Abstract In many quantum quench experiments involving cold atom systems the post-quench phase can be described by a quantum field theory of free scalars or fermions, typically in a box or in an external potential. We will study mass quench of free scalars in arbitrary spatial dimensions d with particular emphasis on the rate of relaxation to equilibrium. Local correlators expectedly equilibrate to GGE; for quench to zero mass, interestingly the rate of approach to equilibrium is exponential or power law depending on whether d is odd or even respectively. For quench to non-zero mass, the correlators relax to equilibrium by a cosine-modulated power law, for all spatial dimensions d, even or odd. We briefly discuss generalization to O(N ) models.
We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass
m(t) that goes from
m0 to zero. We prove that, as anticipated in MSS \cite{Mandal:2015jla}, the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form
∣ψ⟩=
exp[−κ2H−∑n>2∞κnWn]∣Bd⟩. The
Wn (
n=2,3,...,
W2=H) here represent the conserved
W∞ charges and
∣Bd⟩ represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the
κn's as in MSS. We compute exact time-dependent correlators for some specific quench protocols
m(t). The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature
β=4κ2, and chemical potentials
μn=4κn. In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol
m(t) from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all
κn's, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.
We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass m(t) that goes from m0 to zero. We prove that, as anticipated in MSS [1], the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form |ψ〉 = exp[−κ2H − ∑ >2 ∞ κ n W n ]|Bd〉. The W n (n = 2, 3, . . ., W2 = H) here represent the conserved W∞ charges and |Bd〉 represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the κ n ’s as in MSS. We compute exact time-dependent correlators for some specific quench protocols m(t). The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature β = 4κ2, and chemical potentials μ n = 4κ n . In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol m(t) from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all κ n ’s, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.
One of the fundamental principles of statistical physics is that only partial information about a system's state is required for its macroscopic description. This is not only true for thermal ensembles, but also for the unconventional ensemble, known as Generalized Gibbs Ensemble (GGE), that is expected to describe the relaxation of integrable systems after a quantum quench. By analytically studying the quench dynamics in a prototypical one-dimensional critical model, the massless free bosonic field theory, we find evidence of a novel type of equilibration characterized by the preservation of an enormous amount of memory of the initial state that is accessible by local measurements. In particular, we show that the equilibration retains memory of non-Gaussian initial correlations, in contrast to the case of massive free evolution which erases all such memory. The GGE in its standard form, being a Gaussian ensemble, fails to predict correctly the equilibrium values of local observables, unless the initial state is Gaussian itself. Our findings show that the equilibration of a broad class of quenches whose evolution is described by Luttinger liquid theory with an initial state that is non-Gaussian in terms of the bosonic field, is not correctly captured by the corresponding bosonic GGE, raising doubts about the validity of the latter in general one-dimensional gapless integrable systems such as the Lieb-Liniger model. We also propose that the same experiment by which the GGE was recently observed [Langen et al., Science 348 (2015) 207-211] can also be used to observe its failure, simply by starting from a non-Gaussian initial state.
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 a one-dimensional free Fermi gas or Tonk's gas (Bose gas with infinite repulsion).
We study spectral properties and the dynamics after a quench of one-dimensional spinless fermions with short-range interactions and long-range random hopping. We show that a sufficiently fast decay of the hopping term promotes localization effects at finite temperature, which prevents thermalization even if the classical motion is chaotic. For slower decays, we find that thermalization does occur. However, within this model, the latter regime falls in an unexpected universality class, namely, observables exhibit a power-law (as opposed to an exponential) approach to their thermal expectation values.
We study the equilibration properties of observables in long-range field theories after the mass quench in
d=1, 2, and 3 dimensions. We classify the regimes where we expect equilibration or its absence in different dimensions. In addition, we study the effect of the initial state in the equilibration properties of our system. In the case of free field theories, we show that whenever we have equilibration, the long-time limit of correlations can be described by the generalized Gibbs ensemble. We prove that all integrals of motion in our system are nonlocal.