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.
Using higher-dimensional bosonization we study correlation functions of fermions with singular forward scattering. Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)] we consider density-density interactions in
d dimensions that diverge in Fourier space for small momentum transfers |q| as $|\mathbf{q}{|}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}=2(d\ensuremath{-}1).$ In this case the single-particle Green's function shows Luttinger-liquid behavior. However, in contrast to
d=1, in higher dimensions the singularity of the momentum distribution
nk for wave vectors k close to the Fermi wave vector
kF is characterized by a different exponent than the singularity of the density of states \ensuremath{\rho}(\ensuremath{\omega}) for frequencies \ensuremath{\omega} close to zero. We also calculate the irreducible polarization ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\mathbf{q},\ensuremath{\omega})$ for $|\mathbf{q}|\ensuremath{\approx}{2k}_{F}.$ Whereas in the one-dimensional Tomonaga-Luttinger model ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\ifmmode\pm\else\textpm\fi{}{2k}_{F}+q,\ensuremath{\omega})$ exhibits anomalous scaling, we show that in
d>1 the leading singular corrections cancel. Thus, even though singular density-density interactions in
d>1 lead to Luttinger liquid behavior of the single-particle Green's function, gauge invariant correlation functions such as the polarization or the conductivity show conventional Fermi-liquid behavior. We discuss consequences for the effect of disorder on higher-dimensional Luttinger liquids.
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).
In framework of eigen-functional bosonization method, we introduce an imaginary phase field to uniquely represent electron correlation, and demonstrate that the Landau Fermi liquid theory and the Tomonaga-Luttinger liquid theory can be unified. It is very clear in this framework that the Tomonaga-Luttinger liquid behavior of one-dimensional interacting electron gases originates from their Fermi structure, and the non-Landau-Fermi liquid behavior of 2D interacting electron gases is induced by the long-range electron interaction, while 3D interacting electron gases generally show the Landau Fermi liquid behavior.
We study the effect that resistive regions have on the conductance of a quantum wire with interacting electrons which is connected to Fermi liquid leads. Using the bosonization formalism and a Rayleigh dissipation function to model the power dissipation, we use both scattering theory and Green's function techniques to derive the DC conductance. The resistive regions are generally found to lead to incoherent transport. For a single wire, we find that the resistance adds in series to the contact resistance of h / e 2 for spinless electrons, and the total resistance is independent of the Luttinger parameter K W of the wire. We numerically solve the bosonic equations to illustrate what happens when a charge density pulse is incident on the wire; the results depend on the parameters of the resistive and interacting regions in interesting ways. For a junction of Tomonaga-Luttinger liquid wires, we use a dissipationless current splitting matrix to model the junction. For a junction of three wires connected to Fermi liquid leads, there are two families of such matrices; we find that the conductance matrix generally depends on K W for one family but is independent of K W for the other family, regardless of the resistances present in the system.