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 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.
Highlights
For initial states(0) with short range correlations we prove in the appendix, assuming minimal degeneracy of the dispersion relation ωk, that the steady-state obtained from the infinite-time average Γx(∞,y ) is translation invariant up to a small parameter
We have shown in large generality that for large classes of natural initial conditions, local expectation values of systems relaxing under unitary dynamics generated by non-interacting Hamiltonians take the values of translation invariant genaralized Gibbs ensembles
The emerging steady state is parametrized by thermodynamical potentials whose number is intensive, namely of the order of the initial correlation length in units of the lattice spacing
Summary
Over more than a century, it has become clear that the methods of statistical mechanics work incredibly well in a vast range of physical situations. The second component in the case of thermalization is that the equilibrium steady state has no detailed memory of the initial state (beyond, e.g., temperature or chemical potential), namely it is a thermal state It has become clear, that some specific classes of physical systems do not equilibrate [7,8,9,10], at least over the times one can assess in the laboratory. [50] is unnecessary for Gaussification, but we show that the Gaussian state that the system approaches will be time independent This is a proof of equilibration over realistic times for these models, and it proves that the equilibrium state can be described by a generalized Gibbs ensemble (GGE). We consider some possibilities for realizing distinct instances of generalized Gibbs ensembles in optical lattices and systematically studying their stability in the presence of interactions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
