Hilbert space shattering and dynamical freezing in the quantum Ising model

TL;DR

This paper investigates the quantum dynamics of the 2D transverse field Ising model, revealing a phenomenon called Hilbert space shattering into disconnected sectors, leading to localization or thermalization depending on the sector, with a transition between these behaviors.

Contribution

It introduces the concept of Hilbert space shattering in the 2D quantum Ising model, showing how conservation laws cause exponential sectorization and a transition between localized and thermalizing dynamics.

Findings

Hilbert space shattering occurs up to a prethermal timescale.

A transition between weak and strong shattering is observed.

On the weak side, dynamics follow ordinary diffusion.

Abstract

We discuss quantum dynamics in the transverse field Ising model in two spatial dimensions. We show that, up to a prethermal timescale, which we quantify, the Hilbert space 'shatters' into dynamically disconnected subsectors. We identify this shattering as originating from the interplay of a $\mathrm{U}(1)$ conservation law and a one-form $\mathbb{Z}_2$ constraint. We show that the number of dynamically disconnected sectors is exponential in system volume, and includes a subspace exponential in system volume within which the dynamics is exactly localized, even in the absence of quenched disorder. Depending on the emergent sector in which we work, the shattering can be weak (such that typical initial conditions thermalize with respect to their emergent symmetry sector), or strong (such that typical initial conditions exhibit localized dynamics). We present analytical and numerical evidence that a first order-like 'freezing transition' between weak and strong shattering occurs as a function of the symmetry sector, in a non-standard thermodynamic limit. We further numerically show that on the 'weak' (melted) side of the transition domain wall dynamics follows ordinary diffusion.