Continuum mechanics of entanglement in noisy interacting fermion chains

TL;DR

The paper develops a continuum, two-replica description of information scrambling in a weakly interacting Majorana chain, yielding an entanglement membrane with a velocity-dependent line tension. By mapping the replicated problem to a semiclassical SU$(2)$ spin chain and solving coupled equations for fields $z$ and $\bar z$, it shows a bound-state domain-wall structure whose cost per time is $E(v)=s_{ ext{eq}}\mathcal{E}(v)$ and identifies a critical velocity $v_c$ at which the membrane unbinds into travelling waves with speed $v_*=v_c$. The butterfly velocity $v_B$ is identified with $v_c$, and the OTOC exhibits ballistic growth inside the light cone and saturation outside, with a Markov-process perspective corroborating the front propagation. Universality emerges because after rescaling, $\mathcal{E}(v)=v_c g(v/v_c)$ and the long-time scrambling behavior depends only on the crossover length $l=l_{ ext{int}}$ and the entropy density $s_{ ext{eq}}$.

Abstract

We develop an effective continuum description for information scrambling in a chain of randomly interacting Majorana fermions. The approach is based on the semiclassical treatment of the path integral for an effective spin chain that describes "two-replica" observables such as the entanglement purity and the OTOC. This formalism gives exact results for the entanglement membrane and for operator spreading in the limit of weak interactions. In this limit there is a large crossover lengthscale between free and interacting behavior, and this large lengthscale allows for a continuum limit and a controlled saddle-point calculation. The formalism is also somewhat different from that known from random unitary circuits. The entanglement membrane emerges as a kind of bound state of two travelling waves, and shows an interesting unbinding phenomenon as the velocity of the entanglement membrane approaches the butterfly velocity.

Figures (10)

• Figure 1: Plot of $\mathcal{E}(v)$ in terms of the dimensionless function $g(v/v_c)$ in blue, calculated numerically. The dotted orange line shows the asymptote $g_\mathrm{asymp}(a)=\vert a\vert$ which $g(v/v_c)$ approaches as $v\to\pm v_c$. (Here $K\equiv\Delta_I/\Delta_0=0.005$ was used in the numerics).
• Figure 2: Schematic of saddle point solution for the time-evolution operator for different values of $|X/T|$, with space plotted horizontally and time vertically. \ref{['fig:spboundstate']} shows the solution for $|X/T|<v_c$, with a ballistically travelling bound state connecting the cuts at each boundary (solid line). This is the entanglement membrane. \ref{['fig:sptravellingwaves']} shows the solution for $|X/T|>v_c$, with travelling waves propagating from each cut with velocity $v_c$ but remaining isolated from each other (dashed lines). The red dashes show the sections of the boundary which disagree and therefore give an extensive contribution to the boundary action.
• Figure 3: Static solution of the equations of motion with ${\theta(x)=\phi(x)}$, determining the rate of growth of state entanglement after a quench from a weakly-entangled state, and the entanglement membrane tension $\mathcal{E}(v)$ at $v=0$. The exact continuum form of the wall is $\theta(x)=\arctan \exp\left(2 x/ l\right)$ (Eq. \ref{['eq:staticsteadystate']}). This solution is degenerate under spatial translations. Here $l\equiv \Delta_0/\Delta_I$.
• Figure 4: Initial and final boundary conditions of the spin model used to calculate the entanglement purity of the time-evolution operator. Both boundaries are sharp domain walls with all $\mid\uparrow\rangle$ to left and $\mid\rightarrow\rangle$ to the right, with the position of the domain wall translated by $X$ in final boundary compared to the intial one.
• Figure 5: Steady-state solutions of the fictitious dynamics for two velocities in the range $0<v<v_c$. The domain walls are propagating ballistically to the right with velocity $v$, with $\theta$ "lagging behind" $\phi$. Fig. \ref{['fig:movingwall1']} shows $v/v_c=0.2$, where the domain walls in $\theta$ and $\phi$ overlap significantly, while \ref{['fig:movingwall2']} shows $v/v_c=0.9$ where the domain walls propagate almost independently of each other. $x=0$ is the centre of the domain wall which we use to enforce the symmetry between $\theta$ and $\phi$ (Eq. \ref{['eq:movingsymm']}). The length scale $l$ is defined as $l\equiv\Delta_0/\Delta_I$. (Here, $\theta(x)$$\phi(x)$ were calculated numerically with $K\equiv\Delta_I/\Delta_0=0.01$ using the method described in \ref{['sec:solvingequations']}).