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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