Table of contents for Quantum physics in one dimension / Thierry Giamarchi.


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1   Peculiarities of d = 1                                            1
    1.1  Crash course on Fermi liquids                               1
    1.2  One dimension: Failure of perturbation theory               5
    1.3  How to solve                                                   14
         1.3.1 Dzyaloshinskii-Larkin solution                           15
         1.3.2 Renormalization solution                                 21
2   Bosonization                                                        29
    2.1  Spinless model; representation of excitations                  29
    2.2  Physical properties and correlation functions                  37
         2.2.1 Thermodynamics                                           40
         2.2.2 Correlations                                             42
    2.3  Model with spin; charge and spin excitations                   50
         2.3.1 Physical observables                                     53
         2.3.2 Renormalization equations for sine-Gordon Hamil-
                      tonians                                           56
         2.3.3 Phase diagram                                            65
3   Luttinger liquids                                                   70
    3.1  Phenomenological bosonization                                  70
    3.2  Semiclassical and physical interpretations                     81
    3.3  Links with 2D statistical mechanics                            86
         3.3.1 Elastic systems                                          86
         3.3.2 Coulomb gas and XY model                                 91
    3.4  Basics of conformal theory                                     94
4   Refinements                                                        100
    4.1  Long-range interactions                                       100
    4.2  Mott transition                                               106
         4.2.1 Basic ingredients                                       106
         4.2.2 Commensurate case: Luther-Emery solution                111
         4.2.3 Doping; C-IC transition                                 115
    4.3  Effects of magnetic field and magnetic anisotropy         121
         4.3.1 Magnetic field                                          121
         4.3.2 Magnetic anisotropies                                   124
    4.4  Logarithmic corrections of correlation functions          131
5   Microscopic methods                                                137
    5.1  Bethe-ansatz                                                  137
         5.1.1 Spin chain                                              138
         5.1.2 One, two, three                                         139




         5.1.3 Many; Bethe-ansatz                                     143
         5.1.4 Bethe-ansatz and Luttinger liquids                     146
         5.1.5 Partial solution of the equations                      148
    5.2  A zest of numerics                                           153
         5.2.1 Exact diagonalizations                                 154
         5.2.2 Monte-Carlo                                            155
         5.2.3 DMRG                                                   157

6   Spin 1/2 chains                                                   160
    6.1  Physical properties of the spin 1/2 chain                    160
         6.1.1 Hamiltonian                                            160
         6.1.2 Bosonization solution                                  163
         6.1.3 Finite magnetic field                                  170
    6.2  Extensions                                                   175
         6.2.1 Frustrated chains                                      175
         6.2.2 Spin-Peierls                                           177
    6.3  Experimental realization of spin chains                      184
    6.4  Coupled chains                                               188
         6.4.1 Spin ladders                                           189
         6.4.2 Infinite number of chains                              196
7   Interacting fermions on a lattice                                 200
    7.1  Microscopic models                                           200
         7.1.1 Hubbard model                                          200
         7.1.2 t-J model                                              212
         7.1.3 U-V model and beyond                                   215
    7.2  Transport                                                    219
         7.2.1 Conductance, conductivity                              219
         7.2.2 Clean case; persistent currents                        223
         7.2.3 Mott insulator                                         228
8   Coupled fermionic chains                                          238
    8.1  Fermionic ladders                                            239
         8.1.1 Spinless ladders                                       239
         8.1.2 Ladders with spins                                     246
    8.2  Physical realizations of Ladders                             253
    8.3 Infinite number of chains                                     254
         8.3.1 Hopping between chains                                 255
         8.3.2 Two-body hopping                                       258
    8.4  Organic quasi-one-dimensional conductors                     262
9   Disordered systems                                                270
    9.1  Effect of disorder; Anderson localization                    270
         9.1.1 Generalities on disordered systems                     270
         9.1.2 Collective versus single individual pinning        275
    9.2  Many impurities                                              276



        9.2.1 Basics                                                 276
        9.2.2 Physical properties                                    285
        9.2.3 Extensions and pitfalls                                296
    9.3  Quantum wires                                               299
10 Boundaries and isolated impurities                                303
    10.1 Effect of a boundary                                        303
    10.2 Isolated impurities                                         307
         10.2.1 Weak coupling                                        308
         10.2.2 Strong coupling                                      310
         10.2.3 More than one impurity                               318
    10.3 Nanotubes                                                   325
    10.4 Edge states in quantum Hall systems                         328
11 Significant others                                                333
    11.1 Interacting one-dimensional bosons                          333
         11.1.1 Commensurate bosons                                  337
         11.1.2 Disorder: Bose glass                                 339
         11.1.3 Experimental realizations                            342
    11.2 Impurities in Fermi liquids                                 346
         11.2.1 X-ray edge problem                                  -347
         11.2.2 Kondo problem                                        355
         11.2.3 Multichannel Kondo problem                           364
A   Basics of Many-body                                              370
    A.1- Notations and formulas                                      370
    A.2 Digest of many-body                                          371
B   Not so important fine technical points                           376
    B.1 Explicit form of U operators                                 376
    B.2 Completness of Hilbert space                                 377
C   Correlation functions                                            380
    C.1 Path integral                                                380
    C.2 Basic correlations                                           381
    C.3 Analytic continuation                                        387
    C.4 Fourier transform of the retarded correlation function    389
D   Bosonization dictionary                                          391
    D.1 Slinless fermions                                            391
    D.2 Spin chains                                                  392
    D.3 Fermions with spins                                          393
    D.4 Averages                                                     393
    D.5 Babel tower                                                  394
E   Sine-Gordon                                                      396
    E.1 Renormalization                                              396



    E.2 Variational calculation                                      400
    E.3  Semiclassical approximations                                402

F   Numerical solution                                               404
References                                                           408
Index                                                                421





Library of Congress Subject Headings for this publication: Quantum theory, One-dimensional conductors