Time-dependent matrix product ansatz for interacting reversible dynamics

TL;DR

This paper introduces a time-dependent matrix product ansatz for the rule 54 reversible cellular automaton, enabling exact calculations of dynamic properties and revealing coexistence of ballistic and diffusive transport in an interacting deterministic lattice gas.

Contribution

It provides the first explicit tMPA solution for an interacting reversible cellular automaton, allowing exact analysis of its dynamics and transport properties.

Findings

Exact computation of the dynamic structure factor.

Solution of the inhomogeneous quench problem.

Demonstration of coexistence of ballistic and diffusive transport.

Abstract

We present an explicit time-dependent matrix product ansatz (tMPA) which describes the time-evolution of any local observable in an interacting and deterministic lattice gas, specifically for the rule 54 reversible cellular automaton of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)]. Our construction is based on an explicit solution of real-space real-time inverse scattering problem. We consider two applications of this tMPA. Firstly, we provide the first exact and explicit computation of the dynamic structure factor in an interacting deterministic model, and secondly, we solve the extremal case of the inhomogeneous quench problem, where a semi-infinite lattice in the maximum entropy state is joined with an empty semi-infinite lattice. Both of these exact results rigorously demonstrate a coexistence of ballistic and diffusive transport behaviour in the model, as expected for normal fluids.