Emergence of Hilbert Space Fragmentation in the Ising Model with a Weak Transverse Field

The last two decades have witnessed a substantial advance in revealing conditions for quantum many-body systems to thermalize following the experimental progress in quantum simulators. The transverse-field Ising model is one of the fundamental models in quantum many-body systems, yet full understanding of its dynamics remains elusive for higher than in one dimension. Here, we show the emergence of non-ergodicity for the Ising model in a weak transverse field on a square lattice in arbitrary dimension d. Specifically we investigate the effective non-integrable model in the weak-transverse field limit and demonstrate that novel Hilbert-space fragmentation occurs for d>1 as a consequence of only one emergent U(1) conservation law, i.e., domain-wall-number conservation. This conservation law leads to a kinetic constraint in the model and the appearance of frozen regions, which give rise to exponentially many fragmented subspaces in the Hilbert space. Our results indicate nontrivial initial-state dependence for long-lived prethermal dynamics of the Ising models in a weak transverse field.