Bosonization of interacting fermions in arbitrary dimensions

Bosonization of Interacting Fermions in Arbitrary Dimensions......Page 1
Preface......Page 2
Contents......Page 4
Part I. Development of the formalism......Page 8
1.1 Perturbation theory and quasi-particles......Page 10
1.2 A brief history of bosonization in d > 1......Page 12
1.3 The scope of this book......Page 15
1.4 Notations and assumptions......Page 16
2.1 The generic many-body Hamiltonian......Page 18
2.2.1 Definition of the Green’s function......Page 21
2.2.2 Definition of the interacting Fermi surface......Page 22
2.2.3 Landau Fermi liquids......Page 23
2.3 The density-density correlation function......Page 27
2.4 Patching the Fermi surface......Page 29
2.4.1 Definition of the patches and boxes......Page 30
2.4.2 Linearization of the energy dispersion......Page 32
2.4.3 Around-the-corner processes and the proper choice of the cutoffs......Page 34
2.5 Curved patches and reduction of the patch number......Page 35
2.6 Summary and outlook......Page 38
3.1 Grassmannian functional integrals......Page 40
3.2.1 Decoupling of the interaction......Page 43
3.2.2 Transformation of the single-particle Green’s function......Page 45
3.3 The second Hubbard-Stratonovich transformation......Page 47
3.3.1 Transformation of the density-density correlation function......Page 48
3.3.2 Definition of the bosonized kinetic energy......Page 50
3.4 Summary and outlook......Page 51
4. Bosonization of the Hamiltonian and the density-density correlation function......Page 54
4.1 The generalized closed loop theorem......Page 55
4.2.1 The effective action for the φ-field......Page 60
4.2.2 The Gaussian propagator of the φ-field......Page 62
4.2.3 The effective action for the ˜ρ-field......Page 63
4.2.4 The Gaussian propagator of the ˜ρ-field......Page 65
4.2.5 The bosonized Hamiltonian......Page 66
4.3.1 General expansion of the bosonized kinetic energy......Page 67
4.3.2 The leading correction to the effective action......Page 70
4.3.3 The leading correction to the bosonic propagator......Page 73
4.3.4 The hidden small parameter......Page 75
4.3.5 Calculating corrections to the RPA via bosonization......Page 79
4.4 Summary and outlook......Page 81
5.1 The Gaussian approximation with linearized energy dispersion......Page 82
5.1.1 The Green’s function for fixed background field......Page 83
5.1.2 Gaussian averaging: calculation of the Debye-Waller......Page 89
5.1.3 The Green’s function in real space......Page 91
5.1.4 The underlying asymptotic Ward identity......Page 93
5.1.5 The Fermi liquid renormalization factors Z and Zm......Page 97
5.2 Beyond the Gaussian approximation......Page 102
5.2.1 The Green’s function for fixed background field......Page 104
5.2.2 Non-Gaussian averaging......Page 111
5.3.1 The average eikonal......Page 116
5.3.2 The prefactor Green’s functions......Page 117
5.3.3 Connection with lowest order perturbation theory......Page 121
5.4 Summary and outlook......Page 125
Part II. Applications to physical systems......Page 130
6. Singular interactions (f_q  im |q|^{-\\eta})......Page 132
6.1.1 Non-linear energy dispersion......Page 133
6.1.2 The limit of linear energy dispersion......Page 134
6.1.3 Finite versus infinite patch number......Page 136
6.2 The static Debye-Waller factor for linearized energy dispersion......Page 138
6.2.1 Consequences of spherical symmetry......Page 139
6.2.2 The existence of the quasi-particle residue......Page 142
6.2.3 Why the Coulomb interaction is so nice......Page 144
6.2.4 The sub-leading corrections for 0 < η < 2(d − 1)......Page 145
6.2.5 The regime η ≥ 2(d − 1)......Page 147
6.3 Luttinger liquid behavior in d = 1......Page 150
6.4 Summary and outlook......Page 153
7. Quasi-one-dimensional metals......Page 156
7.1 The Coulomb interaction in chains without interchain hopping......Page 157
7.2 Finite interchain hopping......Page 162
7.2.1 The 4-patch model......Page 163
7.2.2 How curvature kills the nesting singularity......Page 171
7.2.3 Anomalous scaling in a Fermi liquid......Page 174
7.2.4 The nesting singularity for general Fermi surfaces......Page 175
7.3 Summary and outlook......Page 176
8. Electron-phonon interactions......Page 180
8.1.1 The Debye model......Page 181
8.1.2 Integration over the phonons......Page 184
8.2 The Debye-Waller factor......Page 186
8.3 Phonon energy shift and phonon damping......Page 187
8.4 The quasi-particle residue......Page 191
8.4.1 Isotropic phonon dispersion......Page 192
8.4.2 Quasi-one-dimensional electrons or phonons......Page 193
8.5 Summary and outlook......Page 195
9. Fermions in a stochastic medium......Page 198
9.1.1 Non-interacting disordered fermions......Page 199
9.1.2 Interacting disordered fermions......Page 202
9.2 Static disorder......Page 204
9.3 Dynamic disorder......Page 207
9.3.1 Gaussian white noise......Page 208
9.3.2 Finite correlation time......Page 210
9.4 Summary and outlook......Page 211
10. Transverse gauge fields......Page 214
10.1.1 The coupled matter gauge field action......Page 217
10.1.2 The effective matter action......Page 220
10.1.3 The effective gauge field action......Page 222
10.2.1 The Green’s function for fixed gauge field......Page 225
10.2.2 Gaussian averaging......Page 226
10.3.1 The transverse dielectric tensor......Page 230
10.3.2 Screening and gauge invariance......Page 232
10.3.3 The transverse dielectric function for spherical Fermi surfaces......Page 233
10.4 The transverse Debye-Waller factor......Page 235
10.4.1 Exact rescalings......Page 236
10.4.2 The relevance of curvature......Page 239
10.4.3 Two-dimensional Maxwell-Chern-Simons theory......Page 242
10.5 Summary and outlook......Page 248
A.1 The non-interacting polarization for spherical Fermi surfaces......Page 252
A.2 The dynamic structure factor for spherical Fermi surfaces......Page 254
A.3.1 The Coulomb interaction in 1 ≤ d ≤ 3......Page 257
A.3.2 General singular interactions......Page 259
A.4 Collective modes for finite patch number......Page 260
References......Page 264