-Y A I! 2:»..3 u b... I. . lit... «:5 v 1 . . . 4 1?. .r? .1 .7... $11.92... Jar. 38%... I-—A_a i. :3... .. . K 63.}... ... a. :1 .A . k ’ t amquflmrm, l..<....u.~r! ‘ g 4. 3:2 ‘ r . .2. v4 oirlot oz: (tr! 5. 1...... «Z! 1 1.5.; , r 4,...” grain... JV; Hat»! A4, «1.3% v.57. f! 3 . . :5. 1. a 1? Jedi: fr r/ y r , >1 . . z .rtfr’ at}; n,“ . r. 3‘. 1 $5...” r... . ii? :33 1.1, . lun‘tnlfiunuflhhsm « vathLPuL‘;|l|L.r~NLUK n5; ‘ r 3.1.... ..r .v._ .3. 7. 11.. r... z; ..:r 253...!» m3 if»; x5}. 1 ,vyv: _L. U, This is to certify that the thesis entitled ,- 3TATlS'TlC'AL HEC-HRNNS 0F 7HG {SAM HUBBRRD Mom—EL, Wm: '35-. "33:: L ' " ’” V. 1:2 . .0-" I . ME; _ :01,» Univczmxtg HnLF-FILLED RANDOM EXCHAk/Cag‘ IslNG CHAINS presented by D fl Kl o C A (5 H3 has been accepted towards fulfillment of the requirements for Ph'D- degreein P14 )’S(\C5 / Major professor Date 3/7/1103 0-7639 ABSTRACT I. STATISTICAL MECHANICS OF THE HALF-FILLED BAND HUBBARD MODEL II. RANDOM EXCHANGE ISING CHAINS By Dario Cabib The two main subjects of this thesis, unrelated as they may seem, can be regarded from a general theoretical point of view as being two different aspects of the same branch of Physics. This branch of Physics studies the pro- perties of systems consisting of interacting particles at finite temperature. Both parts of the thesis are essentially studies of theoretical models for interacting particles, and in both the general methods and concepts of Statistical Mechanics are used. Part A is a study of the Half—Filled-Band Hubbard model. In the introductory chapter we make comment on the history and derivation of the model, we mention the various exact results existing in the literature and we outline the properties of some organic solids that have been recently related to the Hubbard model. In the following chapter we Dario Cabib present the exact results obtained for the four-atom ring: these results are interpreted physically, and used to resolve serious discrepancies existing in the literature; furthermore an attempt is made to extrapolate these results to an infinite one-dimensional system and contact is made with recent ex- periments on the organic solid N-methyl phenazinium tetracyanoquinodimethan.(NMP—TCNQ). The results obtained for the susCeptibility show that the Half-Filled-Band Hubbard model is deficient as to the explanation of these experi- ments. We tried to improve the theory in various ways but our efforts were not completely satisfactory: we show that although one can quantitatively fit the experimental para- magnetic susceptibility of NMP-TCNQ using a temperature dependent Hubbard model, there is still a lack of under- standing of the physical mechanisms responsible for the behavior of such a system. We then describe a high- temperature expansion of the Half-Filled-Band Hubbard hamiltonian, which is valid in the case of small interaction between the electrons compared to the temperature; we give the result for the susceptibility in first order of the ratio interaction/temperature. In the final chapter of Part A we explain and correct a serious error occurring in the liter- ature involving a calculation of the zero frequency conduc— tivity in the single band Hubbard model. We point out the subtleties involved in the symmetry properties of the Dario Cabib current operator as defined with a model hamiltonian such as the Hubbard hamiltonian, and explain how the lack of under- standing of these subtleties were the cause of the above mentioned error. Part B is a study of the Ising model with random exchange interactions. One strong motivation for this work is to understand the effects of the randomness of the inter- actions on the critical behavior at finite temperature. Our calculations refer only to one-dimensional systems; from the analysis of the low-temperature behaviOr of such systems we get some insight on the critical behavior of systems which display phase transitions at finite temperature. We describe extensively this low-temperature behavior, especially as a function of different distributions of the exchange param- eters. Finally the effect of a random highly anisotropic Heisenberg interaction at low and high temperature is studied. The results are compared with the periodic Ising chain. I. STATISTICAL MECHANICS OF THE HALF-FILLED BAND HUBBARD MODEL II. RANDOM EXCHANGE ISING CHAINS By Dario Cabib A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1973 ACKNOWLEDGMENTS It is a great pleasure to express my gratitude to Professor T. A. Kaplan for having given me the opportunity to collaborate closely with him in physics research. This collaboration has been a continuous stimulation for me to learn, especially because of Professor Kaplan's deep insight into the problems of concern and his way of enjoying teaching and doing science.” This thesis would not have been possible without his continuous friendship, his time and effort. I am very grateful to Professor S. D. Mahanti for suggesting the problem discussed in the second part of this thesis and for close collaboration in carrying it through. It has been very exciting to interact with him and to enjoy his friendship throughout my program. I am greatly indebted to Professor T. 0. Woodruff and the Physics Department of Michigan State University for providing me with a research assistantship, and the Commission for Cultural Exchanges between Italy and the U.S.A. for pro- viding me with a Fulbright Travel Grant. I thank Dr. R. A. Bari for helpful discussions in connection with Section AII.2 and Chapter AIII, and Dr. Jill Bonner for helpful suggestions in connection with Chapter BI. TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF FIGURES, PART A LIST OF FIGURES, PART 8.. PART A I. III. INTRODUCTION 1 Preliminaries and Definition of the Model 2. The Atomic Limit . . 3 Exact Calculations . 4. N- -methyl Phenazinium Tetracyanoquinodimethan (NMP- TCNQ) . . . . . . REFERENCES STATISTICAL MECHANICS OF THE HALF- FILLED- BAND HUBBARD MODEL l. Statistical Mechanics of the Half-Filled Band Hubbard Model . 2. Static Properties of the Half— Filled Band Hubbard Model. REFERENCES . 3. Attempts to Explain the Susceptibility of NMP— TCNQ by Modifying the Hubbard Model REFERENCES . 4. Expansion in Powers of U/kT . REFERENCES . . . . . . . . ON THE DEFINITION OF THE CURRENT OPERATOR- APPLICATION TO THE HUBBARD MODEL . l. Introduction . 2. Lack of Translational Invariance of j; Resolution of the Disagreement in the Literature . . 3. How to Define the Current Operator iii N .._I_| _l LCD-h NRO“) 21 56 58 Page 4. Summary . . . . . . . . . . . . 62 REFERENCES . . . . . . . . . . . . . 64 PART B I. ONE- DIMENSIONAL ISING MODEL WITH RANDOM EXCHANGE INTERACTIONS . . . . . 67 1. Introduction . . . . . . . 68 2. The Model and its Solution . . . . . . 73 A. Specific heat . . . . . 76 B. Spin- spin Correlation Function and X4 . . . 77 C. Zero Field Perpendicular Susceptibil iy .. . 84 3. Effect of Random xy Interaction . . . . 87 4. Conclusions . . . . . . . . . . . 89 REFERENCES . . . . . . . . . . . . 9l iv LIST OF FIGURES PART A Figure Page AI.l. TCNQ molecule . . . . . . . . . . . 15 AI.2. NMP molecule . . . . . . . . . . . l6 AI.3. Chains of stacking planar NMP—TCNQ molecules: (a) side view of the chains; (b) front view of the chains . . . . . . . . . . . l6 AII.l. Inverse susceptibility vs. temperature. . . 32 AII.2. Determination of b/U as function of temperature . . . . . . . . . 4l AII.3. kTO/U vs. b/U . . . . . . . . . . . 46 v LIST OF FIGURES PART B Figure Page 1. Specific Heat of random (v=0) and periodic (J=Jm/2) Ising chains vs. kT/ldml . . . . . 7l 2 Spin-spin correlation function L1- = L (r) (r=j-l) as a function of distance r at kT = .OSIJml. . . . . . . . . . . . . 79 3. Spin-spin correlation function LIrI as a function of r: (a) kT/Jm = .5, (b kT/Jm = .3, (c) kT/Jm = .1 8l ' Xll . _ 4. -——§ |Jm| vs. kT/lJml 1n the v-O case . . . . 83 NpB xilJm| 52 ———§—— vs. kT/ |J | in the v=0 case . . . . 86 m NpB vi PART A LA» PART A CHAPTER I INTRODUCTION 1. Preliminaries and Definition of the Model One of the fundamental problems in Solid State Physics is to solve the equation of motion for an arbitrary number of electrons and nuclei interacting via Coulomb forces. This problem is so extremely difficult in the general case that it has been solvedanalytically only in the case of one nucleus interacting with one electron (the hydrogen atom). As the number of particles increases, the difficulties rise rapidly and when the collection of parti- cles is a macroscopic system, a detailed solution to the Schrodinger equation, with all the interactions, is out of the question. 0n the other hand much of the information about the behavior of systems of electrons and nuclei can be attained by the use of general theorems and of different approxi- mations, valid in different physical situations. In many cases these approximations or theories (one may regard the development of a theory as being an attempt to understand nature through a simplified, and therefore approximate, picture) give an accurate account of physical phenomena; therefore one is often confronted with the problem of under- standing the features of a theory and the phenomena that it predicts, to be able to make contact with experimental findings. The way one usually does this is to choose on semi-qualitative grounds a simplified Hamiltonian, and study its properties exactly, if possible, or approximately, otherwise; the results are then compared with experiments. It is in this spirit that Hubbard proposed.I his Hamilton- ian in 1963. Similar to other contemporary work,2 Hubbard was concerned with the study of correlated electrons in solids; his goal was to account for the effects of cor- relations in narrow bands and he suggested that his theory be applied to d-electrons in transition metals. As we shall see in Section AI.4, some authors3 proposed that this theory be applied also to some organic solids; furthermore it has 4 been claimed that this or very similar theories can account for the physics of the benzene ring and of a vast number of magnetic insulators,5 usually thought of as being described by the Heisenberg Hamiltonian (at the end of Section AI.3 we will mention the relationship between the Heisenberg and the Hubbard Hamiltonian). Let us now focus upon the description of Hubbard's model and its properties. It is well known that in a crystal the energy levels of the electrons are grouped in bands. Throughout this work we will restrict ourselves to the case of a crystal of N atoms and an average of N electrons, filling exactly half of one non-degenerate band; we will disregard the presence of all the other bands. (To do so we construct a so called projected Hamiltonian, where part of the full Hamiltonian matrix is completely ignored.) We define an orthonormal complete set of N Mannier functions for this band; the Nannier functions are localized at the lattice sites i.e. each of them is appreciably different from zero only in the neighborhood of a lattice site, and can be occupied by a maximum number of two electrons, allowing for the spin degeneracy. The theory will not depend on the detailed functional form of the Nannier functions, and we will therefore leave them completely general. We then define operators cio+ and c1.O which respectively create and destroy an electron in the Nannier function at site i with spin projection o. The c10+'s and ngls satisfy the usual fermion anticommutation relations + and n. c =C IO 10 id is the number operator of site i and spin 0. The Hubbard Hamiltonian is written in terms of these operators as H = igo'bij(cio+cjo+cjo+cio) + U §n1¢ni+. (l) The bijls and U are constant parameters and have precise physical meanings. For simplicity we will always consider the case b1j=b for i and j nearest neighbors, and zero otherwise. b is often called the transfer or hopping integral. The first summation in (l) is a noninteracting electron term that can be written, in the basis of the Bloch functions, as nk0 are defined as "L0 aha aka (3) where ako+ and ak0 are respectively the creation and annihi- lation operators for an electron in the Bloch function with crystal momentum k and spin 0. ako+ is related to the Nannier functions creation operators by + _ -1/2 154 + a ek are the one-electron energies of the band in question, whose width is proportional to b; in general ik-R.. 13 --1a (5) e =Zb..e k3 (3,,- dimension, for instance, 8k = 2bcosk. is the position vector between sites i and j). In one The second summation in (l) is the interaction term: when two electrons with opposite spins occupy the same site (the exclusion principle forbids two electrons with the same spin to occupy the same site) they repel each other with an energy U. They do not interact if they are on different sites. Hubbard gave a derivation of the Hamiltonian that took his name and studied it in different approximations: the Hartree-Fock approximation,1 a Green function decoupling 6 procedure1 (Hubbard I), and a second approximate solution (Hubbard III), which improved Hubbard I. Since then the Hubbard Hamiltonian has been of great theoretical interest. There are some exact results‘,7T12 but in the general case b,U¢0 at finite temperature TfO they refer only to small one-dimensional systems; the ground state and some of the low lying states have been calculated for infinite chains. The approximate calculations are usually based on Green function decoupling schemesl’6’13’14 which do not give criteria for the estimates of the errors involved and sometimes14 are wrong in the limiting case bij/U + 0; these difficulties were overcome in part by .l5 T. A. Kaplan and R. A. Bar1 with the TSDA. (The TSDA 16 is a variational approximation due to T. A. Kaplan and discussed in some detail by Kaplan and Argyres.]7) Finally the derivation of the Hamiltonian itself given by Hubbard has been recently criticized18 and an attempt is being made to improve that derivation. The fundamental question which arises is: why study such a model, since it appears to be only a crude approxi- mation to reality. This is so because, as we explained before, we take into account only one band and the intrasite Coulomb repulsion: the presence of other bands and the long range of the Coulomb interaction may be nonnegligible. To try to answer this question let us briefly examine the two cases b/U<>U the Hamiltonian (l) may seem unrealistic not only because the repulsion between electrons on different sites becomes important, but also because other bands may start to come into play. On the other hand screening effects may reduce the inter- and intra-atomic interactions appreciably making it conceivable that (l) is still in some cases a good description of reality. In the present work our approach has been to study the Hamiltonian (l) from a phenomenological point of view: in other words we have been interested in the features of the theory as functions of the parameters b and U. This means also that we have not worried about the derivation of the Hamiltonian and its validity for the different physical situations, and the different values of the parameters b and U. Both cases b>>U and b<>U) and a one Bohr magneton Curie law at very low kT< brought about by the for- mation of double occupancies when kT 2 U. This discussion has been given by Kaplan.19 The local mOment is never~ zero,20 even when T-+w because there is always a finite.fi probability (Z 1/2) to find singly occupied sites. To complete this section, we mention that for b#0 but satisfying b/U<>U; on the other hand when b<’ as far as we know, which would clearly show the kind of order. Finally, with the help of the chemical potential LN have argued that at any finite U at T=0 the system is insu- lating and is conducting only at U=0 or away from the half- filled band. 7 Lw's work was the starting point of subsequent exact calculations at T=0, and for this it was of great fundamental importance. In fact, using Lw's method and results Ovchinnikov9 calculated the spectrum of the lowest excitations with total spin 0 and l and Takahashi8 obtained the magnetization and the zero-field susceptibility (both 12 works refer to the half-filled-band case). Shiba extended Lw's and Takahashi's calculations to arbitrary number of electrons (not necessarily half-filled band). Griffiths' 24 work on the magnetization and susceptibility of the infinite antiferromagnetic Heisenberg chain was also used by Takahashi8 and Shiba12; therefore it appears that the 14 same problems of rigor pointed out by Griffithsz4 apply to Takahashi's and Shiba's work as well (the lack of rigor is in the characterization of the lowest energy levels for a given total 52, although there may be very plausible argu- ments in support of the assumptions made). 10 uses.a'differentli The papers by Shiba.and Pincus approach:' they diagonalize by computer the Hamiltonian (l) for chains and rings of up to N=6 atoms, and then they compute the thermodynamics in the Canonical10 Ensemble. As reported in Chapter II this is essentially our approach; our calculations were carried out (both in the Canonical and Grand Canonical Ensemble) simultaneously with and independently of Shiba and Pincus' and concerned rings of four atoms. After our work was presented (Magnetism Conference, Nov. l972), further work within the same approach 11 For more extensive discussion of the motivations, appeared. the checks of the computer program, presentation and interpre— tation of the results and further references we refer here to Section AII.l. 4. N-methyl Phenazinium Tetracyanoquinodimethan (NMP—TCNQ) An extensive and detailed description of the systems that will be the subject of the present Section can be found in the works by Fritchie,25 Epstein et al.,3 Shchegolev26 27 and Heeger and Garito. We report here some of the proper- ties of the TCNQ organic solids (especially of the NMP-TCNQ) l5 because the experiments on NMP-TCNQ have been closely related3’27 to the recent theoretical studies of the Hubbard model. There is a class of so-called "organic charge trans— fer salts” that are characterized by interesting features. First of all they are organic solids composed of two types of molecules, a donor and an acceptor giving rise to the presence of unpaired electrons in the crystal. These un— paired electrons are generally thought3 to be responsible for the magnetic and electric properties of the system (due to the nature of the molecular orbital involved), along with the crystal structure of each solid. Second these salts are highly anisotropic, displaying a very pronounced one-dimensional behavior, the unpaired electrons moving along the chains made up of the acceptor molecules. (The effect of other bands is usually considered negligible.3) The one-dimensionality is clearly displayed by the conduc— 26 tivity measurements by Shchegolev. The anion in these salts is the TCNQ molecule, a planar molecule of the form: CN CN \\ /// \\\ // C = = C /- \_/ CN CN < W8A° > Figure AI.l.--TCNQ molecule. 16 it is capable of combining with other melecules that play the role of cation. The negative charge resides in the lowest n level and is localized near the cyanide groups, because of their strong electron affinity.3 In par- ticular TCNQ can combine with the NMP molecule of the form: Figure AI.2.--NMP Molecule. in this cation the positive charge resides on the nitrogen and carbon atoms involved in the bond between the methyl group and the phenazinium, and all the electrons of the cation are paired. Furthermore the methyl group can be bound to either N at random so that there is also a ran- domness in the position of the positive charge. Both NMP and TCNQ form alternating chains of stacking blanar molecules as shown in the following schematic figure: TCNQ h ' + \\ TCNQ h ' + .:.::.'N, N ..:.::.:., 40%” TCNQ chain—>(a) \\\\ TCNQ chain+(b)0 fl O Figure AI.3.-~Chains of stacking planar NMP-TCNQ molecules: (a) side view of the chains; (b) front view of the chains. l7 As seen in Figure AI.3(a) they stack face to face to each other and the interplanar distance is roughly 3.3 A; in (b) the distance between TCNQ's is roughly 7.8 A. This picture shows that the TCNQ chains are far apart and it is reasonable to think of the electrons as bound to move in one dimension. Each TCNQ can accommodate two extra electrons with opposite spin assuming that it has associated with it one spatial wave function, but the total number of these electrons is equal to the number of TCNQ molecules so that the band in which the electrons move is half filled. Certainly the electrons in the band interact via Coulomb forces, and this complicates the physics of this one-dimensional electron gas, even if we neglect the interactions with the phonons and with the highly polarizable NMP molecules. The experimental measurements3’26 refer to low; temperature specific heat, up to about 20 degrees Kelvin, d.c. conductivity and spin susceptibility up to about 400 degrees K (above which the substance melts) and electric permeability. Epstein et al.3 tried to interpret their experi- mental results with the Hubbard model. The few exact theo- retical results existing at zero temperature8 and a calcu- lation of Hubbard's gap based on Ovchinnikov's excitation spectrum9 were the theoretical basis for establishing the values of U and b appropriate to the NMP-TCNQ. In 18 Sections AII.l and AII.2 we show the serious difficulties this approach runs into. Later Heeger and Garito27 tried to change the theoretical picture allowing for the parameters b and U to vary with temperature; this was done to better account for their experimental data. In Sections AII.3 and AII.4 we will discuss this aspect of the problem in greater detail. REFERENCES CHAPTER I l. J. Hubbard, Proc. Roy. Soc. (London) A 276, 238 (1963). 2. M. C. Gutzwiller, Phys. Rev. Letters 19, 159 E19633; J. Kanamori, Progr. Theoret. Phys. (Kyoto) §_, 275 1963 . 3. A. J. Epstein,et al., Phys. Rev. 55, 952 (1972). 4. P. B. Visscher, and L. M. Falicov, Journal of Chem. Phys. 52, 4217 (1970). 5. J. B. Goodenough, Magnetism and the Chemical Bond, Interscience Monographs of Chemistry, Inorganic Chemistry Section, Vol. l, Interscience Publishers, New York 1963 . J. Hubbard, Proc. Roy. Soc. (London), A 2 l, 6. 401 (1964). 7. E. H. Lieb, and F. Y. Wu, Phys. Rev. Letters 20, 1445 (1968). 8. M. Takahashi, Progr. Theoret. Phys. 3, 1619 (1970). 9. A. A. 0vchinnikov, Sov. Phys. JETP 30, 1160 (l970).[Zh. Experim. i Theor. Fiz. 51, 2137 (1969)]. 10. H. Shiba, and P. A. Pincus, Phys. Rev. 35, 1966 (1972). 11. H. Shiba, Progr. Theoret. Phys. 48, 2171 (1972). 12. H. Shiba, Phys. Rev. B6, 930 (1972). 13. L. M. Roth, Phys. Rev. 184, 451 (1969). 14 . Langer; M. Plischke; and D. Mattis, Phys. Rev. Letters 3, 1448 (1969). NE 19 20 15. T. A. Kaplan, and R. A. Bari, Journal of Applied Phys. _l, 875 (1970). 16. T. A. Kaplan, Bull. Amer. Phys. Soc. 13, 386 (1968). 17. T. A. Kaplan, and P. N. Argyres, Phys. Rev. B1, 2457 (1970). 18. N. Silva,.and T. A. Kaplan, Bull. Amer. Phys. Soc. 18, 450 (1973). 19. T. A. Kaplan, Bull. Amer. Phys. Soc. 18, 399 (1973). 20. T. A. Kaplan, and S. D. Mahanti, Bull Amer. Phys. Soc. 292 (1972); K. Levin, R. Bass, and K. H. Benneman, Phys. Rev. 86, 1865 (1972). 21. P. W. Anderson, Solid State Physics 14, ed. by Seitz and Turnbull (Academic Press, New York 1963): p. 99; L. N. Bulaevskii, Zh. Exp. Theor. Fiz. 51, 230 (1966) [JETP 24, 154 (1967)]. 22. R. A. Bari, and T. A. Kaplan, Phys. Rev. 86, 4623 (1972). 23. C. N. Yang, Phys. Rev. Letters 19, 1312 (1967); H. Bethe, Z. Phyzik ll, 205 (1931). 24. R. B. Griffiths, Phys. Rev. 133A, 768 (1964). 25. C. J. Fritchie, Acta Cryst. 20, 892 (1966). 26. I. F. Shchegolev, Phys. Stat. Sol. (a) 12, 9 (1972). 27. A. J. Heeger, and A. F. Garito, AIP Conf. Proc. No. 10, Magnetism and Magnetic Materials, 1476 (1972). PART A CHAPTER II STATISTICAL MECHANICS OF THE HALF—FILLED-BAND HUBBARD MODEL Statistical Mechanics of the Half-Filled Band Hubbard Model (Phys. Rev. B7, 2199 (1973), with T. A. Kaplan) 21 lx'l PHYSICAL REVIEW B VOLUME 7, NUMBER 5 2199 1 MARCH 1973 =l= Statistical Mechanics of the Half-Filled-Band Hubbard Model D. Cabib and T. A. Kaplan Michigan State University, East Lansing, Michigan 48823 (Received 21 July 1972) We have calculated thermodynamic properties of the half—filled-band Hubbard model for a ring of N=4 atoms. Our results resolve serious discrepancies between similar calculations which have appeared. For weak interactions, a new kind of smooth magnetic transition (non— antiferromagnetic) is found at low temperature. For strong interactions, properties are ap— proximately independent of Nwhen the grand canonical ensemble is used, enabling contact to be made with recent experimental work on N—methyl phenazinium tetracyanoquinodimethan (NM P) (TCNQ); the comparison suggests strongly that the Hubbard model is seriously deficient as a means of description of these experiments. There has been considerable interest recently“3 in the Hubbard model for electrons in a half—filled band. Since exact results are extremely limited, particularly in the intermediate temperature range and for bandwidth b of the order of the Coulomb in— teraction U, we began a study of exact numerical solutions for small numbers of atoms. Since that time three papers“-6 have appeared giving results of similar calculations. Their results disagree with each other in several important qualitative respects: in the region of large b/U one group“ (SP) found one peak in the specific—heat—vs—temper— ature curve, the other group516 (HM) finding three peaks; for b/Uz 1, the groups again disagree as to the number of peaks found. (These statements concern the four—atom ring, the only case common to both groups. ) Here we resolve these important theoretical dis- crepancies. We agree with the number of specific— heat peaks found by HM; however, numerical com— parison is not possible because of inconsistencies in their results. We also disagree with their inter— pretation of these peaks and find instead a new kind of smooth magnetic transition. Further, the extrapolation to large systems as to the existence of the low -temperature peaks for large b/U is shOWn to be not possible on the basis of the four- atom results in disagreement with HM: whenever one-half the number of atoms is even, we show that there is a low-T peak for large b/U which does not scale with the size of the system. The behavior for small b/U does not appear to be spurious in relation to macroscopic systems, and we there- fore carefully examined the susceptibility to com— pare with recent experimental results.7 Whereas the previous calculations were made using the canonical ensemble, we have also made calcula- tions in the grand canonical ensemble, as moti- vated below. We consider a system of four atoms at the cor- 2200 TABLE I. Comparison of results with high—temperature expansion. a U— (H) %U2 fi-U+ 1L1 1/32 25><10'3 0.19987512 1.2x 10‘4 0.12700320 25><10'4 0.01999988 1.2x10'7 0.125 207 55 25x 10-5 0.002 000 00 1.2 x 1040 0.125 02078 ners of a square. As usual the Hubbard Hamil- tonian is written H=Ebuclacja+UZ> NHN“ . (1) Ho 1 We include only nearest-neighbor hoppings (PH 2 b when i andj are nearest neighbors). Unless specified otherwise, bE 1. All energy eigenvalues and eigenfunctions are calculated numerically for several values of U; from these the statistical average of any operator 0 (expressed as a function of the creation and destruction operators c}, and C”), can be calculated either in the grand canoni- cal or canonical ensemble (GCE or CE) according to the equation - 301- 111172) <0):T-Il:r0‘:e H-uNe) 7 (2) where B: l/kT, u = chemical potential. The trace runs over all states in the GCE, and only over states with fixed number of particles Ne in the CE. It turns out that for the half-filled band ((Ne ) = num- ber of atoms) )1 = éU independent of T. The motivation for calculating in both the GCE and the CE is twofold. One point is that in the D. CABIB AND T. A. KAPLAN '1 atomic limit (b/U- 0), any intensive parameter (e.g., the free energy per atom) is independent of the number of atoms N when calculated in the GCE. Therefore the GCE for small N can be expected to give results close to those for N—~ so for small b/ U. The other point is that, since all results for CE and GCE become the same for N —~ 0°, any qualitative feature that we may discover for small Nwill be considered suggestive as to the large—N behavior only if such a feature occurs both in CE and in GCE. The checks of our computer program are: (1) At high temperatures for all U we expanded the expo— nentials in (2) in powers of [3 retaining only terms of the first few orders in B. We compare the nu— merical results with the expansion coefficients. For instance we have computed the following quan— tities for U=4 in the GCE: = U— tBU2+ 0032), Li= 432+ our), (4) where L,,= ((Ni, +N‘.) (N, m, — N, m,)). (Because of symmetry, L,l is independent of i.) The numerical results are given in Table I. We see that U— (H) is about 88 and that éUZB— U+ (H) is of order 82 or higher; similarly 1L11/32 is about i and - 1L11/ Ba+é is of order [3, in agreement with (3) and (4). (ii) In the two cases U=O and U=°°, the various (0) were again calculated analyticallya in GCE and compared with the numerical results. There is agreement in at least the first eight figures. (iii) In the case of large U and low T we checked the magnetic susceptibility against the results of (3) L L N k N k .50 .50 .25- .25- 0 I I I I I I I kT O I I 1 I I 1 RT FIG: 1. Specific heat C .Ol 0.1 LO ID .01 0.1 1.0 IO and Slim-39m correlation functions Ln vs temperature L (0) L (b) in the GCE. (a) U=8; (b) I g n U: 0.7. 0.5 - 1 RT 4 J: 0.5 - 0.4 F ’\ a 2 / \ / \ 0.3 — / ‘ _ / \ \ / 2 I /O\ \ 0.2 ~ I / \ - / ’ \ II, ‘ ’I-s 2\ \ I / ‘\\ \ ‘ 0.1— / 1"o '~\\“ \\2, / ‘ ‘ ~ / ‘1. / / l 1 ) 5 IO U FIG. 2. Temperature at which the specific-heat maxi— ma occur vs U are shown by the continuous lines. The dashed lines labeled by numbers n show the temperatures near which anomalies in L,l occur. Bonner and Fisher9 for the Heisenberg model which is expected to reproduce the behavior of the Hub- bard model under these conditions when the ex— change constant J= — 2122/ U. We find convergence with increasing U of our peak location and height to within about 12 and 6%, respectively, by the time U: 15. The specific -heat vs T is shown in Fig. 1 for U=8 and 0. 7 for the GCE. In qualitative agree— ment with HM we find three peaks in the specific heat at least for 0< U26 both in CE and GCE. For H: 8 there is rough agreement with SP’s results, but disagreement for lower U. Quantitative com- parison with the work of HM is not possible be- cause of inconsistencies in their results. (Figures 1 and 2 of Ref. 5 give appreciably different peak locations.) In Fig. 2 we summarize the tempera- tures at which the peaks in the specific heat occur. To understand the physics of these peaks, we studied the spin—spin correlation function §Lm n = 0, l, 2. We note that the zero—field spin-suscep— tibility X is related to this by x=(kT)"(L,+2L1+L,) . (5) STATISTICAL MECHANICS OF THE HALF-FILLED-BAND. . . 2201 As shown in Fig. 1, L0, -—L1, and L2 undergo a more or less sudden change in correspondence to one or another of the peaks in the specific heat. For clarity, we discuss separately the two regions, U> 6 and U< 6 (where there are two and three spe— cific -heat peaks, respectively). For U> 6 we see from Fig. (1a) that lLll and L2 simultaneously decrease sharply at temperatures near TI=Tn, the low-T peak in the specific heat, while LO remains essentially constant through this temperature region. Aside from the lack of any mathematical singularity in these functions, this behavior is very similar to the well-known anti- ferromagnetic transition in large three-dimension- al systems. We will therefore adopt the terminol- ogy, used in the liter‘atux‘e,“‘6 which calls TI: T" the Néel temperature. We note that this tempera- ture : sz/U, as expected from the relation between the Hubbard and the Heisenberg model mentioned above. In the small—U region, we note a remarkable fact. Although X has a peak near the lowest tem— perature peak (T1) in C, L2 is seen in Fig. 1(b) to have an essentially constant value different from zero up to the temperature (Tu) at which the middle peak in C occurs, and above this temperature it goes rapidly to zero. 1L1], On the other hand, is seen to start to decrease sharply near TI. The fact that |L1| and L2 do not start to decrease sharply near the same temperature is in marked contrast to typical behavior at a magnetic transi- tion. Hence the characterizationS'6 of T, as a Néel temperature is misleading and unacceptable. We also note that L0 is essentially constant near TI, and decreases rapidly near T“, for U small. The relation of the high—T C peak (at Tm) to a characteristic change in L0 has already been noted.4'6'10 We see [Fig. 1(b)] an additional effect at small U, namely, L1 also shows an anomaly near Tm, which somewhat surprisingly disappears at a value of U roughly equal to one. This plus the other anomalies in L,l are indicated by the numbers accompanying the curves in Fig. 2. We consider the significance of the unusual re—_ sults obtained, namely, the low—T peaks in C for small U and their physics. In fact, one cannot ex- pect these effects to continue to exist as the num- ber of atoms N—~ co because of the following reason. Consider first the four-atom four-electron system. For U: 0, the ground state is sixfold degenerate, including a triplet and three singlets. This de- generacy is seen by considering the occupancy of the one—electron levels Eh: 2b cosk, k: 0, i $11, 11. The minimum, which occurs either at k =0 or k: 11, accomodates two electrons; but the other two elec- trons can occupy four one-electron states (k: 1%11 spin up and down) all with the same energy. The existence of the triplet among these ground states 2202 implies, of course, that x will exhibit Curie—law behavior at low T. Furthermore, C will show a low-T peak when U increases from zero because of the splitting induced in this ground level. Clear- ly, this effect occurs whenever %N is even, but it will become negligible as N-ow; e. g. , the Curie- law term in x will approach zero since the total magnetic moment is always from a triplet, and will not increase with N. On the other hand, when %N is odd, the ground state for U=0 is a singlet, so that the above effect will not occur. Clearly, for N: 2 or 6, the first excited state lies above the ground state by an en- ergy of the order of b for U small, so that no low- T peak (at kT << b) in the specific heat will occur. Hence, in these very small systems, there is no “band antiferromagnetism” (for which, by defini- tion, the Néel temperature - 0 with decreasing U). One cannot conclude from this that such antiferro- magnetism does not occur for macroscopic sys- tems, since for large N the separation of the low— lying states is O(b/N) for U: 0. (It might be that as N increases for small U the peak splits, with the lower-T peak moving to low temperature.) Although as we have just seen, one clearly can— not use the four—atom results to guess about large systems for small U, this is not so for large U. In fact, when U=oo, we have noted above that the GCE results for small N give the large-N behavior exactly. Furthermore, the qualitative behavior that we find (a Neel-like smooth transition at kTN z 2bz/U, a highly correlated nonmagnetic system for m << kT<< U with << =i, these correlations decreasing markedly as kT becomes '2 U) is what we expected on the basis of earlier work.’"11 There3 essentially the same physical picture was found for large U on the basis of a variational single-determinant approximation, in which the best one-electron states were found to be the Wannier functions for all T. Therefore we felt that one should look carefully at x vs T for a sign of the leveling off of x'1 found by Epstein et at.” at high T (~ 200 °K). Using their values b= 0. 021, U/b= 8, we looked closely in the D. CABIB AND T. A. KAPLAN 1 region of temperature corresponding to the experi— mental anomaly. We found no such effect. Fur- thermore, the location of the minimum in x'1 (at kTo: ZbZ/UE 60 °K for the above values of b and U) occurs at much higher temperature (by a factor of about 3) than the temperature at which a rounding off occurs in the experiment? We can get a sug— gestion as to whether To might reduce by the needed factor when N increases from 4 to 00 from the results on the Heisenberg chain,9 and from comparison with the easily solved N= 2 Hubbard model. For the Heisenberg chain, To decreases by about 20% when N goes from 4 to no, and for the Hubbard model by about 10% when N goes from 2 to 4. Thus it seems unlikely that To for N: 00 will be low enough. Furthermore, we expect the qualitative behavior to be similar to that for the Heisenberg model, for which x“ shows a minimum and then levels to a finite nonzero value at T: 0.9 In support of this ex- trapolation, we note that the minimum value of X4 in the Heisenberg model is insensitive to N for N 14 and that in the Hubbard model the exact value12 of x" at T=0 lies well above this minimum calcu— lated for N: 4 (for U=8, b = 1); this is consistent with an extrapolated X(T)'l, which is qualitatively similar to that found for the Heisenberg chain.9 Such qualitative behavior is very different from the experimental results. In view of this discrepancy at low T and the above failure to find the experi- mentally observed leveling off in X4 at high T, one is led to suggest that major modifications of the Hubbard model are needed to explain essential fea- tures of the high-T transition (called a “metal—in— sulator transition” by Epstein et al.) and the low— T antiferromagnetic behavior. We thank Professor S. D. Mahanti for valuable discussions. Note added in Droof. For additional aspects of the comparison with experiment on (NMP) (TCNQ) and the extrapolations see D. Cabib and T. A. Kaplan, AIP Conference Proceedings No. 5, Mag— nelism and Magnetic Materials, edited by C. D. Graham, Jr. and J. J. Rhyne (AIP, New York, 1972). *Work supported by the National Science Foundation. ‘J. Hubbard, Proc. Roy. Soc. (London) £73, 238 (1963). 2E. H. Lieb and F. Y. Wu, Phys. Rev. Letters Q, 1445 (1968). 3T. A. Kaplan and R. Bari, J. Applied Phys. 4_1, 875 (1970); Proceedings of the Tenth International Confer- ence on Physics of Semiconductors, edited by S. P. Keller, J. C. Heusel, and F. Stern (U. S. AEC Div. of Tech. Information, Springfield, Va., 1970), p. 301. 4H. S. Shiba and P. Pincus, Phys. Rev. B§, 1966 (1972). 5K. H. Heinig and J. Monecke, Phys. Status Solidi (b) :12, K139 (1972). 6K. H. Heinig and J. Monecke, Phys. Status Solidi (b) 4_9, K141 (1972). 7A. J. Epstein, S. Etemad, A. F. Garito, and A. J. Heeger, Phys. Rev. 13g, 952 (1972). 8The U: 0° results were obtained in Kaplan and Argy— res, Phys. Rev. Bl, 2457 (1970), App. A. 9J. Bonner and M. Fisher, Phys. Rev. 135, A640 (1964). 10But the view of Ref. 4 that the local moments disap— pear at high T is untenable; see Kaplan and Mahanti, '_7_ STATISTICAL MECHANICS OF THE HALF-FILLED-BAND... 2203 Bull. Am. Phys. Soc. u, 292 (1972). 12M. Takahashi, Progr. Theoret. Phys. (Kyoto) 4_3, 1‘See, also, R. A. Bari and T. A. Kaplan, Phys. Rev. 1619 (1970). 3 g, 4623 (1972). 27 2. Static Properties of the Half-Filled Band Hubbard Model [AIP Conference Proceedingg, Magnetism and Magnetic Materials,’Vol. 10, 1504 (1972)], with T. A. Kaplan Exact calculations on the 4 atom ring Hubbard model1 had resolved serious confusion existing in the literature 2,3 on the subject. More specifically the qualitative behavior of the specific heat agrees with that found by Heinig and Monecke,3 (even though the quantitative results are appreciably different) and disagrees with Shiba and Pincus2 (for weak to intermediate interactions). The results obtained with the grand canonical ensemble1 do not differ qualitatively from those obtained with the canonical ensemble; but the use of the GCE is important for the extrapolation to a large number of atoms, N, because in this case when the ratio of hopping integral to Coulomb repulsion, b/U+0, any intensive parameter is independent of N (we take U and b > 0). The specific heat has three peaks for 0 where 512 is the z-component of spin at site i, are very illuminating on the physical significance of these specific heat peaks: the various anomalies in Ln were shown to occur always at temperatures very close to the ones at which one or another specific heat peak occurs. Since the large U/b region had been essentially understood 28 previously4’5 and since we agree qualitatively with the existing resultsz’3’4’5 in this region, we will focus for the moment On the small U/b region. Here the specific: heat has three peaks at temperatures T < T I II < TIII‘ (T1 and TII + O as U/b + 0.) We found that -b1 decreases rapidly near TI’ approaching a constant different from 0, and again near TIII’ approaching 0. L2 decreases from a constant value to 0 near TII‘ From this picture we cannot charac- terize TI as being similar to a Néel temperature (as HM do) because the first and second-neighbor correlations L1 and L2 do not have anomalies at the same temperature. This type of transition had not been found previously. HM argued, by extrapolation, that the low—T specific heat peaks will occur for large N, but we showed1 that this Was_wrong. ,Let us summarize here the reason. First of all this phenomenon occurs only when N/2 is even; in fact in this case the one- particle energy levels for U=0 are given by Ek = -2b cos k k = O, i——, -—, . . . , n and the N—electron ground state (half-filled band) fills completely the states with k=0, iZW/N, . . . , (2n/N)-(N/4 - l) leaving 2 more electrons the possibility of occupying the four states k=in/2(sz=il/2) which all correspond to the same one-electron energy. This gives rise to a six—fold degener- acy of the ground state which is removed when a small U is 29 turned on, explaining the existence of the low—T peaks in the specific heat per atom.6 .This.degeneracy will remain six-fold for all even N/2 and its effect will therefore become negligible when N+w, the height in the above peaks decreasing as l/N. One is naturally led to ask what the situation is for a chain. In this case the hopping integral b is taken to be the same for every nearest neighbor pair of sites, the end-sites having only one-sided hopping. The one- electron energies are fit Ek = -2b cos k, k = ———, 2 = l, 2, . . . , N N+l Therefore, when the number of electrons is even, the ground state is non-degenerate. Hence, when U increases from zero, no appreciable low-T peak in C will occur in contrast to the behavior discussed above for the ring. By appreciable we mean f(C/T) dT integrated over the peak is 2 kB/N and by low-T we mean the peak location, T0<< b/N and T +0 as U+0 O (we have small N in mind). In other words, we cannot expect to find in this case the new type of smooth magnetic transi- tion which was found for the ring. HM7 also stated that_the low-lying specific heat peaks should not occur for the chain, but gave an incorrect argument. Namely they said that the vanishing of the conduc- tivity o for the chain implied the vanishing of the peaks in C, presumably because they had established a causal relation 30 between anomalies in o with those same low-T peaks in C. A reason for the incorrectness of this argument is that their calculation of o is seriously in error, as we now show. HM's results imply that o(w=0)¢0 at zero temperature for any U/b. A calculation from Kubo's formula for the conductivity yields: ( ) Tl l _Bfim _Bfin 0 w = — Z ' ~ Z nm IJ—nm|2 —————————————— 6(w + Enm) nm n ~ "BE - 2 + — B E e n [ll | . 2 nm + 3 Ii 1 ] 5(w) Z n mfin nm m degenerate with n Z is the partition function. Z' is extended to states such that EnfEm (En is the energy eigenvalue corresponding to the n-th energy eigenstate in zero electric field), Ek=Ek—unk (nk=number of electrons in the state corresponding to Ek), inm is the matrix element of the current operator, . _ _. _ + i — 1e Zbij(3i Bj) C10 Cjo' Clearly for a small system (with discrete energies) only the sum involving the square brackets contributes to 0(0). States with an even number of electrons can always be chosen to give a zero contribution to the sum involving the first term in square brackets. (For an even number of particles 31 the time reversal operator 9 and H can be simultaneously diagonalized and since i changes sign under time reversal, inn=0 in this basis.) Since in our case the ground state has an even number of particles, this sum must vanish at zero T. Hence the only contribution to 0(0) at zero T is the second sum in square brackets. For it to be different from zero at zero temperature it is necessary that the ground state be degenerate. Explicit calculations for four atoms showed that the ground state is non-degenerate for finite U and hence the HM result is incorrect. The error can be traced to the paper by Monecke,9 upon which the conductivity calculations for 4 atoms are based. Since the 4-atom calculations for U/b>>l do not give special results whose significance is restricted to small systems, and the GCE for b=0 does not depend on N, we1 tried to compare zero magnetic field susceptibility' calculations with the experimental measurements by Epstein et al.10 (see Figure AII.l). (To this end we discuss the extrapolation to large N. First of all, it is known that for large enough U/b and kT<2, we expect larger changes in behavior in going from say N=2 to 4 than from N=4 to 6 or 6 to 8, etc. 36 3. Attempts to Explain the Susceptibility of NMP-TCNQ by Modifying the Hubbard Model In the previous two Sections we were concerned, among other problems, with the comparison of the theory with the experiments. It had been claimed by some authors3 (the Penn group), that the organic solid NMP—TCNQ can be fairly well represented by the one—dimensional half—filled-band Hubbard model if we appropriately choose the values of the parameters b and U. Later the same group27 suggested that the experimental data would be better explained by letting the parameter b increase with temperature. The needed change would be of roughly a factor of two on going from low T (bm.02 ev.) to Tm200 °K (bm.05 ev.) so that the ratio U/b goes from m8 to m4. However, these conclusions were based on admittedly crude theoretical considerations, so that we did not expect them to be taken seriously. In Sections AII.l and AII.2 we showed the serious difficulties encountered by the claim3 that a T—independent b and U theory represents the behavior of NMP-TCNQ and we stressed the necessity of drastically changing the model. In what follows we shall describe the attempts we made to explain the experiments by changing the model dis- cussed in Sections AII.l and AII.2. They were of two types: in the first attempt we modified the Hamiltonian, but we considered only temperature-independent parameters; in the 37 second, following later emphasis put by the Penn group28 on the temperature dependence of b, we tried to qualitatively fit the experimental susceptibility (X) using the half- filled-band Hubbard model but allowing the hopping integral to change with temperature. Let us now describe the first attempt. We tried a canonical calculation of X for a system of four atoms on the corners of a rectangle, instead of a square as described in Section AII.l: that is, there are two hopping integrals b and b‘. Furthermore the number of electrons is fixed and equal to 2 to account for the total number of unpaired electrons in a TCNQ chain. The justification for trying such a model is essentially the idea that each TCNQ molecule in the organic solid NMP—TCNQ may contribute two spatial orbitals (instead of the one proposed by Epstein et al.3), that can be filled by a maximum of four electrons (because of the spin), and in our calculation the four atoms on the rectangle correspond to the four cyanide groups on two adjacent TCNQ's (see Figure AI.3). The picture given by Epstein et al.3 describes the electrons as occupying a n orbital which is concentrated mainly at the two terminal cyanide groups in a TCNQ molecule; furthermore, according to the same authors, if two electrons happen to be simul— taneously on the same molecule they will be located at the two far ends of the molecule so as to minimize the Coulomb repulsion. But this picture is actually impossible with a 38 single molecular orbital (because in the two-electron state constructed in this way, there is high probability of finding the two electrons at the same end). Hence our model seems more reasonable because we let each cyanide group in a TCNQ molecule contribute a spatial orbital that can be occupied by a maximum of two paired electrons (so that the whole molecule contributes a total of four one-electron states). If these two spatial orbitals overlap appreciably, they would make up two bands very different in energy from each other: in this case one could neglect the higher energy band and end up with Epstein et al.'s single E—orbital picture. 0n the other hand if the two orbitals do not over- lap very much the two bands will be nearly degenerate and there is no justification for neglecting one of them. A calculation of a 4—atom Hubbard model on a rectangle with two electrons as described above, would hopefully account for the physics of this latter case, apart from the usual problems of extrapolating the results to an infinite system. We tried a few sets of parameters: i) b=l, b'=.5, U=8, ii) b=l, b'=2, U=8, iii) b=b'=l, U=l, 4, 8. b represents the hopping between the two orbitals within the same molecule, b' the hopping between two orbitals on near cyanide groups on neighboring molecules, U is the Coulomb repulsion of two electrons on the same orbital. Unfortunately the results were discouraging; none of the main features of the experi- mental X-1 curve vs. temperature were recovered: the 39 high-temperature leveling off is absent in the theoretical curve, and there still remains a minimum (the location of the minimum is shifted toward higher temperatures compared to the previous results showed in Figure l of Section AII.2); finally the slope in the Curie-Weiss region of temperature is unchanged with respect to the same previous results. Especially this slope should not depend on the size of the system as explained in Section AII.2. Thus, these seemingly reasonable generalizations of the Hubbard model apparently do not overcome its deficiencies vis a vis the experiments. As for the second attempt, let us now describe its conceptual significance and our results. To let the parameters of the Hamiltonian change with temperature has very subtle conceptual implications. What is usually done to test the validity of a theory is to calculate the thermodynamic properties of the Hamiltonian with fixed parameters and then compare with experiments. Furthermore in general one expects a Hamiltonian to be able to fit any experimental results, if it is a function of a sufficiently large number of temperature~dependent parameters. On the other hand the concept of a temperature- dependent Hamiltonian is not completely extraneous to a theory of the Hubbard type. In fact we have to bear in mind that the Hubbard Hamiltonian is an approximate one: it has to be justified in terms of a derivation from the exact N-electron and N-nucleus Hamiltonian. Important 40 physical effects may occur in a real system that are com— pletely neglected in a temperature-independent theory of the Hubbard type such as the ones induced by electron-phonon, other bands or long-range electron—electron interactions. As a matter of fact a derivation of the Hubbard Hamiltonian based on Bogoliubov's variational principle18 would give a temperature dependent theory; and it would also give the best Hubbard Hamiltonian which approximates the exact Hamiltonian of the crystal. In any case if one is forced to let the parameters of his approximate theory vary ap- preciably with temperature, to be able to make contact with experiments, that suggests at least that the neglected interactions are not negligible, i.e. the starting Hamiltonian has to be appreciably changed. 27,28 We tried to check Heeger and Garito's (HG) suggestion of a band broadening with temperature; we found that their data on NMP-TCNQ could be fit quantitatively with a temperature dependent bandwidth as follows. We plot the experimental inverse susceptibility S'] = (xU/Nu§)—1 as a function of kT/U; then we plot on the same graph the l theoretical S_ for different ratios of b/U: when b/U is in the range l/8ml/4 we use our 4-atom results (Section AII.l) and Shiba'sH 6—at0m results and extrapolation. In Figure AII.2 theoretical curves (for N=w) vs. kT/U for U/b=4 and 35have been added to Figure AII.l, and the plot 41 is changed (xb+xU, kT/b+kT/U) because U is presumed28 a constant independent of temperature. 'U/b=3.5 I EXPERI"‘ 4 " \ 2 -1 — ( (XU/NUB) U/b=4 .2 _ ,. 4-ATOM RING /,::<‘:’ _‘\.(U/b=8 ,, x” 0 ,,,~//" KsCURIE WEISS (HEISENBERG) " l 1 1 I kT/U 2(b/U)2 .062 .125 Figure AII.2.-—Determination of b/U as function of temperature. As seen from Figure AII.2, the U/b=8 and the U/b=4 curves intersect the experimental curve respectively at T=0 and kT/Um2(b/U)2; the U/b=3 curve agrees roughly with experiment above kT/Um.l Furthermore the minimum value of 3'1 increases with b/U and shifts to higher temperatures: when U/b=0 the theoretical curve is at infinity. The fact that the minimum shifts to higher temperatures is very plausible (because kT m bZ/U for b<< U and kTminm b/U for b>> U). Therefore, min at least in principle, one can determine b/U as a function of T by determining the intersection points between the 1 experimental S' and the various theoretical 5'] curves: to each temperature there corresponds a theoretical curve 42 with a particular b/U that crosses the experimental curve at that temperature. This correspondence gives b/U as a function of T. From Figure AII.2 we see that within this scheme b/U does indeed increase by roughly a factor of two on going from 0°K to m200°K in agreement with HG's suggestionzg; but we also see that at every b/U the minimum in the theoretical S'1 as a function of T lies to the right of the intersection point with the experimental 3']. Since this minimum is the ordering temperature which marks a smooth but rapid transi— tion from weakly correlated spins (|l<< ’ ifj) lZ jz . N 2 . . 29 to strongly correlated spins (|[—|I, ifj) , the interpretation of the experimental results is totally 28: the different from that given by Epstein et al.3 and HG spin variables Siz are strongly correlated in the region of temperature 5 200°K, instead of being uncorrelated. In other 1 words the straight-line portion of 5' below 200°K in the NMP-TCNQ would not correspond at all to a Curie-Weiss behavior above the ordering temperature as suggested3’28 , where the spins are weakly correlated, but would correspond to a long and appreciable short-range order of the electron spins. Clearly, a crucial experiment for testing the model is a measurement of the spin-spin correlation function, which at least in principle should be possible via neutron scattering. REFERENCES Section 3 J. Hubbard, Proc. Roy. Soc. (London) A 276, 238 2. M. C. Gutzwiller, Phys. Rev. Letters 15, 159 63); J. Kanamori, Progr. Theoret. Phys. (Kyoto) 30,275 63 . 3. A. J. Epstein, et al., Phys. Rev. 55, 952 (l972). 4. P. B. Visscher, and L. M. Falicov, Journal of Chem. Phys. 52, 42l7 (l970). 5. J. B. Goodenough, Magnetism and the Chemical Bond, Interscience Monographs of Chemistry, Inorganic Chemistry Section, Vol. l, Interscience Publishers, New York l963 . J. Hubbard, Proc. Roy. Soc. (London), A 2 l, 6. 401 (1964). 7. E. H. Lieb, and F. Y. Wu, Phys. Rev. Letters 25, 1445 (l968). M. Takahashi, Progr. Theoret. Phys. 43, l6l9 (1970). A. A. 0vchinnikov, Sov. Phys. JETP 5Q, 1160 (1970). lo. H. Shiba, and P. A. Pincus, Phys. Rev. 55, l966 (l972). ll. H. Shiba, Progr. Theoret. Phys. 48, 2l7l (l972). l2. H. Shiba, Phys. Rev. 55, 930 (l972). 13. L. M. Roth, Phys. Rev. 184, 451 (1969). Z l4. Langer; M. Plischke; and D. Mattis, Phys. Rev. Letters 23, l448 (l969). 43 i} 1‘ .‘IIH’IIW ‘1 [5’15 3“! a. 44 l5. T. A. Kaplan, and R. A. Bari, Journal of Applied Phys. 4l, 875 (l970). l6. T. A. Kaplan, Bull. Amer. Phys. Soc. 15, 386 (1968). l7. T. A. Kaplan, and P. N. Argyres, Phys. Rev. 51, 2457 (l970). l8. N. Silva, and T. A. Kaplan, Bull. Amer. Phys. Soc. 15, 450 (l973). l9. T. A. Kaplan, Bull. Amer. Phys. Soc. 15, 399 (1973). 20. T. A. Kaplan, and S. D. Mahanti, Bull Amer. Phys. Soc. 292 (l97 2); K. Levin, R. Bass, and K. H. Benneman, Phys. Rev. 55, l865 (l972). 2l. P. M. Anderson, Solid State Physics 14, ed9 9by Seitz and Turnbull (Academic Press, New York l96377p L. N. Bulaevskii, Zh. Exp. Theor. Fiz. 51, 230 (l966) [JETP 54, l54 (l967)]. 22. R. A. Bari, and T. A. Kaplan, Phys. Rev. 55, 4623 (l972). 23. N. Yang, Phys. Rev. Letters 15, l3l2 (1967); H. Bethe, Z. Phyzik 7l, 205 (l93l). 24. R. B. Griffiths, Phys. Rev. 133A, 768 (1964). 25. C. J. Fritchie, Acta Cryst. 55, 892 (l966). 26. I. F. Shchegolev, Phys. Stat. Sol. (a) 15, 9 (1972). 27. E. Ehrenfreund; E. F. RybacZewski; A. F. Garito; and A. J. Heeger, Phys. Rev. Letters 55, 873 (l972). 28. A. J. Heeger, and A. F. Garito, AIP Conf. Proc. No. l0, Magnetism and Magnetic Materials, l476 (l972). 29. Note that even in the limit of zero interactions, U= 0, these spin observables S 2 are strongly correlated for T>U but not necessarily greater than b (in the case U<>U). So far we have carried out the first term of the expansion and up to the first order in BU and first order in U/b the result agrees with RPA31; the second order term is under study. If we define H0, 20 and xo/NuBZ to be the Hamiltonian, partition function, and susceptibility per particle respec- tively, when U=0 (see Equation (l) of Chapter I), we have, in first order in BU: 2 _ 2 _ 2 x/NuB — l/NuB (x0 + x1) -(9 B/4NZ0) trIeXP[-B(H0-uNe)]° 2 2 '[i("k+‘"k+)] (1 - BH1 - Z1/Zo)} + 0[(BU) J (2) Since 2 _ 2 2 xO/NuB -(g B/4NZO) tr epr-B(H0 - uNe)] [E(nk,-nk,)l = 928/4 x [1 — (2/N)Efk] (3) Equation (2) defines X], N = Z n , u is the chemical e k0 ko potential, H1 is the Coulomb repulsion + + H1 ‘ (U/N) k k? qak+q.+ ak,+ ak'-q.l ak'.+ (4) 49 written in the Bloch—function basis. 21/20 is defined as: Z1/Z0 = -8 tr exp[-B(H0 - uNe)] H1 (5) Finally in (3) fk is the Fermi function f(Ek) at wave- vector k. (2) and (5) are obtained taking into account the non-commutativity of H0 with H]. If we restrict ourselves to the half-filled-band case where the average number of electrons is equal to N, u=U/2 and is independent of temperature. Expanding the exponential factor in (2) and (5), Equation (2) becomes, up to first order in BU, 2 2 X/NUB = ' Z1X0/ZONUB + (928/4Nzo) tr EXp(-BH0)- -[1 +(BU/2)Ne - 8H,] [E(nk, - nk,)12 (6) Using repeatedly the rule that the thermal average w1th the Ham1lton1an H0 15 equal to ... if (ki’Oi) # (k. J,oJ.) for each X1/Nu32 = (ngzU/8) [1 - (Z/NlikaZ = (2U/gz) (xo/Nu32)2 (7) It is easy to see that this first order correction does not affect t if we assume that dzx /dT2| = #0, which we 0 TO, U 0 0 checked numerically. Therefore in the expansion (l), a0=0, 50 and the intercept of the straight line asymptote with the coordinate axes coincides with the origin. We can also write X/NpBZ = XO/Nsz (1 + ZUXO/NuBzgz) + 0((BU)2) (8) and it is easily seen that if U is treated as a perturbation parameter, in first order in U (8) agrees with the RPA expression obtained by Hubbard and Jain.31 Therefore the RPA is exact for the Hubbard model in this order. (For a more general type of interaction the exact and the RPA results need not agree in first order in the interaCtion,) In conclusion, in this Section we have calculated the susceptibility of the half-filled—band Hubbard model to first order in BU, and, concentrating on the U< are zero for n f m if the basis set {|n>} is chosen in such a way that each |n> is a simultaneous eigenfunction of the Hamiltonian and the lattice translation operator. Cabib and Kaplan8 56 showed that the conductivity calculations by Heinig and 9 for the ring of 4 atoms, based on Monecke's work,7 Monecke are wrong, by showing that the zero temperature d.c. conductivity vanishes, in contrast to the result given by those authors.9 A very simple and direct argument that shows the incorrectness of Monecke's analysis is that from his equation for on pg. 372, it immediately follows that the diagonal elements are also zero. Hence his argument must lead him to the conclusion that the operator 1 of eq. (3) above is zero, a conclusion which is evidently wrong. However, finding the explicit error was subtle and the result somewhat surprising, so we will now present the arguments. The essential point is that Monecke's proof7 of = 0 for n f m was based on the assumption that 1 is invariant under the lattice translations. In the next Section we show that this assumption is incorrect. 2. Lack of Translational Invariance of j; Resolution of the Disagreement in the Literature Let us first consider the case of a ring. Here the lattice translation operation is an N-fold rotation about the axis perpendicular to the plane and passing through the cen- ter of the ring, with N equal to the number of atoms. If T is the operator which performs this rotation 57 -‘l N T1T = -ie 2 t i.:1=1 o + 13(31 ' —j io jo _ _ . _ + ‘ ‘6 i3 t1-1,j-1(31-1 B--j-1) cio cjo O = -ie 2 t ij 0 + 1j(31-1 ‘ Bj-l) cio Cjo Thus [Tai] # 0 (5) because clearly Bi-l - Ej-l f 51 - Ej' (These two vectors have the same length but different direction). In the case of a chain of N atoms with periodic + _ + = = N+i,o ‘ C10 and tN,N+1 tN,1 t12 because the Mannier functions W(:'3i) and w(5—5N+i) are equal, making H translationally invariant. However, we now boundary conditions c point out that despite this, 1 as in (4) is not translation- ally invariant. This is so because there appears a term (we take nearest neighbor hopping for simplicity) . + + . ' ‘e t12(3N ‘ 31) (0N0 C10 ' C10 CNo)’ (6) clearly 5N - 5] f 5N - R and this is sufficient to spoil —N+1’ the translational invariance. (As stated in the Introduction the failure of Monecke's argument in either case is in the 58 fact that he incorrectly assumed that 1 as calculated from eq. (3) is translationally invariant.) If one simply replaces, ad hoc, the term (6) in eq. . + + . (4) by -1e t]2(5N - 3N+l) (cNO c1O — c1O CNo)’ then we obta1n a new operator 5, which is translationally invariant. In fact one can see that it is this 5 that has been used in 2-7 in connection with the calculation of the literature the current-current correlation function (with the reser- vation noted above in connection with ref. 7). This con- cludes the resolution of the disagreement between the results of different workers who began with apparently the same definitions. 3. How to Define the Current Operator? There remains the question, how should one define the current operator? That there is ambiguity stems from the fact that the operations of commuting the polarization operator and the Hamiltonian (to find the current) and projecting these operators on to the space of functions spanned by the single-band states, do not commute. That is, suppose P is this projection operator, e5 E 5 e51 and 1 H = 2 pi 2/2m + V(. . rj. .) are the polarization and Hamiltonian respectively. Then commutation followed by projection gives 1 eP P1gP = P— [eLH] P = —— (5H - HX) P (7) i i 59 whereas projection first gives10 1 .i = T [peim PHP] = (pip HP - PH PAP); (8) l e 7 the two expressions on the right of (7) and (8) are seen to be unequal in general (they are unequal if H and 5 connect single-band states with at least one common state which is orthogonal to all the single-band states). In the case of a linear chain with periodic boundary conditions, the lack of translational invariance of 1 is an unsatisfactory property. Apart from the fact that this lack of symmetry is contrary to the usual concept of current (19), it is easy to see in the simple special case of U=0, that the expectation value of 1 in the Bloch-function energy eigenstates is zero for finite N (also contrary to the usual expectation). In our opinion, 5 is a satisfactory choice at least from the viewpoint of satisfying the right symmetry (again for the chain with periodic boundary conditions). For our nearest neighbor example, with N > 2, we have (for t12 real) + + e N i = T 312 1:12 E 0 (C10 Ci+1,o ' Ci+1,o 010) (9) 1 i=1 2 Furthermore, under rather loose restrictions as specified below, 5 can be chosen to be the same as P 19 P, in which 60 case the definition is a natural one. To see this we note that we can write (in the absence of an external vector potential): I . + P' P = Z c. . e = — Z n (10) m ko 59 where nk0 are the Bloch-function occupation-number operators, and 5 is the electron momentum. Also from (9) we have 5 = e EWE e£)n£0. (11) where e = l 2 e1£R —ij t-- (12) ‘h N i,j 13 Clearly (10) and (11) are equal if 1 — <£|RIE> = Vk 8k, (13) m a relation found in many solid-state texts. Since this 8k must be differentiated in order to obtain (11), it must—be defined for continuous 5; to every choice of the 5, (to within crystal lengths) there cor- responds a different function ck; all of these functions 61 have identical values at the discrete wave-vectors in the Brillouin zone, but they differ in their derivatives at these wave-vectors. The particular choice of the 51 leading to (9) yields that continuous function 8k which would be approached by the set of discrete values when N+w, whereas other choices will lead to different 6k and different cur- rents (e.g., if 3N+l is replaced by 51—then 5 will be replaced by 1).1].. Equation (13) will not be true for arbitrary Bloch- functions |5> and energies ck, since the derivation12 depends on the assumption that |5> and ck are eigenstates and eigenvalues of a Hamiltonian of the form 2 P h = - + V(L) (l4) 2m where V(L) is a periodic k—independent local potential.12 Nevertheless it is clear that there exists a variety of V(L) and their resulting 15> and ck which will provide a sufficiently rich variety of sets of parameters tij = eih'gijek to probably be able to achieve (wi. hwj) = zl—a [FM any Hubbard Hamiltonian one desires; for any of these the above discussion shows that the remaining ambiguity in the definition of the current operator (the choice between 5 and P1gP) does not exist. 62 4. Summary In the above discussion we first reviewed the general definition of the current operator as it is used by differ- ent authors in the calculation of the electric conductivity in the one-band Hubbard model. He pointed out the disagree- ment existing in the results of different authors, one7 finding free electron behavior (absorption only at w=0), the 2,6 others finding also high frequency absorption (w=U in the atomic limit); we settled the controversy by showing that Monecke's discussion7 leads him to the obviously incorrect conclusion that 1 E 0. In Section 2 we showed that his assumption, that 1 as defined in (3) is transla- tionally invariant, is wrong in both a ring and a chain of N atoms with periodic boundary conditions. The other authorsz-6 used a definition much closer to 5 than to 1: if we presume they had in mind periodic boundary conditions, then their definition was identical to 5; if they had in mind a finite crystal, then their definition would differ from 5 by only surface terms (whose effect becomes negli- gible for macroscopic crystals), in contrast to the dif- ference between 5 and the definition 1 for periodic boundary conditions,H as used by Monecke.7 Also, theyz-6 did not use the commutator (3) subsequent to obtaining an explicit form for the current. Since the lack of trans- lational invariance in 1 makes this 1 an unsatisfactory current operator, in Section 3 we approached the problem 63 of the definition of a ”good'I current operator; after having explained that there is still an ambiguity in this defi- nition due to the fact that the projection of the product of two operators is not in general equal to the product of the respective projections, we showed that the ambiguity disappears under broad conditions: the “good" current is proportional to Z (Vk ek)nkc provided that we properly define ko the one-particle energies ck at those vectors k away from the . 2n£ discrete values ——— . N REFERENCES CHAPTER III 1. A. Isihara, Statistical Physics, Academic Press (1971) Chapt. 13. 2. K. Kubo, J. Phys. Soc. Japan 51, 30 (1971). 3. R. Bari; D. Adler; and R. Lange, Phys. Rev. 55, 2898 (1970). 4. R. Kubo, and N. Ohata, J. Phys. Soc. Japan, 55, 1402 (1970). 5. F. Brinkman, and T. M. Rice, Phys. Rev. 55, 1324 (1970). R. Bari, and T. A. Kaplan, Phys. Rev. 55, 4623 (1972). 7. J. Monecke, Phys. Stat. Sol. (b), 51, 369 (1972). 8. D. Cabib, and T. A. Kaplan, AIP Conference Proc., Magnetism and Magnetic Materials Vol. 10, 1504 ' (l972). . H. Heinig and J. Monecke, Phys. Stat. Sol. (b), 59, K117 (1972). 10. For simplicity we use in Eq. (8) the same symbol 1 as in Eq. (3), although P 5 P differs from 5 by off-diagonal elements of 5 between Wannier functions (which were argued to be small in ref. 3) 11. It is interesting to note that, even though for large N there are very few terms involved in the difference 5 - 1, (presuming short-range tj' of course), there contri- bution is as large as that of al the other terms. In our nearest—neighbor example, the function (12) appropriate to 1 is Ek = (2t/N)[(N—1) cos ka + cos (N - 1) ka], as compared to 6k = 2t cos ka, appropriate to 5. It is easy to verify that the term cos(N-l) ka gives as large a contribution to 64 65 dEk/dk as that of (N - 1) cos ka. Furthermore, (dEk/dk)k = 5&5 with l integral, becomes physically meaningless in the limit of large N, in contrast to the well-behaved dek/dk. 12. See, e.g. J. C. Slater, Insulators, Semicon- ductors and Metals, Quantum Theory of Molecules and Solids, Vol. 3, McGraw-Hill, 1967, App. 13. Not only does the standard derivation assume this locality, but one can show that locality is necessary for the result Eq. (13), a fact which we believe has not been noted previously. The proof is based on the fact that the velocity operator v for electrons in a periodic potential is equal to (p = momentum operator, m = electron mass) only when the pos1tion operator 5 commutes with the potential: in the case of a non local potential this commutativity does not hold. PART B 66 PART B CHAPTER I ONE-DIMENSIONAL ISING MODEL WITH RANDOM EXCHANGE INTERACTIONS 67 68 1. Introduction Thermodynamic properties of a spin system with random exchange interaction have been of considerable interest recently.1'4 There are two important aspects in the study of the properties of a random spin system: the effect of randomness of the exchange interaction on the sharpness of the phase transition and on the low temperature behaviour of the thermo- dynamic quantities such as specific heat (C) and susceptibility (X) McCoy and Wu]’2 have studied the effect of randomness on the sharpness of the phase transition of a two dimensional rectangular Ising lattice with a special type of randomness. In their model, the horizontal interactions (J1) between all the spins are the same, but the vertical interactions (J2) , which connect spins of the jth row with those of the j+lth row are the same for all the spins in the jth row, but vary randomly from one row to the other. Using a narrow distribution P(J of width 0(N'1) , (N being 2) the total number of spins), they find that the critical temperature is shifted by an amount O(N-1) and the specific heat deviates by an amount 0(1) for large N from Onsager's value only for T - TC ~ O(N'Z) However, for a finite but narrow width w of P(J2) , in the 2 thermodynamic limit (N+ 00) , McCoy and Wu find that ’44, l 69 the logarithmic divergence in the specific heat is absent. In fact, C is an infinitely differentiable function of ' temperature at TC even though it is not an analytic function. For (T - TC)>> w2 , it approaches Onsager's value. The second interesting aspect in the study of a random spin system is the behaviour of the thermodynamic properties such as specific heat, susceptibility, away from the region of the phase transition. In one dimen- sion, since there is no phase transition, this second aspect, in particular the low and high temperature behaviour of C and X becomes more relevant. Fan and McCoy4 have used the method of McCoy and Wu1 to analyze the thermodynamic properties of a one dimensional Ising model with random nearest neighbor exchange and with an external magnetic field. They have studied a system where the distribution of exchange P(J) has a width of order N-l In this paper, we have studied in detail the properties of the one-dimensional random Ising model for various types of distribution P(J) in the case of zero magnetic field. In particular we have analyzed the case of constant distri- bution, P(J) = constant for J in the interval [0, Jm] and P(J) = 0 otherwise for both the ferro (Jm>0) and the antiferromagnetic (Jm<0) cases. (In the present paper, Jm is always a finite number and this distribution will be referred to as “constant distribution”.) In 70 addition, the effect of distributions of the form P(J) = AlJlV for J in the interval [0,Jm] and P(J) = 0 otherwise, on the low temperature (kT<<|Jm|) behaviour of C and X , has been studied. We find that the low temperature behaviour of X11 , XL and C is very sensitive to the nature of the distribution P(J) . In the case of constant distribution, C(T) « VT for low T . This behaviour is quite similar to that of a more general random spin system interacting via a long range RKKY interaction.5 The physical origin of this linear behaviour can be ascribed to the finite density of low lying excited states present because of non-zero P(O) Furthermore a direct comparison (see Fig. l) of the random chain C—curve with the periodic C-curve with J = Jm/2 shows that the height of the peak (occurring at about .25 Jm in the former and .2 Jm in the latter case) is lower 1 (~ 40%) and broader in the former. The same behaviour is expected in the more general cases of v > 0 , but as v increases the effect will be less pronounced. The effect of randomness on the temperature dependence of the parallel susceptibility X11(T) is quite dramatic. For a periodic Ising chain, it is known6 that X11(T) goes to zero with T exponentially in the antiferromagnetic (AF) case and diverges exponentially in the ferromagnetic case (F) . However, in the random case with constant P(J) , x (0) we find that ~ii§—- = 193—; in the AF case and NpB IJm1 71 .+Ee_\ex .m> meeeeo meemH AN\Ecnnv oeuOWng use Aon>v Eon:S eo pew: owewumamii._ weaned o.~ m... o._ no 0 1 H _ _ oo.o no.0 / Eoucom / I ON.O HN/O _ _ / __ , __ 1 8.0 , _ o_uo:on_\¢) _ , _ / 2 I and I O¢.O , 2 C I 8.0 72 2 diverges as [JhI/(kT) in the F case. This indicates that the essential singularity of xl|(T) at T = 0 present in the periodic system is removed by the random— ness: X||(T) is either regular or has a pole singularity at T = 0 , in the random system. The major effect of the randomness is the removal of the gap that is present for a periodic system. This then alters the low lying excita— tions significantly. The results for more general distri- butions are discussed in the next Section. In contrast to X1|(T) , XI(T) does not depend upon the sign of Jm and therefore its value is the same in the F and AF cases. We have found that the value of XL(O) is enhanced from its value in the periodic case7 by the amount 2 log 2 in the case of constant P(J) . In addition, a remarkable effect of the randomness is to remove the peak in XL(T) as a function of T which is present in the periodic case. Again, these effects can be understood by noting that the gap in the excitation spectrum that occurs in the periodic case is absent in the random system. At high T i.e. kT>>|Jm1 , XL exhibits a Curie like behaviour. The Meiss part in X1_] is absent due to the absence of the interaction along the x and y directions. We have made some attempts to study the effect of an interaction of the type -2; J4 (Sixsi+lx + Sini+1y) (we will refer to this operator as xy inter- action) on the thermodynamic properties of the system. 73 We have carried out a high temperature expansion to the leading order in J+ /kT and Jm/kT and found that the inclusion of the xy interaction leads to a Weiss term in X1—1 . The low temperature behaviour of the random Ising system with small xy interaction is quite interesting but difficult. For an arbitrary distribution of J+ , P(Ji) , the ground state is not known. It is known8 that for a special class of P(Ji) , namely J+ = y Ji' and 00 . However, in the general case J1 = y, Jll , the ground state is more complex. The nature of the ground and low-lying excited states is presently under study. In Section 2, we present all our results for the Ising chain. Section 3 deals with the effect of the xy interaction on various physical-and thermodynamic properties. 2. The Model and its Solution The Hamiltonian describing a system of N Ising spins interacting via nearest neighbour exchange Jij in the —) presence of an external magnetic field H is given by: 9M H: _ l __B I I J--0- 0- Z 1j 12 jz 2 1 CH 22+ (1) Jijls are a set of random numbers with a distribution P(J) . He will be concerned with a distribution P(J) different from zero for J 0 , J max <0) . In this paper we shall m1n be concerned with cases (a) and (b) only and report the results for different distributions P(J) of the exchange parameters Jij for a one dimensional system. Fan and McCoy4 have studied the thermodynamics of the system given by eqn. (1) in the one-dimensional case with H = O . P(J) is taken in their work as a peaked -1 function with width proportional to N , and the correc- tions to the thermodynamic quantities of the periodic chain are calculated in different orders of NT] Our approach in this paper is to generate ensembles of random spin systems by computer, calculate the thermo- dynamic properties of each member of the ensemble and then take an average. We have limited ourselves to finite chains consisting of 500 - 2000 spins. We have found that the spin correlation functions and other thermodynamic quantities as calculated for these finite chains already exhibit the N» w behaviour in the sense that increasing the number of spins does not alter the numerical results in significant figures. The partition function in the absence of the external . . . . 9 f1eld can be obta1ned by u51ng the transfer matr1x method 75 and is given by: cosh K, + "N sinh K1) (2) l where K, = B Ji/Z and N is the number of spins, and we have assumed periodic boundary conditions. The second term inside the bracket of eqn. (2) would not appear if we had used the open end boundary conditions. All our numerical calculations have been performed in the thermodynamic limit (N+ 00) , where the second term in eqn. (2) does not contribute. In the presence of a magnetic field H , one cannot obtain a closed form expression for Z by using the same method as for the H = 0 case. This is so because the transfer matrices T1.10 whose matrix elements are given by §(J. 00' + h(o+o')) (0|T110')= e2 l (3) where h = % guB HZ and o,o' = t 1 , do not commute in general if h # 0 . Therefore one cannot diagonalize all Ti 5 simultaneously. In order to obtain the partition function in this case, one has to first multiply N (2 x 2) nondiagonal matrices and then diagonalize the resulting (2 x 2) matrix. An alternate procedure is to use the method given by McCoy and Wu.1 We now present the results for various thermodynamic quantities in the absence of magnetic field. 76 A. Specific heat Using eqn. (2) for the partition function, we can obtain the specific heat per spin 2 N _ 5_ ,2 2 where we have omitted terms which approach zero in the thermodynamic limit. In terms of the distribution function P(J) for the random variable J1 , we can replace the summation in eqn. (4) by an integration for large N and eqn. (4) becomes 82 Jmax 2 2 C/Nk = 4‘ 1J P(J) J sech K dJ (5) min where K = BJ/2 and Jmax 1, dJ P(J) =1 (6) min defines the proper normalization condition for P(J) The specific heat C/Nk has been computed by using eqns. (5) and (6) for P(J) given by: \)+ V |J| , 0 0 (F) P(J) = or Jm : J i o for Jm< 0 (AF) (7) 0 otherwise 77 For the sake of comparison we have plotted C/Nk for v = 0 in fig. 1 together with C/Nk for the periodic chain with J = Jm/2 . The important effects of random- ness can be summarized as follows: at low temperatures (kT< is easily obtained by differentiating10the partition function L1. = <0, 0. > = % gS-i 23K 3K 3K for i the following expression: Jm T11) = fl 1 .1" tanh B’—J dJ (12) J 0+1 0 2 For T+ O , L(l) behaves asymptotically as J “— e _ V“ _m Lo) (1 bv 1 WI (13) with b\) ___ 2(\)+1)F X:}) (14) IJml In the special case v = O , L(r) can be written as %1 ln cosh J )r (15) m L(r) = (2 _m ZkT We have calculated L(r) using eqn. (15) and we display 1t in Fig. 3 at different temperatures. One can see from the figure that the range of correlation increases with decreasing temperature, and decreases exponentially with r for a fixed temperature. For the sake of comparison, we have also plotted L(r) for the periodic case at kT/Jmax = 0.1. The effect of randomness is to drastically reduce the range of spin correlation at low temperatures. However as the temperature is increased, the effect of exchange fluctuation is not as important because thermal fluctuations are large. From eqn. (11) , realizing that % Z, is equivalent L(r) 81 \ \ \ \ \ \ \\ \ \\ \ “~\ . . c \ Pernodlc \ \ . x \\ \ I J 35 40 Figure 3.—-Spin—spin correlation function L(r) as a function of r: (a) kT/Jm = .5, (b) kT/Jm = .3, (c) kT/Jm = .1. The continuous and dashed lines refer to the random (v=0) and periodic (J=Jm/2) ferromagnetic Ising chains respectively. 82 to taking an ensemble average, we obtain: Eit = 33. LILLLI (16) The parallel susceptibility is shown in Fig. (4) in the F and AF cases for v = 0 . The very interesting results at low temperatures can be summarized as follows: 2 . . . . lo 2 N 15 f1n1te 1n the AF case and e ual to W “B ‘1 1,14,1— at T = 0 ; it behaves as 12 in the F case, contrary to the usual periodic Ising Aodel where the existence of a gap makes XII/Nu: vanish (AF) , or blow up (F) exponentially at T = 0 . Furthermore, these results generalize in the following way for P(J) of eqn. (7). For Jm < 0 (AF) , X11 9 bu (kT)V (17) Nu: For Jm>0 (F) , X11 . 92 1 (18) N 2 2 b 0+2 “B v (kT) From eqns. (l7) and (18) we can see that the low temperature behaviour depends crucially on the distribution of the exchange interactions. The absence of a gap for any 0 3 0 makes XII (AF) vanish and X|1(F) blow up only as a power in T instead of exponentially. In fact it is instructive to note that we can recover the exponential behavior of X11 at low temperature if we allow P(J) to be a constant 2 B X” “ml/NP: 83 X...( F, Random) 7.0 - . / 613 — 5J0 - 1413 - 213 - /X"( AF, Random) L 1 o 1 1 0.5 1.0 kT/IJmI '01 ll Figure 4.-~E—? |Jm| vs. kT/|Jm| in the v=0 case. “B 84 between J1 and J2 and 0 otherwise (when J1 , J2 are both <0 (AF) or >0 (F)) C. Zero Field Perpendicular Susceptibility A detailed study of the perpendicular susceptibility 7 for various one-and two—dimensional was made by Fisher periodic Ising lattices. Two interesting features of one dimensional periodic Ising systems are the finiteness of XL at T = 0 , and the smooth peak in the susceptibility as a function of temperature. We have analyzed the effect of randomness on these features. To obtain XI , we intro- duce a field HX in the x-direction. The Hamiltonian is given by .. l_ l. H ‘ ‘ 2 4 J1 Oiz Oi+12 ‘ 2 9“B Hx 4 0ix (19) Taking into account the noncommutativity of the second term in the Hamiltonian with the first term (HO) , we obtain, 8 ——-— = sf. 2 I ojx101> <0 (20) “.2 4N ij 0 11,18 -BHO BHO where Oix(y) = e Oix e and refers to the usual zero field thermal trace. It is easily seen that only the terms with i = j contribute to XL . After taking the trace and carrying out the y integration, we 85 have X N _1__2 = 2 2 K1. tanh K1. - K1._1 tanh K1._] (2,) Nu 4NkT _ 2 2 B 1"] K1 - Ki-1 Equation (21) in the periodic limit (K1._1 = Ki = K) reduces to7 ‘ X Nl‘é'l i = g: (tanh K + K sech2 K) (22) 2 40 From eqns. (21) and (22) it is seen that XL is independent of the sign of J, and therefore its behaviour is the same in the ferro and antiferromagnetic cases. In Fig. (5), we have plotted X1lJml/NUB for random and periodic chains in the case of constant distribution.2 The two pronounced effects of randomness are: (1) Xi(0) for the random system is enhanced by an amount log 2 over XL(O) for the periodic system with J = Jm/2 . (2) The peak in XL(T) as a function of T that occurs for a periodic chain is completely washed out. However we believe that XL(T) has still an essential singularity at T = 0 in the random case. In addition to the low and intermediate temperature (kT f Jm) behaviour of XI that we have discussed above, we find that for kT>>Jm LL 1_ ~ (23) Nug T 813 7.0 6nO 513 4u0 3X) 21) 86 XL(Peflodic) I'O xJ_(Rondom) O I I 24441 (15 L0 1.5 kT/IJmI XllJml Figure 5.-- — 2 vs. kT/lJml in the v=0 case. NIIB (A) 87 The absence of a Neiss term can be ascribed to the absence of exchange interaction along x and y directions. The effect of incorporating such inter— actions on Xi will be discussed in the next Section. Effect of Random xy Interaction In the presence of an xy interaction with random exchange J, , the total Hamiltonian is given by H = HI + H (24) where HI is the Ising Hamiltonian given in eqn. (1) which includes the interaction with an external magnetic field and ny lS 91ven by _ 1_ ny ‘ ' 2 A Ji (Oix Gi+1x + Oiy Oi+ly) (25) Ji is the random exchange interaction along the x and y directions. An exact solution of the Hamiltonian H for an arbitrary distribution Ji is difficult. However, the effect of including ny on the high temperature behaviour of XI and X11 can be obtained by using a perturbation theory. For the random Ising model when v=0 , we have seen that when kT+ w , XI/NUS ~ AT and from eqn. (16), one can show that XII/Nug ~ lT +(1 )2(Jm/2). In order to see kT the effect of ny on the high temperature behaviour of these susceptibilities, we expand X11 and XI up to 88 order (BH)2 Let M be the magnetic moment operator (M = % 908 Z 01). In the presence of the total 1 Hamiltonian H = HI + H , one has xy H H (26) = Tr e'B M/Tr e-B Expanding the right hand side of eqn. (26) up to terms 0f order (BH)2 , we have <fi> = <%>m [—B 00 + %E_m] (27) Where 0° denotes the infinite temperature trace of any operator X . In obtaining eqn. (27), we have made use of the fact that °° = 0° = 0 . From eqn. (27), we can obtain Xi by taking the magnetic field H along the x—direction and calculating dMX/dHX at HX = O . We obtain XI 2 1. “2=B+fi—TZJ- (28) N08 1 From eqn. (28), one finds that the Curie Weiss constant 0 is iven b 1 9 Y kei = [1— Z oi = fP(JJ—) 1L doL (29) 89 A similar expansion can be performed for X11 by taking the field along the z—direction and one finds that Xll ___2 = g + B Z J. (30) NuB i from the above equation, we see that up to order B2 there is no contribution to XII/N1182 from ny , and therefore the Curie Weiss constant 811 given by 1 k0 = — J. = JP J dJ 11 N11 1 H is unaffected by the inclusion of ny The ground and low lying excited states of the system described by the Hamiltonian H + H depend crucially I xy on the probability distributions P(J) and P(J ) However, for the ferromagnetic coupling between spins with O