Hilbert space shattering and dynamical freezing in the quantum Ising model

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 U(1) conservation law and a one -form Z2 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 nonstandard thermodynamic limit. We further numerically show that on the weak (melted) side of the transition domain wall dynamics follows ordinary diffusion.

Automated summary

We discuss the quantum dynamics in the transverse field Ising model in two dimensions and show that the Hilbert space shatters into dynamically disconnected subsectors up to a quantified prethermal timescale. This shattering arises from the interplay of a U(1) conservation law and a one-form Z2 constraint. The number of disconnected subsectors is exponential in system volume, and a subspace with exponential growth within the system volume exhibits exactly localized dynamics.