THE ONE-DIMENSIONAL HUBBARD MODEL

By JESSICA ANDERSON

CONTENTS INTRODUCTION ON THE ORIGIN OF THE HUBBARD MODEL THE HUBBARD MODEL – A PARADIGM IN CONDENSED MATTER PHYSICS EXTERNAL ?ELDS CONCLUSIONS APPENDICES TO CHAPTER 1 A RESPONSE TO EXTERNAL ?ELDS THE HUBBARD HAMILTONIAN AND ITS SYMMETRIES THE HAMILTONIAN SYMMETRIES 2 A THE STRONG COUPLING LIMIT 2 B CONTINUUM LIMITS THE BETHE ANSATZ SOLUTION THE HAMILTONIAN IN ?RST QUANTIZATION SOLUTION OF THE TWO-PARTICLE PROBLEM MANY-PARTICLE WAVE FUNCTIONS AND LIEB-WU EQUATIONS SYMMETRY PROPERTIES OF WAVE FUNCTIONS AND STATES THE NORM OF THE EIGENFUNCTIONS 3 A SCALAR PRODUCTS AND PROJECTION OPERATORS 3 B DERIVATION OF BETHE ANSATZ WAVE FUNCTIONS AND LIEB-WU EQUATIONS 3 C SOME TECHNICAL DETAILS 3 D HIGHEST WEIGHT PROPERTY OF THE BETHE ANSATZ STATES WITH RESPECT TO TOTAL SPIN 3 E EXPLICIT EXPRESSIONS FOR THE AMPLITUDES IN THE BETHE ANSATZ WAVE FUNCTIONS 3 F LOWEST WEIGHT THEOREM FOR THE ?-PAIRING SYMMETRY 3 G LIMITING CASES OF THE BETHE ANSATZ SOLUTION STRING HYPOTHESIS STRING CON?GURATIONS STRING SOLUTIONS AS BOUND STATES TAKAHASHI’S EQUATIONS COMPLETENESS OF THE BETHE ANSATZ HIGHER-LEVEL BETHE ANSATZ 4 A ON DEVIATIONS FROM THE STRING HYPOTHESIS 4 B DETAILS ABOUT THE ENUMERATION OF EIGENSTATES THERMODYNAMICS IN THE YANG-YANG APPROACH A POINT OF REFERENCE: NONINTERACTING ELECTRONS THERMODYNAMIC BETHE ANSATZ (TBA) EQUATIONS THERMODYNAMICS IN?NITE TEMPERATURE LIMIT ZERO TEMPERATURE LIMIT 5 A ZERO TEMPERATURE LIMIT FOR ?? 1() 5 B PROPERTIES OF THE INTEGRAL EQUATIONS AT T = GROUND STATE PROPERTIES IN THE THERMODYNAMIC LIMIT DE?NING EQUATIONS GROUND STATE PHASE DIAGRAM DENSITY AND MAGNETIZATION SPIN AND CHARGE VELOCITIES SUSCEPTIBILITIES GROUND STATE ENERGY 6 A NUMERICAL SOLUTION OF INTEGRAL EQUATIONS 6 B GROUND STATE PROPERTIES IN ZERO MAGNETIC ?ELD 6 C SMALL MAGNETIC ?ELDS AT HALF ?LLING: APPLICATION OF THE WIENER-HOPF METHOD EXCITED STATES AT ZERO TEMPERATURE ZERO MAGNETIC ?ELD AND HALF-?LLED BAND ROOT DENSITY FORMALISM SCATTERING MATRIX ‘PHYSICAL’ BETHE ANSATZ EQUATIONS FINITE MAGNETIC ?ELD AND HALF-?LLED BAND ZERO MAGNETIC ?ELD AND LESS THAN HALF-?LLED BAND FINITE MAGNETIC ?ELD AND LESS THAN HALF-?LLED BAND EMPTY BAND IN THE IN?NITE VOLUME 7 A RELATING ROOT-DENSITY AND DRESSED-ENERGY FORMALISMS 7 B LOWER BOUNDS FOR ?N(0), N ?2 AT HALF ?LLING IN A ?NITE MAGNETIC ?ELD FINITE SIZE CORRECTIONS AT ZERO TEMPERATURE GENERIC CASE – THE REPULSIVE HUBBARD MODEL IN A MAGNETIC ?ELD SPECIAL CASES FINITE SIZE SPECTRUM OF THE OPEN HUBBARD CHAIN RELATION OF THE DRESSED CHARGE MATRIX TO OBSERVABLES 8 A WIENER HOPF CALCULATION OF THE DRESSED CHARGE ASYMPTOTICS OF CORRELATION FUNCTIONS LOW ENERGY EFFECTIVE ?ELD THEORY AT WEAK COUPLING CONFORMAL ?ELD THEORY AND ?NITE SIZE SCALING CORRELATION FUNCTIONS OF THE ONE-DIMENSIONAL HUBBARD MODEL CORRELATION FUNCTIONS IN MOMENTUM SPACE CORRELATION FUNCTIONS IN THE OPEN BOUNDARY HUBBARD CHAIN 9 A SINGULAR BEHAVIOUR OF MOMENTUM-SPACE CORRELATORS SCALING AND CONTINUUM LIMITS AT HALF-?LLING CONSTRUCTION OF THE SCALING LIMIT THE S-MATRIX IN THE SCALING LIMIT CONTINUUM LIMIT CORRELATION FUNCTIONS IN THE SCALING LIMIT CORRELATION FUNCTIONS IN THE CONTINUUM LIMIT FINITE TEMPERATURES 10 A CURRENT ALGEBRA 10 B TWO-PARTICLE FORM FACTORS 10 C CORRELATION FUNCTIONS IN THE GAUSSIAN MODEL UNIVERSAL CORRELATIONS AT LOW DENSITY THE HUBBARD MODEL IN THE GAS PHASE CORRELATION FUNCTIONS OF THE IMPENETRABLE ELECTRON GAS THE ALGEBRAIC APPROACH TO THE HUBBARD MODEL INTRODUCTION TO THE QUANTUM INVERSE SCATTERING METHOD SHASTRY’S R-MATRIX GRADED QUANTUM INVERSE SCATTERING METHOD THE HUBBARD MODEL AS A FUNDAMENTAL GRADED MODEL SOLUTION OF THE QUANTUM INVERSE PROBLEM ON THE ALGEBRAIC BETHE ANSATZ FOR THE HUBBARD MODEL 12 A A PROOF THAT SHASTRY’S R-MATRIX SATIS?ES THE YANG-BAXTER EQUATION 12 B A PROOF OF THE INVERSION FORMULA 12 C A LIST OF COMMUTATION RELATIONS 12 D SOME IDENTITIES NEEDED IN THE CONSTRUCTION OF THE TWO-PARTICLE ALGEBRAIC BETHE ANSATZ-STATES 12 E AN EXPLICIT EXPRESSION FOR THE FERMIONIC R-OPERATOR OF THE HUBBARD MODEL THE PATH INTEGRAL APPROACH TO THERMODYNAMICS THE QUANTUM TRANSFER MATRIX AND INTEGRABILITY THE HEISENBERG CHAIN SHASTRY’S MODEL AS A CLASSICAL ANALOGUE OF THE 1D HUBBARD MODEL DIAGONALIZATION OF THE QUANTUM TRANSFER MATRIX ASSOCIATED AUXILIARY PROBLEM OF DIFFERENCE TYPE DERIVATION OF NON-LINEAR INTEGRAL EQUATIONS INTEGRAL EXPRESSION FOR THE EIGENVALUE NUMERICAL RESULTS ANALYTICAL SOLUTIONS TO THE INTEGRAL EQUATIONS 13 A DERIVATION OF TBA EQUATIONS FROM FUSION HIERARCHY ANALYSIS 13 B DERIVATION OF SINGLE INTEGRAL EQUATION THE YANGIAN SYMMETRY OF THE HUBBARD MODEL THE VARIABLE-RANGE-HOPPING HAMILTONIAN CONSTRUCTION OF THE YANGIAN GENERATORS 14 A YANGIANS S-MATRIX AND YANGIAN SYMMETRY IN THE IN?NITE INTERVAL LIMIT PRELIMINARIES PASSAGE TO THE IN?NITE INTERVAL YANGIAN SYMMETRY AND COMMUTING OPERATORS CONSTRUCTING N-PARTICLE STATES EIGENVALUES OF QUANTUM DETERMINANT AND HAMILTONIAN 15 A SOME USEFUL FORMULAE HUBBARD MODEL IN THE ATTRACTIVE CASE HALF-?LLED CASE THE GROUND STATE AND LOW LYING EXCITATIONS BELOW HALF ?LLING INTERACTION WITH MAGNETIC ?ELD PHASE DIAGRAM CRITICAL BEHAVIOR 16 A APPENDIX A 16 B APPENDIX B MATHEMATICAL APPENDICES USEFUL INTEGRALS THE WIENER-HOPF METHOD REFERENCES