One-dimensional quantum liquids: Beyond the Luttinger liquid paradigm

Abstract

For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. We review the recent progress in understanding 1D quantum fluids beyond the low-energy limit, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi edge singularity and the Fermi liquid theory, perturbation theory in the interaction strength, and new ways of treating finitesize properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. We review existing results for various dynamic correlation functions, in particular the dynamic structure factor and the spectral function. Moreover, we show how a dispersion nonlinearity leads to finite particle lifetimes, and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and we explain the relation between the two.

Prethermalization and thermalization of a quenched interacting Luttinger liquid

“…dispersive) corrections to a perfectly linear dispersion around the Fermi energy [9,10,58,59]. Although the fermionic band curvature is irrelevant in the sense of the renormalization group (RG) [58] and does therefore not modify the static infrared behavior of the fermions, it is visible in dynamic observables, such as the fermionic spectral function or the dynamical structure factor [47,[60][61][62][63][64]. In this work we will show that in a nonequilibrium situation, the quasi-particle scattering induced by the band curvature leads to a dynamical redistribution of energy and allows the system to relax towards a thermal state.…”

“…[64]) and place the present approach into this context. At zero temperature and without band curvature, long wavelength physics of the interacting fermion model can be exactly mapped to the quadratic Luttinger model and therefore has well-defined, sharp phononic excitations, expressed by a spectral function of the phonons A q,ω = i(G R q,ω − G A q,ω ) = 2πδ(ω − u|q|).…”

Coulomb drag

“…In recent years, a lot of the attention was devoted to 1D liquids with nonlinear dispersion [for reviews on this topic see Deshpande et al (2010); Imambekov et al (2012);and Matveev (2013) ]. In the TLL theory, the curvature of the quasiparticle spectrum is described by an irrelevant operator (in the renormalization group sense).…”

Time-Dependent Impurity in Ultracold Fermions: Orthogonality Catastrophe and Beyond

“…Being one of the very few known examples of a solvable nonequilibrium many-body problems, the OC also provides a conceptual framework for understanding several fundamental phenomena in solid-state physics, including the Kondo effect [17], resonant tunneling in mesoscopic structures [18][19][20][21][22][23], 1D quantum physics beyond the Luttinger-liquid paradigm [24][25][26], and the motion of a heavy particle in a Fermi gas [27]. For a recent review of the OC, see Ref.…”