Scrambling and thermalization in a diffusive quantum many-body system

TL;DR

The paper investigates how information scrambling and thermalization unfold in a diffusive, incoherent quantum many-body system by simulating the non-integrable 1D Bose-Hubbard model at high temperature with matrix-product-operator methods. It finds that OTOCs exhibit ballistic light-cone spreading of information despite diffusive, quasi-particle-free transport seen in time-ordered correlators, revealing a separation between scrambling and global thermalization timescales. Two experimentally feasible protocols—global and local interferometry—are proposed to measure both time-ordered and OTO correlators in bosonic/fermionic lattice systems. The results illuminate how information propagates independently of hydrodynamic relaxation and suggest avenues for testing holographic-inspired ideas in more realistic lattice models, including extensions to higher dimensions and other Hamiltonians.

Abstract

Out-of-time ordered (OTO) correlation functions describe scrambling of information in correlated quantum matter. They are of particular interest in incoherent quantum systems lacking well defined quasi-particles. Thus far, it is largely elusive how OTO correlators spread in incoherent systems with diffusive transport governed by a few globally conserved quantities. Here, we study the dynamical response of such a system using high-performance matrix-product-operator techniques. Specifically, we consider the non-integrable, one-dimensional Bose-Hubbard model in the incoherent high-temperature regime. Our system exhibits diffusive dynamics in time-ordered correlators of globally conserved quantities, whereas OTO correlators display a ballistic, light-cone spreading of quantum information. The slowest process in the global thermalization of the system is thus diffusive, yet information spreading is not inhibited by such slow dynamics. We furthermore develop an experimentally feasible protocol to overcome some challenges faced by existing proposals and to probe time-ordered and OTO correlation functions. Our study opens new avenues for both the theoretical and experimental exploration of thermalization and information scrambling dynamics.

Figures (11)

• Figure 1: Dynamical correlation functions in the incoherent transport regime. (a) Out-of-time ordered (OTO) correlators measure the scrambling of information across a quantum state. We compute OTO correlators $\mathcal{F}_{ij}(t)=\langle c_j^\dag(t) c_i^\dag c_j(t) c_i \rangle$ in the 1D Bose-Hubbard model at high temperature $T=4J$ for interactions $U=J$, chemical potential $\mu=0$, and system size $L=30$. In the high temperature regime, well-defined quasiparticles cease to exist. However, the OTO correlator $\mathcal{F}_{ij}$ exhibits a light-cone spreading of information. (b) The breakdown of well-defined quasiparticles is demonstrated by the one-particle Green's function $\mathcal{G}_{ij}(t)=\langle c_j^\dag(t) c_i \rangle$, which quickly decays to zero within $\tau J \sim 0.6$. The lifetime is thus shorter than the hopping rate, indicating a regime of incoherent transport.
• Figure 2: Light-cone spreading of quantum information. Contour plots of the reduced OTO correlator $\mathcal{F}^r_{ij}(t)\sim |\mathcal{F}_{ij}(t) - \langle \hat{n}_i \hat{n}_j \rangle|/\langle \hat{n}_i \hat{n}_j \rangle$ as a function of time and distance for interaction strength $U=J$, chemical potential $\mu=0$, and temperature (a) $T=2J$ and (b) $T=16J$, respectively. The spreading of quantum information forms a light-cone pattern. The contour lines indicate changes of $\mathcal{F}^r_{ij}(t)$ by 0.1.
• Figure 3: Characterizing the speed of information propagation. (a) Reduced OTO correlators $\mathcal{F}^r_{ij}(t)$ are shown as a function of time for different distances $|i-j|$, interaction strength $U=J$, and temperature $T=4J$. We introduce the light-cone velocity $v_\text{lc}$ by the space-time region, where $\mathcal{F}^r$ surpasses a small threshold and the butterfly velocity $v_\text{b}$ where it attains a large fraction of order one. (b) The light-cone velocity $v_\text{lc}$ grows with temperature and is bounded from below by the zero temperature Luttinger liquid velocity (colored arrows). By contrast, the butterfly velocity $v_\text{b}$ is systematically smaller than $v_\text{lc}$ and approximately independent of temperature $T$. The data is shown for two values of the interaction strength $U=J$ and $U=3J$.
• Figure 4: Lyapunov exponent. The reduced OTO correlator $\mathcal{F}^r_{ij}(t)$ is expected to grow exponentially on a timescale set by the butterfly velocity $v_\text{b}$ with a rate that defines the Lyapunov exponent $\lambda_L$. In our system, the regime of exponential growth is restricted to a rather small time range, see also Fig.\ref{['fig:lyapunov']}. Our data suggests that the Lyapunov exponent $\lambda_L$ is parametrically smaller than the conjectured upper bound $2\pi T$ and increases slowly as the temperature $T$ is lowered. The data is shown for interaction strength $U=\{1, 3, 9\}J$.
• Figure 5: Thermalization in closed quantum systems. Conserved quantities restrict the approach of a closed quantum system to global equilibrium, thus, rendering global thermalization a slow process. In the Bose-Hubbard model the total particle number is conserved leading to diffusive power-law tails in the connected density correlator $C_{n}(x,t) = \mathop{\mathrm{\hbox{Re}}}\nolimits [ \langle \hat{n}_x(t) \hat{n}_0 \rangle -\langle \hat{n}_x\rangle \langle \hat{n}_0\rangle ]$. (a) At low temperatures ($T=J$), where quasiparticles are reasonably well defined, the density correlator does not reach the diffusive regime within the accessible simulation time but is dominated by ballistic sound peaks. (b) By contrast, for high temperatures ($T=10J$) the crossover to diffusion becomes apparent. (c) For temperatures $T\gtrsim 4J$ the local density correlator $C_{n}(0,t)\sim 1/\sqrt{Dt}$, where $D$ is the diffusion constant. By contrast, at low temperature $T=J$ the diffusive regime has not yet been reached within the numerically accessible times and the correlations rather decay ballistically $C_{n}(0,t)\sim 1/{t}$. The slow relaxation of the hydrodynamic modes leads to the global thermalization time scale $t_\text{th} \sim L^2/D$ that is parametrically larger than the scrambling time scale $t_\text{scr} \sim L/v_\text{b}$ of quantum information.