Chapter 11. The Fermi Liquid and the Luttinger Model

Giuliano Orso, Evgeni Burovski, Th. Jolicœur

Joshua D. Baktay

This thesis focuses on the development and application of tensor networks methods for simulating one-dimensional quantum many-body systems. The general aim is to unveil numerical evidence for a new universality class of 1D interacting fermion systems referred to as quasi-Fermi liquids (qFLs). Realizing such systems beyond a purely analytical context and elucidating their peculiar characteristics is the principle value of this thesis.Chapter 1 begins by introducing the essential physics of one-dimensional systems, known as Luttinger liquids relative to their higher dimensional counterparts. Then it introduces non-linear Luttinger liquids juxtaposed against the defining quasi-Fermi liquid (qFL) field theory that motivates the following lattice studies. Chapter 2 provides an in-depth review of matrix product states (MPS), the main methodology used to explore quasi-Fermi liquids. The chapter serves as a pedagogical introduction to the field and details those novel algorithmic developments necessary for the study of fermionic systems. The central result of this chapter is the section on finite entanglement scaling techniques applied to dynamic correlation functions. Chapter 3 begins the study of lattice models that exhibit quasi-Fermi liquid characteristics. Here we explore the ground states exclusively by computing their static correlation functions. The central results include Luttinger parameter phase space diagrams which demonstrate the existence of the quasi-Fermi liquid across multiple lattice models. Chapter 4 studies the excitations of the qFL lattice models. We begin with results from non-MPS based methods which were essential for benchmarking our novel application of MPS to fermionic spectral functions. The central results of the chapter are the power-law exponents that demonstrate the low-energy universality of the qFL and our analysis of the dynamic structure factor. Chapter 5 concludes with a recapitulation of the main results and an outlook on future work. The Appendix contains those tensor diagrams necessary for a numerical implementation of those algorithms introduced in Chapter 2.--Author's abstract

C. Di Castro, Roberto Raimondi

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.

Heinz J. Schulz, Gianaurelio Cuniberti, P. Pieri

Henrik Bruus, Karsten Flensberg

Abstract The 1D interacting electron gas is very different from its higher dimensional counterparts. The main difference being the breakdown of Fermi liquid theory in 1D. In Chapter 15 we saw that the main achievement of the Fermi liquid theory is that it enables us to picture the elementary excitations in terms of quasi particles. As we shall see in this chapter, the quasi particle excitations are in 1D electron systems replaced by the Luttinger–Tomonaga excitations, a sort of density waves, having a completely different nature. The corresponding electron liquid is denoted the Luttinger liquid.