Slow dynamics from a nested hierarchy of frozen states

Abstract

We identify the mechanism of slow heterogeneous relaxation in quantum kinetically constrained models (KCMs) in which the potential energy strength is controlled by a coupling parameter. The regime of slow relaxation includes the large-coupling limit. By expanding around that limit, we reveal a \emph{nested hierarchy} of states that remain frozen on time scales determined by powers of the coupling. The classification of such states, together with the evolution of their Krylov complexity, reveal that these time scales are related to the distance between the sites where facilitated dynamics is allowed by the kinetic constraint. While correlations within frozen states relax slowly and exhibit metastable plateaus that persist on time scales set by powers of the coupling parameter, the correlations in the rest of the states decay rapidly. We compute the plateau heights of correlations across all frozen states up to second-order corrections in the inverse coupling. Our results explain slow relaxation in quantum KCMs and elucidate dynamical heterogeneity by relating the relaxation times to the spatial separations between the active regions.

Figures (5)

• Figure 1: Slow relaxation in the XPX model. Panel (a) shows a hierarchy of correlations $\overline{c_t}(\boldsymbol{s})=t^{-1} \int_0^t{\rm d} \tau c_\tau(\boldsymbol{s})$, Eq. \ref{['eq:correlation']}, averaged over the computational-basis states $\ket{\boldsymbol{s}}$, frozen on time scales $t\sim\Delta^k$ (the so-called "level-$k$ states"---see Fig. \ref{['fig:hierarchy']}). The correlation function averaged over the rest of states instead decays rapidly (dashed line). Panels (b) and (c) show correlations in all frozen states, as well as their average (black line). The initial plateau is more pronounced at larger $\Delta$. In all plots the system size is $L=14$.
• Figure 2: Nested hierarchy of states.$\mathcal{C}_k$ contains the states frozen by the truncation $H^{(k)}$ of the effective Hamiltonian $H$. Level-$k$ states are those in $\mathcal{C}_k$ that are only frozen by $H^{(\ell)}$ for $\ell\le k$.
• Figure 3: Frozen states and scaling of Krylov sectors. Panel (a) shows the scaling of the number of frozen states $|\mathcal{C}_k|$ (black) and the number of Krylov sectors $\mathcal{N}_{\mathrm{Krylov}}$ (red) with order $k$ of the truncation, for system size $L = 16$. The values of $|\mathcal{C}_k|$ computed from Eq. \ref{['eq:number-frozen-states']} match the numerically computed ones shown in the plot. Panel (b) depicts the scaling of $|\mathcal{C}_k|$ with system size $L$ for various $k$. The dashed lines show the asymptotic prediction from Eq. \ref{['eq:asymptotics']}. In both panels the coupling was set to $\Delta = 4$.
• Figure 4: Krylov complexity and correlation functions in frozen states. Panels (a)--(c) show time-averaged Krylov complexity for a level-$k$ initial state, additionally averaged over all initial states in the level. Notice the collapse in the rescaled time $t/\Delta^k$. Panels (d)--(f) show time-averaged correlation functions in level-$k$ states, averaged across the level. The dashed lines show the plateau-height estimates given in Eq. \ref{['eq:plateau-value']}. The $O(\Delta^{-2})$ correction in Eq. \ref{['eq:plateau-value']} suggests that the two-staged plateaus (for $\Delta=4$) become single-staged as $\Delta\to\infty$. In all plots the system size is $L=14$.
• Figure S1: Krylov complexity of frozen states for the quantum Fredrickson-Andersen model. Time-averaged Krylov complexity for a level-$k$ initial state, for $k=1,3$, additionally averaged over all initial states in the level. Notice the collapse in the rescaled time $t/\Delta^k$, where $\Delta=e^s/6$. Levels $k=2, 4$ in this model are empty, so we omit them. In both plots the system size is $L=12$.