Dynamics of entanglement and transport in one-dimensional systems with quenched randomness

Quenched randomness can have a dramatic effect on the dynamics of isolated 1D quantum many-body systems, even for systems that thermalize. This is because transport, entanglement, and operator spreading can be hindered by “Griffiths” rare regions, which locally resemble the many-body-localized phase and thus act as weak links. We propose coarse-grained models for entanglement growth and for the spreading of quantum operators in the presence of such weak links. We also examine entanglement growth across a single weak link numerically. We show that these weak links have a stronger effect on entanglement growth than previously assumed: entanglement growth is subballistic whenever such weak links have a power-law probability distribution at low couplings, i.e., throughout the entire thermal Griffiths phase. We argue that the probability distribution of the entanglement entropy across a cut can be understood from a simple picture in terms of a classical surface growth model. We also discuss spreading of operators and conserved quantities. Surprisingly, the four length scales associated with (i) production of entanglement, (ii) spreading of conserved quantities, (iii) spreading of operators, and (iv) the width of the “front” of a spreading operator, are characterized by dynamical exponents that in general are all distinct. Our numerical analysis of entanglement growth between weakly coupled systems may be of independent interest.

Figure 1

Schematic: entanglement across a cut through bond at position $x$.

Figure 2

Surface growth picture for entanglement $S\left(x\right)$ across a cut at $x$ in a finite chain, following Eq. (6). The lines show successive equally spaced times. Weak links with larger rates $\mathrm{\Gamma }$ are successively “dominated” by weaker links with smaller $\mathrm{\Gamma }$. [This sample was generated with $a=0$, see Eq. (2).]

Figure 3

Random circuit model: each bond $x$ receives Haar-random unitaries at its own rate $\mathrm{\Gamma }\left(x\right)/2$.

Figure 4

(Left) Maximum possible slope for the entanglement profile in the quantum circuit model, $\partial S/\partial x=1$. (Right) A staircase with a defect. After coarse-graining, a finite density $\rho$ of defects results in a slope $\partial S/\partial x=1-2\rho$.

Figure 5

Growth at a weak link location (random circuit model). Unitaries are applied at the weak link at a small rate ${\mathrm{\Gamma }}_1$. Each such event increases the height at the origin by two units. Two “defects” then travel up the sides of the double staircase at an average speed $\mathrm{\Gamma }/2$.

Figure 6

(Left) Entanglement growth around a weak link characterized by a small rate ${\mathrm{\Gamma }}_1$. (Right) Entanglement growth around a pair of weak links, showing how the weaker link dominates at late times. The figure shows eight equally spaced times.

Figure 7

Minimal cut configurations determining $S(x,t)$ in an infinite system with a single weak link of strength ${\mathrm{\Gamma }}_1$. The grey patch represents the unitary circuit and the thick line represents the (coarse-grained) minimal cut: (left) for $x<\frac{\mathrm{\Gamma }t}{4}\frac{(1-4{\mathrm{\Gamma }}_1/\mathrm{\Gamma })}{(1-2{\mathrm{\Gamma }}_1/\mathrm{\Gamma })}$ and (right) for $x>\frac{\mathrm{\Gamma }t}{4}\frac{(1-4{\mathrm{\Gamma }}_1/\mathrm{\Gamma })}{(1-2{\mathrm{\Gamma }}_1/\mathrm{\Gamma })}$.

Figure 8

Definition of dynamical exponents $z_O$ and $z_W$ governing the size of a spreading operator and the size of the front [as measured using the OTO correlator $C(x,t)$, Sec. 3]. The front is only well-defined in the weak-disorder regime ($a>1$), where $z_O=1$ and $z_W<1$.

Figure 9

Operator spreading caused by passage through a weak link (schematic). (Top) The spreading of the “wave packet” $\rho (x,t)$ [Eq. (16)]. (Bottom) The corresponding spreading of the front of the commutator $C(x,t)$ [Eq. (14)].

Figure 10

Two types of microscopic weak link. Left: an infinite spin chain with a weak bond connecting spins $b_1$ and $b_2$. (Top right) A chain in which the $Z$ component of spin $b$ is almost conserved. This can be mapped to a chain with a weak bond by duplicating $b$ (bottom).

Figure 11

Setup for the numerical simulation. A chain of $L=2l+1$ spin-$\frac{1}{2}$ degrees of freedom is subjected to Floquet dynamics with Ising couplings and transverse and longitudinal fields. The $Z$ component of the central spin is almost conserved due to the weak transverse field.

Figure 12

$S_{\text{vN}}$ as a function of position in systems with $L=21$ and $\varepsilon =0.3$. The different curves show times from $t=0$ to 500 in increments of $\begin{array}{c}\\ \mathrm{t}\\ ·\end{array}=10$ (the subsystems are separately entangled prior to $t=0$).

Figure 13

As Fig. 12, but for the second Renyi entropy $S_2$.

Figure 14

The von Neumann (top) and second Renyi (bottom) entropies scaled by their maximal value at large time $S_{\max }$ vs the rescaled time ${\mathrm{\Gamma }}_{n}t/S_{\max }$ for different values of $L$ and for $\epsilon =0.2$. The value of $S_{\max }$ is determined numerically.

Figure 15

The initial growth rates ${\mathrm{\Gamma }}_n$ as a function of $\epsilon$ (calculated using $L=21$). The stars (blue) and circles (red) correspond to von Neumann and second Renyi entropy, respectively.

Figure 16

(a) The solid lines show the normalized deviation from maximal entropy $\mathrm{\Delta }/S_{\text{max}}$ defined in Eq. (69) vs time $t$. The different colors correspond to different values of $\epsilon$ ranging between $0.1,0.2,...,0.7$. The dashed curves are the fits to the linear form: $1-\mathrm{\Gamma }t/S_{\text{max}}$ (red dashed) and to the exponential form: $1-exp\left[-{\mathrm{\Gamma }}^{\prime }\left(t-t_{*}\right)\right]/S_{\text{max}}$ (black dashed). The red dots mark the crossover points between linear behavior, at short times, and exponential, at long times. These crossover points are used to extract the timescale $t_l$ is extracted. (b) Comparison between the extracted crossover time $t_l$ (red dots) and the expected form Eq. (71) (blue solid curve).