Equilibration towards generalized Gibbs ensembles in non-interacting theories

Abstract

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 longwavelength 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.

Relaxation to Gaussian generalized Gibbs ensembles in quadratic bosonic systems in the thermodynamic limit

“…Moreover, the GGE (2.11) is a simple Gaussian state: only the quadratic gauge invariant termsÎ k =b † kb k contribute to the GGE, as long as the initial (not necessarily Gaussian) state fulfills a few physical conditions, where the presence of anomalous correlations is allowed. In contrast to the previous works [16,[34][35][36][37], which proved the Gaussification for large but finite systems, we have directly analyzed the systems in the thermodynamic limit, with continuous spectra. This greatly simplifies the analysis, and in this picture, the mechanism for the equilibration is due to the Riemann-Lebesgue lemma and the diagonal singularity.…”

“…The key to the proofs is the Lieb-Robinson bound [11]. In addition, power-law relaxation towards Gaussian GGE was studied for similar bosonic/fermionic lattice mod-els with quadratic Hamiltonians, by making use of the Kusmin-Landau bound [36] and the stationary-phase approximation [37] under the clustering condition on the initial state. See also Ref.…”

Complexity and entanglement for thermofield double states

“…[92][93][94] and further developed in refs. [74,[94][95][96][97]. In this picture, independent entangled pairs propagate in a ballistic fashion following an effective group velocity.…”

Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

“…Finally, we address subleading terms in the asymptotic expansion of log χ(λ). Let us start with the approximation of the integrals in (14). For any regular function f (E) the principal value integral can be approximated as…”

“…A natural way to test such concepts is to consider evolution of the physical system after some parameters of the Hamiltonian have been rapidly changed -this is the so-called quantum quench setup [1][2][3][4]. After a long time a quenched integrable system equilibrates to a steady state that can be described in the quench action formalism [5,6] or by a complete generalized Gibbs ensemble (GGE) [7][8][9][10][11][12][13][14].…”