Local factorisation of the dynamics of quantum spin systems

TL;DR

This paper proves that the dynamics of quantum spin systems can be locally approximated with exponentially decaying error, aiding understanding of entanglement and stability in quantum many-body physics.

Contribution

It introduces a local factorization method for the dynamics of quantum spin systems, providing bounds on approximation errors related to surface area.

Findings

Error decays almost exponentially with neighborhood size

Approximation error scales with the square of the boundary area

Supports analysis of entanglement area laws

Abstract

Motivated by the study of area laws for the entanglement entropy of gapped ground states of quantum spin systems and their stability, we prove that the unitary cocycle generated by a local time-dependent Hamiltonian can be approximated, for any finite set $X$, by a tensor product of the corresponding unitaries in $X$ and its complement, multiplied by a dynamics strictly supported in the neighbourhood of the surface $\partial X$. The error decays almost exponentially in the size of the neighbourhood and grows with the square of the area~$\vert \partial X\vert^2$.