Exact Thermalization Dynamics in the “Rule 54” Quantum Cellular Automaton

Abstract

We study the out-of-equilibrium dynamics of the quantum cellular automaton known as "Rule 54." For a class of low-entangled initial states, we provide an analytic description of the effect of the global evolution on finite subsystems in terms of simple quantum channels, which gives access to the full thermalization dynamics at the microscopic level. As an example, we provide analytic formulas for the evolution of local observables and Rényi entropies. We show that, in contrast to other known examples of exactly solvable quantum circuits, Rule 54 does not behave as a simple Markovian bath on its own parts, and displays typical nonequilibrium features of interacting integrable many-body quantum systems such as finite relaxation rate and interaction-induced dressing effects. Our study provides a rare example where the full thermalization dynamics can be solved exactly at the microscopic level.

Entanglement dynamics in Rule 54: Exact results and quasiparticle picture

“…Indeed, while the states considered in Ref. [31] all relax (locally) to the Gibbs state with infinite temperature, here we show that exact results can be obtained also for states relaxing to richer Gibbs ensembles (characterised by an arbitrary chemical potential). This allows us, for instance, to study exactly inhomogeneous quenches giving rise to a non-trivial (generalised) hydrodynamic regime at late times [40,41].…”

“…The objective of this paper is to extend the exact results presented in Ref. [31] to a larger class of initial states. This is the second of two papers dedicated to this task.…”

“…Even though physically very relevant, these quantities are notoriously hard to compute. This is especially true when considering interacting integrable models, where, up to very recently [31], they could be accessed only in the limit t A. Indeed, as pointed out in Ref.…”

“…On the contrary, in certain special cases it leads to exact results. In particular, concerning the entanglement dynamics, it provides exact results in dual-unitary quantum circuits [9][10][11]24] and in Rule 54 [31].…”

“…The situation changed recently, when Ref. [31] presented an exact characterisation of the growth of entanglement in the quantum cellular automaton Rule 54, which can be considered one of the simplest examples of interacting integrable models (see also [32][33][34][35][36][37][38]). The result was again based on a time-channel approach and lead to an exact characterisation of the growth of entanglement when the system is initialised in a particular class of initial states.…”