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arXiv:1203.0914 · Physical Review B

Generalized Gibbs ensemble and work statistics of a quenched Luttinger liquid

Balázs Dóra, Ádám Bácsi, Gergely Zaránd

Published Oct 22, 2012• 63 citations
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Abstract

We analyze the probability distribution function (PDF) of work done on a Luttinger liquid for an arbitrary finite duration interaction quench and show that it can be described in terms of a generalized Gibbs ensemble. We construct the corresponding density matrix with explicit intermode correlations, and determine the duration and interaction dependence of the probability of an adiabatic transition and the PDF of nonadiabatic processes. In the thermodynamic limit, the PDF of work exhibits a non-Gaussian maximum around the excess heat, carrying almost all the spectral weight. In contrast, in the small system limit most spectral weight is carried by a delta peak at the energy of the adiabatic process, and an oscillating PDF with dips at energies commensurate to the quench duration and with an exponential envelope develops. Relevance to cold atom experiments is also discussed.

DOI
10.1103/physrevb.86.161109

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We analyze the probability distribution function (PDF) of work done on a Luttinger liquid for an arbitrary finite duration interaction quench and show that it can be described in terms of a generalized Gibbs ensemble. We construct the corresponding density matrix with explicit intermode correlations, and determine the duration and interaction dependence of the probability of an adiabatic transition and the PDF of nonadiabatic processes. In the thermodynamic limit, the PDF of work exhibits a non-Gaussian maximum around the excess heat, carrying almost all the spectral weight. In contrast, in the small system limit most spectral weight is carried by a delta peak at the energy of the adiabatic process, and an oscillating PDF with dips at energies commensurate to the quench duration and with an exponential envelope develops. Relevance to cold atom experiments is also discussed.
Motivated by recent experiments on interacting cold atoms, we analyze interaction quenches in Luttinger liquids (LLs), where the interaction is ramped from zero to a finite value within a finite time. The fermionic single particle density matrix reveals several regions of spatial and temporal coordinates relative to the quench time, termed as Fermi liquid, sudden quench LL, adiabatic LL regime, and a LL regime with a time-dependent exponent. The various regimes can also be observed in the momentum distribution of the fermions, directly accessible through time of flight experiments. Most of our results apply to arbitrary quench protocols.
We study the escort probability distribution function of work done during an interaction quantum quench of Luttinger liquids. It crosses over from the thermodynamic to the small system limit with increasing aaa, which is the order of the escort distribution, and depends on the universal combination $[|{K}_{i}\ensuremath{-}{K}_{f}{|/({K}_{i}+{K}_{F})]}^{a}$, with Ki{K}_{i}Ki​, Kf{K}_{f}Kf​ the initial and final Luttinger liquid parameters, respectively. From its characteristic function, the diagonal R\'enyi entropies and the many-body inverse participation ratio (IPR) are determined to evaluate the information content of the time-evolved wave function in terms of the eigenstates of the final Hamiltonian. The hierarchy of overlaps is dominated by that of the ground states. The IPR exhibits a crossover from Gaussian to power-law decay with increasing interaction quench parameter.
Using a Luttinger liquid theory we investigate the time evolution of the particle density of a one-dimensional fermionic system with open boundaries and subject to a finite duration quench of the interparticle interaction. We provide analytical and asymptotic solutions to the unitary time evolution of the system, showing that both switching on and switching off the quench ramp create light-cone perturbations in the density. The post-quench dynamics is strongly affected by the interference between these two perturbations. In particular, we find that the discrepancy between the time-dependent density and the one obtained by a generalized Gibbs ensemble picture vanishes with an oscillatory behavior as a function of the quench duration, with local minima corresponding to a perfect overlap of the two light-cone perturbations. For adiabatic quenches, we also obtain a similar behavior in the approach of the generalized Gibbs ensemble density towards the one associated with the ground state of the final Hamiltonian.
We study the nonequilibrium dynamics of the Luttinger model after a quantum quench, when the initial state is a finite temperature thermal equilibrium state. The diagonal elements of the density matrix in the steady state show thermal features for high temperature initial states only, otherwise they retain highly nonthermal character. The time evolution of Uhlmann fidelity, which measures the distance between the time evolved and initial states, is evaluated for arbitrary initial temperatures and quench protocols. In the long time limit, the overlap between the time evolved and initial system decreases exponentially with the temperature with a universal prefactor. Within perturbation theory, the statistics of final total energy and work are numerically evaluated in the case of a sudden quench, which yield identical distributions at zero temperature. In both statistics, temperature effects are more significant in small systems. The Dirac-$\ensuremath{\delta}$ peak at the adiabatic ground state energy remains present in the probability distribution of the total energy, but disappears from the work distribution at nonzero initial temperatures.
The statistics of work performed on a system by a sudden random quench is investigated. Considering systems with finite dimensional Hilbert spaces we model a sudden random quench by randomly choosing elements from a Gaussian unitary ensemble (GUE) consisting of Hermitian matrices with identically, Gaussian distributed matrix elements. A probability density function (pdf) of work in terms of initial and final energy distributions is derived and evaluated for a two-level system. Explicit results are obtained for quenches with a sharply given initial Hamiltonian, while the work pdfs for quenches between Hamiltonians from two independent GUEs can only be determined in explicit form in the limits of zero and infinite temperature. The same work distribution as for a sudden random quench is obtained for an adiabatic, i.e., infinitely slow, protocol connecting the same initial and final Hamiltonians.
We present a comprehensive analysis of the relaxation dynamics of a Luttinger liquid subject to a sequence of sudden interaction quenches. The critical exponent β governing the decay of the steady-state propagator is expressed as an explicit functional of the switching protocol. At long distances β depends only on the initial state while at short distances it is also history dependent. Continuous protocols of arbitrary complexity can be realized with infinitely long sequences. For quenches of finite duration we prove that there exists no protocol to bring the initial non-interacting system in the ground state of the Luttinger liquid, albeit thermalization occurs at short distances. The adiabatic theorem is then investigated with ramp switchings of increasing duration and several analytic results for both the propagator and the excitation energy are derived.