Symmetry re-breaking in an effective theory of quantum coarsening

TL;DR

This work addresses out-of-equilibrium quantum coarsening in 2D and explains two key experimental observations—the speeding up of coarsening near the phase boundary and persistent order-parameter oscillations after quenches within the ordered phase—using a minimal classical-spin framework derived from the transverse-field Ising model. By combining energy-conserving classical dynamics with mean-field and Gaussian-fluctuation analyses, the authors show that the observed phenomena arise from purely semiclassical dynamics: a dual role of the kinetic term enhances coarsening deep in the ordered phase while the vanishing domain-wall tension near criticality slows it, and mean-field oscillations are captured by a two-dimensional Lipkin-Meshkov-Glick dynamics with a dynamical separatrix at $g_{\mathrm{dyn}}=2$. The central new concept, symmetry re-breaking, emerges when spatial fluctuations drive local MF trajectories across the MF separatrix, creating domains that coarsen within the ordered phase and leave the final magnetization sign determined by the oscillation count. These results suggest that semiclassical approaches can disentangle genuine quantum effects from dynamical features in quantum coarsening and point to broader applicability of symmetry re-breaking in symmetry-broken Hamiltonian systems across experimental platforms.

Abstract

We present a simple theory accounting for two central observations in a recent experiment on quantum coarsening and collective dynamics on a programmable quantum simulator [T. Manovitz et al., Nature \textbf{638}, 86 (2025)]: an apparent speeding up of the coarsening process as the phase transition is approached; and persistent oscillations of the order parameter after quenches within the ordered phase. Our theory, based on the Hamiltonian structure of the equations of motion in the classical limit of the quantum model, finds a speeding up already deep within the ordered phase, with subsequent slowing down as the domain wall tension vanishes upon approaching the critical line. Further, the oscillations are captured within a mean-field treatment of the order parameter field. For quenches within the ordered phase, small spatially-varying fluctuations in the initial mean-field lead to a remarkable long-time effect, wherein the system dynamically destroys its long-range order and has to coarsen to re-establish it. We term this phenomenon \emph{symmetry re-breaking}, as the resulting late-time magnetization can have a sign opposite to the initial magnetization.

Figures (5)

• Figure 1: Phase diagram and global dynamics in the classical transverse-field Ising model. (a) The Binder cumulant $B \!=\! 3(1 - \ev*{m^4}/3\ev*{m^2}^2)/2$ distinguishes between the ordered ($B\!=\!1$) and disordered ($B\!=\!0$) phases. The red line shows the states accessed with the areal speed simulations. The black line shows the post-quench states accessed in the oscillation experiments which exhibit either ferromagnetic (solid), paramagnetic (dotted), or symmetry re-breaking (dashed) behaviors. The square dot marks the thermodynamic transition at $g=g_c$, and the round dot the dynamical transition at $g=g_{\mathrm{dyn}}$ in the mean-field model. The data was obtained via Monte Carlo simulations at a fixed temperature, for a 50$\times$50 lattice, and plotted as a function of the excess energy density $\varepsilon$. (b) The Bloch sphere of a classical spin $i$. Switching on a transverse field misaligns the net (exchange+applied) local field with the spin direction, leading to spin precession.
• Figure 2: Areal speed of coarsening. An initial "up" circular domain, with the opposite magnetization as compared to its surroundings, is eroded as the system evolves with Hamiltonian dynamics [$g\!=\!2$ and times $t\!=\!0$,250,500 for panels (a),(b),(c) respectively]. (d--e) The speed $v_a$ at which this happens increases with $g$ starting from $g\!=\!0$, until it decreases in the proximity of the phase transition. The excess energy density is fixed to $\varepsilon\!=\!0.2$, as marked in Fig. \ref{['fig:phase_diag']}. A lattice of size 400$\times$400 is used for the simulations.
• Figure 3: Post-quench oscillations. (a--d) After the quench, the magnetization (colored lines) oscillates at the frequency set by the mean-field solution (pale grey lines), while getting damped. The plots are respectively for $g=1.8,2.0,2.2,2.5$; the thermodynamic phase transition is at $g_c \simeq 2.4$. (e) Comparison of the oscillation frequencies with the mean-field prediction in gray, showing excellent agreement. The frequency minimum does not correspond to the Ising transition (vertical dotted line). Notice the absence of damping $\gamma$ in the mean field theory. (f) Bloch sphere representation of each spin in the uniform state, showing a trajectory confined to one half of the Bloch sphere, one crossing to the other half of the sphere, and the separatrix between the two. A lattice of size 200$\times$200 is used for the simulations.
• Figure 4: Symmetry re-breaking. (a) After the magnetization oscillations die out, the snapshot shows that the system is locally ordered, but globally broken into domains (colorbar identical to Fig. \ref{['fig:speed']}). These domains coarsen in time, eventually leading to the reestablishment of a global magnetization at one of its two equilibrium values. (b--c) The sign of the steady-state magnetization $m_\infty$ is determined by the parity of the number of oscillations the system undergoes before they are damped out. This number increases with decreasing energy density of the pre-quench state. A lattice of size 200$\times$200 is used for the simulations.
• Figure 5: Role of fluctuations. Order parameter oscillations (a) and growth of fluctuations (b) for the 2D $\phi^4$ theory, Eq. \ref{['eq:phi4']}, for a quench $r_i\!=\!-3.9$ to $r\!=\!-2$. The exact dynamics (solid blue lines) agrees with the gaussian decoupling (Eq. \ref{['eq:gaussian']}, pale gray lines) until the fluctuations become of the same order of the mean field. Panels (c) and (d) similarly show a quench into the symmetry re-breaking region ($r_i\!=\!-4.1$, $r\!=\!-2$), for a case where the asymptotic value of the order parameter has the opposite sign. A lattice of size 201$\times$201 is used for the simulations.