Note: Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- General bibliography -- Books -- Review articles -- Reprint volumes -- Instead of a reading guide -- 1 Introduction -- 1.1 On the origin of the Hubbard model -- 1.2 The Hubbard model - a paradigm in condensed matter physics -- 1.2.1 Integrable models -- 1.2.2 Bethe ansatz solution of the Hubbard model -- 1.2.3 The one-dimensional Hubbard model and experiments -- 1.3 External fields -- 1.3.1 External fields in three dimensions -- 1.3.2 External fields in one dimension -- 1.4 Conclusions -- Appendices to Chapter 1 -- 1.A Response to external fields -- 1.A.1 The current operator -- 1.A.2 Linear response -- 1.A.3 Optical conductivity, Drude weight and f-sum rule -- 2 The Hubbard Hamiltonian and its symmetries -- 2.1 The Hamiltonian -- 2.2 Symmetries -- 2.2.1 Permutations -- 2.2.2 Spatial symmetries -- 2.2.3 The momentum operator -- 2.2.4 More discrete symmetries -- 2.2.5 SO(4) symmetry -- 2.3 Conclusions -- Appendices to Chapter 2 -- 2.A The strong coupling limit -- 2.A.1 Projectors -- 2.A.2 Second order perturbation theory around an energy level -- 2.A.3 The Hubbard model in the strong coupling limit -- 2.A.4 Heisenberg spin chain and Mott transition -- 2.A.5 Neglecting the three-site terms -- 2.A.6 The t-0 model -- 2.A.7 An overview over the strong coupling effective models related to the Hubbard model -- 2.B Continuum limits -- 3 The Bethe ansatz solution -- 3.1 The Hamiltonian in first quantization -- 3.2 Solution of the two-particle problem -- 3.2.1 Separation of variables -- 3.2.2 The centre of mass motion -- 3.2.3 The relative motion -- 3.2.4 Eigenstates on the infinite interval -- 3.2.5 Periodic boundary conditions -- 3.2.6 The Eta-pair -- 3.3 Many-particle wave functions and Lieb-Wu equations -- 3.3.1 The symmetric group. , 3.3.2 Many-particle wave functions and Lieb-Wu equations -- 3.4 Symmetry properties of wave functions and states -- 3.4.1 Symmetries under permutations -- 3.4.2 SO(4) multiplets -- 3.5 The norm of the eigenfunctions -- 3.5.1 An action for the Lieb-Wu equations -- 3.5.2 The norm formula -- 3.6 Conclusions -- Appendices to Chapter 3 -- 3.A Scalar products and projection operators -- 3.B Derivation of Bethe ansatz wave functions and Lieb-Wu equations -- 3.B.1 The Bethe ansatz wave function -- 3.B.2 Equations for the amplitudes -- 3.B.3 Consistency -- 3.B.4 Periodic boundary conditions -- 3.B.5 Algebraic solution of the spin problem -- 3.B.6 Summary -- 3.C Some technical details -- 3.C.1 Yang-Baxter equation -- 3.C.2 The consistency problem -- 3.D Highest weight property of the Bethe ansatz states with respect to total spin -- 3.D.1 Spin operators in fermionic and in spin chain representation -- 3.D.2 Action of spin operators on Bethe ansatz states -- 3.D.3 su(2) invariance of the spin problem -- 3.E Explicit expressions for the amplitudes in the Bethe ansatz wave functions -- 3.F Lowest weight theorem for the Eta-pairing symmetry -- 3.G Limiting cases of the Bethe ansatz solution -- 3.G.1 Strong coupling limits -- 3.G.2 Continuum limit and Bethe ansatz solution of the model of electrons with delta-function interaction -- 3.G.3 Weak coupling limit -- 4 String hypothesis -- 4.1 String configurations -- 4.1.1 k-Lambda strings -- 4.1.2 A composition principle -- 4.1.3 Lambda strings -- 4.2 String solutions as bound states -- 4.2.1 k-Lambda strings -- 4.2.2 Lambda strings -- 4.3 Takahashi's equations -- 4.4 Completeness of the Bethe ansatz -- 4.5 Higher-level Bethe ansatz -- Appendices to Chapter 4 -- 4.A On deviations from the string hypothesis -- 4.B Details about the enumeration of eigenstates -- 4.B.1 Simple examples: 2 and 4 site lattices. , 4.B.2 Counting eigenstates -- 5 Thermodynamics in the Yang-Yang approach -- 5.1 A point of reference: noninteracting electrons -- 5.2 Thermodynamic Bethe Ansatz (TBA) equations -- 5.3 Thermodynamics -- 5.4 Infinite temperature limit -- 5.5 Zero temperature limit -- 5.5.1 Dressed energies -- 5.5.2 Root densities -- 5.5.3 Dressed momenta -- 5.5.4 Zero temperature limit in zero magnetic field -- Appendices to Chapter 5 -- 5.A Zero temperature limit for… -- 5.B Properties of the integral equations at T = 0 -- 6 Ground state properties in the thermodynamic limit -- 6.1 A point of reference: noninteracting electrons -- 6.2 Defining equations -- 6.3 Ground state phase diagram -- 6.4 Density and magnetization -- 6.4.1 Fixed B -- 6.4.2 Fixed mu -- 6.4.3 Fixed nc -- 6.5 Spin and charge velocities -- 6.6 Susceptibilities -- 6.6.1 Phases I and III -- 6.6.2 Phase II -- 6.6.3 Phase IV: matrix notations -- 6.6.4 Phase V -- 6.7 Ground state energy -- Appendices to Chapter 6 -- 6.A Numerical solution of integral equations -- 6.B Ground state properties in zero magnetic field -- 6.B.1 Half filled band -- 6.B.2 The almost half filled band -- 6.B.3 Low density -- 6.C Small magnetic fields at half filling: application of the Wiener-Hopf method -- 6.C.1 General structure -- 6.C.2 Solution of the equation for y0(Lambda) -- 6.C.3 Equation for y1(Lambda) -- 6.C.4 Dressed energies -- 7 Excited states at zero temperature -- 7.1 A point of reference: noninteracting electrons -- 7.1.1 'Single-particle' excitations -- 7.1.2 'Particle-hole' excitations -- 7.2 Zero magnetic field and half-filled band -- 7.2.1 Elementary excitations -- 7.2.2 Holon and spinon band widths -- 7.2.3 Spin and charge velocities at half filling -- 7.2.4 Two-particle sector -- 7.2.5 2N particle sector -- 7.3 Root-density formalism -- 7.3.1 The half-filled ground state -- 7.3.2 General excited states. , 7.3.3 Charge singlet excitation -- 7.3.4 Charge triplet excitation -- 7.3.5 Spin singlet excitation -- 7.3.6 Spin triplet excitation -- 7.3.7 Spin-charge scattering state -- 7.4 Scattering matrix -- 7.4.1 Charge sector -- 7.4.2 Spin sector -- 7.4.3 Scattering of spin and charge -- 7.5 'Physical' Bethe ansatz equations -- 7.6 Finite magnetic field and half-filled band -- 7.6.1 Ground state -- 7.6.2 Excitations of the spin degrees of freedom -- 7.6.3 Excitations involving the charge sector -- 7.7 Zero magnetic field and less than half-filled band -- 7.7.1 Charge-neutral excitations -- 7.7.2 Charged excitations -- 7.8 Finite magnetic field and less than half-filled band -- 7.9 Empty band in the infinite volume -- Appendices to Chapter 7 -- 7.A Relating root-density and dressed-energy formalisms -- 7.B Lower bounds for… -- 8 Finite size corrections at zero temperature -- 8.1 Generic case - the repulsive Hubbard model in a magnetic field -- 8.1.1 Finite-size corrections to Takahashi's equations -- 8.1.2 Finite-size corrections to the energy -- 8.1.3 The dressed charge matrix -- 8.2 Special cases -- 8.2.1 Zero magnetic field -- 8.2.2 Partially filled spin-polarized band -- 8.2.3 The half-filled band -- 8.2.4 Strong coupling limit -- 8.3 Finite-size spectrum of the open Hubbard chain -- 8.3.1 Bethe Ansatz equations for the open Hubbard chain -- 8.3.2 Surface energy of the open Hubbard chain -- 8.3.3 Ground-state expectation value of n1 -- 8.3.4 Finite-size corrections to the energy of the open Hubbard chain -- 8.4 Relation of the dressed charge matrix to observables -- 8.4.1 Zero magnetic field -- 8.4.2 Half-filling -- 8.4.3 Generic case -- Appendices to Chapter 8 -- 8.A Wiener Hopf calculation of the dressed charge -- 8.A.1 Weak coupling limit of the dressed charge in zero magnetic field -- 8.A.2 Solution of the strong coupling equations for small B. , 9 Asymptotics of correlation functions -- 9.1 Low energy effective field theory at weak coupling -- 9.1.1 Continuum limit -- 9.1.2 Bosonization and separation of spin and charge degrees of freedom -- 9.1.3 Bosonization results for correlation functions -- 9.2 Conformal field theory and finite size scaling -- 9.2.1 Universality classes -- 9.2.2 Low-lying excitations and correlation functions -- 9.2.3 Extension to models with several critical degrees of freedom -- 9.3 Correlation functions of the one-dimensional Hubbard model -- 9.3.1 Zero magnetic field -- 9.3.2 Half-filled band -- 9.3.3 Magnetic field effects in the strong coupling limit -- 9.4 Correlation functions in momentum space -- 9.4.1 Spectral function -- 9.4.2 Dynamical structure factor -- 9.5 Correlation functions in the open boundary Hubbard chain -- 9.5.1 Friedel oscillations -- 9.5.2 Orthogonality catastrophe -- Appendices to Chapter 9 -- 9. A Singular behaviour of momentum-space correlators -- 10 Scaling and continuum limits at half-filling -- 10.1 Construction of the scaling limit -- 10.2 The S-matrix in the scaling limit -- 10.2.1 Massive charge sector -- 10.2.2 Massless spin sector -- 10.2.3 Scattering between spin and charge -- 10.3 Continuum limit -- 10.3.1 Bosonization -- 10.4 Correlation functions in the scaling limit -- 10.4.1 Spin-charge factorization of correlation functions -- 10.4.2 Spectral representation of two-point functions in the charge sector -- 10.4.3 Form factors -- 10.4.4 Optical conductivity -- 10.4.5 Single particle Green's function -- 10.4.6 Spectral function -- 10.4.7 Density response function -- 10.4.8 Spin correlation functions -- 10.5 Correlation functions in the continuum limit -- 10.5.1 Optical conductivity -- 10.5.2 Single particle Green's function -- 10.5.3 Tunneling density of states -- 10.5.4 Momentum distribution function. , 10.5.5 Density-density response function.

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