Entanglement Membrane in Chaotic Many-Body Systems

In certain analytically tractable quantum chaotic systems, the calculation of out-of-time-order correlation functions, entanglement entropies after a quench, and other related dynamical observables reduces to an effective theory of an “entanglement membrane” in spacetime. These tractable systems involve an average over random local unitaries defining the dynamical evolution. We show here how to make sense of this membrane in more realistic models, which do not involve an average over random unitaries. Our approach relies on introducing effective pairing degrees of freedom in spacetime, describing a pairing of forward and backward Feynman trajectories, inspired by the structure emerging in random unitary circuits. This viewpoint provides a framework for applying ideas of coarse graining to dynamical quantities in chaotic systems. We apply the approach to some translationally invariant Floquet spin chains studied in the literature. We show that a consistent line tension may be defined for the entanglement membrane and that there are qualitative differences in this tension between generic models and “dual-unitary” circuits. These results allow scaling pictures for out-of-time-order correlators and for entanglement to be taken over from random circuits to nonrandom Floquet models. We also provide an efficient numerical algorithm for determining the entanglement line tension in $1+1\mathrm{D}$.

Popular Summary

Statistical physics is a lens on the microscopic world. At its highest resolution, an inconceivable number of particles interact via the laws of quantum mechanics. On zooming out to the macroscopic level, this daunting throng can oftentimes be described by a simple classical theory. This is the power of “coarse graining.” We use this lens to inspect the flow of quantum information and find that an unconventional hydrodynamic theory of membranelike objects emerges. In this work, we make sense of such membranes in generic quantum circuits.

To detect information spreading, we construct an interference protocol in which quantum circuits are duplicated an even number of times. A constructive interference phenomenon forces these copies to form pairs. The membrane—a line bisecting a circuit in a way that optimizes information flow across it—marks the boundary separating different ways to pair among the duplicated circuits. Imperfections slightly thicken the membrane, but a redefinition of the flow rate can account for it after coarse graining to macroscopic scales. We derive hydrodynamic equations to solve the flow rate.

The power of the coarse graining lies in the fact that only macroscopic data are needed to determine quantum information spreading. The membrane picture can have many extensions—for instance, in understanding a phase transition point beyond which information stops flowing. Intriguingly, the membrane has close connections to quantum gravity, where a membranelike object encircling a black hole measures the quantum information the black hole has engulfed. In the future, it may be possible to probe the properties of the membrane in many-particle quantum systems in the laboratory.

Figure 1

The membrane picture of entanglement and other dynamical quantities. (a) Example of a membrane for evaluating Rényi entropy growth in a 1D quench. Minimizing the membrane tension gives the entanglement. The tension is a function of local velocity $v=\dot{x}$. (b) Schematic line tension function for a chaotic system with parity symmetry. $\mathcal{E}(v)$ is convex and tangential to $|v|$ at $(\pm v_B,v_B)$.

Figure 2

Thick membrane (schematic). Except in the special case of Haar-averaged unitaries, the membrane (a domain wall between permutations $\sigma$ and ${\sigma }^{\prime }$) is thickened by clusters where the effective spin takes the value “$\perp$.” In the multilayer tensor network for $U^{(N)}$, this value represents the propagation of states orthogonal to paired states. (In general, the interior of the thick membrane also contains paired states. For $N>2$, these may be distinct from $\sigma$ or ${\sigma }^{\prime }$ [11, 14].)

Figure 3

The structure of the circuit with gates acting on nearest-neighbor sites. In the Floquet models we study, each gate is identical. In the random unitary case, each gate is independently sampled from the Haar ensemble.

Figure 4

Domain walls in (a) $Z_{0\perp }$ (or in the Haar-averaged circuit) and (b) $Z$, taking into account $\perp$ states. In (a), the width of the domain wall is 1. The width of the domain wall in (b) is argued to be of order 1.

Figure 5

(a) A domain wall configuration with a $\perp$ cluster. The weight is nonzero only when the domain wall passes through the $\perp$ cluster. (b) The average of the square of (a) in terms of the $S_4$ spin model. The spin states outside the cluster are fixed.

Figure 6

Domain walls on a tilted square lattice. Left: In the replica approach, two domain walls have an attractive interaction when they both visit two sides of a plaquette (starred). Right: This replica treatment describes a single domain wall subject to a random potential on the vertical bonds. The domain wall in the figure is affected by the potentials on the two dashed bonds.

Figure 7

Numerical computation of $\mathcal{E}(v)$ with $t_{\max }$ up to 8 for (a) the reflection-symmetric gate in Eq. (72) with $x=0.8$; (b) a generic nonsymmetric gate (specified in Appendix pp2). Black dots mark estimates of $v_B^{L/R}$ from an independent calculation of the OTOC in Appendix pp2-s3.

Figure 8

Numerical calculations of $\mathcal{E}(v)$ for Floquet Ising models. (a) shows the parameters in Eq. (80). (b) shows the “dual-unitary” parameter values in Eq. (81). The bottom arrow in case (b) illustrates the convergence to the expected limiting form $\mathcal{E}(v)=1$. The intersections with the line $\mathcal{E}(v)=|v|$ are in good agreement with the known value $v_B=1$ (top arrows).

Figure 9

The spacetime structure of $W(x,t)/Z(x,t)$ for (a) the generic reflection-symmetric gate in Eq. (71) [Fig. 7] and (b) the maximally entangling dual-unitary gate specified in Eq. (81) [Fig. 8]. The cross marks the coordinate (0,0). In our construction, only steps with even $x+t$ are possible.

Figure 10

Weakly entangling gate with $x=0.5$ [Eq. (72)]: Heat map showing the relative importance of long steps.

Figure 11

Weakly entangling gate with $x=0.5$ [Eq. (72)]: (a) Successive approximations to $\mathrm{\Gamma }(s)$ (see Sec. 4a). For some small $t$ values, $\mathrm{\Gamma }(s)$ is negative around $s=0$. (b) Approximations to $\mathcal{E}(v)$ obtained by Legendre transform of (a). Note the small $\mathcal{E}(0)=\mathrm{\Gamma }(0)$.

Figure 12

Heat map of $\tilde{Z}(x,t)=\exp (s_{\mathrm{eq}}{v}_{E}t)Z(x,t)$. (a) Floquet Ising model as in Fig. 8. $\tilde{Z}(x,t)$ is consistent with the solution of a parabolic equation. For example, the decay at $x=0$ is compatible with the $t^{-1/2}$ scaling in Eq. (83): This scaling is demonstrated in Appendix pp2-s6. (b) Dual-unitary model as in Fig. 8. Unlike the previous model, $\tilde{Z}(x,t)$ is approximately constant in the interior of the light cone and zero outside.

Figure 13

The partition function $Z_{\mathrm{op}}(x,t)$ for a dual-unitary circuit. (a) $q^{2L}{Z}_{\mathrm{op}}(x=1,t=5)$. By unitarity [the first identity in Eq. (98)], we can remove the unitaries outside the light cone, giving $q^{2t}{Z}_{\mathrm{op}}(x=1,t=5)$ as expression (b). Then, Eq. (98) allows the remaining gates to be removed.

Figure 14

The membrane picture for the out-of-time-ordered commutator $\tilde{C}(x,t)$. The top boundary condition creates two domain walls. The bottom boundary condition favors $-$, providing an outward force on the lower end points of the two domain walls. The right domain wall thus moves with the optimal speed $v_B$. Here, the left wall is forced by the dual state $|{-}^{*}⟩$ at the operator insertion point to travel with $v>v_B$.

Figure 15

Domain wall (membrane) in Floquet evolution without a circuit structure. The spin variables $+$, $-$, and $\perp$ denote the three choices of projection operators inserted on the horizontal bars beneath them; see Eqs. (102) and (103). Figures show samples of thick domain walls for (a) two-site and (b) one-site insertion.

Figure 16

Numerical calculations of $\mathcal{E}(v)$ for Floquet evolution with the square wave $h_x(t)$ [Eqs. (104) and (105)] for (a) two-site and (b) one-site schemes.

Figure 17

Numerical calculations of $\mathcal{E}(v)$ for Floquet evolution with the sinusoidally varying $h_x(t)$ in Eq. (107) for the two-site scheme.

Figure 18

(a) Pauli weight for the kicked-Ising model with parameters in Eq. (80). (b) Estimation of $v_B$ by fitting $t_{\text{arrival}}$ versus $d$. “Chaotic Ising” is the kicked-Ising model with the parameters in Eq. (80), while “Symmetric” and “Random” refer to the generic parity-symmetric gate with $x=0.8$ [Eq. (72)] and the fixed random parity-asymmetric gate (see Sec. 5a and Appendix pp2-s2). For the asymmetric gate, we fit separately for the left and right butterfly velocities. For the dual-unitary model (not shown), $v_B=1$ exactly.

Figure 19

The approximate entanglement velocity $v_E$ computed by truncating to times $\le t$, for $t$ up to 8. The first four curves are for the four circuits studied in Sec. 5 [the first three are also the subject of Fig. 18 for $v_B$]. The fifth shows the generic symmetric gate [Eq. (72)] at a weaker coupling strength. The two curves with $w=0$ and $w=0.3$ are the entanglement velocities for the Floquet Hamiltonian with sinusoidal field in Eq. (107). Because of the absence of circuit structure, their starting values at $t=1$ are not identical to that obtained from the random average, while the others are.

Figure 20

The decay of $|W|$ and $Z$ at $x=0$, 1 and $\exp (-s_{\mathrm{eq}}{v}_{E}t)$ for (top) reflection symmetric gate in Eq. (72), (center) Floquet Ising model in Eq. (80), and (bottom) dual-unitary gate in Eq. (81).

Figure 21

The diffusive decay of $\tilde{Z}(x,t)$ at $x=0$ for the chaotic Ising model in Eq. (80). Both axes are logarithmic.