Facilitated Spin Models of Dissipative Quantum Glasses

We introduce a class of dissipative quantum spin models with local interactions and without quenched disorder that show glassy behavior. These models are the quantum analogs of the classical facilitated spin models. Just like their classical counterparts, quantum facilitated models display complex glassy dynamics despite the fact that their stationary state is essentially trivial. In these systems, dynamical arrest is a consequence of kinetic constraints and not of static ordering. These models display a quantum version of dynamic heterogeneity: the dynamics toward relaxation is spatially correlated despite the absence of static correlations. Associated dynamical fluctuation phenomena such as decoupling of time scales is also observed. Moreover, we find that close to the classical limit, quantum fluctuations can enhance glassiness, as recently reported for quantum liquids.

Figure 1

Quantum facilitated models. A lattice of two-level systems with coherent and incoherent excitation or deexcitation. Interaction is via kinetic constraints $f_i$ which make the rates on site $i$ dependent on the state of its neighbors. The stationary state is trivial and non-interacting, while the dynamics is highly correlated and glassy when $\kappa \gg \Omega$, $\gamma$.

Figure 2

Kinetic constraints. For the classical models, the upper panel shows three sites in a configuration, assuming all other sites are in the 0 state. For the quantum models, the bottom panel shows three sites of a product basis state for the density matrix, where we use the notation $\mathit{u}\equiv |u⟩⟨u|$ and $\mathit{e}\equiv |e⟩⟨e|$. Red (dark on white) symbols represent sites that cannot evolve. Green (white on dark) symbols represent sites where the state of neighboring sites allows for their evolution.

Figure 3

Structure in trajectory space and dynamic heterogeneity. Quantum trajectories of the qFA, qEast, and unconstrained models, from QJMC simulations for $N=10$ spins and periodic boundaries. We plot $⟨P_i^{(e)}⟩(t)\in [0,1]\equiv [\mathrm{\text{white, black}}]$ for all sites $i=1$ to $N$. The rates ($\kappa$, $\Omega$, $\gamma$) for the trajectories shown are: (1, 0.25, 0) for the qFA and (1, 0.4, 0) for the qEast and the unconstrained ones. Space-time correlations are evident in the trajectories of the constrained models and absent in the unconstrained system.

Figure 4

Glassy slowing down and dynamical correlations. (a) Average relaxation time ${\tau }_{\mathrm{relax}}$ as a function of $\Omega /\kappa$ (at $\gamma =0$) in the qFA model. The dots are results from QJMC simulations. The line is the theoretical expectation for relaxation via activated diffusion of excitations: since the initial state has a single excitation, in a lattice of size $N$ with periodic boundary conditions we expect ${\tau }_{\mathrm{relax}}\approx (2/N)\sum _{l=1}^{N/2}\tau (l)$. Inset: ${\tau }_{\mathrm{relax}}$ for the qEast model. (b) Fluctuations in waiting times between quantum jump events for the same qFA system, quantified via $Q_{\mathrm{Mandel}}$.

Figure 5

Enhanced glassiness due to weak quantum fluctuations. Effective diffusion constant $D$ in the qFA model at $\gamma \ne 0$. Panels (a) and (b) show $D$ for the cases of $\gamma ={10}^{-2}\kappa$ and ${10}^{-1}\kappa$. As $\gamma$ and $\Omega$ become larger compared to $\kappa$, the analytic approximation (6) becomes less quantitatively accurate, as expected, but reentrance is still evident from the exact result. Panel (c) shows similar reentrance behavior of the branching rate $B$. $D_0$ and $B_0$ are the classical values of the diffusion constant and the branching rate, respectively.