Quantum quench and thermalization of one-dimensional Fermi gas via phase-space hydrodynamics

Abstract

By exploring a phase space hydrodynamics description of one-dimensional free Fermi gas, we discuss how systems settle down to steady states described by the generalized Gibbs ensembles through quantum quenches. We investigate time evolutions of the Fermions which are trapped in external potentials or a circle for a variety of initial conditions and quench protocols. We analytically compute local observables such as particle density and show that they always exhibit power law relaxation at late times. We find a simple rule which determines the power law exponent. Our findings are, in principle, observable in experiments in an one dimensional free Fermi gas or Tonk's gas (Bose gas with infinite repulsion). 1 manas.kulkarni@icts.res.in 2 mandal@theory.tifr.res.in 3 morita.takeshi@shizuoka.ac.jp of thermalization of local observables in a so-called free systems or integrable systems where the thermal ensemble is often characterized by an infinite number of chemical potentials, corresponding to an infinite number of conserved quantities. Such an ensemble is called a GGE (generalized Gibbs ensemble). These issues have been summarized in a number of articles in recent years; for a partial list of references, see [1,2,3].The issue of thermalization in integrable models [4,5,6,7,8,9,10,11] has been discussed in detail some time ago in Ref [12], where thermalization has been shown to occur under some general assumptions. In spite of such general arguments, it is important to see if one can obtain some explicit results about time evolution of observables, especially at long times. For quantum quenches leading to gapless Hamiltonians, such explicit results were obtained at long times in Ref.[13] which rigorously showed thermalization of the reduced density matrix of subsystems of a large system. Fully exact time-dependence and subsequent results on thermalization were obtained for free scalars and Fermions in 1+1 dimensions for mass quenches ending at zero mass in [14].In this paper, we will discuss quantum quench in a free Fermi gas in one space dimension, by using a large-N technique 4 . This subject was dealt with earlier in a paper [17] by two of the present authors, where it was shown how moments of the Fermion density approached thermalization from particular sudden quenches. Apart from studies of thermalization, several other dynamical aspects, such as the onset of shock fronts, have been studied in large-N non-interacting as well as interacting Fermi gases [18,19,20,21,22] (see also the Appendix A) ; finite-N corrections have been studied in [23].The importance of the large-N limit is that the Fermions are described by a semiclassical fluid in the phase space. As it turns out, for simple phase space configurations, the equations describing such phase space fluids boil down to equations of conventional hydrodynamics. Thus, it is tempting to think that thermalization can be understood somehow in terms of the equations of conventional hydrodynamics, which would be of significant interest. However, conventional hydrodyna...

Real-time spin-charge separation in one-dimensional Fermi gases from generalized hydrodynamics

“…We stress that the formalism above is similar to the phase-space hydrodynamics used to study pulse propagation in free Fermi gases [59,[64][65][66][67], non-linear Luttinger liquids [68], (spin-)Calogero [69][70][71][72] and Lieb-Liniger models [73][74][75]. As a main difference, however, here the velocities are dressed by the interactions, which are also responsible for the non-trivial interplay between spin and charge dominating the dynamics at short times.…”

Entanglement entropies of inhomogeneous Luttinger liquids

“…Refs. [20,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] for recent works in that direction. We focus on inhomogeneous Luttinger liquids, namely on quantum systems whose effective field theory description corresponds to a free boson theory in a curved metric and with a spatially varying coupling strength or Luttinger parameter K [4,20,22,23,28,43,44]; the theory will be briefly reviewed in Sec.…”

Phase space holography with no strings attached

“…In that regard, particularly interesting are the non-stationary configurations representing particles released from a confining potential which gets suddenly switched on/off [111][112][113].…”