Hilbert space shattering and dynamical freezing in the quantum Ising model

Oliver Hart, Rahul Nandkishore

Published 2022-03-11Version 1

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 numerical evidence that a weak-strong 'freezing transition' 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.