Localization and Melting of Interfaces in the Two-Dimensional Quantum Ising Model

We study the nonequilibrium evolution of coexisting ferromagnetic domains in the two-dimensional quantum Ising model—a setup relevant in several contexts, from quantum nucleation dynamics and false-vacuum decay scenarios to recent experiments with Rydberg-atom arrays. We demonstrate that the quantum-fluctuating interface delimiting a large bubble can be studied as an effective one-dimensional system through a “holographic” mapping. For the considered model, the emergent interface excitations map to an integrable chain of fermionic particles. We discuss how this integrability is broken by geometric features of the bubbles and by corrections in inverse powers of the ferromagnetic coupling, and provide a lower bound to the timescale after which the bubble is ultimately expected to melt. Remarkably, we demonstrate that a symmetry-breaking longitudinal field gives rise to a robust ergodicity breaking in two dimensions, a phenomenon underpinned by Stark many-body localization of the emergent fermionic excitations of the interface.

Figure 1

(a) Example of a convex bubble of “down” spins (unshaded square) in a sea of “up” spins (shaded square) (see main text for details). (b) One of the maximally split configurations of the bubble in panel (a). (c) Schematic representation of the allowed transitions in $H_{\mathrm{PXP}}$ preserving the total domain-wall length. (Other possible orientations are not displayed.) (d) Mapping of an interface corner onto a fermionic chain.

Figure 2

Snapshots of the time evolution of two corners separated by a large distance $L$, for $g=1$, $h=0.24$. The black initial profile evolves periodically as indicated by the various colors, reaching the yellow line at half-period $t=\pi /(2|h|)$ before receding toward the initial condition. Inset: Time evolution of the density profile $⟨{\psi }_x^{†}(t){\psi }_x(t)⟩$ of the fermionic chain, starting from the state $|{\mathrm{\Psi }}_0⟩$ corresponding to the right corner of the main figure.

Figure 3

Imbalance, Eq. (7), for $J=4$ (solid lines) and $J=\infty$ (dashed line) as a function of $h<0$, with $g=1$. The blue dots represent the imbalance extrapolated to $L\to \infty$ using the ansatz $I(L)=I_{\infty }+A/L$ (data for $L=6$,10,14 not shown in the plot). For $|h|<1$, the selected points correspond to the locations of the local maxima of $I(J=\infty )$, and are compatible with zero within 1 (at most 2) standard deviations.