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arXiv:1601.00671 · Physical Review Letters

Equilibration via Gaussification in Fermionic Lattice Systems

Marek Gluza, Christian Krumnow, M. Friesdorf, Christian Gogolin, Jens Eisert

Published Nov 4, 2016• 67 citations
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Abstract

In this Letter, we present a result on the nonequilibrium dynamics causing equilibration and Gaussification of quadratic noninteracting fermionic Hamiltonians. Specifically, based on two basic assumptions-clustering of correlations in the initial state and the Hamiltonian exhibiting delocalizing transport-we prove that non-Gaussian initial states become locally indistinguishable from fermionic Gaussian states after a short and well controlled time. This relaxation dynamics is governed by a power-law independent of the system size. Our argument is general enough to allow for pure and mixed initial states, including thermal and ground states of interacting Hamiltonians on large classes of lattices as well as certain spin systems. The argument gives rise to rigorously proven instances of a convergence to a generalized Gibbs ensemble. Our results allow us to develop an intuition of equilibration that is expected to be more generally valid and relates to current experiments of cold atoms in optical lattices.

DOI
10.1103/physrevlett.117.190602

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In this Letter, we present a result on the nonequilibrium dynamics causing equilibration and Gaussification of quadratic noninteracting fermionic Hamiltonians. Specifically, based on two basic assumptions-clustering of correlations in the initial state and the Hamiltonian exhibiting delocalizing transport-we prove that non-Gaussian initial states become locally indistinguishable from fermionic Gaussian states after a short and well controlled time. This relaxation dynamics is governed by a power-law independent of the system size. Our argument is general enough to allow for pure and mixed initial states, including thermal and ground states of interacting Hamiltonians on large classes of lattices as well as certain spin systems. The argument gives rise to rigorously proven instances of a convergence to a generalized Gibbs ensemble. Our results allow us to develop an intuition of equilibration that is expected to be more generally valid and relates to current experiments of cold atoms in optical lattices.
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Even after almost a century, the foundations of quantum statistical mechanics are still not completely understood. In this work, we provide a precise account on these foundations for a class of systems of paradigmatic importance that appear frequently as mean-field models in condensed matter physics, namely non-interacting lattice models of fermions (with straightforward extension to bosons). We demonstrate that already the translation invariance of the Hamiltonian governing the dynamics and a finite correlation length of the possibly non-Gaussian initial state provide sufficient structure to make mathematically precise statements about the equilibration of the system towards a generalized Gibbs ensemble, even for highly non-translation invariant initial states far from ground states of non-interacting models. Whenever these are given, the system will equilibrate rapidly according to a power-law in time as long as there are no long-wavelength dislocations in the initial second moments that would render the system resilient to relaxation. Our proof technique is rooted in the machinery of Kusmin-Landau bounds. Subsequently, we numerically illustrate our analytical findings by discussing quench scenarios with an initial state corresponding to an Anderson insulator observing power-law equilibration. We discuss the implications of the results for the understanding of current quantum simulators, both in how one can understand the behaviour of equilibration in time, as well as concerning perspectives for realizing distinct instances of generalized Gibbs ensembles in optical lattice-based architectures.
We present an elementary, general, and semiquantitative description of relaxation to Gaussian and generalized Gibbs states in lattice models of fermions or bosons with quadratic Hamiltonians. Our arguments apply to arbitrary initial states that satisfy a mild condition on clustering of correlations. We also show that similar arguments can be used to understand relaxation (or its absence) in systems with time-dependent quadratic Hamiltonians and provide a semiquantitative description of relaxation in quadratic periodically driven (Floquet) systems.
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Integrable quantum many-body systems are considered to equilibrate to generalized Gibbs ensembles (GGEs) characterized by the expectation values of integrals of motion. We study the dynamics of exactly solvable quadratic bosonic systems in the thermodynamic limit, and show a general mechanism for the relaxation to GGEs, in terms of the diagonal singularity. We show analytically and explicitly that a free bosonic system relaxes from a general (not necessarily Gaussian) initial state under certain physical conditions to a Gaussian GGE. We also show the relaxation to a Gaussian GGE in an exactly solvable coupled system, a harmonic oscillator linearly interacting with bosonic reservoirs.
A variety of recent studies have shed light on the far from equilibrium behavior of quantum systems. Research suggests that this away from equilibrium approach may lead to the realization of unfamiliar dynamical quantum states. For instance, we see ballistic transport at finite temperature, an absence of thermalization, and development of long-range entanglement from initially uncorrelated states. It has been shown that in some cases a time-evolving quantum state is equivalent to the ground state of an "emergent" Hamiltonian where the time enters as a param- eter. This mapping gives us insight into the nature of correlations between particles. For high temperatures, we expect no long-range correlations to develop. However, the emergent model demonstrates that the time-evolved state can be realized as a low energy eigenstate, which naturally possesses long-range correlations. Here, we study the dynamical behavior of both noninteracting and interacting particles in a one-dimensional lattice under various conditions. As the system un- dergoes time evolution, we look at the distribution of particles as well as correlations between particles in the system. We then compare the actual time-evolving state of the system to the ground state of the emergent Hamiltonian to study how long this description is valid. We generally expect that the description is valid in a given region as long as the boundary effects do not propagate into the region. We first verify and reproduce results in the literature (see Vidmar et al. [1]). We proceed to study the effect of interaction strength in a particular interacting model that shows both ballistic and diffusive transport. We then apply this formalism to calculate infinite temperature dynamics of a noninteracting system. Interestingly, this system also develops long-range correlations over time in spite of starting out from an infinite temperature state.