Solution of a Minimal Model for Many-Body Quantum Chaos

We solve a minimal model for an ergodic phase in a spatially extended quantum many-body system. The model consists of a chain of sites with nearest-neighbor coupling under Floquet time evolution. Quantum states at each site span a $q$-dimensional Hilbert space, and time evolution for a pair of sites is generated by a $q^2\times q^2$ random unitary matrix. The Floquet operator is specified by a quantum circuit of depth two, in which each site is coupled to its neighbor on one side during the first half of the evolution period and to its neighbor on the other side during the second half of the period. We show how dynamical behavior averaged over realizations of the random matrices can be evaluated using diagrammatic techniques and how this approach leads to exact expressions in the large-$q$ limit. We give results for the spectral form factor, relaxation of local observables, bipartite entanglement growth, and operator spreading.

Popular Summary

In classical chaotic systems, a small change in the initial conditions leads to a big difference in the final outcome—a behavior known as the butterfly effect. A fundamental question in statistical mechanics is how the quantum analog of a chaotic system evolves over time. One widely used and elegant description of chaotic quantum systems is based on the statistics of large matrices with random elements. However, this fails to capture the structure of a spatially extended system, so it cannot describe the propagation of quantum information underlying the butterfly effect. We show how to use random matrices to describe chaotic quantum systems that are spatially extended and composed of many particles—the first solvable model of its kind.

Our model describes a lattice of quantum spins or qubits, each coupled to its neighbors and evolving coherently in time. We use two simplifications to make the model tractable: the system is driven periodically and the couplings are random. We develop diagrammatic techniques that allow us to compute the time-dependent many-body quantum state in the limit where the number of degrees of freedom at each site is large. We calculate observables including the growth of nonlocal correlations between parts of the system and the spread of quantum information.

We expect this model will pave the way to a new class of analytically solvable Floquet models for quantum chaotic systems, which will help to elucidate the dynamics of quantum information in generic many-body quantum systems.

Figure 1

Illustration of the model for Floquet time evolution studied in this paper. Space and time are represented by the horizontal and vertical directions. Lattice sites are indicated by filled dots, and the coupling of pairs of sites under time evolution is shown with rectangles.

Figure 2

Illustration of $\mathrm{Tr}[O(t)O(0)]$ for $t=2$. $U$’s and $U^{†}$’s are represented by rectangles, respectively, shaded in light and dark gray. The local operator $O_x$ is represented by a square. The curled lines at the top and bottom edge indicate closed loops which denote the trace operation. (a) Initial expression. (b) After cancellation of pairs of $U$ with $U^{†}$ wherever possible. The boundaries of the regions within which $U$’s and $U^{†}$’s remain form light cones with a velocity $v=2$ set by the construction of the model. (c) Illustration of $\mathrm{Tr}[O(t)O(0)]$ for $t=2$ formed by folding Fig. 2 (right) so that timelines run upward for $W$ and downward for $W^{†}$.

Figure 3

Adaptations to diagrammatic notation made in order to show time evolution of an individual site. The shaded rectangles representing $U_{i,i+1}$ and $U_{i,i+1}^{†}$ are replaced by pairs of double lines, with an arrow directed from the column label to the row label of the matrix. To record the distinction between odd and even $i$, the double lines are blue in the first case and red in the second. This color convention allows us to represent single-site diagrams without ambiguities (see, e.g., Fig. 7).

Figure 4

Alternative diagrammatic representations of $\mathrm{Tr}[W(t=2)]\mathrm{Tr}[W^{†}(t=2)]$, which gives $K(t=2)$ after ensemble averaging using the notation of the left and right sides of Fig. 3.

Figure 5

Single-site diagrams associated with $\mathrm{Tr}[W(t=2)]\mathrm{Tr}[W^{†}(t=2)]$ (left) and its ensemble average $K(t=2)$ (right).

Figure 6

The $T$ and $U$ loops (right) associated with a contracted diagram for $\mathrm{Tr}[W(t=2)]\mathrm{Tr}[W^{†}(t=2)]$ (left in this figure and top middle in Fig. 5). The associated algebraic factors are ${\mathcal{A}}_T(G_i)=q^2$ and ${\mathcal{A}}_U(G_i)=(V_{1,1}{)}^{1/2}$.

Figure 7

(a) The site diagram associated with $\mathrm{Tr}[W(t)]\mathrm{Tr}[W^{†}(t)]$. It is color coded as follows: For an odd site $i$, the blue dots and double lines are contributions from $U_{i,i+1}$ or $U_{i,i+1}^{†}$ and the red ones are from $U_{i-1,i}$ or $U_{i-1,i}^{†}$. For an even site, this coding is reversed. (b)–(d) Three examples from a total of $t$ leading-order diagrams for $K(t)$ at $t>0$.

Figure 8

(a),(b) Site diagrams associated with $[O(x,t=2)O(x){]}_{\mathrm{av}}$ at sites $i\ne x$ and sites $i=x$, respectively. Color coding is as in Fig. 7. (c) The leading contraction of the site diagram for $x\ne i$. (d),(e) Two alternative contractions of the site diagram for $[O(x,t=2)O(x){]}_{\mathrm{av}}$ at site $x$. (d) gives a contribution $\mathcal{A}(G_x)\propto \mathrm{Tr}{O}_x=0$. (e) has $\mathcal{A}(G_x)\ne 0$ but forces similar contractions on adjacent sites via the bond constraint, and these contributions vanish as $q\to \infty$.

Figure 9

Summary of the steps that convert the evaluation of the average purity into a combinatorial statistical mechanics problem. (a) Illustration of $e^{-S_2(t)}$ for $t=1$. Open circles represent a product state in the site basis and other conventions are as in Fig. 2. (b) Top: Folded representation of $\exp [-S_2(t)]$ obtained by folding (a) so that timelines in $W(t)$ run upward and those in $W^{†}(t)$ run downward. The large blue box highlights a stack of four unitary operators (see main text). (b) Bottom left: Site diagrams in regions $A$ and $B$. Note in particular how the timelines of sites in the regions $A$ and $B$ connect differently at the top of these diagram because of the structure of traces in Eq. (9). (b) Bottom right: The two possible local contractions made within a block, which we label $A$ and $B$. Only leading-order diagrams have such contractions. We represent these contractions in two alternative ways: either explicitly for a single site, as on the left side of the equivalences, or schematically for a pair of sites and a stack of unitaries, as on the right side of the equivalences. Note that the top boundary of the diagram can be treated as equivalent to blocks of such contractions. (c) Leading-order diagrams for $⟨e^{-S_2(t)}⟩$ at large $q$ are minimal-length domain-wall (DW) diagrams. Their weight is $q^{-h_{AB}}$ with $h_{AB}$ being the length of horizontal DW between $A$ and $B$ contractions. The labels $A$ and $B$ on blocks in this figure indicate the type of local contraction. Top: A minimal-length DW diagram for $⟨\exp [-S_2(t)]⟩$ at $t=4$ and $L>6$. Bottom: One of the two minimal-length DW diagrams at $t>2$ and $L=6$. These two diagrams have the DW directed solely to the left or right.

Figure 10

Diagrammatic representation of OTOC. Left: Folded diagram for $\mathrm{tr}[O(x,t)O(y)O(x,t)O(y)]$ with $|x-y|=2$ and $t=2$. The blue box highlights four unitary operators, which can be represented in the block representation in a similar way as the ones in Fig. 9. Right: An apparent minimal-length horizontal DW diagram that vanishes due to $\mathrm{tr}O(y)\equiv 0$. The red blocks are unitary pairs that cannot be annihilated. The dark gray blocks represent the boundary conditions. $A^{\prime }$ represents an effective contraction $A$ [Fig. 9] with operators along the loops, and similar for $B^{\prime }$.

Figure 11

Enumeration of all distinct ways in which a contraction may be added to a site diagram. Columns from left to right are as follows. (i) Eight distinct combinations of one or two loops, before addition of a contraction, with associated contributions to $\mathcal{O}(G_i^{(m-1)})$. Three types of loop appear, containing zero, one, or two local operators $O$ indicated using open squares. (ii) The same after adding a contraction shown as a dashed line and redrawn purely as loops, with associated contributions to $\mathcal{O}(G_i^{(m)})$. (iii) Values of ${\mathrm{\Delta }}^{(m)}$ arising from each contraction obtained as the ratio of contributions in columns (ii) and (i).

Figure 12

An example of order determination of a diagram using the method of contraction addition. Arrows giving an orientation to loops are omitted for clarity.

Figure 13

Top: The block representation of a leading DW diagram for $⟨\exp [-S_2(t)]⟩$. The top row (the zeroth row) denotes the effective blocks that represent the structure of traces in subregions $A$ and $B$ [Fig. 9 bottom left]. The dashed line represents a light cone outside of which all unitary pairs can be annihilated as described in Fig. 2. The red line represents costly boundaries (between blocks of different contraction types; see Appendix pp2-s2 and Table 1). Bottom: The correspondence between blocks and the symbols used in the bottom right of Fig. 9.

Figure 14

Contractions in a portion of a site diagram for $⟨\exp [-S_2(t)]⟩$ showing all seven possible block types.

Figure 15

Examples of contractions in a portion of a site diagram for $⟨\exp [-S_2(t)]⟩$. Left sides of (a) and (b): Two neighboring blocks along site $i$ shown in the space-time plane with the time axis vertical. Right sides of (a) and (b): Equivalent site diagrams with the time axis vertical; open-ended dashed lines are contracted nonlocally with an external block. Red lines represent sections of boundary (of unit length in the left-hand pictures). (a) Right: The boundary crosses four times a loop of length at least 2 at the red crosses, so $\omega (a,b)=q^{-1}$. (b) Right: The boundary has three costly crossings, so $\omega (b_1,a_2)=q^{-3/4}$.

Figure 16

A generic configuration of an odd row of boundaries denoted as $(a{d}_1{d}_2{d}_3,\dots ,d_{k}b{)}_{\mathrm{wall}}$. Costly boundaries (DWs) are drawn in red. Costless boundaries are drawn in black.

Figure 17

The six local contractions for a block appearing in $⟨\exp [-S_3(t)]⟩$ with the same notation as the one in Fig. 14.

Figure 18

The recursive rules for enumerating the $p+1$th row of the leading-order diagrams given the $p$th row of the leading-order diagram. These rules are derived from the fact that, in a leading diagram, the difference between the DW positions at even row $p$ and odd row $p+1$ is exactly one lattice spacing.

Figure 19

A leading-order DW diagram in the block representation for $⟨\exp [-2S_3(t)]⟩$ in the large-$q$ limit. The order is given by $q^{-2h_{ab}-h_{a{j}_i}-h_{j_{i}b}}$, where $h_{c_1{c}_2}$ counts the horizontal walls between $c_1$ and $c_2$ contractions in red.

Figure 20

A partially contracted diagram for the OTOC at $t=2$ in the block representation. The choice of an $A$ contraction (or a $B$ contraction) of blocks on the leftmost bond forces blocks in the upper (lower) light cone (dashed lines) to have the same type of contraction. Otherwise, the diagram has an order bounded by $q^{-1}$.