The Luttinger Model and Its Solution: An Exactly Soluble Model of a Many-Fermion System (J M Luttinger) Exact Solution of a Many-Fermion System and its Associated Boson Field (Daniel C Mattis and Elliott H Lieb) Lattice, Dynamical and Non Linear Effects: Luttinger Model and Luttinger Liquids (Vieri Mastropietro) The Luttinger Liquid and Integrable Models (J Sirker) Long Time Correlations of Nonlinear Luttinger Liquids Rodrigo (G Pereira) An Expanded Luttinger Model (Daniel C Mattis) Applications and Experimental Tests: Quantum Hall Edge Physics and Its One-Dimensional Luttinger Liquid Description (Orion Ciftja) A Luttinger Liquid Core Inside Helium-4 Filled Nanopores (Adrian Del Maestro) Some Experimental Tests of Tomonaga - Luttinger Liquids (T Giamarchi) Bosonization and Its Application to Transport in Quantum Wires (Feifei Li) Generalizations to Higher Dimensions: Fermions in Two Dimensions, Bosonization, and Exactly Solvable Models (Jonas De Woul and Edwin Langmann) Luttinger Liquid, Singular Interaction and Quantum Criticality in Cuprate Materials (C Di Castro and S Caprara) Luttinger Model in Dimensions d > 1 (Daniel C Mattis).
The Luttinger Model and Its Solution: An Exactly Soluble Model of a Many-Fermion System (J M Luttinger) Exact Solution of a Many-Fermion System and its Associated Boson Field (Daniel C Mattis and Elliott H Lieb) Lattice, Dynamical and Non Linear Effects: Luttinger Model and Luttinger Liquids (Vieri Mastropietro) The Luttinger Liquid and Integrable Models (J Sirker) Long Time Correlations of Nonlinear Luttinger Liquids Rodrigo (G Pereira) An Expanded Luttinger Model (Daniel C Mattis) Applications and Experimental Tests: Quantum Hall Edge Physics and Its One-Dimensional Luttinger Liquid Description (Orion Ciftja) A Luttinger Liquid Core Inside Helium-4 Filled Nanopores (Adrian Del Maestro) Some Experimental Tests of Tomonaga - Luttinger Liquids (T Giamarchi) Bosonization and Its Application to Transport in Quantum Wires (Feifei Li) Generalizations to Higher Dimensions: Fermions in Two Dimensions, Bosonization, and Exactly Solvable Models (Jonas De Woul and Edwin Langmann) Luttinger Liquid, Singular Interaction and Quantum Criticality in Cuprate Materials (C Di Castro and S Caprara) Luttinger Model in Dimensions d > 1 (Daniel C Mattis).
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
Introduction. Solvable Models, Rigorous Results, and Advances in Formalism: The Hubbard Model: Some Rigorous Results and Open Problems E.H. Lieb. On the Bethe Ansatz Soluble Degenerate Hubbard Model H. Frahm, A. Schadschneider. Infinite in all Dimensions: Large Coupling, High Dimensions, Many Components: The Mott Transition in Infinite Dimensions: Old Ideas and Some Surprises G. KIotliar, M.J. Rozenberg. The Hubbard Model with Infinite Interaction: Magnetic Properties V.Ya. Krivnov, et al. Fermi Liquid versus Luttinger Liquid: Two Particle Scattering and Orthogonality Catastrophe in the Hubbard Model W. Metzner. Conservation Laws in Normal Metals: Luttinger Liquid vs. Fermi Liquid C. Di Castro, et al. Perturbative, Mean Field, Variational, and Numerical Studies: Hartree-Fock and RPA Studies of the Hubbard Model G. Guinea, et al. The Wavefunction Renormalization Constant for the One- and Two-band Hubbard Hamiltonians in Two Dimensions E. Louis, et al. Experiments and Physical Applications: Electron Spectroscopy and Hubbard: Issues and Opportunities J.W. Allen, et al. On Electrical Properties of Chalcogenide Glassy Semiconductors in the Framework of Hubbard Model with Negative Correlation Energy S.D. Savransky. 34 additional articles. Index.
This volume provides a detailed account of bosonization. The first part of the book examines the technical aspects of bosonization including one-dimensional fermions, the Gaussian model, the structure of Hilbert space in conformal theories, Bose-Einstein condensation in two dimensions, non-Abelian bosonization, and the Ising and WZNW models. The second part presents applications of the bosonization technique to realistic models including the Tomonaga-Luttinger liquid, spin liquids in one dimension and the spin-1/2 Heisenberg chain with alternative exchange. The third part addresses the problems of quantum impurities. Chapters cover potential scattering, the X-ray edge problem, impurities in Tomonaga-Luttinger liquids and the multi-channel Kondo problem.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
Non-Fermi liquid behavior is found for the first time in a two-dimensional (2D) system with non-singular interactions using Haldane’s bosonization scheme. The bosonized system is solved exactly by a generalized Bogoliubov transformation. The fermion momentum distribution, calculated using a generalized Mattis-Lieb technique, exhibits a non-universal power law in the vicinity of the Fermi surface for intermediate interaction strengths. In recent years, there has been great interest in possible non-Fermi-liquid-like behavior in the ground state of two-dimensional (2D) strongly correlated fermionic systems [1]. It is well-known that in one dimension the ground state of a fermionic system with arbitrarily weak interactions is not a Fermi liquid but a state usually known as a Luttinger liquid [2]. For the 2D case, many studies have suggested that, in the limit of weak interactions, this behavior does not appear [3]. Studies of systems with a small number of particles also suggest Fermi liquid behavior in 2D [4]. Although Khveshchenko et al have found non-Fermi-liquid behavior in 2D, they worked only with a singular long-range current-current interaction [5]. The one-dimensional Luttinger liquid solution can be obtained by a bosonization