Emergent statistical mechanics of entanglement in random unitary circuits

We map the dynamics of entanglement in random unitary circuits, with finite onsite Hilbert space dimension $q$, to an effective classical statistical mechanics, and develop general diagrammatic tools for calculations in random unitary circuits. We demonstrate explicitly the emergence of a “minimal membrane” governing entanglement growth, which in 1+1 dimensions is a directed random walk in spacetime (or a variant thereof). Using the replica trick to handle the logarithm in the definition of the $n\mathrm{th}$ Rényi entropy, $S_n$, we map the calculation of the entanglement after a quench to a problem of interacting random walks. A key role is played by effective classical spins (taking values in a permutation group) which distinguish between different ways of pairing spacetime histories in the replicated system. For the second Rényi entropy, $S_2$, we are able to take the replica limit explicitly. This gives a mapping between entanglement growth and a directed polymer in a random medium at finite temperature (confirming Kardar-Parisi-Zhang scaling for entanglement growth in generic noisy systems). We find that the entanglement growth rate (“speed”) $v_n$ depends on the Rényi index $n$, and we calculate $v_2$ and $v_3$ in an expansion in the inverse local Hilbert space dimension, $1/q$. These rates are determined by the free energy of a random walk and of a bound state of two random walks, respectively, and include contributions of “energetic” and “entropic” origin. We give a combinatorial interpretation of the Page-like subleading corrections to the entanglement at late times and discuss the dynamics of the entanglement close to and after saturation. We briefly discuss the application of these insights to time-independent Hamiltonian dynamics.

Figure 1

Left: the time-evolved wave function represented as a unitary circuit (schematic). Time runs vertically. The circles at the bottom represent an initial product state. The legs on the top carry the wave function's spin indices. Right: the quantities we study (e.g., the second Rényi entropy, shown) are contractions of powers of the circuit (blue) and its conjugate (red). The replica trick requires $k$ copies of this stack (not shown).

Figure 2

By averaging a stack with $N$ layers of $U\left(t\right)$ and $N$ layers of $U{\left(t\right)}^{*}$ (left), we obtain an effective classical model of interacting spins (right) with three-body interactions. These spins take values in the permutation group $S_N$. The important configurations involve domain walls between different spin values; see, e.g., Fig. 3.

Figure 3

In the calculation of $S_2$ for $n=2$ (diagram on left), each replica contains a single elementary walk (domain wall). For $n>2$, e.g., $n=3$ (right), each replica contains multiple walks, which can form a bound state with a finite typical width. To calculate averages involving $S_n$, we must use $k$ replicas (which multiplies the number of elementary walks by $k$) and take $k\to 0$ at the end of the calculation.

Figure 4

The structure of the random circuit. Each two-site random gate is shown as a four-leg block. $x>0$ corresponds to region A. Time evolution is going upward.

Figure 5

$\overline{Z_2^k}$ maps to $k$ directed walks on the tilted square lattice, with attractive interactions of order $1/q^4$ (vertical direction is physical time). The figure shows $k=2$. Left: This configuration of the walks has no interaction contribution at this order. Right: In this configuration, there is an interaction contribution from the square with a star; see Sec. 6.

Figure 6

Outcome of a splitting event. Left: Two commutative domain walls have two ways to split: exchange or passing through. Right: Two noncommutative domain walls have three ways to split. Red (dashed) represents (12), blue (solid) represents (23), and green (dot-dashed) represents (13).

Figure 7

At leading order in $q$, interactions between replicas arise from a particular real-space “Feynman diagram” in which the world line's particle types have the combinatorial structure shown. (This continuum figure is only a diagram: In our calculation, the “Feynman diagram” is a particular local configurations of paths on the lattice.)

Figure 8

Contraction of the permutation states by counting the number of cycles (loops).

Figure 9

Graphical representation of the unitary averages in Eq. (18). Each blue square with four legs is the two site local unitary gate and each red square is its complex conjugate. The ellipsis represents a total of $N=nk$ copies of each. On the right, the two top legs are ${|\tau \rangle }_i{|\tau \rangle }_{i+1}$ and the two bottom legs are ${}_i\langle \sigma |{}_{i+1}\langle \sigma |$. We associate spins $\sigma$ and $\tau$ with the vertices.

Figure 10

Left: $\overline{tr\left({\rho }_A^n\right)}$ represented as a lattice magnet. The upper boundary is contracted with the boundary state $\langle \tilde{\tau }|=\langle {\tau }_{n,k}|$ for region $A$ and $\langle \mathbb{I}|$ for region $B$, the bottom boundary is identical to the top for the operator entanglement and free for the state entanglement. Each four-leg block is the tensor in Eq. (18). Right: The domain wall representation on the triangular lattice after integrating out the $\tau$ spins in each center of the down-pointing triangle (green).

Figure 11

Definition of domain walls. Left: Domain wall labeling convention. Right: $\sigma$ spins on the triangular lattice and domain walls on the dual lattice. The figure shows the splitting of two commutative elementary domain walls.

Figure 12

Spins on the spatial boundary. Left: The leftmost legs of the unitary gates act on the same boundary site, so they are effectively connected, as shown by the dashed line. The $\tau$ spin on the boundary still connects to 3 $\sigma$ spins, and form a tilted triangle. Right: The boundary triangle on the triangular lattice. The red line is the top link; the blue lines are the bottom left and right links of the down-pointing triangle.

Figure 13

Reduction from triangular lattice to square lattice. Pairs of consecutive steps are combined into a single step on the square lattice. Each blue (red) step on the left corresponds to a blue (red) step on the right. We also refer in the text to “visits” to vertical bonds like the one indicated by the dashed line. According to our definition, the vertical bond indicated here is only visited by the blue walk and not the red (so there is no special hexagon interaction between these two walks).

Figure 14

An example of the half special hexagon interaction at the boundary. (Left) The special hexagon in the bulk gives an interaction of order $\frac{1}{q^4}$, while the half hexagon at the bottom boundary gives an interaction of order $\frac{1}{q^2}$. Hence the boundary interaction will dominate the early time fluctuation. (Right) A domain wall configuration of the half special hexagon. Since there are only two extra legs, it is of order $\frac{1}{q^2}$.

Figure 15

Fluctuations of $S_2$ for different $q$ and number of layers of network. Depending on the position of the entanglement cut, the entanglement increases either in odd or even time steps. For each $t$, we have placed the cut so that the final layer of unitaries can create entanglement. The lines are the analytic lowest order result $\frac{1}{q}\sqrt{\frac{2}{4^t}\left(\begin{array}{c}2(t-1)\\ (t-1)\end{array}\right)}$.

Figure 16

Bipartite operator Rényi entanglement entropy for two-site gate. This is the simplest one-layer random tensor network. The domain wall diagrams correspond exactly to those in Eq. (64). We verified the strength $\frac{1}{2q^4}$ in Eq. (64) as well as the pairwise nature of the interaction.

Figure 17

Bipartite operator Rényi entanglement entropy of four sites. The tensor network has three layers: The first and last layers have one gate and middle layer has two gates. The interaction still fits $\frac{k(k-1)}{2}\frac{5}{9q^4}$, which is predicted by only considering the special hexagon interaction.

Figure 18

Domain wall paths for $t\gg L/2v_2$. Left: entanglement of half of a chain with two boundaries. There are two possible paths at the leading order in $q$. They exit the system through a tilted triangle on the boundary. Right: entanglement of half of a chain with periodic BCs. The two entanglement cuts give two domain walls, ${\tau }_{n,1}$ and its inverse. They meet at a down-pointing triangle either in region $A$ or in $B$ again giving two possibilities.

Figure 19

Possible bubble diagrams. Each line represents an elementary domain wall. Configurations (a), (b), (c), and (d) have 0 weight: (a) and (b) vanish because the tip of the hexagon is $J(\mathbb{I},\mathbb{I};\left(12\right))=0$; (c) and (d) vanish because on the top $K_{\wedge }=0$. Configuration (e) is an order $\frac{1}{q^6}$ correction. Configuration (f) can be an order $\frac{1}{q^6}$ correction if the dashed loop is an elementary domain wall. It is an order $\frac{1}{q^4}$ correction if it is a special hexagon as in Eq. (63).