Emergent Glassy Dynamics in a Quantum Dimer Model

We consider the quench dynamics of a two-dimensional quantum dimer model and determine the role of its kinematic constraints. We interpret the nonequilibrium dynamics in terms of the underlying equilibrium phase transitions consisting of a Berezinskii-Kosterlitz-Thouless (BKT) transition between a columnar ordered valence bond solid (VBS) and a valence bond liquid (VBL), as well as a first-order transition between a staggered VBS and the VBL. We find that quenches from a columnar VBS are ergodic and both order parameters and spatial correlations quickly relax to their thermal equilibrium. By contrast, the staggered side of the first-order transition does not display thermalization on numerically accessible timescales. Based on the model’s kinematic constraints, we uncover a mechanism of relaxation that rests on emergent, highly detuned multidefect processes in a staggered background, which gives rise to slow, glassy dynamics at low temperatures even in the thermodynamic limit.

Figure 1

Dynamical phase diagram of the QDM (schematic). Quenches within and across the columnar phase are accompanied by fast relaxation of local observables to thermal expectation values, with a rate that is fixed by the microscopic parameter $J$. Quenches across the first-order transition show a “melting” character, with thermalization times $\sim \xi$ given by the length scale of the staggered domains. Within the staggered phase, the dynamics appears glassy with relaxation times bounded by ${\tau }_{\mathrm{eq}}\gtrsim \exp [c\log (V/J){\xi }^4]$. Arrows are indicative of the different quench protocols considered.

Figure 2

Thermalization of columnar initial states. (a) Relaxation of the columnar order parameter ${\widehat{\varphi }}_c$ both within and across the columnar phase. (b) Long-time averaged values of ${\widehat{\varphi }}_c^2$ for different system sizes, starting from a columnar initial state and quenched to finite values of $V/J$. Included are the corresponding thermal values for $L=8$. Agreement between the two becomes less accurate around the phase transition at $V_c/J\approx -3.1$, but is excellent deep within the phases. (c) The crossing in the long-time averaged Binder cumulant $U_B$ of ${\widehat{\varphi }}_c$ marks the corresponding finite-$T$ phase transition. (d) The time-averaged dimer correlation functions $C(r)$, starting from a columnar initial state, agree well with the corresponding equilibrium values, even deep within the VBS phase.

Figure 3

Localization of staggered initial states. (a) Long-time averaged values of the staggered order parameter ${\widehat{\varphi }}_s^2$ starting from a staggered initial state. (b) In a quench across the transition to the disordered regime (here, $V/J=0$), the dynamics of ${\widehat{\varphi }}_s$ scales with $L^{-1}$. (c) The state $|p_A⟩$. Excitations from this state containing four plaquettes are expected to delocalize along the diagonals of the system. (d) The thermal expectation value of the potential energy landscape at an energy matching the quench in (e). (e) The late-time plaquette densities obtained after quenching $(1/\sqrt{2})(|p_A⟩+|p_B⟩)$ to $V/J=3$ differ strongly from the thermal values in (d) at large distances from the center.

Figure 4

Cost for local equilibration. (a) Starting configuration of pyramids of length $\xi$, with the orientation of the central plaquettes indicated in blue. (b) Thermalization of local observables requires the formation of larger pyramids of size ${\xi }^{\prime }$, which creates defects (red) with respect to the original background at the boundaries.