Integrable Trotterization: Local Conservation Laws and Boundary Driving

TL;DR

This paper introduces an integrable Trotterization method for lattice models, deriving local conservation laws and boundary-driven nonequilibrium steady states, with potential applications in quantum simulation experiments.

Contribution

It presents a novel integrable Trotterization scheme for lattice models, including boundary driving and exact steady state solutions, extending integrability to discrete-time dynamics.

Findings

Derived local conservation laws from an inhomogeneous transfer matrix.

Constructed explicit nonequilibrium steady state density matrices.

Demonstrated potential for experimental simulation in quantum systems.

Abstract

We discuss a general procedure to construct an integrable real-time trotterization of interacting lattice models. As an illustrative example we consider a spin-$1/2$ chain, with continuous time dynamics described by the isotropic ($XXX$) Heisenberg Hamiltonian. For periodic boundary conditions local conservation laws are derived from an inhomogeneous transfer matrix and a boost operator is constructed. In the continuous time limit these local charges reduce to the known integrals of motion of the Heisenberg chain. In a simple Kraus representation we also examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries. We show explicitly, how an exact nonequilibrium steady state density matrix can be written in terms of a staggered matrix product ansatz. This simple trotterization scheme, in particular in the open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of trapped ion and atom optics setups.