Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics

Abstract

In these four lectures I describe basic ideas and methods applicable to both classical and quantum systems displaying slow relaxation and non-equilibrium dynamics. The first half of these notes considers classical systems, and the second half, quantum systems. In Lecture 1, I briefly review the glass transition problem as a paradigm of slow relaxation and dynamical arrest in classical many-body systems. I discuss theoretical perspectives on how to think about glasses, and in particular how to model them in terms of kinetically constrained dynamics. In Lecture 2, I describe how via large deviation methods it is possible to define a statistical mechanics of trajectories which reveals the dynamical phase structure of systems with complex relaxation such as glasses. Lecture 3 is about closed (i.e. isolated) many-body quantum systems. I review thermalisation and many-body localisation, and consider the possibility of slow thermalisation and quantum non-ergodicity in the absence of disorder, thus connecting with some of the ideas of the first lecture. Lecture 4 is about open quantum systems, that is, quantum systems interacting with an environment. I review the description of open quantum dynamics within the Markovian approximation in terms of quantum master equations and stochastic quantum trajectories, and explain how to extend the dynamical large deviation method to study the statistical properties of ensembles of quantum jump trajectories. My overall aim is to draw analogies between classical and quantum nonequilibrium and find connections in the way we think about problems in these areas. Contents

Trajectory phase transitions in noninteracting spin systems

“…Our findings here fit with recent results for few body problems [23,24] where singularities in long-time trajectory ensembles were not a consequence of interactions in the dynamics in the large size limit (like in most other systems displaying LD transitions [21,22]) but of the properties of the dynamical observables probed. We focused on spin systems and observables where tilted generators displayed a duality symmetry, which we exploited to identify precisely the location of the dynamical transitions.…”

“…Interesting phase behaviours in trajectory space are encoded in the analytic properties of the SCGF and its singularities are indicative of dynamical phase transitions. The rate function ϕ(o) and the SCGF θ(s) are related by a Legendre-Fenchel transform [19][20][21][22]…”

Emergent Glassy Dynamics in a Quantum Dimer Model

“…Moreover, for such initial states consisting of a set of staggered dimer domains of length scale ξ, we analytically derive a lower bound on the local thermalization timescale that is exponential in ξ 4 . Induced by highly detuned processes involving 'defects' seperated by a distance ξ, this reveals a mechanism similar to proposals of fractonic systems at low temperatures [21][22][23], leading to arrested quantum dynamics at T = 0 and slow relaxation at finite energy densities reminiscent of the physics of classical glasses [24].…”

Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems

“…(2)(3)(4). The large deviation (LD) theory of stochastic dynamics is concerned with the behaviour of observables that are time-integrated over trajectories, for some long time τ [43]. At level 2.5 these observables fall into two main classes [26][27][28][29][30]: empirical fluxes q i τ (ψ, ψ ), corresponding to the number of jumps from ψ to ψ per unit time in a trajectory (i.e., empirical transition rates), and the empirical measure µ τ (ψ), corresponding to the fraction of time that the system spends in ψ.…”