevent memory-space wastage, transposes of already-stored matrices are indicated by a pointer to the latter. Operators are not stored as matrices indexed by the full Hilbert space, but since each corresponds to a specific change in quantum numbers, only those blocks which are non-zero are stored, thus avoiding huge blocks of zeros. As an example, suppose that one wishes to measure (n,), whererc,-= ]C<r 4<T ;<T C  a n  d  (• • •) represents an operator's expectation value in the eigenstate of interest, and that T,(ij) L j > + ' c  a  c  n  C -  occurs in the Hamiltonian. This would require that matrices represent-  ing the operators TI,- for all i and c]^ (and thereby its transpose c ) for i representing a i(T  site at the leading edge of the block of contiguous sites being built up (and for each cr), be maintained (i.e. truncated and transformed at each iteration). When the Hilbert space has as each basis state an eigenstate of the operators N and S , i.e. in which each basis z  state has these two as good quantum numbers, one can use the fact that c  ia  states of quantum numbers related by (AiV, AS ) Z  links only  = (—1, —a/2), wherein we mean j = 1  and 1= — 1 . Thus, we group together states sharing the same good quantum numbers and store only matrix elements for specific pairs of these state groups. The Hamiltonian,  Chapter 3. The Density-Matrix Renormalization Group Method  of course, has (AiV,A<S ) = (0,0). z  Chapter 4  T h e One-Dimensional P a i r - H o p p i n g M o d e l  4.1  Introduction  In this chapter, we consider the first of the two models of strongly correlated electron systems studied in the thesis, namely the "pair-hopping model" which is defined by the Hamiltonian H = -t £ [clc  + h.c] - V E H T ^ S I S T + ' -] H C  ja  {ij)tr  {ij)  in the notation introduced in Chapter 2. While the familiar first term, involving t, represents the nearest-neighbour hopping of single conduction electrons, the second, involving V, represents the nearest-neighbour hopping of on-site singlet pairs of electrons, which are created by the combined operator cj|cjj. So this Hamiltonian models a competition between these two hopping terms. As t —> —t is a symmetry of H (this is shown in Section 4.2), we take t > 0, and consider both signs of the interaction V. This model has some relevance to the question of high temperature superconductivity for three separate reasons. First, experiment shows that the cuprates are strongly type-II; in particular their superconducting correlation length £ is very small, namely a few lattice spacings. This indicates that the Cooper pairs of electrons with opposite spin are quite small, especially in relation to the completely overlapping large pairs found in "conventional" superconductors. The pair-hopping model treats an obvious limit of this behaviour, namely that  38  39  Chapter 4. The One-Dimensional Pair-Hopping Model  in which the pairs are actually on-site singlets.  1  Secondly, we consider the presence or absence of a spin gap from the ground state of the model system, a property which is important in determining the strength of the superconducting susceptibility of weakly-coupled chains. In high temperature superconductivity, the weak coupling of copper-oxide planes is thought to be an important element in the mechanism of the phase transition; evidence also continues to mount for the presence of a spin "pseudogap" above the superconducting temperature [43]. Certainly the spin sector of any alleged explanation of these materials must be quite interesting, as the proximity (as a function of oxygen-doping) of an antiferromagnetic phase to the superconducting one implies. Recently Emery, Kivelson, and Zachar [23] have proposed a mechanism for high temperature superconductivity based on a spin-gap proximity effect involving weakly interacting one-dimensional hole-rich "stripes" forming walls between narrow antiferromagnetic domains; in fact it is a pair-hopping interaction which forms a basic element of their theory. In one of two scenarios they consider, the hopping of pairs between the stripes and the antiferromagnet domains is responsible for the formation of a spin gap and subsequently the superconductivity. And thirdly, since members of the tight-binding class of Hubbard-type models are clearly very applicable to the description of the electronic properties of the cuprates, it is important to consider each of the possible interactions. Of the full range of nearestneighbour interactions, the hopping of on-site spin-singlet pairs has not been well studied to date. Two goals of the work reported in this chapter are to widen the range of parameters under which the pair-hopping model has been studied and to clarify its properties where it has been considered before. Since the pair-hopping model has only these rather vague connections to the problem T h e pairs in this purely one-dimensional system are of course just s-wave pairs; experimental evidence, however, indicates that the superconductivity of the cuprates is of the d 2_ 2 form. 1  x  y  Chapter 4. The One-Dimensional Pair-Hopping Model  40  of high temperature superconductivity, it is certainly not our imagination—and much less belief—that it is a model which will contribute concretely to any full description of the phenomenon. Rather, our approach is to clarify the behaviour of this particular model as a member of the general class of correlated electron systems. The phase diagram of the pair-hopping model in one dimension, particularly for positive V, has been, and indeed continues to be, the subject of some controversy. The model was introduced by Penson and Kolb[44, 45] in 1986. In two closely related papers they presented exact diagonalization calculations on short chains (L < 12) which appeared to demonstrate a spin gap transition at a finite value of V « 1.42; below this critical value, they claimed there is no spin gap, and above it there is, and an essential singularity divides the two phases. In 1988, Affleck and Marston [46] studied the model with bosonization and the renormalization group method, concluding that the pair-hopping model for small positive V should be identical to the weak-coupling attractive Hubbard model, and that therefore a spin gap should be present for all positive V, turning on exponentially from V = 0. Five years later, Hui and Doniach [47] reported a subsequent broader numerical study corroborating the work of Penson and Kolb, and accompanied this work with a presentation of renormalization group arguments countering the claims of Affleck and Marston and supporting the previous numerical work and their own. In 1995, Affleck and I [48] presented density-matrix renormalization group results on much longer chains, L < 60, together with a study of the analytical renormalization group picture, concluding that there is in fact a spin gap for all positive V. (Subsequent works agree with [49] and extend [50, 51] our results.)  This does not mean, however,  that the phase diagram of the one-dimensional pair-hopping model is uninteresting; on the contrary, there are phase transitions at both positive and negative values of V. Away from, but close to, half-filling, a phase-separation transition occurs at finite positive V ; at half-filling and negative V, there are at least two distinct phases, one exhibiting a charge  Chapter 4. The One-Dimensional Pair-Hopping Model  41  gap and the other a spin gap. (A concurrent [52] and followup [51] study examined this negative-V phase transition in more detail, confirming our results, and finding in addition that the critical value is filling-dependent and that the spin-gap phase exhibits so-called ^-superconductivity, in which the Cooper pairs have total momentum 7r instead of zero.) This chapter details these arguments. It should be emphasized from the outset that we do not present a complete study on all aspects of the pair-hopping model. Various other interactions which would certainly be applicable to any experimental situation could be added to the Hamiltonian, and these interactions would be important in any attempted theoretical explanation of observed phenomena. Hui and Doniach [47] consider in addition to the pure pair-hopping model what they name the "Penson-Kolb-Hubbard model" which includes an on-site repulsive Hubbard term. A more general conduction band including next-nearest-neighbour terms could be considered as well as a nearest-neighbour Coulomb interaction. For example, Refs. [53], [54], and [55] study a larger class of one-dimensional lattice models to determine soluble points in the phase diagram. Rather we concentrate on the pure pair-hopping model in an effort to determine some of the physics of the pair-hopping interaction alone.  4.2  S p i n G a p and Phase Separation f r o m the S t r o n g C o u p l i n g L i m i t  The pair-hopping model is easily shown to exhibit a spin-gap as well as a phase-separated phase at large \ V\/t.  Before demonstrating this, it is important to consider the symmetry  properties of the Hamiltonian. Changing the sign of t in the pair-hopping Hamiltonian of Eq. (4.1) is equivalent to the operator redefinition c  ia  -  (-1)'^  (4.2)  Chapter 4. The One-Dimensional Pair-Hopping Model  42  on one of the two sublattices in the bipartite lattice, which preserves all of the anticommutation relationships. No similar redefinition of these operators corresponds to changing the sign of V.  4.2.1  Spin G a p  At large \V\/t, all sites are doubly occupied or empty (assuming an even number of electrons) and the model becomes equivalent to one involving spinless fermions[44, 45]. In particular there is a large gap, of 0(V), to any excited state with non-zero spin. This is true for either sign of V, but it is important to note that V —> —V is not a symmetry of the model, unlike in the Hubbard model's duality relationship in which changing the sign of U is equivalent to interchanging operators for charge with those for spin [46]  Ui-l*^  2s\.  (4.3)  We now analytically examine the large |V| limit for an open chain, noting that nothing essential will change in going to periodic or infinite length chains. As shown in Refs. [44] and [45], setting t = 0 results in a ground state involving only empty and doubly occupied sites, so the on-site pairs are effectively non-interacting spinless fermions. Using this fact, the ground state for an open chain of even length L is easily computed to be the half-filled band of spinless fermions with energy 2  E* = -2\V\ V c o s - ^ - = |y| ~i  L+l  V  ( 1 - c s c — ( 4 . 4 ) 2L + 2J y  J  There is no charge gap, since adding an on-site singlet pair of electrons, i.e. adding a spinless fermion in this limit, requires only an energy proportional to 1/L, but adding a single electron produces a necessarily immobile site (because t = 0), effectively breaking the chain. The energy will depend on the location of the break, and is easily shown to 2  N o t e that we take the chemical potential to be zero.  Chapter 4. The One-Dimensional Pair-Hopping Model  43  be minimized if the break is at an end of the chain, in which case the energy is that of | pairs hopping on an open chain of length L — 1, namely E, = -2\V\ £ cos H  =  | | ( i _ ot J L ) . V  (4.5)  C  So the single-particle gap for the open chain is A,  =  p  =  Ei-Eo | y |  [  (4.6)  C S C  2iT2-  C O t  s]  ( 4  '  7 )  So for t = 0, we have a model equivalent to free spinless fermions, corresponding to a spin gap proportional to |V| but no charge gap. To see whether this situation persists for finite |V|/i, we can do perturbation theory in the lattice model in t/V.  This is very  similar to the well-known results on the large-C/ Hubbard model. In this case we project out singly occupied sites. A single application of t takes us into the high-energy subspace with two singly occupied sites. In second order perturbation theory we generate an effective interaction of 0(t /V) 2  in the spinless fermion model. This simply corresponds  to a nearest-neighbour interaction of the spinless fermions. This interaction is known to be exactly marginal leading to a critical line with vanishing gap. Thus there is a spin gap for large \ V\/t.  3  As there is no spin (or charge) gap for V — 0,  there must be some transition. Finite-size numerical work[44, 45, 47] has been performed on this model in one dimension for positive V, suggesting a phase transition at which the spin gap (or single-particle excitation gap) opens, aAV/tm  1.4. Two different analytical  renormalization group (RG) analyses have been applied to the model. One[46] suggested the existence of a spin gap A for all V > 0, with s  A  oc e~*  t/v  s  (4.9)  T h i s is shown, for example, for the XXZ spin chain which is identical to interacting spinless fermions in Ref. [26]. 3  Chapter 4. The One-Dimensional Pair-Hopping Model  44  as V/t —»• 0 , and no transition for any positive V; the other[47] suggested that there is +  a transition at V/t « 1.4, consistent with the numerical work. Previous numerical work has used chains of length L < 12. We present data for much longer chains, L < 60, using the density matrix renormalization group technique, thereby countering the dominance of finite-size effects. We also discuss the subtleties involved in trying to extract information about the phase diagram from a low-order analytical R G calculation. On the basis of a reliable interpretation of the analytical R G equations (presented in Section 4.3) and careful consideration of and comparison with numerical results (presented in Section 4.4.2), we conclude that the positive V transition occurs at V = 0 instead of at some finite | V | . We show numerically that for small positive V, the behaviour of the single-particle gap is of the form predicted by the R G flows in the numerically accessible region of phase space. It was argued in Ref. [46] that there is a phase transition, corresponding to the appearance of a spin gap, for some finite negative V. (The case V < 0 was not studied in Refs. [44], [45], or [47].) In Section 4.4.3 we present our numerical calculations which find a spin-gap transition at V  c  4.2.2  — 1M.  Phase Separation  There has been considerable interest of late in phase separation in the Hubbard and t-J models in one and higher dimensions [56, 57, 58]. The pair-hopping model provides a simple example of a model where it is easy to see that a phase-separation transition must occur at some finite critical coupling, with a non-zero total magnetization. Consider the model at t = 0 with a total magnetization S  z  — + M / 2 , corresponding  to an excess of M spin-up electrons. Since the ground state involves as many on-site singlet pairs as possible, all the spin-down electrons are paired with spin-up electrons, leaving the M spin-up electrons immobile (since with t = 0 only pairs may hop). Thus  45  Chapter 4. The One-Dimensional Pair-Hopping Model  the model is equivalent to spinless fermions with vanishing hopping terms to M sites. Equivalently we have an XY spin chain with vanishing exchange coupling to M sites, corresponding to M nonmagnetic impurities. These impurities simply have the effect of breaking the chain up into chainlets. The above analysis in the free spinless fermion model for an added electron at t = 0 shows that in the case t = 0, a single unpaired electron sits at a chain end; it is clear that added electrons of the same s will clump at the chain ends as well. The energy is lowest z  when all M impurities are next to each other, leaving an XY chain oiL—M sites. That is, at any finite magnetization, the chain phase separates at \V\/t 3> 1: one part of the chain assumes the net magnetization. (It is important to note that this is not a peculiarity of the open chain; in the periodic case as well, at t = 0, added polarized electrons cut the chain and the chain-breaking energy is clearly minimized by clumping them together.) Since going from t = 0 to some large but finite \V\/t introduces only a marginal operator, it is clear that this phase-separated phase will persist from | V | = oo to some critical values of V, which will probably have different absolute values for opposite signs of V because the Hamiltonian is not symmetric under V —> —V. The renormalization group analysis of the model at weak coupling, \V\/t <C 1, indicates behaviour similar to that of the Hubbard model, with no phase separation. This suggests that phase-separation transitions should occur at finite values ofV/t (one for positive V and one for negative V"). In Section 4.4.4, we present evidence for such a transition near half-filling at V = V i « 3.52, but we have c  not examined the one at V = V  c2  < 0.  46  Chapter 4. The One-Dimensional Pair-Hopping Model  4.3  A n a l y t i c a l R e n o r m a l i z a t i o n G r o u p Studies of t h e Phase D i a g r a m  The R G analyses of Ref. [46] (hereafter A M ) and Ref. [47] (hereafter HD) came to quite different conclusions. Here we would like to explain the reasons for this and give arguments in favour of the former approach. We use essentially the notation of H D , which is taken from the review article of S61yom[27]. Taking the continuum limit of the pair-hopping model we obtain a general Hamiltonian H = J dx[Ho + n- },  (4.10)  mt  where Ho and H t are the dimensionless kinetic energy density and interaction Hamilm  tonian density. The kinetic energy is that given by Eq. (2.30); the interaction term can be written as Wi„  =  t  T « P {-\9 JLJR  - 2 & J L • JR  p  - ^ M I ^ ^ R ^ R * -\94[JLJL-IJL-JL  + (L "  R)]  (4.11)  + (L^R)]}.  We have chosen to write the Hamiltonian in a manifestly SU(2)-invariant way, using the charge and spin currents (or densities) defined in Eqs. (2.48) and (2.49). The last term can also be written as JLJL  - | J L • JL  = JLJL - 4 J £ J £ = 4 J J , L T  L i  (4.12)  where Jha  —  V'LVLC*  (repeated index not summed),  and we have used the fact that SU(2) symmetry implies J  L  (4-13)  • JL = 3J£J£. To the first  non-vanishing order in V, the bare couplings have the values v = 2t; g = -g F  p  s  = g = g =. 2V/irv . 3  4  F  (4.14)  Chapter 4. The One-Dimensional Pair-Hopping Model  47  To cubic order, the R G equations are given by [47] = 9s +  ~ ~lU  + 54k  (4-15)  2  (4-16)  +  =  =  9 9z + \(9 + 9 3 - 2g g )g  =  l(9 9l-9 )-  (4.17)  2  2  P  p  P  4  3  (4-18)  3  P  s  Here I = — In A, where A is an ultraviolet cut-off. As we lower the cut-off to study the long-distance behaviour, I increases. Part of the discrepancy between the conclusions of A M and H D arises from the treatment of the # coupling. If we bosonize the theory, then 4  JLJL pL-fi  where 0  C]S  = 4JlJt  = ~(9M  (4.19)  2  =  ^(9+<M ,  (4-20)  2  are charge and spin bosons, and d± = d/dt — d/dx. Hence g± simply shifts  the velocities of charge and spin excitations to v  c  v  s  = v (l + g /2) 4  (4.21)  = v (l-g /2).  (4.22)  F  F  4  A common approach to Luttinger liquids is to simply set v and v to their renormalc  s  ized values and drop g$ from the R G equations. This approach was used by A M . The R G equation for g then decouples from the g and g ones. This arises from the fact that, s  p  3  upon bosonizing, the corresponding operators involve only the spin boson and only the charge boson respectively. We then see that g — 0 is not a stable fixed point: if g < 0, s  s  as is the case for V > 0, g will flow away to strong coupling. This is usually taken to s  indicate that the system is in a phase with a gap for spin excitations.  Chapter 4. The One-Dimensional Pair-Hopping Model  48  On the other hand, a quite different conclusion can be reached if g± is kept in the R G equations. Then, according to Eq. (4.15), g = 0 becomes a stable fixed point from s  the negative side provided that <? < —2. The nature of this putative phase can be 4  understood by also rewriting the free electron kinetic energy in terms of spin and charge currents. Setting all coupling constants to zero except <?, the full Hamiltonian density 4  can be written H = ±*v [{l - ± p ) J i A + (1 + igJJiJl] F  4  + (L <-> R).  (4.23)  We see that for # < —2, the spin part of the Hamiltonian becomes unstable. That is, 4  J[(x) and Jji(x) tend to become large, necessitating the keeping of higher order terms in the Hamiltonian. On the other hand, the condition of zero total magnetization requires J dx[J[ + J«] = 0.  (4.24)  A possible interpretation of this phase (which occurs in other known cases) is a ferromagnetic phase. The condition of zero total magnetization forces a domain structure, i.e. phase separation, to occur. One side of the system has positive polarization and the other half negative. H D integrate the cubic R G equations including # , using the initial values of Eq. (4.14) 4  [plus the 0(V ) 2  corrections which are not important at small V]. The result was that for  0 < V/t < 1, a fixed point was reached with # « —2.5 and g = 0. 4  s  Whether or not <? is included, for V/t < 1, #3 renormalizes to zero and g to some 4  p  small positive value which depends on V/t, corresponding to zero gap for charge excitations. A M identify this phase as having a spin gap since g does not flow to zero. H D assume s  this phase with g = 0 and <? < — 2 has no gap for single-particle excitations. Since s  4  these excitations have spin | and charge 1 this would imply, from the usual Luttinger  49  Chapter 4. The One-Dimensional Pair-Hopping Model  liquid viewpoint that there is neither a charge gap nor a spin gap. We do not find this calculation convincing. It is not possible to argue rigorously that g renormalizes to a 4  value less than —2 using only the cubic order R G equations. If this actually happened, as H D claim, this would presumably imply a transition into a ferromagnetic phase (or possibly some other more exotic phase characterized by the harmonic spin Hamiltonian of Eq. (4.23) becoming unstable) for arbitrarily small V. No direct numerical evidence for ferromagnetism (or other exotic behaviour) at small V/t has been presented. Although earlier numerical work in Refs. [44], [45], and [47] saw indications of vanishing spin gap in this region of parameters, the numerical results presented here in Sec. 4.4.2 based on much longer chains (L < 60 instead of L < 12) find a non-zero spin gap. HD claim a different phase is reached for V/t > 1 with a non-zero g at the fixed s  point, corresponding to a spin gap as do A M . However A M and H D now disagree about the behaviour of the charge couplings, g and g . Note that the R G equations (4.16) and p  (4.17) imply that g — ±g 3  3  are separatrices (for <? = 0). For g > 0, if \g \ < g , g  p  4  p  3  p  3  flows to zero (see Fig. 4.1), corresponding to a harmonic gapless effective Hamiltonian for charge. Outside this region both g and g flow off to values of 0(1). This is normally 3  p  interpreted as a phase with a charge gap. It is a remarkable feature of the pair-hopping model that, to 0(V), g = g : the system lies on a separatrix. It is necessary to calculate p  3  the bare couplings to 0(V ) 2  to deduce whether or not g flows to zero. Both papers 3  agree that these G(V ) terms place the bare couplings in the basin of attraction of the 2  g = 0 critical line, for small V. A M assume (on the grounds of simplicity) that the 3  system remains in this basin of attraction for all V > 0. On the other hand, H D use the expression for the bare couplings to 0(V ) 2  for arbitrarily large V to deduce that  the bare couplings moved outside this basin of attraction at a critical V « t (the same critical point at which g and <? change). The cubic R G equations predict a fixed point s  a t  4  Q = 93 — —2, which H D assume corresponds to vanishing charge gap. P  50  Chapter 4. The One-Dimensional Pair-Hopping Model  (a) Flow in the charge sector, for small #3 and g . p  -2  0  S  s  (b) Flow in the spin sector. Figure 4.1: Third-order renormalization group flow diagrams, ignoring g . 4  Chapter 4. The One-Dimensional Pair-Hopping Model  51  We caution that it is not possible to tell from these low order calculations of the bare couplings and the R G equations whether or not the bare couplings ever leave the domain of attraction of the # = 0 critical line. Furthermore, if they did, this phase 3  would normally be identified as having a charge gap, which we know does not occur for small or large V. (The existence of an apparent, finite coupling fixed point of the cubic R G equations at couplings of 0(1) does not necessarily signal the existence of a different critical point. It could disappear upon keeping higher order terms.) In Sec. 4.2 we gave analytic arguments implying that, for large | V | / t , there is a spin gap but no charge gap. By ignoring <? (i.e. absorbing it into velocity renormalizations) and making a plausible 4  assumption about the behaviour of bare coupling constants at large V, we obtain simple behaviour requiring no phase transition for any V > 0. There is always a spin gap and no charge gap. On the other hand, by including the renormalization of p and using weak coupling 4  results at strong coupling H D obtain two different phases: a bizarre small V phase with an unstable harmonic spin Hamiltonian and a large V phase which would likely correspond to a charge gap, in contradiction with the expected large V result. H D applied the same R G analysis to the positive U Hubbard model, giving a small U phase with <? > 2, corresponding to a negative harmonic Hamiltonian in the charge 4  sector and a large U phase with a non-zero g which would normally correspond to a s  spin gap. As they pointed out themselves, this is in contradiction with the expected behaviour which is a charge gap and no spin gap for all positive U (at half-filling).  Chapter 4. The One-Dimensional Pair-Hopping Model  4.4  52  Density-Matrix Renormalization G r o u p Results  4.4.1  DMRG  Details  We use the infinite system D M R G method described in Section 3.2.2, treating open chains of even length up to 60, and maintaining 64 (and in some cases, where necessary, 128) states in each block. The ground state has total spin 0 and is at half-filling; we add a single electron (pair of electrons of opposite spin) to compute the spin (charge) gap for each 4  length. These results are extrapolated to infinite length taking into account truncation error uncertainties.  The figures summarize the results of our D M R G calculations, as  explained in this section.  4.4.2  S p i n G a p for V > 0  We find that the spin gap does not vanish for any V > 0, as shown in Fig. 4.2. In comparing its dependence on V with that predicted from the analytical R G of Ref. [46], namely A « te~*  (4.25)  t/v  s  which is valid for small V/t, it should be kept in mind that the numerical work is not dependable for V/t < 1 because there the expected correlation length £ « v /2A becomes F  of order the system size L. The finite-size gap alone is A expect to measure A/t lower than A s A F  ~  « -KV /L SO that one cannot  F S  F  2-K/L K, 0.1 for L = 60.  The R G flow equations to two-loop order, after dropping p as explained in Sec. 4.3, 4  give for the spin coupling g  s  P . -  1  - ^  - ^ l ± | ^  1  2  4  1+  = l n f o  (4.26)  L  T h i s assignment is valid since in this case there is no charge gap wherever there is a spin gap.  Chapter 4. The One-Dimensional Pair-Hopping Model  53  v/t The open squares and dashed line are the charge gap, and the filled squares and solid line are the spin gap. A clear phase transition is evident near V = V m —1M, but for positive V, the spin gap opens up from V = 0. The error bars indicate uncertainty in extrapolating L —• 0; the lines are to guide the eye. (The error bars on the charge gap for positive V are large simply because there was no need to accurately compute these points which are firmly established to be zero on analytical grounds.) c  _  1  Figure 4.2: Summary of numerical results for the pair-hopping model.  54  Chapter 4. The One-Dimensional Pair-Hopping Model  where L is an initial length scale (A = L is the ultraviolet cut-off) and the initial spin 1  0  coupling is [47] (4.27) We take the spin gap to be the energy (inverse length) scale at which g enters the regime s  of strong coupling, specifically where g = a = 0(—1), resulting in s  (4.28) where a = O(-l)  is used as the criterion for \g \ becoming large. Fig. 4.3 shows that the s  numerically computed spin gap is indeed of the form predicted by the R G flow equations with <? dropped. 4  While the above comparison of the R G flows are to numerical results on open (not periodic) chains, we believe these results (and in particular the non-vanishing of the gap for V > 0) constitute a reliable estimate of the situation in the thermodynamic limit. However the situation is somewhat different in the phase-separated region (V > V 3.52 and V < V  c2  cl  w  < 0) than in the non-phase-separated region at smaller |V|. In the  non-phase-separated region, the excitation which we study is concentrated in the bulk of the chain as discussed in Sec. 4.4.4 and Figs. 4.4-4.6. Thus we expect that its excitation energy is not affected significantly by the boundary conditions for sufficiently long chains. However, in the phase-separated region, the excitation lives near the ends of the chain and its energy may well be strongly affected by the boundary conditions. In this case, the energy which we measure is still a lower bound on the bulk gap. This follows because the state which we study is the lowest energy one with these quantum numbers. If the bulk gap were lower, we would expect a lower energy state to exist, localized far from the chain ends. Thus our results give strong evidence for a spin gap for all V > 0 but only give a reliable estimate of the size of the gap for V  c2  < V < V \ « 3.52, except for  magnitudes less than the finite-size gap as discussed above.  c  Chapter 4. The One-Dimensional Pair-Hopping Model  55  2  0 -to  < 2  4  -  0  0.5  1  1.5  2  t/V Fitting the data for only two points, namely V/t = 1,2, to the form given by Eq. (4.28), the dashed lines are the upper and lower limits of the resulting fitted curves taking the numerical error bars into account. The lower limit extrapolates well over the range t < V < At, which is the expected region of validity. (Thefitis not expected to be valid for V < t because of the finite-size gap [see text], while V > K i ~ 3.5t is the phase-separated region.) Figure 4.3: Fitting the DMRG-computed spin gaps to the prediction from renormalization group flows.  56  Chapter 4. The One-Dimensional Pair-Hopping Model  10  20 Site Index i  30  Figure 4.4: Expectation values (S (i)) for for different values of V/t, for one electron added relative to half-filling. The unpaired electron delocalizes into the chain near V/t = 3.5. (The L = 60 chain is symmetric about its central link.) z  Chapter 4. The One-Dimensional Pair-Hopping Model  57  Figure 4.5: Spin at a chain end (open squares) and net spin in the centre half of the chain (filled squares) as a function of V/t for a single added electron.  Chapter 4. The One-Dimensional Pair-Hopping Model  Figure 4.6: Spin at a chain end (open squares) and net spin in the centre chain (filled squares) as a function of V/t for two added electrons.  Chapter 4. The One-Dimensional Pair-Hopping Model  4.4.3  59  P h a s e T r a n s i t i o n at V = V « —1.5* c  As discussed in Ref. [46], for small V < 0 the pair-hopping model is identical to the positive U Hubbard model. Thus we expect a charge gap but no spin gap in this region. It was also argued that there should be a phase transition at finite V < 0 because at V/t —> —oo there is no charge gap but a spin gap. In Fig. 4.2 we present D M R G results confirming this prediction, with the transition occurring at V = V ~ —1.52. c  Our numerical results are consistent with the spin gap appearing at the same critical coupling at which the charge gap disappears; however, the presence of two distinct critical couplings cannot be ruled out. It is unclear to us whether this critical point (or points) simply corresponds to the renormalized couplings g and 5-3 passing through zero, or to s  some more exotic critical point.  4.4.4  P h a s e Separation at V — V \ « 3.52 c  To demonstrate the phase-separation transition, we examine the behaviour of wave functions obtained using the D M R G at L = 60 in the sector of one electron added relative to half-filling.  Specifically, we plot in Fig. 4.4 the expectation value of S (i) z  for sites  i = 1,... ,30 (the chain is symmetric about the central link) for different values of V/t. For large V, the excess spin is localized at the chain ends, and as V is reduced, the spin extends further into the bulk. As V/t drops from 4 to 3, looking at the wave function near the centre of the chain shows that near these values of V/t the spin becomes unbound from the chain end and is rapidly and fully delocalized into the bulk of the chain, leading us to consider V i « 3.52 as a phase-separation critical point. This conclusion is further c  verified by examining the spin on the chain end as a function of V/t, as well as the total spin in the centre half of the chain, as shown in Figs. 4.5 and 4.6. Due to the fact that we have employed the infinite system method, instead of the finite  60  Chapter 4. The One-Dimensional Pair-Hopping Model  system method[30], these wavefunctions are not expected to be precise particularly near the phase-separation transition and at the chain ends. However, we expect that the results are accurate to within a few per cent at worst, certainly not affecting the qualitative behaviour of our figures which clearly demonstrate the phase-separation transition. While a priori this phase transition could occur at a different value of V than the bulk phase separation, the simplest scenario would have both transitions occurring at the same point: essentially the bulk transition drives the boundary transition. The numerical evidence on one and two added electrons seems to indicate that for low net magnetization, V i is constant. c  This phase-separation transition will occur for finite V i in the periodic and infinite c  chain as well (though not necessarily at the same value of V i as for the open chain): c  added unpaired electrons will still break the chain into chainlets and the energy will be minimized if they clump together. However, it will be more difficult to detect in a periodic chain since the ground state is usually translationally invariant.  Chapter 5  T h e One-Dimensional Heisenberg-Kondo L a t t i c e M o d e l  5.1  Introduction  The magnetic and electronic properties of conductors having embedded magnetic impurities continue to be shrouded to quite some degree in mystery, even though they have been studied for over thirty years. Their simplest theoretical description is via the singleimpurity Kondo model together with the assumption that the magnetic impurities do not interact with one another. The Hamiltonian for the single-impurity Kondo model is [59] ^ =E ^ 4 A where d  + K5- -O,  (5-1)  J  S  is the creation operator for a conduction electron of wavevector k and spin  kc  a, e£ is the conduction electron dispersion relation, S is the spin operator of the single magnetic impurity, and so = E 4 4 ^ 0 / 3 = E "0  4J^ fc'/?  ( - )  c  5  2  Wa/3  is the conduction-electron spin operator at the location i — 0 of the impurity. A more fundamental Hamiltonian, from which the Kondo Hamiltonian can be derived via the Schrieffer-Wolff transformation [60], is that of the Anderson model [61]  # = E Wl k<r c  +  E V*Af. + h.c.  +  e / E fife  (5-3)  kcr  where fl is the creation operator for an / electron of spin a which hybridizes with the conduction electrons at the impurity site via the transition matrix element V f, and c  61  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  62  e/ is its energy level. Usually one considers only one / electron hybridizing with the conduction electrons, so the restriction YJ„  flf  c  < 1 is employed.  Since the only length scale in the single-impurity Kondo model is £ ~ ae  1/pjK  K  (where  a is the lattice spacing and p is the electronic density of states per spin at the Fermi surface) as obtained from the renormalization group viewpoint [29], the non-interacting approximation should apply for impurity densities n  Because £K 3>  i m p  < 1/&  (5.4)  and in fact typically £K ~ 1000a, the single-impurity Kondo model  should apply only for very small impurity concentrations. Surprisingly, it seems to work for quite large impurity concentrations as well, while the sample purities sufficient to probe the true low density limit are not experimentally realizable. The same length scale £  K  has also recently been seen to arise in a scaling theory of the Kondo screening cloud  [62, 63, 64], countering an alternative explanation of the linear scaling with impurity density of measured quantities such as resistivity and impurity susceptibility, namely that the screening takes place on short length scales. Heavy fermion (or more specifically, but less popularly, "heavy electron") compounds include a rare earth or actinide element such as cerium (Ce), ytterbium (Yb), or uranium (U) in which the / shell is partially filled. In the high-temperature range (which is room temperature, for example), experiment indicates that these / electrons behave as freely fluctuating localized magnetic moments which are weakly coupled to a conduction band containing non-/ electrons. On the other hand, at low temperatures the materials exhibit properties approximately mimicing a Fermi liquid but one in which the fermions have large effective mass, on the order of 10 to 10 times the electron mass. As in the case of 2  3  impurities, a basic question remains as to whether the local magnetic moments—which in this case are located at every unit cell—are either (a) screened from one another by being  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  63  quenched by the conduction electrons or are otherwise independent and non-interacting or (b) participate in the formation of a collective magnetic state. Experimentally, many heavy fermion materials do exhibit magnetic ground states, albeit often of surprising form or magnitude. Because it is widely believed that the quenching of the local moments into a heavy Fermi liquid can somehow be based on the Kondo effect, theoretical models of the Kondo lattice type have been, and continue to be, proposed and studied in attempts to understand this magnetic behaviour, quite apart from the superconductivity which often closely accompanies or coexists with the magnetism. In these models, local / electrons are present at each lattice site, instead of just at impurity sites, and interact with the electrons in the conduction band. In the first of these, Doniach actually neglected the charge degrees of freedom of the conduction electrons altogether by introducing the Kondo necklace model [65]:  #KNM = J  ±  +  " "]  5  + JKE&-  5  (-) 5  5  where in the notation adopted in this thesis 5, and S; represent the site-i s p i n - | operators for the / and conduction electrons respectively, obeying the standard commutation relations (5.6)  [5?, 5}] c  E^; o.  (5.7) (5.8)  This model consists of two spin-| chains with an anisotropic nearest-neighbour Heisenberg coupling between those representing the conduction electrons and an isotropic (interchain) coupling between the two types of electrons.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  64  A n extension of the standard Kondo necklace model of Eq. (5.5) was considered by Strong and Millis [66, 67]. Their model was also a Heisenberg spin ladder with 1  generally anisotropic intrachain couplings on both / and conduction electron chains and an isotropic interchain coupling:  i w = E{^K4 +"4] + ii ^} x  5  J  s  (ij)  + Y,{J As?s* + s?s]] + jfs;s;} f  +J ££W,.  (5.9)  K  i  A more general model including the charge degrees of freedom of the conduction electrons, but still assuming the / electrons manifest themselves only in their spin, is the Kondo lattice model, which is basically the Kondo model except that a magnetic moment is now located at every lattice site [69]: # K L M = ~t £ (ij)cr  c\ C a  j(T  +  h.C.  + J Y^SiK  Si.  (5.10)  i  Here c and s are the operators for conduction electrons as denned in Chapter 2; we take the Kondo coupling to be antiferromagnetic ( J K > 0) as usual. The / electrons are considered to be completely localized and we assume that each site has exactly one / electron; therefore, in the Hamiltonian we have represented the / electron on each site just by its single spin-| operator.  Any hybridization that would occur between these  / electrons and the conduction electrons, via an Anderson interaction term, is assumed to have been taken care of by the presence of the (on-site) Kondo interaction. A t halffilling, the Kondo lattice model has both charge and spin gaps for all non-zero coupling strengths [69, 70, 71]. Away from half-filling, it is well established that at large Kondocoupling there is an incompletely ferromagnetic phase [72, 40], but not much is known 1  T h i s model is a specialization of two-chain Heisenberg systems studied by Schulz [68].  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  65  about the smaller coupling case except that the ground state appears to be paramagnetic and consistent with a Luttinger-liquid picture with the / electrons contributing to the Fermi surface [40, 73]. We would like to understand the magnetic properties of the Kondo lattice model away from half-filling; in particular we would like to know if there is a spin gap in some region of the phase diagram. 5.2  T h e H a m i l t o n i a n of the H e i s e n b e r g - K o n d o L a t t i c e M o d e l  We consider in addition to the Kondo interaction in the standard Kondo lattice model a direct Heisenberg coupling J H between the localized / spins. That is, we take as Hamiltonian #H-KLM = ~ * £  <h ja c  +  H  (ij)cr  - C  + JKJ2 i  • Si +  •  Sj.  (5.11)  (ij)  Considering this as being a coupled two-chain system, the / spins may be thought of as residing on one chain, and the conduction electrons on the other. The lattice and coupling constants are depicted in Fig. 5.1, and a typical configuration of spins and electrons in Fig. 5.2.  spin chain K t  conduction electrons  Figure 5.1: The lattice and coupling constants for the one-dimensional Heisenberg-Kondo lattice model. In general, there will always be some effective Heisenberg interaction between the / spins, even in the pure Kondo lattice model of Eq. (5.10), such as a Ruderman-KittelKasuya-Yosida ( R K K Y ) interaction mediated by the conduction electrons. This R K K Y  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  66  Figure 5.2: A typical configuration of spins and conduction electrons in a general state of the Heisenberg-Kondo lattice model. Double (single) arrows represent / (conduction-electron) spins. term would be weaker and of longer range than our direct Heisenberg term which we consider to come from a different superexchange process.  In some organic materials  which are potentially described by the above model, the direct and R R K Y terms have been experimentally estimated; this will be described in Section 5.3. Throughout, we restrict attention to the case of antiferromagnetic couplings J , J K  H  (5.12)  > 0.  The Heisenberg-Kondo lattice model retains the particle-hole symmetry of the free conduction-electron hopping term discussed in Chapter 2 upon the extension of the transformation to include its effect Q — (Q  x  c?y Q \ _ W z  qx  qy  qz\  (5.13)  on the / spins to counter the corresponding sign changes in the components of the conduction electron operator  under P H given in Eq. (2.20). This means that every  result determined for below half-filling will correspond directly to one above half-filling. For simplicity, we therefore consider only the case of half-filling or below. While this Hamiltonian as written can be considered in any number of dimensions, we consider it mainly on a one-dimensional lattice, although several aspects will be relevant to higher-dimensional cases as well.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  67  The Heisenberg-Kondo lattice model is motivated by approximate experimental realizations, related theories of high temperature superconductivity, a mapping to a modified t-J model, and its accessibility to bosonization and numerical analyses. These will be discussed in turn in the following sections.  5.3  Relation to Experiment  A n explicit nearest-neighbour Heisenberg spin coupling J  H  in addition to the pure one-  dimensional Kondo lattice model could be generated by conduction-electron exchange via the R K K Y mechanism. Alternatively, or in combination with this, a direct exchange coupling of some other origin is possible. In a series of experiments on organic chain conductors, Ogawa et al. [74, 75] and Quirion et al. [76] showed that two related quasi-one-dimensional conductors exhibit low-temperature transitions caused by the strong coupling of conduction electrons to a regular array of local electronic moments. These compounds are Cu(tatbp)I, where "tatbp" represents triazatetrabenzporphyrinato, and Cu(pc)I, where "pc" represents phthalocyanine. The copper sites are seen to remain in a definite valence of C u 2  2 +  without  fluctuations, allowing the possibility (quite unusual in transition metals) of a Kondo-type coupling between the copper and the conduction electrons. The copper spins also have a direct Heisenberg coupling to one another. There is a high degree of one-dimensionality, as the ratio of inter- to intra-chain couplings is much less than 10~ [74]. 4  For Cu(pc)I, the direct C u - C u 2 +  2 +  coupling is determined [74] from its spin-lattice  relaxation rate and that of its insulating parent compound Cu(pc) to be J H ~ 0.6 K-/CB; 3  T h i s is seen in that the magnetic susceptibility is in excellent agreement with a temperatureindependent part due to conduction electrons plus a Curie-Weiss part giving the value of | for the copper spin [74, 75]. T h i s is done by comparing the slope of characteristic times I i [for Cu(pc)] and Tis [for Cu(pc)I] versus frequency u with spin chain calculations, together with the ratio 7 / ? n of electronic and nuclear gyromagnetic ratios and geometrical coefficients determined from the molecular structure. 2  3  e  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  68  their total coupling, including that mediated by the conduction electrons via the Kondo coupling, is estimated from the Curie-Weiss temperature and the E P R (electron paramagnetic resonance) linewidth data to be about 12 K-k . B  Estimates of Kondo coupling J  depend on the theoretical model considered; one model [77] predicts J 4  K  K  ~ 1600 K-fc B  The hopping coefficient is estimated to be t m 3600 K-A; from thermoelectric power B  measurements [75]. In both of these compounds, the magnetic interactions between the copper atoms together with their coupling to the conduction electron spins are believed to cause their observed strange low-temperature behaviour, such as an sharp enhancement of the anomalous p-value (gyromagnetic factor) measured by E P R below about 8 K . There is no good theoretical basis for this, as the authors point out, since there had been no well-studied models of this type of system. Caron and Bourbonnais [77] claim that the Kondo interaction in these materials has to be a non-local one, because at the filling n = 1/3 which is the case in these materials the local coupling cannot give, via the R K K Y interaction alone, a Neel temperature as observed but only a Curie temperature (i.e. a negative Neel temperature).  They  therefore consider a model with a Kondo coupling whose g-dependence is of the form J (q) K  — JKQ[1 + acos^a]. Their computations, proceeding by way of a random phase  approximation, are able to account for the behaviour of the anomalous g-value down to about 8 K but not its subsequent sharp rise. Our study of the Heisenberg-Kondo lattice model will hopefully be helpful in the development of a theory to describe these organic chain compounds with greater certainty and accuracy. T h i s assumes its authors consider the Kondo interaction normalized according to 2 J yjV Si • Si as do the experimental papers they refer to. 4  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  5.4  69  R e l a t i o n t o Theories of H i g h T e m p e r a t u r e S u p e r c o n d u c t i v i t y  In the high-critical-temperature cuprate superconductors, magnetic interactions are widely believed to play an important role. Certainly the proximity as a function of doping of the antiferromagnetic phase to the superconducting one gives credence to this. The direct exchange interactions between the magnetic copper sites ( C u ) are what lead to this 2 +  antiferromagnetism. A general Anderson-like tight-binding model[78] for high-T superc  conductors includes one copper orbital and two oxygen orbitals per unit cell. The basic parameters in the model include a C u - 0 hopping parameter, on-site Hubbard repulsion terms for copper and oxygen orbitals, and a single electron energy difference between copper and oxygen orbitals. In the limit where the energy difference and the copper Hubbard repulsion are large, the occupation of the copper sites is frozen at one and the mobile holes reside exclusively on the oxygens, resulting in a Heisenberg-Kondo lattice model. While it is unclear that the copper oxides are in this parameter range, such models have been studied by a number of authors [79, 80, 81, 82, 83, 84, 85]. Of course, the work in this thesis relates only to the one-dimensional model. The mechanism for high temperature superconductivity proposed in October 1996 by Emery, Kivelson, and Zachar [23] was referred to briefly in the introduction of the previous chapter (i.e. Section 4.1). In addition to having a pair-hopping interaction leading to spin-gap formation, their model is also closely related to one-dimensional Kondo lattice models such as our Hamiltonian (without the direct Heisenberg term) in a formal way, involving a comparison of Toulouse points in the spin sectors of the Hamiltonians. One5  dimensionality plays an important role in this model as well, via the charged stripes formed by phase separation.  Studying the one-dimensional Heisenberg-Kondo lattice  model will hopefully help to clarify the spin-gap behaviour of the models and therefore 5  A Toulouse point is a particular solvable point made manifest upon a unitary transformation.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  70  possibly of the high temperature superconductors as well.  5.5  C o n n e c t i o n W i t h the t-J M o d e l a n d F e r r o m a g n e t i s m  In this section, I will demonstrate that for large Kondo coupling J , the HeisenbergK  Kondo lattice model in any dimension and at any conduction-electron band-filling away from half-filling is equivalent to a modified t-J model except in the small J H limit . 6  Because of the particle-hole symmetry in the model, we show this for fillings below half. The starting point for the establishment of this connection is the fact [72] that the Kondo lattice model at strong coupling below half-filling has an effective Hamiltonian involving the hopping of unpaired / electrons on a background of on-site singlet pairs consisting of an / electron and a conduction electron. This effective Hamiltonian was derived in one of a series of papers [87, 70, 72, 88, 89] which further demonstrated, by a double application of the Perron-Frobenius theorem, the origin of ferromagnetism in the one-dimensional Kondo lattice model in this limit. (This I also show to be true in any dimension by a simpler argument in Appendix A.) We can then determine that there is a region of the phase diagram of the Heisenberg-Kondo lattice model in which there definitely is a spin gap, based on numerical results on the one-dimensional t-J model [57, 90, 91, 92, 93, 58]. While the only degree of freedom for the / electrons which we are considering in our model is their spin, we may still choose to represent them by means of fermionic creation and annihilation operators fl and f , with the restriction a  ia  E/i/«r = l 6  Vi  (5.14)  T h i s statement, which was first made in Ref. [86], will become more precise later in this section.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  71  of single occupancy, and the spin is expressed as 3 =  ( - ) 5  1 5  a/3  This representation will prove to be helpful in understanding the large-JK limit. Below half-filling, the t = 0 limit of the Kondo lattice model has a ground state consisting of independent sites, as many as possible being on-site singlet pairs of an / electron and a conduction electron  | g ) =  7  l i H J t )  s  MzM\  0)  (5. ) 18  and the rest being unpaired / electrons of either spin I ft) or | 4)-  (5-17)  The ground state of the Kondo lattice model for t = 0 in the sector of Hilbert space corresponding to a particular filling below half-filling is therefore degenerate as it involves linear combinations, with unconstrained coefficients, of wave functions of the form |s,fr,jL,s,s,^,s,fr,ir,s,^,...).  (5.18)  This huge degeneracy is broken by including the conduction electron hopping term t. This term will allow the effective motion of unpaired / spins on the background of on-site singlet pairs. The effective Hamiltonian for the Kondo lattice model below half-filling for large Kondo coupling in any dimension is, as obtained by Sigrist et al.[72],  {i))cr  where ]}„ = (1 - n^fl,  (here  = T, n a  ia  = y j ^ c f ^ is the site-i number operator for  conduction electrons) is an operator representing the creation of an unpaired / electron. Here I introduce double arrows to represent / electrons and single arrows to represent conduction electrons, as in F i g . 5.2. 7  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  72  The first term in this effective Hamiltonian represents the hopping of unpaired / electrons on a background of on-site singlet pairs of / and conduction electrons. Of course, it is still really the conduction electrons which are hopping, but because of the large Kondo coupling, these are always absorbed into on-site singlet pairs. The factor of \ enters simply because when the conduction electron hopping occurs, it can only involve that component of the on-site singlet pair which contains the conduction electron of the spin necessary to form an on-site singlet pair with the unpaired / electron on the site to which it hops. For example, (S,fr  |#  K L  MI  IT, s> = ^=(U,1t I H 4 | C ) l 1t,U)-j= = - * / 2 . 2 i  (5.20)  Because of the earlier-demonstrated particle-hole symmetry, the effective Hamiltonian above half-filling describes / electrons together with singlet pairs of conduction electrons hopping on a background of on-site / - and conduction-electron singlet pairs. The terms of order 2 / J K were demonstrated by Sigrist et al.[72] to be responsible for the formation 2  of a ferromagnetic ground state in the one-dimensional Kondo lattice model away from half-filling. To this Hamiltonian, we consider the addition of a direct Heisenberg coupling between the / spins. For large enough Kondo coupling J K , we choose this extra coupling J  H  to  be such that JK>^,^H  (5.21)  so that we are still in the strong-Kondo-coupling limit of the model, and t /J 2  K  < Jn  (5.22)  so that the terms in the effective Hamiltonian responsible for ferromagnetism are no longer able to secure a ferromagnetic state.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  73  I now show that the resulting Hamiltonian is equivalent to a modified t-J model plus small corrections. The standard t-J model is H  TJ  = -UJ  4c  £  a  j(r  + h.c. +  J  £[£  tJ  (ij)"  (5.23)  • Sj - IriiUj}.  (ij)  Here the c and s operators are both for conduction electrons. Since the t-J model is usually thought of as having been derived from a large-t/ Hubbard model (in which case J  = At jlU), the usually-implicit assumption, even when the model is being considered 2  tJ  in its own right, is that the sites are restricted from being doubly-occupied. This is enforced either by a direct restriction on the Hilbert space of each site, or indirectly by rewriting the hopping term as  -  * * J £  ( !  -  ni-MoCjA^  ~  3&) + h-c,  n  ( 5 - 2 4 )  (ij)cr  or equivalently, using the fermion anticommutation relations, - UJ  £  {c.-Mclc^-A-.)  + h.c,  (5.25)  (ij)"  where ' T, <r=l  I,  (5.26)  *=T.  In our study of the Heisenberg-Kondo lattice model in its large Kondo-coupling limit, we will not compare directly to the standard t-J model, but rather to a "modified t-J model" which does not have the nearest-neighbour Coulomb repulsion term —J j 12(ij) niUj/A of t  the t-J model. This will entail a careful consideration of the effect of this term on the spin-gap region of interest. In adding the Heisenberg term to the large-coupling Kondo lattice model we couple not the conduction electrons but the / spins to each other. If the coupling existed only between those unpaired f spins which hop with matrix element —t/2, the connection to  74  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  the modified t-J model would be direct. In fact, the Heisenberg coupling is ineffectual 8  on / spins which are already strongly coupled with the conduction electrons in an on-site singlet pair. Denoting by P the projection into the low-energy subspace of the large-J  K  Kondo model, we simply have on each site PS\S) = PS[\ 1U) - I m  = 0,  (5.27)  where S is the /-electron spin operator for that site, as can be readily seen by considering the three components S , S , and S~ separately. This means that we need only consider z  +  the Heisenberg interaction between unpaired / spins in this limit. To complete the connection with the large-JK Heisenberg-Kondo lattice model, it is important to remember that the conduction electrons of the t-J model correspond to the unpaired / electrons of the Heisenberg-Kondo lattice model. Denoting by n and n j the conduction electron t  band-fillings in the Heisenberg-Kondo lattice and t-J models respectively, the correspondence is through n  tJ  = |1 — n\ (n = 1 is the symmetric half-filling point), keeping in mind  that t  tJ  = t/2 and J  tJ  The one-dimensional t-J model, including the  = J .  (5.28)  H  UiUj  term mentioned above, is known  from numerical and analytical study to have a spin-gap phase for low density n j and t  J  > 2t j. Two slightly different numerical determinations of the phase diagram are  tJ  t  depicted in Figures 5.3 and 5.4; an analytic determination of the spin gap in the twoelectron case is repeated later in this section. We focus on the spin-gap region, which apparently includes a convex region of phase space connecting the vertices (n j,J j/t j) t  t  t  = (0,2), (0.23,3.0), and (0,2.7).  On the  boundary of this region with the (gapless) Luttinger liquid phase, the spin gap is zero, and N o t e also that the single-occupancy restriction is automatically enforced simply because of the fact that each site has one / spin. 8  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  75  Figure 5.3: The phase diagram of the one-dimensional t-J model, determined by projecting the true ground state from a wave function determined by a variational Monte Carlo method (adapted from Ref. [58]; original figure courtesy of C. Stephen Hellberg).  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  76  Figure 5.4: A second version of the one-dimensional t-J model's phase diagram, based on a variational Monte Carlo method (adapted from Ref. [93]; original figure courtesy of Kenji Kobayashi). Note that the axes are interchanged from those of Fig. 5.3.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  there are indications that the spin gap increases with increasing J  and with decreasing  tJ  n j [94], so that it is likely a maximum near the point (n j, Jtj/Uj) t  t  Since the term containing  UiUj  77  — (0,2.7).  in the t-J model involves only a charge interaction, it  may be expected that dropping this term will not diminish the spin-gap in this region, and possibly cause the region to expand, especially into the phase-separated region which is most strongly affected by this term [94]. More careful consideration of the effect of the t-J model's Coulomb repulsion term indicates that the spin-gap region will actually be shifted to higher values of J j if it is neglected. This can be seen by comparing the case t  of two conduction electrons in the t-J model [92] and in the modified t-J model which determines the J j-intercept of the boundary separating the spin-gap and Luttinger-liquid t  phases. For sufficiently large J j, in either the t-J or the modified t-J model, the ground state t  in the sector of two electrons will clearly involve a singlet state. Taking as our ansatz the wavefunction oo  l*> = Z>" il°>>  (5-29)  6  71 = 1  where a is a constant to be determined and  ^^4=E^44 „ +  (5.30)  i/3  is a singlet state creation operator giving two electrons separated by n sites. Here ' 1,  ^  (a,/?) = ( U )  = 1 " I , K/?)= ( U ) 0,  (5-31)  otherwise  is the totally anti-symmetric Levi-Civita tensor used to enforce the Fermi statistics. Since the application of the t-J Hamiltonian on the singlet state is . (-2t blJ b\)\0), n= 1  #^1|0)  =T  tJ  tJ  ,  - 2 * ( & i - i + &i+i)|0>, t J  n>l  (5-32)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  78  we obtain by comparing coefficients of &JJ0) in the Schrodinger equation H\ty) = E\V) that a = 2t j/Jt.j and E = —J j — 4t j/J j.  Evidently, this solution is valid only for  2  t  t  a < 1, i.e. J  t  > 2t j, or the wave function Eq. (5.29) would not be normalizable. In this  tJ  t  case, the ground state is a singlet bound state with approximate size l / l n ( J /2t j). tJ  A  t  simple estimate of the density-dependence of the boundary separating the spin-gap and Luttinger-liquid phases of the t-J model is to determine at what point the bound states overlap. That is, we equate 2/n j with the effective pair size, which can be taken to t  be some constant a as 1 times this. This gives for small J j/2t j t  boundary n  tJ  = 4a(J /2t j tJ  t  t  — 1 the approximate  — 1), or more transparently J j/t j t  = n /2a  t  + 2. The  tJ  numerical work in the literature[58, 95] is of insufficient accuracy to test this prediction. The number of points measured in the low-density region is quite small: only three data points in Ref. [95] (which are used to determine the phase diagram of Ref. [58] depicted in Fig. 5.3) are at n j < 0.1, and with error bars too large to allow a reasonable measurement t  of a. This could be the subject of a more precise D M R G calculation. We now consider the triplet states. In this case, the  riiUj  term cancels the  S; • Sj  term  in the Hamiltonian and we are left with a system of two non-interacting spinless fermions. On the infinite chain, the energies are minimized by giving each fermion maximum spatial freedom, each having the dispersion relation  = —2t jCosk, for a total ground state  energy of — 4t j. Hence the spin gap when J  t  t  tJ  > 2t j is — 4t + J t  tJ  tJ  +  4t j/J j. 2  t  Now if we consider the modified t-J model, i.e. without the rtiUj term, we note that the only effect is that J j is replaced by 3 J j / 4 in the single matrix element involving J t  (  tJ  in the &t|0) representation for the singlet ground state, and in the triplet case an explicit Coulomb repulsion J j/4 is added to the Pauli exclusion. Thus the bound state is only t  present for J  tJ  > 8t j/3 and the gap in this case is now — At + 3 J j / 4 + t  tJ  t  16t j/3J . 2  tJ  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  79  This scaling of the spin-gap/Luttinger-liquid phase boundary is in this way demonstrably valid only for a two-electron system; however, we assume that to a first approximation the factor of 4/3 is applicable to general low-density systems. We now consider the boundary between the spin-gap and phase-separation phases. Chen and Lee [92] estimate this boundary by considering two variational states. In the spin-gap phase, they consider a non-interacting gas of singlet pairs, as discussed above, and in the phase-separated phase, they consider the completely phase-separated situation of having the n L electrons completely condensed into a Heisenberg chain, which has an tJ  energy —nLJ \n2  (here L is the length of the chain). Thus the boundary occurs at the  tJ  solution of (5.33) which gives J j = 3.218t j. This is actually an upper bound on the boundary at small t  electron density n  t  tJ  < 1 [94], and numerical estimates of the boundary range as low as  Ju — 2.7t j. t  Without the riiUj term, i.e. in the modified t-J model, we have instead the requirement (5.34) which gives J j = Q.255t j, more than double the estimate in the standard t-J model. t  t  This implies, even though this too is a lower bound, that the spin-gap phase in the modified t-J model is much larger than that of the standard t-J model, but apart from this rough estimate, the precise boundary is not known. In fact, because of the strong finite-size effects which will enter (because even for long chains, the number of electrons participating in the fully condensed phase will be quite small for small fillings n j ~ 1/8) t  we will not be able to probe this phase separated region with any confidence at this stage.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  80  In the Heisenberg-Kondo lattice model, we are looking for a region in the phase diagram which has a spin gap and no charge gap. The full phase diagram of the HeisenbergKondo lattice model is three dimensional, with parameters Jx/t,  Ju/t, and the filling n.  Since the filling is the most difficult parameter to vary in the numerical D M R G study, we choose a particular filling n = 7/8 to ensure that in the large J through the spin-gap region by varying J - At large J H  J  K  K  limit we can pass  we expect, then, as we increase  from zero, to find first a small ferromagnetic region until the Heisenberg term be-  H  comes comparable to the 0(t /JK) 2  fluctuations  responsible for ferromagnetism, then a  Luttinger liquid region, followed by a spin-gap region containing at least the approximate range 2.9 = (4/3)2.2 < J / ( £ / 2 ) < 6 (for n = 7/8), and finally a phase separated region H  [we have used the relation t j = 2t from Eq. (5.28)]. For larger intermediate couplings t  where J  H  is no longer much less than J , the physics is not predictable in this way, but K  we will have to numerically determine the extent of the spin-gap phase in the complete phase diagram of the Heisenberg-Kondo lattice model.  5.6  Bosonization Analysis  It is instructive to think of the Heisenberg-Kondo lattice model as a two-chain system (or two-leg ladder in the terminology of Section 1.2). In particular, the direct Heisenberg term had to be added to the Kondo lattice model in order to bring bosonization to bear on the problem [96, 86]. In this section, I discuss the formation of spin gaps in a 9  series of related two-chain systems, the Heisenberg ladder, the zig-zag Heisenberg ladder, and the Heisenberg-Kondo lattice model at and away from half-filling. I also consider spontaneous dimerization in the pure spin systems and its concealed variant in systems with charge degrees of freedom, as well as related work on the pure Kondo lattice model A n o t h e r way of using bosonization, that of Ref. [97], does not require a JH term but has other requirements; this is discussed in Section 5.6.4. 9  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  •  — •  •  i i i  i i i  i  *  —  iJ  i  *  *  i i i  , i  *  81  9  »  i i i  i i i  * •  Figure 5.5: The two-leg Heisenberg spin-^ ladder, in a strongly anisotropic limit. 5.6.1  T h e Heisenberg Ladders  In the single Heisenberg spin-| chain having the Hamiltonian H^ = JY^Si'S  (5.35)  i+lt  i  we may represent the spins under non-Abelian bosonization by Sj = J + J + const, x {-l) Tr(dg), j  L  K  (5.36)  a sum of constant and alternating parts. Here J L and JR. are SU(2) currents and g is an SU(2) matrix field, with corresponding scaling dimensions dim(J  )  =  1  (5.37)  dim(^)  =  \.  (5.38)  LiR  As represented in Fig. 5.5, we may couple two spin-| chains together to form a spin ladder with Hamiltonian #SL = h E S ? ' • Sil\ + h E  • Sf\  (5.39)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  +  ^2  i  •  •  •  •  •  leg 1  *  • i+l  i  82  * leg 2  •  Figure 5.6: The two-leg Heisenberg zig-zag spin-| ladder. where J i is the interchain coupling and J the intrachain coupling. To understand the 2  low energy physics for Ji <C J we consider the most relevant interaction term this gives 2  rise to, i.e. the term with the smallest scaling dimension, upon bosonization. The most relevant interaction term in J\S^ •  is, using the representation (5.36) and the scaling  dimensions (5.37) and (5.38), J [ ( - l ) T r ( ^ ) ) ] • [(-l)*Tr(cV )] = JiTr(ag^) i  1  • Tr(dg^),  2)  1  (5.40)  which has the scaling dimension \ + i = 1 of a relevant operator, and therefore produces a (spin) gap proportional to J\ plus logarithmic corrections arising from other (marginal) operators. Consider now instead of perpendicular rungs a staggered interchain Heisenberg coupling, depicted in Fig. 5.6, having the Hamiltonian  #ZZL = h  E  > •  + Ji  ia  E Sl  1]  • $ + SH\}. 2)  (5.41)  i  For this zig-zag ladder, studied in Ref. [86], in determining the most relevant operator, we no longer have the cancellation ( —1)'(—1) = 1 of signs which gave a relevant interchain J  J  interaction in the standard Heisenberg ladder's non-Abelian bosonization representation, but we now have an exact cancellation of terms: ( —1)'[(—1)' + (—1) ' ] = 0. So in this J  J  J +1  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  83  JJ  0  Figure 5.7: Renormalization group flow of the marginal interchain coupling J i in the zig-zag spin ladder. case the most relevant interaction term is H  M  =  MJT  + JT) • (Jf  =  M ? » - JT  + JV  + JV)  (5.42)  + JI"  • 4  2)  •4  2)  + 4"  • 4'),  (5.43)  which has the dimension 1 + 1 = 2 of a marginal operator. The first two terms, each involving only one of L and R, only serve to renormalize the Fermi velocity and we therefore assume that they can be neglected. We are left with an interaction Hamiltonian involving four independent SU(2) currents: #int =  JI(JV  • JV +  • Jf ).  (5.44)  ]  One may now simply interchange the labelling for  and J ^ and the result is propor2)  tional to the non-interacting Hamiltonian which can be written HO = EMJV  + JV) , 2  (5-45)  a  whose expansion is identical to that of H  mt  after dropping its velocity-renormalizing  terms. Under renormalization group flow, the marginal interaction involving Ji flows to zero coupling for J\ < 0 and to strong coupling for J\ > 0 as depicted in Fig. 5.7. Therefore for ferromagnetic interchain coupling Ji < 0 this gives a gapless system, while antiferromagnetic couplings J\ > 0 there is a spin gap A oc e ~ s  a j 2 / J l  ,  (5.46)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  /  /  / /  /  .  /  -.  /  /  •• /  /  /  /  .  /  /  /  /  /  /  /  t  84  /  /  /  /  Figure 5.8: The dimerization found in the two-leg Heisenberg spin-| zig-zag ladder. where a is a positive constant. In addition to the formation of a spin gap in the zig-zag ladder upon introducing any infinitesimal antiferromagnetic interchain coupling J i , an interesting form of dimerization also occurs[86]. Consider the dimerization operator, depicted in Fig. 5.8, Di = Sl  1]  • [ S f - SH\}.  (5.47)  In the continuum limit this corresponds to, using the non-Abelian representation Eq. (5.36), Di oc Tr(a ») • T r ( ^ (1  5  ( 2 )  ).  (5.48)  It has been shown by the density-matrix renormalization group method that (Di)  is non-  zero. That is, a discrete symmetry is broken by the selection of a particular ground state. In the non-Abelian bosonization language, this discrete symmetry is 9  (a)  considered separately for a = 1,2.  -  -9  (5-49)  {a)  In the Abelian bosonization picture, we must of  course also have a spontaneously broken discrete symmetry, and observe the same order parameter. This can be obtained by  Tr(cV ) oc 2i(sin v ^ r 0 i , cos V ^ i " ' , sin V 2 ^ ) , a)  a )  a )  (5.50)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  85  for which we used the translation  <?<x  (5.51)  a )(  _ -iv^^ e  0 )  e  -iv^4  a )  Now, in the ordinary two-leg spin ladder, Di is the interchain interaction, but in the case of the zig-zag spin ladder, it is the order parameter in the dimerized phase. Putting Eq. (5.50) into Eq. (5.48), we obtain Di oc - cos \/47r0i  +)  + cos VATT^  +  2 cos v 47T0i" , /  )  (5.52)  where we have written (5.53)  = ^=J^.  A n interesting and useful observation about the dimerization order parameter Di is that both terms involving <fi and terms involving 0 occur [see Eq. (5.52)]. Schulz [68] discusses the duality between <fi and 0 as an analogy with that of the two-dimensional Ising model. In that model, there is a duality between two order parameters; one is non-zero below the critical temperature and the other (usually called the "dis-order parameter") is non-zero above it [98]. In this particular case in which we have both 0 and 0, it is not clear a priori which of these actually contribute to the full dimerization order parameter as measured by the density-matrix renormalization group method; however, using a symmetry argument we are able to determine which terms in Eq. (5.52) actually develop expectation values.  In the non-Abelian picture, the three terms in the dot  product of Eq. (5.48) must, because of the still-unbroken SU(2) symmetry, each contribute the same amount to the dimerization order parameter's expectation value. That is, (Tr^y^Tr^V ))) 2  (5.54)  must be independent of a. Writing this in Abelian bosonization language, using Eq.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  86  (5.50), this means that (cos v ^ 0 i - c o s \ / 4 7 r 0 i ) = (cos V ^ ^ ' + c o s _ )  + )  V4~K<j>[ ) +)  = (cos v 47T0*" -cos \ / 4 7 r 0 i } , /  )  +)  (5.55) from the first equality of which we can immediately conclude that  {cosV^4>i ) +)  = °-  (5.56)  For the dimerization to be non-zero, as found by DMRG[86], this implies that ( c o s v ^ ^ ) + 0,  (5.57)  and since it cannot be the case that both a boson field <fi and its dual field 4> have an expectation value, we therefore have ( c o s v ^ ^ ' ) = 0.  (5.58)  7  So the only non-zero expectation values are (cos v ! ? ^ ' ) = -(cos 7  V^4>[ ) ± +)  0.  (5.59)  Comparing with Eq. (5.52), we conclude that the dimerization can be written simply as (Di) oc ( c o s V l T r ^ ) .  5.6.2  (5.60)  T h e Heisenberg-Kondo L a t t i c e M o d e l  For the Heisenberg-Kondo lattice model as defined by the Hamiltonian of Eq. (5.11) we proceed to an analogous non-Abelian bosonization. We write Sj « j[(x) + Jl(x) + const, x (-l) Ti{ag j  f  (x))  (5.61)  for the /-spin chain (as for each leg of the Heisenberg ladder in the previous section), and for the conduction electrons Sj « Jl(x) + 4(z) + const, x [e* * Ti(ag )e **+° + h.c.]. ik  x  c  iy/  (5.62)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  87  These operators have the same scaling dimensions as in the previous section, with the additional operator representing the charge degrees of freedom having din^e ^ ^) = 1  At exactly half-filling, we have e  2  \.  (5.63)  = ( — a n d so in this case we obtain as our most  2lkFX  relevant interaction term from J ^ Si • s*; K  i  (Trag ) • (Trc?p )cos(v 27r0 ), /  c  /  c  (5.64)  which has dimension | + 5 + | = | , a relevant operator. However, away from half-filling, e  2lkFX  ^ ( — a n d so the commensurate sign can-  cellation occurring in this interaction term is replaced by an averaging to zero. The most relevant interaction term in the Hamiltonian is, then, # i « t = MJl + 4) • (j[ + 4),  (5.65)  just as in the case of the zig-zag spin ladder. In this case, however, we have two different spin-wave velocities: [96] v  =  v  = TTJH/2,  c  f  2*sin(7m/2)  (5.66) (5.67)  where n is the average electron density in the conduction electron chain, so as before, for ferromagnetic interchain (Kondo) coupling J < 0 we have no gap, but for antiferroK  magnetic coupling J K > 0 we have a spin gap A  S  oce-  C T / j K  where c is a positive constant and v = v* + v . c  ,  (5.68)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  88  This indicates that there should be a spin gap in the Heisenberg-Kondo lattice model away from half-filling for infinitesimal Kondo coupling, due to the presence of a marginally relevant operator. It is not clear a priori how this spin gap phase survives upon the increase of the Kondo coupling. In fact, it could also be very difficult to numerically determine the presence or extent of this phase due to the exponentially small gap size. In Section 5.7.6 we present D M R G calculations which quantitatively verify the relation Eq. (5.68). 5.6.3  Concealed Dimerization Order Parameters  In this section, we make some predictions about correlation functions in the HeisenbergKondo lattice model based on an analogy with the zig-zag spin ladder. Before proceeding to this, it is useful to consider a related analogy between a spin-chain and a model of strongly correlated electrons. The Hamiltonian of the Heisenberg second-nearest-neighbour spin-| chain is H = J £S -S y  i  i+1  i  + J"£ S l  S  i  (5.69)  i+2i  i  where we consider J, J' > 0, and that of the negative U Hubbard model, which we will consider at half-filling, is  ff = - t E i < w icr  i  +  tf5>T*i.  where U < 0. The former model has a spin gap and related dimerization due to a marginally relevant spin coupling g . The latter also has a spin gap (but no charge a  gap) as can be seen easily at large negative U where singlet pairing occurs, and at small negative U where there is a duality transformation to the U > 0 Hubbard model which has a charge gap but no spin gap. We will show that the dimerization order parameter found in the former model corresponds to one which is concealed in the latter model, but not entirely without any noticeable effect.  ( - °) 5 7  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  89  In the spin chain, the marginally relevant operator occurring in the Hamiltonian here is -  | Jj_ J L J£ + h.c. « -Jj_ cos v 87T0  (5-71)  /  S)  while J [ | J £ J R ~ (d 0) serves only to renormalize the boson radius. 2  M  The two non-  equivalent minimizing solutions of this are <p = 0, \J^/2 since the compactification radius s  here is 1 / y ^ . 7  The dimerization order parameter is Di = S  2l  • (S +i - S -i) ~ Tr(<?) = cos V^(f) . 2i  s  2i  (5.72)  Because of the fact that the operator cos V 8TT</ S occurring in the Hamiltonian is marginal /  )  and obtains an expectation value, the dimerization order parameter also obtains a corresponding non-zero expectation value since then (cos\/2^>s} = ± 1 .  (5.73)  We now make the following conjecture to present an analogy between the negative U Hubbard model at half-filling and the second-nearest-neighbour Heisenberg spin-^ chain.  Since (a) both the Hubbard model and the spin chain have a spin gap, (b)  dimerization accompanies the spin gap phase in the spin chain case, (c) the spin chain's spin gap and dimerization can both be seen to arise from closely related operators when expressed in bosonized form, and (d) the spin gaps in these two models arise from the same operator when expressed in bosonized form, therefore one would expect that some form of dimerization should be present and potentially observable in the negative U Hubbard model at half-filling as well. While it may seem that one may just employ the same already-bosonized operator as in the spin chain, it will not be that simple.  Instead, our putative dimerization  order parameter will have to be written in terms of the original operators of the model at hand. In particular, one fundamental difference between the two Hamiltonians under  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  90  consideration in this analogy will have to be reckoned with, namely that only the Hubbard model has charge operators. In general, its local operators will involve both charge and spin sectors of the Hilbert space. Therefore one will have to find a microscopic local operator in the Hubbard model which, after taking the continuum limit and bosonizing, has as its spin part the same operator which leads to dimerization in the spin chain. For the U < 0 Hubbard model, one may consider whether the itinerant spins dimerize in the same sense as the spins of the spin chain. That is, consider the operator (5.74)  (U = s • (s i - s -i), 2i  2i+  2i  i.e. the alternating part of s i-s 2  2i+1  =  (ipl^dip ) • (ip i^ip )  cx  [J + (-l) *Tr(0a)cosv 2^0c] • [J - (-l) Tr(go)cosV2^4> }- (5.76)  2i  2i+  (5.75)  2i+1  2  /  2i  c  Now, since J • Ti(ga) = Tig in the sense of the operator product expansion, the ground 10  state expectation value of this alternating part is (di)  oc ((Tr^cosv^r^c)  (5-77)  (Tr0)(cosv27r0c>-  (5.78)  =  /  This is exactly the same as what we had in the spin chain case in Eq. (5.72), except that here we have to include a factor which is the expectation value of a charge operator. Although the spin part's contribution here is (Teg)  o  (5.79)  as in the spin chain case, the gaplessness of the charge sector of the U < 0 Hubbard model leads to charge part's expectation value being (cos \/27r0c) = 0 10  T h i s is discussed, for example, in Ref. [26].  (5.80)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  91  which gives  (di) = 0  (5.81)  for the itinerant spin dimerization. The presence of massless charge modes in the U < 0 Hubbard model has effectively concealed its otherwise non-zero spin dimerization. Saying this another way, the dimerization is identical to that in the spin chain apart from a multiplicative factor; this factor, however, happens to be zero by virtue of the behaviour of the other (charge) sector of the model. While this local operator's expectation value vanishes, the would-be dimerization makes its effect known on correlation functions. That is, the non-zero expectation value of Trg actually has an impact, though not in terms of the real dimerization order parameter. For example, the dimerization correlation function is (didA  oc (Trpcosv^0c(^)Trpcosv 27r(56c(2/)} /  =  (Tr5) (cosv^7r0c(^)cos  =  constant x i  2  V^Mv))  .  (5.82) (5-83) (5.84)  Here the contribution of the spin part to this correlation function is simply a constant due to the presence of the spin gap, and as a result its power-law decay is slower (i.e. the power 77c is smaller than it would have been if (Trg) = 0). Similarly one can demonstrate that the U < 0 Hubbard model has quasi-long-range order (qLRO) singlet superconductivity, since the correlation function corresponding to the pair operator ~ cos V2~-(j> cos \/27r0  e^ipLaipRp  c  s  (5.85)  has, due to the fact that (cos\/27r0 ) ^ 0, slower power-law decay than it would if the s  spin modes were not gapped. Turning from the analogy between the dimerized next-nearest neighbour Heisenberg spin chain and the U < 0 Hubbard model at half-filling, we now consider an analogy  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  92  between the zig-zag spin ladder and the Heisenberg-Kondo lattice model to determine if it predicts the presence of a concealed dimerization order parameter in the latter. First of all, consider the zig-zag spin ladder. The dimerization pattern corresponds to (Tr(p c?) • Tr(gWo)) ^ 0 but (Tig®) = 0. Thus the symmetries (1)  9  - -9  9  - -9  W  (5.86)  W  and {2)  (5-87)  {2)  considered separately are spontaneously broken but the symmetry W _ _W  g  together with g  {2)  g  -g  {2)  (5.88)  is not. This "protects" the operators Tig^ from developing expectation values. That is, there is a phase in which, by symmetry, they don't get expectation values. In the Heisenberg-Kondo lattice model we can consider two spin-singlet pairing operators. On the lattice they are O  e  =  E^/3e  a / 3  = E ^  i  T  i  ^ a n d  i  Oo = E C A r [ 2 M r U  (5.89)  ( - °) 5 9  Under site-parity, which transforms P :TA  -  P :3  -  S  S  V-i  (5-91) (5.92)  we see that P :O  e  -  O  P :0  o  -  -00,  s  s  e  (5.93) (5.94)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  i.e. O is even under site-parity, while 0 e  is odd because ip o ax T  o  93  — —X o' i?V f °  2  T  2  ;  r  a n  y  two-spinors ip and %. In the continuum, their most singular parts are O  e  0  o  =  iplia ipR and  =  V L * 2WR  (5.95)  2  • T^(g a).  ' (5.96)  f  In this form, we can see that the site-parity is as above, noting that P  s  : Ti(gfa) -  Tr(/<f).  (5.97)  and  (5.98)  In completely bosonized form they are O  e  «  Trg e ^  0  o  «  Tr(p a) • T r ^ ^ e * ^ .  c  2i  /R  c  (5.99)  2  That this form for 0 is precisely the dimerization order parameter for the zig-zag ladder o  times a charge operator is no accident: we intentionally constructed an operator on the lattice which reduced to this in bosonized form and which had the correct symmetry properties.  In fact, we could write any operator on the lattice which reduced to this  bosonized form without regard for symmetry. For example, we could consider  0 = 52tf<T aib -Si,  (5.100)  2  i+1  i  whose most singular part in the continuum is precisely that of 0 . G  As a parenthetical comment, we point out that Zachar et al.[97] predict power law behaviour for both of these order parameters in the Kondo lattice model. Before we decide conclusively that the operator 0  Q  11  is the dimerization operator for  the spin-gap phase of the Heisenberg-Kondo lattice model, we must consider its strongcoupling limit which maps onto the modified t-J model. We will see later that D M R G n  R e f . [97] labels the operator V L ^ R  spin-singlet  pairing operator 00.  a  triplet pairing operator; it is one term of our odd-parity  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  94  calculations demonstrate that our model has a spin gap in a phase which extends to this large-JK limit. The t-J model has a simple spin-singlet pairing operator ipJio ipi i,  which  2  +  is link-parity even. To get an operator which maps in the continuum limit to the operator iplioipR.,  we must take the site-parity even part, i.e. ipJio [ipi i + ipi^i\. This corresponds 2  +  to the same operator in both the Heisenberg-Kondo lattice model and the t-J model, as can be seen most easily by performing a particle-hole transformation on the t-J model. However, if we construct the site-parity odd and link parity-even part of tpja aip i 2  i+  • Si  which upon bosonization becomes ip1cr di/jR • Tvg^a, this has no mapping into the site2  parity even pair operator in the t-J model and so does not exhibit quasi-long-range order. This phase corresponds to (g ) ^ 0 but (g ) — 0 and (Trg o • Trg d) = 0. Presumably if c  f  c  f  (Tig?) ^ 0, there would be genuine observable spontaneously broken parity symmetry, i.e. dimerization. Thus, the phase of the strong Kondo coupling limit of the Heisenberg-Kondo lattice model which has this dimerization does not correspond to the phase which occurs in the zigzag spin ladder. It may correspond to a phase which might in principle occur in some more general zigzag spin ladder in which one of the two legs has dimer order but not the other and not the zigzag terms. Such a phase would have a parity symmetry about a vertical line through a link in the dimerized leg and a site on the undimerized one. This symmetry could "protect" the other two order parameters. Possibly such a phase would occur if one of the chains had a frustrating next nearest neighbour coupling while the other one didn't. If this remaining parity (about a site in the undimerized leg) is unbroken, then the Lieb-Schulz-Mattis theorem [99] as discussed in Ref. [100] implies that there is no spin gap. It is possible, however, that this t-J phase does not occur everywhere in the HeisenbergKondo lattice model where there is a spin gap.  While this certainly is the simplest  assumption, and is apparently consistent with the bosonization renormalization group  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  analysis, if this were not the case, then indeed the operator 0  95  may be the dimerization  D  order parameter in the small-JK case. 5.6.4  A S t r o n g l y A n i s o t r o p i c L i m i t of the K o n d o L a t t i c e M o d e l  Zachar, Kivelson, and Emery [97] performed a study of a particular portion of the phase diagram of the pure Kondo lattice model, considering conduction electrons hopping on a lattice with lattice constant a (which is taken to the continuum limit) coupled to a periodic array of / spins spaced a distance b apart, one located at every b/a — 1/c sites. After breaking the full rotational symmetry to write the Kondo interaction as J\\S*s* + |Jj_(5'^ Sj~ f  +  S^sf)  with  J|_,  Jy >  they then further break down these two  components into forward- and back-scattering parts, indicated by superscripts f and b. After bosonizing the conduction-electron part and considering the Toulouse point at which the forward-scattering z-component of the Kondo coupling is reset to  = TTV  F  where v is the Fermi velocity (of the conduction electrons), the charge and spin parts F  are seen to decouple, and a spin gap A  s  -^[cj[/v f  ~  3  V  F  (5.101)  (X  results in the limit in which the / spins are dense, as determined by the criterion b <C •up/A , which is the fermionic correlation length. s  Since in our Heisenberg-Kondo lattice model we consider / spins located on each lattice site, we are certainly in a regime in which this density criterion could be attained. While it certainly is not clear a priori how strongly the presence of the spin gap depends on the strong anisotropy and/or the particular Toulouse point studied here, this work presents the possibility that for the Kondo lattice model away from half-filling a small spin gap exists with a power-law dependence on the Kondo coupling. This remains a question for numerical determination.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  5.7  96  D e n s i t y - M a t r i x R e n o r m a l i z a t i o n G r o u p Results  The previous three sections demonstrate the existence or possible existence of a spin gap in the Heisenberg-Kondo lattice model in three areas of its phase diagram. For large Kondo coupling, the mapping onto the modified t-J model demonstrates that between two finite values of the Heisenberg coupling, a spin gap should be present. For small Kondo coupling which we consider as a weak coupling between a Heisenberg s p i n - | chain and a conduction electron chain below half-filling, bosonization demonstrates that a spin gap should turn on exponentially.  Finally, a Toulouse point in the strongly-  anisotropic (Jj. <C J\\ ~ t) version of the Kondo lattice model holds up the possibility of a spin gap existing in the Kondo lattice model away from half-filling. The density-matrix renormalization group technique is applied in an attempt to more clearly resolve these matters. Of particular interest is whether the regime known to have a spin gap at large Kondo coupling extends to weak-Kondo-coupling.  5.7.1  S u m m a r y : P r e d i c t i o n of the Phase D i a g r a m  We begin by presenting our prediction of the form of the phase diagram of the HeisenbergKondo lattice model in Figures 5.9, 5.10, and 5.11. following. We know that for J  H  > 0 the large J  K  These predictions are based on the  limit maps onto a modified t-J model,  whose phase diagram is reasonably well-known for the full range of band fillings. For J  H  = 0 and large J , it is well established that there is a ferromagnetic phase away from K  half-filling, which will likely survive for J  < t /J . Our density-matrix renormalization 2  H  K  group calculations, reported in detail in the remainder of this chapter, give finite values of spin gaps at many points, and in particular verify (at two values of J ) our theoretical H  picture, via bosonization, in the limit of small JK- The bosonization preduction is then taken to be correct for all values of JH- Outside of the spin-gap region, the system is  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  9 7  Figure 5 . 9 : Phase diagram of the one-dimensional Heisenberg-Kondo lattice model at a particular filling which has a spin gap for J —> co. K  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  98  Spin Gap  0 Figure 5.10: Reducing the rilling factor by a small amount will adjust the phase boundaries from the solid lines to the dashed lines.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  99  Figure 5.11: Phases in the H - K L M for a filling which does not give a spin gap in the limit JK —> co.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  100  found at points studied by D M R G to be gapless in the thermodynamic limit.  5.7.2  O v e r v i e w of C a l c u l a t i o n Details  A l l of the D M R G calculations for the Heisenberg-Kondo lattice model reported here were done using the finite system method. Gaps are reliable extrapolations to infinite chain length from several different chain lengths commensurate with the filling n = 7/8 admitting S  = 0, namely lengths 16, 32, 48, and 64. The calculations were done on  z  an I B M RS/6000 workstation equipped with 64MB R A M ; typical processing times are on the order of 10 hours of C P U time per ground state calculation for a chain of length 64 keeping 180 states in two finite system method sweeps, although such accuracy was not required for all points calculated. Error bars, where shown, are approximate only; they indicate a measure of how much we could reasonably expect the values to change by keeping all states. As a general rule, the discarded weights (sums of the density-matrix eigenvalues neglected in the basis truncation at each step) in this case are usually on the order of 10~ for large Kondo coupling J 8  K  and up to 1 0  - 4  for small JK or near phase  boundaries.  5.7.3  D e t e r m i n a t i o n of the Ferromagnetic State  As mentioned in Section 5.5, Sigrist et al. demonstrated that the one-dimensional Kondo lattice model (i.e. our model at J  H  = 0) away from half-filling has a ferromagnetic ground  state in the limit of large J . For an average of n — N/L conduction electrons per site, the K  large-J ferromagnetism predicted in the Kondo chain away from half-filling is incomplete K  in that as many as possible (nL) on-site singlet pairs between / and conduction electrons are formed, leaving the remaining (1 — n)L unpaired / spins to develop the ferromagnetic state of total spin S = (1 — n)L/2.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  101  Numerical verifications of this analytical result are quite limited to date. Tsunetsugu et al [89] exactly diagonalized the Kondo lattice Hamiltonian on open chains with 6, 7, 8, and 9 sites at (commensurate) electron densities between n = 2/9 and n — 7/9, finding that the critical value of J  K  above which ferromagnetism results varies approximately  linearly with density and is not noticeably size-dependent. Extrapolating their results to a filling of n = 7/8 for which our D M R G calculations were done leads to an estimate of a critical coupling of J £ ?» 3.5t. Troyer and Wiirtz [101] studied the Kondo lattice model at fillings n = 1/3 and n = 2/3 using the quantum Monte Carlo technique. They found, considering periodic and antiperiodic boundary conditions on chains of lengths up to L — 24, that for n — 1/3 the ferromagnetic transition appears to lie between JK = t/2 and 2t; for n = 2/3, the best they could say was that the transition may lie between J  K  = t/2 and At. Within these  ranges of couplings, the system appears to be quite complicated, in addition to causing severe sign problems plus the possibility that the smallest finite temperature employed may still be too high for the true ground state behaviour to be accessible. Moukouri and Caron [40] study the Kondo chain at a filling of n — 0.7 by D M R G , showing that the transition to ferromagnetism occurs somewhere between J /t K  = 1  and 4 (Tsunetsugu et al [89] using only short chains predicted the transition at this filling to be at about J /t K  = 2.7). Moukouri, Caron, and Chen [102] then report that  the ferromagnetic state persists upon the addition of a sufficiently small Heisenberg interaction (of the t-J type) between the conduction electrons. While we are interested in the general case of the Heisenberg-Kondo lattice model, it would be beneficial to obtain more information on the pure Kondo lattice model as well. Therefore we attempt to determine the transition to ferromagnetism using the D M R G method by setting J  H  = 0.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  102  We study the ferromagnetic transition by examining the finite-size scaling of the gap A where E (Q,S ) Z  L  = E (nL, (1 - n)L/2) - E (nL, 0),  L  L  (5.102)  L  is the energy of the ground state of the chain of L sites in the sector  with total charge Q and spin projection S .  In the ferromagnetic phase with total spin  z  quantum number S — <S = (1 — n)L/2, one should find that F  = 0 (since there is an  exact degeneracy for the states with —<S < <S ); outside of this phase, the gap should F  F  scale like 1/L. We find that for J  K  > 2t the ground state is ferromagnetic at n — 7/8 by finding  AL = 0 to better than expected for the numerical uncertainty. For J  K  < 2t, our current  D M R G calculations have technical difficulties; we are unable to determine accurate gaps for this range of parameters. We conclude that a more intensive D M R G calculation would be required to accurately determine the phase boundary for ferromagnetism in the Kondo lattice model at the filling n = 7/8, and the character of the ground state for Kondo couplings below this transition. This phase transition is by all accounts quite a complicated one. If we assume the most likely scenario, namely that below the transition the ground state is that of a Luttinger liquid, this may explain the difficulty. In this case both phases have, in the thermodynamic limit, high degrees of degeneracy within a sizable neighbourhood of the transition. The ferromagnetic phase has 2<S + 1 = (1 - n)L + 1 equal-energy states, one F  for each of S = — «S ,... ,S . z  F  F  The energies of lower total spin S < <S may be quite close F  to the ferromagnetic energy. The spin sector of the paramagnetic phase, if it is gapless, will have a spectrum of states separated only by energies on the order of 1/L. Adding to these complications is the fact that n = 7/8 could have finite size effects (for example, there are only eight holes in a system of length 64) of magnified intensity in the vicinity of a phase transition of this type.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  103  Our predicted phase diagram for the Heisenberg-Kondo lattice model, depicted in Fig. 5.9 for this filling, may help us understand the difficulty in determining the phase transition to ferromagnetism in the pure Kondo lattice model (i.e. without the J  H  term).  We see that adding in the Heisenberg term to the Kondo lattice model can give rise to a spin gap. It may be that the proximity of this spin-gap phase complicates the situation even in the pure Kondo lattice model, possibly because R K K Y interactions between the / spins may have an effect similar to the direct Heisenberg coupling.  5.7.4  B o u n d a r y B e t w e e n Spin-Gap a n d L u t t i n g e r L i q u i d Phases  Our D M R G calculations verify that there are points in the phase diagram which have a spin gap, and points which do not. Where these are close enough together, we can estimate the location of the phase boundary. Based our our results, we find this boundary along the line JK = I0t lies between J it lies between J  K  = 5t and J  K  H  = 1.25£ and 1.65£, and along the line JH = 1.25t  = I0t (close to J  K  = 7.5t).  As shown in Fig. 5.12, the phase boundary along the line J 5.7.5  K  = 2t is near J  H  = 0.6t.  C h a r a c t e r i z a t i o n of the Spin-Gap Phase  The ground state of the one-dimensional Heisenberg-Kondo lattice model in the spin-gap phase is that of a charge-only Luttinger liquid. This conclusion is reached by numerically examining the charge sector of the Hamiltonian in the vicinity of the filling of interest. First, we find numerically that the charge-transfer gap A ( n , L) = E (nL L  ct  where E (Q,S ) Z  L  + 2,0) + E (nL L  - 2,0) - 2E (nL, L  0),  (5.103)  is the ground state of the length-L chain in the sector of charge Q and  total spin projection S , is zero in the thermodynamic limit in this phase. (This chargez  transfer gap is just the charge gap for an appropriate selection of the chemical potential,  104  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  Figure 5.12: The spin gap in the Heisenberg-Kondo lattice model along J function of JR shows a transition near J = 0.6t.  K  H  = 2t as a  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  105  which selection we have not been concerned about since we have always worked within a sector of fixed charge.) That the charge-transfer gap is zero is actually a by-product of what we find in the finite-size scaling of the ground state energies for various charges. A conformal field theoretic analysis predicts that in a charge-only Luttinger liquid, the coefficient of Q in the energy of the ground state in the sector of charge Q should 2  be  where • • • represents terms of higher order in 1/L, including 1/L In L terms as well. Selecting a point (JK,</H) = (3r, 1.65£) at which the spin gap is reasonably large, and therefore the prediction is most readily testable, I calculated EL(Q,0)  by the D M R G  method for Q = 24, 26, 28, and 30 (for L = 32) as well as for Q = 48, 50, 52, 54, 56, 58, and 60 (for L = 64). The dominant contributions to these energies are linear in Q, but a clear quadratic term emerges as well; excellent quadratic fits for both lengths L = 32 and 64, with residuals less than or equal to the numerical uncertainty of the D M R G calculations, were found. These fits are depicted in Figures 5.13 and 5.14.  Comparing  the results with the conformal field theory prediction of Eq. (5.104), we find that the finite-size scaling is well verified; both different chain lengths studied give = 0.082 ± 3 %  (5.105)  where fp = 2£sin(7m/2) is the Fermi velocity in the non-interacting case.  5.7.6  S p i n G a p B e h a v i o u r for S m a l l K o n d o C o u p l i n g  The bosonization analysis of Section 5.6.2 predicted, for small Kondo coupling J , a spin K  gap of the form A  OC - [ * ( ™ / 2 ) + 7 r J ' H / 2 ] / J K c  s  e  2  sin  7r  (5.106)  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  106  Figure 5.13: Quadratic fit of ground state energies for a 32-site chain, at (JK, Jn) = (3£, 1.6'5i), as a function of charge. Residuals, together with numerical uncertainties, are plotted on the greatly-expanded right-hand scale.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  50  55  107  60  N u m b e r of C o n d u c t i o n E l e c t r o n s Q Figure 5.14: Quadratic fit of ground state energies for a 64-site chain, at (JK, Jn) = (3t, 1.65£), as a function of charge. Residuals (together with numerical uncertainties) are plotted on the right-hand scale.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  108  A test of this prediction is depicted in Fig. 5.15. For the smallest J , the numerical K  approach is not reliable since the gaps are comparable to the finite size gap; for large J  K  the exponential prediction is not applicable and in fact the gap turns over in the approach to the large J  K  limit which maps onto a modified t-J model. The resulting approximate  linearity in the range t < JK < 3i is good verification of the bosonization prediction. The slope of the resulting line is to be compared with — c[2£sin(7rn/2) + irJn/2}; since c is an unknown constant, we note that the ratio of the slopes for the two values of J is H  predicted to be 0.86, while it is numerically determined to be 0.83 ± 10%. The predicted value of c determined from the D M R G calculations is 1.1 ± 10%.  Chapter 5. The One-Dimensional Heisenberg-Kondo Lattice Model  109  Chapter 6  Conclusions  6.1  T h e One-Dimensional P a i r - H o p p i n g M o d e l  We conclude that there is a finite spin gap for all positive V in the half-filled pair-hopping model in one dimension, and that to accurately describe its behaviour as a function of V, one must neglect the coupling g in the renormalization group flows. 4  We conclude that there are phase-separation transitions in the pair-hopping model, one at positive V and one at negative V. In one dimension at low doping from half-filling, for V > V i fa SM polarized electrons clump together. c  We conclude that there is a new critical point at (or possibly two critical points near) V — V fa —1.5t at which, proceeding from weak coupling, a spin gap opens and the c  charge gap closes at half-filling.  6.2  T h e One-Dimensional Heisenberg-Kondo L a t t i c e M o d e l  There is a spin gap for the one-dimensional Heisenberg-Kondo lattice model for small Kondo coupling of the form predicted by bosonization renormalization group arguments. In this phase, the low-energy spectrum is that of a charge-only Luttinger liquid. This spin-gap phase extends to strong coupling in a range of fillings for which a spin gap exists in a modified one-dimensional t-J model. We predict the existence of an even-parity spin-singlet pairing operator corresponding to a concealed dimerization order parameter in the spin gap phase. This may exist for  110  Chapter 6. Conclusions  111  the entire spin-gap phase, or be replaced by an odd-parity composite spin-singlet pairing operator for small Kondo coupling. The presence or absence of a spin gap in the pure Kondo lattice model away from half-filling remains uncertain. We have presented evidence, however, that indicates there may be no spin gap in this model unless a direct Heisenberg coupling between / spins is included. We have found that the Kondo lattice model for J  K  > 2t has a ferromagnetic ground  state. We believe ferromagnetism persists upon the addition of a sufficiently small additional Heisenberg coupling.  Appendix A  F e r r o m a g n e t i s m i n the strong-coupling K o n d o lattice m o d e l  That the ground state of the strong-coupling Kondo lattice model in any dimension away from half-filling is incompletely ferromagnetic is demonstrated in a simple and intuitive way in this Appendix. The Hamiltonian for the Kondo lattice model is H = H + tH , K  (A.l)  t  the sum of Kondo and nearest-neighbour hopping terms  H  = JKESW,  K  (A.2)  i  H = -£cUv + h-c t  (A-3)  Here we write the spin operators for the / and conduction electrons as, respectively, Si  (A.4)  = a/3  ?i  =  Y, \cc¥«p ip a/3 c  ( -)  c  A  5  where /* (c£) is a creation operator for an / (conduction) electron of spin a =T,| at lattice site i. The Kondo lattice model considers only localized / electrons: yj<r  fLfio-  —  1  for each i. A.l  Zero H o p p i n g Case  We first consider the case t = 0, for which we have an independent-site Hamiltonian, with each site having the spectrum indicated in Fig. A . l . A t half-filling, the ground 112  Appendix A. Ferromagnetism in the strong-coupling Kondo lattice model  JK/4  —  I M>, I U), |T> = ^ (| U) +  I  ^  T  113  »  o — — |iV>,|J|>,|im>,UU>  |S) = ^ ( | f r | ) - | 4 T ) )  -3J /4 K  Figure A . l : Spectrum of the Kondo term for a single site. Here double (single) arrows represent / (conduction) electrons; for example | fl-f J.) = /|c|c||0). state is non-degenerate as each site is occupied by a singlet pair composed of an / and a conduction electron, for an energy of  —3J /4 K  per site. Away from half-filling, the ground  state will have a number of sites in one of the states of the zero-energy quadruplet, and there is therefore a large degeneracy. This degeneracy will be split by the hopping term H leaving a ferromagnetic state as the ground state when JK > J , where J t  K  K  is some  critical Kondo coupling, which possibly depends on the filling. While Ref. [88] computes a complicated effective Hamiltonian to 0(t /'JK) and twice invokes the Perron-Frobenius 2  theorem to show this, this appendix demonstrates it in simpler and intuitive way. Because of the particle-hole symmetry of the Hamiltonian, it is sufficient to consider the filling to be below half.  A.2  Small Hopping Limit  To first order in the perturbation H there can be no distinction between ferromagnetic t  or antiferromagnetic order as H is a nearest-neighbour hopping operator only. In second t  order, we need only consider the mixing of the 12 otherwise-degenerate three-site states on two neighbouring bonds in any dimension containing a single on-site singlet pair and two remaining unpaired / spins either aligned (S  z  = 1) or anti-aligned (S  z  = 0); H  conserves the total S . I will show that this mixing lowers the energy of an S = 1 triplet z  Appendix A. Ferromagnetism in the strong-coupling Kondo lattice model  114  more than that of any S = 0 singlet, leading to the conclusion that for strong Kondo coupling, the ground state of the non-half-filled Kondo lattice model is incompletely ferromagnetic. This consideration of only two neighbouring bonds completes the analysis to second order in perturbation theory since a nearest-neighbour interaction can have only a second-nearest neighbour effective Hamiltonian in this order. Consider first the case of aligned spins. H mixes the following three three-site states t  (and identically for ff<-*-!]•), which each have an energy —3JK/4: |s,ir,fr)JiT,s,fr>,|fr,fr,s>  (A.e)  Degenerate perturbation theory prescribes that we compute the matrix elements of H 1  t  between these. With the states in the above order, we have / H\  t 1  =  0  1/2  0  1/2  0  1/2  0  1/2  V  ^  (A.7)  ,  0 ,  whose (non-degenerate) eigenvalues €i and corresponding normalized eigenvectors (total spin indicated in brackets) are = o:  C l  e = ±i/V2-. 1  |o}o = ^(|fr,ir,s)-|s,fr,fr))  \s = i)  |o}  ( 5 = 1)  ±1/  ^ = ± i ( | f r , r r , s ) + |s,fr,fr)) + ^|fr,s,fr)  The ground state energies are, to first order in the hopping term, —3JK/4 + e\t. The second-order corrections e^t are then given by 1  e = MHxQ*^\ Q*HA*)« 2  (A.9)  Hi  where, following the notation of Ref.  [103],  E° = —3J /4 is the energy of the degenerate a  K  states under consideration and the projector out of the low-energy subspace is Qo= L  E El£° )<£°a| a  For second order perturbation theory in quantum mechanics, see, for example, Ref. [103].  (A.10)  Appendix A. Ferromagnetism in the strong-coupling Kondo lattice model  115  where E° are different energies of the other degenerate states \E°a) labeled by an index a.  We easily compute the matrix elements for the three states listed in A.6 under  consideration as ^ 3 0 1^ HtQo  E°„ - Hi  -QoH  t  4J  (A.H)  0 6 0 K  0 3  1 leading to -1/2J ,  ei = 0  -5/4J ,  €I =  K  e = 2  K  (A.12) ±1/V2  so the energies of these perturbed S = 1 states are, to second order in the hopping term, z  - 3 J / 4 - t /2 2  K  J  K  - 3 J / 4 ± t/y/2K  (A.13) 5t /4J 2  K  A similar procedure mixing the six states involving anti-aligned spins s,  iu>, | fr, s,  | rr, Jl, s), |s, JL, fr), |  s,  n), |  JJ-, t,  s)  (A.14)  gives matrix elements of H  t  (  t\o  H  0  1/2  0  1/2  0  1/2  0  1/2  0  -  0  y  \ 0 (A.15) 0  1/2  0  1/2  0  1/2  0 1 / 2 0  y  which gives the same first-order corrections ei as for the aligned spins; in this case,  Appendix A. Ferromagnetism in the strong-coupling Kondo lattice model  116  however, they are doubly-degenerate. The corresponding normalized eigenvectors are ei = 0 :  |0>& = ^ ( | S , ^ > - | 1 U , S »  |o) = ^(|s,j;,fr)-U,fr,s)) 2  C l  (A.16)  = ±l/y/2 : \0)l - = ± | (| fr, J|, S) + |S, IT, J | » + ^ | 1>, S, J|) 1/V2  |o)  2 ± 1 /  ^ = ± j (| JI, fr, s) + |s, JI, fr» +  Jl, s, fr).  For the second-order correction splitting these states, we have  HtQo  El -  H  QQHI  3  0 -1  0  0  2  0  6  0  0  0  0  -1  0  3  2  0  0  0  0  2  3  0 -1  0  0  0  0  6  0  2  0  0  -1  0  3  4J,<  K  V  (A.17)  the resulting energies and eigenstates to second order are C l  = 0:  e = -3/2J :  ^(|0)J - |0) ) 2  (5 = 0)  6 = -l/2J :  ^(|0)U|0) )  ( 5 = 1)  e = -5/4J :  ^(l >L/v^ + l  2  2  € l  = ±l/y/2:  2  e 2  K  2  K  0  K  0  = -3/4J :  ^(|0)i  K  ,-|0)  1 / v  0  ^ )  ( =^  ,)  (5 = 0).  2 ± 1 / v  (A.18)  S  These eigenstates are the degenerate eigenstates of the t = 0 Hamiltonian, to which the states split apart by second-order perturbation in H settle in the t —> 0 limit. 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