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arXiv:0903.0895 · Europhysics Letters (EPL)

Quantum quench from a thermal initial state

Spyros Sotiriadis, Pasquale Calabrese, John Cardy

Published Jul 1, 2009• 140 citations
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Abstract

We consider a quantum quench in a system of free bosons, starting from a thermal initial state. As in the case where the system is initially in the ground state, any finite subsystem eventually reaches a stationary thermal state with a momentum-dependent effective temperature. We find that this can, in some cases, even be lower than the initial temperature. We also study lattice effects and discuss more general types of quenches.

DOI
10.1209/0295-5075/87/20002

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We consider a quantum quench in a system of free bosons, starting from a thermal initial state. As in the case where the system is initially in the ground state, any finite subsystem eventually reaches a stationary thermal state with a momentum-dependent effective temperature. We find that this can, in some cases, even be lower than the initial temperature. We also study lattice effects and discuss more general types of quenches.
We study properties of isolated integrable quantum systems after a sudden quench starting from thermal states. We show that, even if the system is initially in thermal equilibrium at finite temperature, the diagonal entropy after a quench remains a fraction of the entropy in the generalized ensembles introduced to describe integrable systems after relaxation. The latter is also, in general, different from the entropy in thermal equilibrium. Furthermore, we examine the difference in the distributions of conserved quantities in the thermal versus the generalized ensembles after a quench and show that they are also, in general, different from each other. This explains why these systems fail to thermalize. A finite-size scaling analysis is presented for each quantity, which allows us to make predictions for thermodynamically large lattice sizes.
We study sudden quantum quenches in which the initial states are selected to be either eigenstates of an integrable Hamiltonian that is nonmappable to a noninteracting one or a nonintegrable Hamiltonian, while the Hamiltonian after the quench is always integrable and mappable to a noninteracting one. By studying weighted energy densities and entropies, we show that quenches starting from nonintegrable (chaotic) eigenstates lead to an ``ergodic'' sampling of the eigenstates of the final Hamiltonian, while those starting from the integrable eigenstates do not (or at least it is not apparent for the system sizes accessible to us). This goes in parallel with the fact that the distribution of conserved quantities in the initial states is thermal in the nonintegrable cases and nonthermal in the integrable ones, and means that, in general, thermalization occurs in integrable systems when the quench starts form an eigenstate of a nonintegrable Hamiltonian (away from the edges of the spectrum), while it fails (or requires larger system sizes than those studied here to become apparent) for quenches starting at integrable points. We test those conclusions by studying the momentum distribution function of hard-core bosons after a quench.
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.
We consider a quantum quench in a noninteracting fermionic one-dimensional field theory. The system of size LLL is initially prepared into two halves L\mathcal{L}L $([\ensuremath{-}L/2,0])$ and R\mathcal{R}R ([0,L/2])([0,L/2])([0,L/2]), each of them thermalized at two different temperatures TL{T}_{\mathcal{L}}TL​ and TR{T}_{\mathcal{R}}TR​, respectively. At a given time, the two halves are joined together by a local coupling and the whole system is left to evolve unitarily. For an infinitely extended system $(L\ensuremath{\rightarrow}\ensuremath{\infty})$, we show that the time evolution of the particle and energy densities is well described via a hydrodynamic approach which allows us to evaluate the correspondent stationary currents. We show, in such a case, that the two-point correlation functions are deduced, at large times, from a simple nonequilibrium steady state. Otherwise, whenever the boundary conditions are retained (in a properly defined thermodynamic limit), any current is suppressed at large times, and the stationary state is described by a generalized Gibbs ensemble, which is diagonal and depends only on the post-quench mode occupation.
We study the problem of a quantum quench in which the initial state is the ground state of an inhomogeneous Hamiltonian, in two different models, conformal field theory and ordinary free field theory, which are known to exhibit thermalization of finite regions in the homogeneous case. We derive general expressions for the evolution of the energy flow and correlation functions, as well as the entanglement entropy in the conformal case. Comparison of the results from the two approaches in the regime of their common validity shows agreement up to a point further discussed. Unlike for the thermal analogue, the evolution in our problem is non-diffusive and can be physically interpreted using an intuitive picture of quasiparticles emitted from the initial time hypersurface and propagating semiclassically.
Abstract. We study the problem of a quantum quench in which the initial state is the ground state of an inhomogeneous hamiltonian, in two different models, conformal field theory and ordinary free field theory, which are known to exhibit thermalisation of finite regions in the homogeneous case. We derive general expressions for the evolution of the energy flow and correlation functions, as well as the entanglement entropy in the conformal case. Comparison of the results of the two approaches in the regime of their common validity shows agreement up to a point further discussed. Unlike the thermal analogue, the evolution in our problem is non-diffusive and can be physically interpreted using an intuitive picture of quasiparticles emitted from the initial time hypersurface and propagating semiclassically.