Locality, Quantum Fluctuations, and Scrambling

Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators. Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs). However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-$N$ models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front. In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians. We construct a random time-dependent Hamiltonian model featuring a large-$N$ limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front. We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front. At small $N$, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques. To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a time-independent Hamiltonian model are also consistent with a diffusively broadened wave front.

Popular Summary

Unitarity and complexity are generic features of quantum dynamics, implying that the information of the initial state is never lost but gradually spreads over the entire system, becoming “scrambled.” As a result, after a long time, local observables depend only on the macroscopic properties of the state. Scrambling was originally studied in black holes but has recently been studied in condensed-matter physics and atomic physics because of its relation to quantum thermalization and chaos. Here, we propose a unified mathematical picture of scrambling.

We construct a model where the space-time profile of information in a quantum system can be tracked. When the number of spins is very large, the model gives results from holographic models (which are toy models for black holes); when there are few spins, it gives results from random-circuit models, which are models of condensed matter that describe randomly coupled spins. Our model also provides the missing link between these two limits.

Information in this model propagates like a wave—quantum fluctuations drive the physics from holographiclike to random circuitlike, turning the information wave front from sharp to diffusively broadened. This is further confirmed by large-scale state-of-the-art numerical simulations of realistic spin chains. The conclusion is that all chaotic 1D Hamiltonians exhibit a universal form of scrambling and chaos growth.

The work opens many new research directions including understanding the mechanisms generating quantum noise in nonrandom systems. The implication of the diffusive broadened information wave front on black-hole dynamics also appears to represent a novel quantum gravity effect.

Figure 1

The Brownian coupled cluster model. Spins within the same cluster interact with each other and also interact with the spins in the neighboring clusters. The intracluster coupling $J$ and the intercluster coupling $\tilde{J}$ are random in both space and time.

Figure 2

The scrambling “phase diagram” in space-time. An overall unimportant spatial offset is omitted. The bold line marks the wave front. The space-time is roughly divided into three regions. In the diffusive region, the squared commutator exhibits a diffusively broadening wave front, quantitatively different from the exponential growth in the large-$N$ limit. Away from the wave front, there is a region where the squared commutator grows exponentially with a modified Lyapunov exponent. Further ahead of the wave front, the chaotic region gives way to the perturbative region. As $N$ increases, the chaotic region expands as indicated by dashed arrows, and it eventually dominates the wave-front behavior in the infinite-$N$ limit.

Figure 3

Comparison between the butterfly velocity $v_B$ (a) and the wave-front diffusion constant $D$ (b) obtained from MPS simulation of the probability distribution to the small-$N$ and large-$N$ analysis. When $N$ is small, $v_B$ and $D$ agree with the small-$N$ result, relying on the local equilibrium assumption. As $N$ increases, the approximation breaks down, and both $v_B$ and $D$ cross over to the large-$N$ result.

Figure 4

The diffusive broadening wave front in the mixed-field Ising chain. (a) The contour of $\log C(r,t)=-20$ and $\log C(r,t)=-45$. The system size and the time limit allow the front to travel through about 200 sites. The contours obtained from MPO simulation with bond dimensions 8 and 16 are identical, showing the excellent convergence of our method. (b) The spatial distance $\delta x$ between the two contours increases with time, showing definitive broadening of the wave front. In the inset, we plot $\delta x$ with $t$ on a log-log plot. The slope of the curve approaches $1/2$, strong evidence of diffusive broadening, and the broadening exponent $p=1$.

Figure 5

The same figure as Fig. 4 but for a transverse-field Ising model describing noninteracting fermions. The slope of the curve on a log-log plot approaches $1/3$, agreeing with exact results for this model.

Figure 6

(a) The difference of $v_B$ and ${\lambda }_L$ between the discrete infinite-$N$ BCC and the continuum approximation. (b) Direct simulation of the discrete model by numerically solving the differential equation (b1). The spatial distance between two contours of $\log C(r,t)$ saturates in the late time, showing that the front is sharp. On a log-log plot, the slope of the curve decreases to zero. This result is in sharp contrast to the local Hamiltonian systems, where the slope increases to $1/2$ and $1/3$ for the chaotic case and noninteracting case, respectively, as shown in Figs. 4 and 5.

Figure 7

Procedures for measuring local observables in a MPS in canonical form (top) and a S-MPS in canonical form (bottom).

Figure 8

(a) Up to $N=10$ spins in each cluster and $L=200$ clusters, the numerical data for the squared commutator converge excellently with the bond dimension of the S-MPS. Bond dimension $\chi =32$ and bond dimension $\chi =48$ give rise to almost identical results for all timescales. This result allows us to access the entire scrambling behavior from early growth to late-time saturation. (b) Near the wave front, $C(r,t)$ perfectly agrees with the fitting function $C_{\mathrm{fit}}(r,t)=1+\mathrm{erf}\mathbf{(}(v_{B}t-r-r_0)/\sqrt{4Dt}\mathbf{)}$ for all $N$.