Classical Lieb-Robinson Bound for Estimating Equilibration Timescales of Isolated Quantum Systems

We study equilibration of an isolated quantum system by mapping it onto a network of classical oscillators in Hilbert space. By choosing a suitable basis for this mapping, the degree of locality of the quantum system reflects in the sparseness of the network. We derive a Lieb-Robinson bound on the speed of propagation across the classical network, which allows us to estimate the timescale at which the quantum system equilibrates. The bound contains a parameter that quantifies the degree of locality of the Hamiltonian and the observable. Locality was disregarded in earlier studies of equilibration times, and it is believed to be a key ingredient for making contact with the majority of physically realistic models. The more local the Hamiltonian and observables, the longer the equilibration timescale predicted by the bound.

Figure 1

Left: Visualization of a typical weighted graph $H_{jk}$ for a local Hamiltonian. The top nodes (red) correspond to nonequilibrium states, and the bottom nodes (blue) are equilibrium states. The thickness of the edges is determined by $|H_{jk}|$. Right: Scaling of the equilibration timescale $t_{jk}^{*}$ from Eq. (9) with the system size $L$ for particularly local and nonlocal choices of the parameters [$g=\parallel H\parallel /L$ and $g=\parallel H\parallel (L-1)/L$, respectively].

Figure 2

Contour plots of Lieb-Robinson bound $B_{jk}(t)$ in Eq. (8) for parameter values $L=10$ and $\parallel H\parallel =\parallel O\parallel =k=r=1$. We use $g=\parallel H\parallel /L$ (left) for a strongly local system and $g=\parallel H\parallel (L-1)/L$ (right) for a nonlocal system. The time $t_{jk}^{*}$ in Eq. (9), which defines a light cone, is shown as a solid red line.

Figure 3

Comparison of our bounds with exact results for the transverse-field Ising model (12) on $L=8$ lattice sites. Left: The bound $B_{jk}$ (solid line) compared to ${\mathrm{\Lambda }}_{jk}$ (dashed line) as a function of time for $\alpha =10$. Right: Comparison of the timescale $t_{jk}^{*}$ and the estimated equilibration time $t_{\mathrm{eq}}$, determined according to the protocol described in the main text, as a function of $\alpha$. Error bars indicate standard errors resulting from averaging over 10 randomly chosen nonequilibrium initial states.