HUBBARD OPERATORS IN THE THEORY OF STRONGLY CORRELATED ELECTRONS
CONTENTS CHAPTER 1 HUBBARD MODEL AS A SIMPLEST MODEL OF STRONG ELECTRON CORRELATIONS 1 1 HAMILTONIAN OF THE HUBBARD MODEL AND ITS SYMMETRY 1 2 TIME INVERSION 1 3 ELECTRON HOLE SYMMETRY 1 4 PSEUDOSPIN SYMMETRY FOR HALF-FILLED CASE 1 5 THE BAND LIMIT 1 6 THE ATOMIC LIMIT, HUBBARD 1 SOLUTION 1 7 THE HUBBARD MODEL IN ONE AND IN?NITE DIMENSION CASES 1 8 THE T–J MODEL 1 9 THE HUBBARD MODEL IN X-OPERATOR REPRESENTATION 1 10 STUDY OF THE ELECTRONIC STRUCTURE BY THE ANGLE RESOLVED PHOTOEMISSION SPECTROSCOPY CHAPTER 2 MULTIELECTRON MODELS IN X-OPERATOR REPRESENTATION 2 1 THE ISOTROPIC HEISENBERG MODEL 2 2 THE HEISENBERG MODEL WITH SINGLE-ION ANISOTROPY 2 3 THE ATOMIC REPRESENTATION FOR S–D(F) EXCHANGE MODEL 2 4 THE PERIODIC ANDERSON MODEL 2 5 THE MULTIBAND MODEL OF TRANSITION METAL OXIDES CHAPTER 3 GENERAL APPROACH TO THE QUASIPARTICLE DESCRIPTION OF STRONGLY CORRELATED SYSTEMS 3 1 THE DE?NITION OF FERMI- AND BOSE-TYPE QUASIPARTICLES 3 2 THE EXACT INTRACELL LOCAL GREEN’S FUNCTION 3 3 THE GENERALIZED TIGHT-BINDING METHOD FOR QUASIPARTICLE BAND STRUCTURE CALCULATIONS 3 4 THE CONCENTRATION DEPENDENCE OF THE QUASIPARTICLE BAND STRUCTURE 3 5 AB INITIO APPROACH TO THE QUASIPARTICLE BAND STRUCTURE OF STRONGLY CORRELATED ELECTRON SYSTEMS 3 6 THE GENERALIZED TIGHT-BINDING METHOD AND THE EXACT LEHMANN REPRESENTATION CHAPTER 4 UNITARY TRANSFORMATION METHOD IN ATOMIC REPRESENTATION 4 1 UNITARY OPERATORS AND TRANSFORMATION LAWS IN ATOMIC REPRESENTATION 4 2 DIAGONALIZATION OF TWO-LEVEL AND QUASI-TWO-LEVEL FORMS 4 3 THE DIAGONALIZATION OF THREE-LEVEL FORMS 4 4 THE DIAGONALIZATION OF THE N LEVEL HAMILTONIAN CHAPTER 5 DIAGRAM TECHNIQUE IN THE X-OPERATORS REPRESENTATION 5 1 GREEN’S FUNCTION IN THE X-OPERATORS REPRESENTATION 5 2 WICK’S THEOREM FOR THE HUBBARD OPERATORS 5 3 AVERAGING THE DIAGONAL X-OPERATORS 5 4 EXTERNAL AND INTERNAL OPERATORS: SIMPLEST VERTEXES 5 5 A HIERARCHY OF INTERACTION LINES AND A TOPOLOGICAL CONTINUITY PRINCIPLES 5 6 CALCULATION OF THE SIGN FOR AN ARBITRARY DIAGRAM 5 7 DIAGRAMS WITH OVALS 5 8 GENERAL RULES FOR ARBITRARY ORDER DIAGRAM 5 9 THE LARKIN EQUATION 5 10 THE SELF-ENERGY AND THE STRENGTH OPERATOR 5 11 LOW TEMPERATURE PROPERTIES OF ANISOTROPIC FERROMAGNET 5 12 E?ECT OF THE THREE-SITE CORRELATED HOPPING ON THE MAGNETIC MECHANISM OF DX2?Y2-SUPERCONDUCTIVITY IN THE T–J?MODEL CHAPTER 6 SPECTRAL PROPERTIES OF ANISOTROPIC METALLIC FERROMAGNETS 6 1 THE HAMILTONIAN OF THE ANISOTROPIC S?F MAGNET IN ATOMIC REPRESENTATION 6 2 THE GREEN FUNCTION AND DISPERSION EQUATION 6 3 THE FERROMAGNETIC METAL WITH SINGLE AXIS ANISOTROPY 6 4 THE METALLIC FERROMAGNETS WITH THE SA OF CUBIC SYMMETRY CHAPTER 7 PECULIARITIES OF THE DE HAAS-VAN ALPHEN E?ECT IN STRONGLY CORRELATED SYSTEMS WITH MAGNETIC POLARON STATES 7 1 THE HAMILTONIAN OF A STRONG CORRELATED NARROW BAND ANTIFERROMAGNET 7 2 THE HAMILTONIAN IN THE NON-COLINEAR PHASE: UNITARY TRANSFORMATION 7 3 CONSTRUCTION OF THE ATOMIC REPRESENTATION BASIS 7 4 THE HAMILTONIAN OF THE NARROW BAND ANTIFERROMAGNET IN THE NON-COLINEAR PHASE IN ATOMIC REPRESENTATION 7 5 THE GREEN’S FUNCTIONS AND DISPERSION RELATIONS 7 6 THE SPECTRUM OF HOLES NEAR THE SPIN-?IP TRANSITION 7 7 THE PECULIARITIES OF THE DE HAAS-VAN ALPHEN E?ECT IN ANTIFERROMAGNETIC SEMIMETAL WITH MAGNETIC POLARON STATES 7 8 THE TEMPERATURE QUANTUM OSCILLATIONS CAUSED BY NON-FERMI LIQUID E?ECTS CHAPTER 8 THE ELECTRONIC STRUCTURE OF COPPER OXIDES IN THE MULTIORBITAL P–D MODEL 8 1 THE EXACT DIAGONALIZATION OF THE CUO4 CLUSTER 8 2 THE CONSTRUCTION OF THE WANNIER FUNCTIONS AND X-REPRESENTATION OF THE MULTIBAND P–D MODEL 8 3 THE EVOLUTION OF THE BAND STRUCTURE AND ARPES DATA WITH DOPING FROM THE UNDOPED ANTIFERROMAGNETIC INSULATOR TO OPTIMAL DOPED AND OVERDOPED METAL 8 4 COMPARISON OF THE ELECTRONIC STRUCTURE AND SUPERCONDUCTIVITY OF CUPRATES AND RUTHENATES CHAPTER HAMILTONIAN OF THE HUBBARD MODEL AND ITS SYMMETRY THE HUBBARD MODEL IS A PARTICULAR LIMIT OF A GENERAL MODEL OF INTERACT- ING ELECTRONS WITH THE BAND STRUCTURE GIVEN IN THE TIGHT-BINDING APPROACH (FIG 1 1) AS COMPARED TO THE GENERAL TIGHT-BINDING MODEL OF ELECTRONS WITH COULOMB INTERACTION THE HUBBARD MODEL HAS TWO IMPORTANT SIMPLI?CA- TIONS FIRSTLY, ONLY ONE ELECTRON ORBITAL AT EACH SITE IS CONSIDERED AND ALL OTHER ORBITALS ARE ASSUMED TO BE UNIMPORTANT IN THE LOW ENERGY PHYSICS SECONDLY, ACCOUNT IS MADE OF THE INTRA-ATOMIC COULOMB MATRIX ELEMENT U ONLY THE HAMILTONIAN OF THE HUBBARD MODEL IS GIVEN BY ˆH ?µ ˆNE = F? (? ?µ) NF? + 2 UNF?NF,?? + FG? (TFGA+ F?AG? + H C ), (1 1) WHERE NF = A+ F?AF?, AF? IS THE ANNIHILATION OPERATOR OF THE ELECTRON AT SITE F WITH SPIN PROJECTION ? = ±1/2, ? IS THE SINGLE ELECTRON ENERGY IN TOT HUBBARD OPERATORS IN THE THEORY OF STRONGLY CORRELATED ELECTRONS ELECTRON HOLE SYMMETRY THE HOLE REPRESENTATION IS GIVEN BY THE TRANSFORMATION A+ F? ?AF,??, RESULTING IN THE FOLLOWING TRANSITION OF THE HAMILTONIAN H(T) ?(µ ??)NE ?H(?T) ?[U ?(µ ??)]NE + (2? ?2µ + U)N THE PARTITION FUNCTIONS IN THE ELECTRON AND THE HOLE REPRESENTATIONS ARE CONNECTED VIA THE RELATION: Zµ(T) = EXP[??(U ?2µ + 2?)N]ZU?µ(T) HUBBARD OPERATORS IN THE THEORY OF STRONGLY CORRELATED ELECTRONS TAKING THE DERIVATIVE OF µ IN BOTH PARTS OF THIS EQUATION, ONE GETS PSEUDOSPIN SYMMETRY FOR HALF-FILLED CASE IN THE CASE ?NE?= N AND FOR THE ALTERNANT LATTICE,1 THE HUBBARD MODEL HAS ADDITIONAL PSEUDOSPIN ISOTROPY (YANG 1989) THE PSEUDOSPIN VECTOR IS DE?NED SIMILAR TO EQ (1 2): R = 2?+ ?(R)? ????(R), (1 5) WHERE ? = (? (X), ? (Y), ? (Z)) IS THE VECTOR CONSTRUCTED WITH THE PAULI MATRICES AND THE TWO COMPONENT SPINOR ??(R) IS GIVEN BY ?(R) = A?(R) EXP(IQ R)A+ ?(R) (1 6) HERE, VECTOR Q = (?, ?, ) IS A D-DIMENSIONAL VECTOR WITH ALL COMPONENTS EQUAL TO ? THE EXPLICIT FORM OF THE PSEUDOSPIN OPERATOR IS AS FOLLOWS: RX(R) = 2 EXP(IQR)(A+ ?(R)A+ ?(R) + A?(R)A?(R)), RY(R) = 2 EXP(IQR)(A+ ?(R)A+ ?(R) ?A?(R)A?(R)), RZ(R) = 2(N?(R) + N?(R) ?1) THESE OPERATORS FORM AN SU(2) ALGEBRA AND FOLLOW THE USUAL SPIN COMMU- TATION RULES THE TOTAL PSEUDOSPIN OPERATORS OF THE SYSTEM ARE EQUAL TO R± TOT = R (RX(R) ± IRY(R)), RZ TOT = R RZ(R), (1 7) 1LATTICE THAT CAN BE SEPARATED IN TWO SUBLATTICES A AND B WITH EACH SITE A HAVING ALL NEAREST NEIGHBORS FROM B AND VICE VERSA HUBBARD OPERATORS IN THE THEORY OF STRONGLY CORRELATED ELECTRONS TOT = THE BAND LIMIT WITH U = 0, THE HAMILTONIAN (1 1) CAN BE DIAGONALIZED WITH THE HELP OF THE FOURIER TRANSFORMATION AK? = Z = 2M ??F IN GENERAL CASE, WITH ARBITRARY DEPENDENCE OF ?(P) ON P , THE FORM OF THE FERMI SURFACE MAY BE RATHER COMPLICATE FOR EXAMPLE, IN TWO DIMENSIONAL SQUARE LATTICE IN THE TIGHT BINDING METHOD, WITH ACCOUNT ONLY FOR NEAREST HUBBARD OPERATORS IN THE THEORY OF STRONGLY CORRELATED ELECTRONS THE ATOMIC LIMIT, HUBBARD 1 SOLUTION IN THE ABSENCE OF HOPPING, TFG = 0, THE MODEL DESCRIBES A SYSTEM OF INDI- VIDUAL SITES, EACH CAN BE IN ONE OF THE FOLLOWING EIGENSTATES: |0??|VACUUM?; |??= A+ F?|0?, ? = ±1/2; |2?= A+ F,+A+ F,?|0? (1 13) WITH EIGENVALUES ?0 = 0, ?? = ? ?µ ??1, ?2 = 2(? ?µ) + U THE NUMBER OF ELECTRONS PER SITE IS GIVEN BY N = N F ?ˆNF?= N F? ?A+ F? AF??= N+ + N?+ 2N2, (1 14) HUBBARD OPERATORS IN THE THEORY OF STRONGLY CORRELATED ELECTRONS