Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos

The spreading of entanglement in out-of-equilibrium quantum systems is currently at the center of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of “maximally chaotic,” periodically driven, quantum spin chains. Specifically, we consider the so-called “self-dual” kicked Ising chains initialized in a class of separable states and devise a method to compute exactly the time evolution of the entanglement entropies of finite blocks of spins in the thermodynamic limit. Remarkably, these exact results are obtained despite the maximally chaotic models considered: Their spectral correlations are described by the circular orthogonal ensemble of random matrices on all scales. Our results saturate the so-called “minimal cut” bound and are in agreement with those found in the contexts of random unitary circuits with infinite-dimensional local Hilbert space and conformal field theory. In particular, they agree with the expectations from both the quasiparticle picture, which accounts for the entanglement spreading in integrable models, and the minimal membrane picture, recently proposed to describe the entanglement growth in generic systems. Based on a novel “duality-based” numerical method, we argue that our results describe the entanglement spreading from any product state at the leading order in time when the model is nonintegrable.

Popular Summary

Entanglement is one of the most distinctive features of quantum mechanics—particles acquire correlations among themselves that have no analog in our everyday experience. Despite its elusive nature, physicists suspect that entanglement may provide a key to not only future quantum technologies but also our fundamental understanding of the physical world. To that end, researchers would like to better understand how entanglement spreads in an ensemble of many interacting quantum particles. Here, we fill a specific gap in that understanding by calculating how entanglement spreads in a quantum chaotic system that, by definition, cannot be reduced to a quasiparticle description that is customary in low-energy condensed-matter physics.

We provide exact results on the dynamics of the entanglement entropy in a nonintegrable quantum many-body system, which can be interpreted as a maximally chaotic system with local interactions. Our results agree with previous general statements on the linear growth of the entanglement entropy and its saturation to the thermodynamic entropy, but they apply exactly for any fixed time and any size of a block of contiguous spins. We develop a new method that can be generally applied to determine the dynamics of correlation measures in self-dual models, in which certain aspects look the same if space and time are exchanged—a key property that allows for exact computation.

Our results provide the first exact computation of dynamics of entanglement for individual quantum chaotic models. In the future, we would like to identify more general classes of models for which similar results could be achieved.

Figure 1

The partition of the spin chain considered in the calculation of the entanglement.

Figure 2

Plot of the exact result of all Rényi entropies given by Eq. (23) for the $\theta =\pi /2$ case. The expressions for $\theta =0$ are delayed by one period.

Figure 3

Pictorial representation of the duality relation (27) fulfilled by the Floquet operator (8). Traces of powers of the Floquet operator correspond to the partition function of a classical Ising model on a $t\times L$ lattice. The column-to-column transfer matrix is given by $U_{KI}[\mathit{h}]$, while the row-to-row transfer matrix between the ($j-1$)th, and the $j$th row is given by ${\tilde{U}}_{KI}[h_j\tilde{\mathbf{1}}]$. Note that self-duality condition implies that both transfer matrices are unitary.

Figure 4

Schematic representation of $\mathrm{tr}[({\rho }_A(t){)}^3]$ according to the expression (31). The six different cylinders corresponding to the partition functions (38) are schematically represented as rectangles. The spin subchains $A$ and $A^c$ connected with the colored belts share identical spin configurations.

Figure 5

Schematic depiction of $\mathrm{tr}[({\rho }_A(t){)}^3]$ written according to Eq. (44). Positive and negative time sheets are, respectively, on the left and on the right. Vertices connected by the colored lines are coupled by the transfer matrices, in analogy with Fig. 4. Blue-shaded horizontal planes denote the spatial transfer matrices, specifically, the operator ${\mathbb{T}}_{\theta ,\varphi }[h]$ for the physical sites corresponding to the block $A$ of $N$ spins and the operator ${\mathbb{R}}_{\theta ,\varphi }[h]=\mathbb{P}{\mathbb{T}}_{\theta ,\varphi }[h]{\mathbb{P}}^{†}$ for the sites corresponding to $L-N$ spins in $A^c$.

Figure 6

Pictorial representation of the arrangement of the dual quantum spin degrees of freedom adopted in the tensor product space ${\mathcal{H}}_t^{\otimes 2n}$.

Figure 7

The second Rényi entropy for a kicked Ising system of $L=30$ spins evolving from “tilted” initial states (12). Top and bottom two panels have, respectively, $N=9$ and $N=13$. The two panels on the left report results for translational-invariant initial states. The blue and green curves correspond, respectively, to transverse and longitudinal separating states [cf. Eqs. (17) and (18)]. Other curves correspond to the initial state ${\theta }_j={\varphi }_j=1$ and different magnetic fields as indicated in the legend. The two panels on the right correspond to the maximally disordered cases, where the spins at each site point in a random direction, and the magnetic fields $h_j$ are either random (purple) or zero (yellow). In the cases with random parameters, we show the average values for a sample of eight realizations using a continuous line and indicate a standard deviation of one realization by a shaded area.

Figure 8

Time evolution of the second Rényi entropy for a subsystem of $N=11$ spins in a kicked Ising system of $L=30$ at the integrable point $\mathit{h}=\mathbf{0}$ for different translationally invariant initial states. The main panel shows the rescaled curves, which are close to Eq. (94) (black dashed line) given by the quasiparticle picture. In the inset, we show the nonrescaled version, where it is apparent that the saturation value depends on the initial state. Note a recurrence after the time $t=10$, consistent with the quasiparticle picture.

Figure 9

The von Neumann entropy [cf. Eq. (16)] for a subsystem of $N=9$ spins in a kicked Ising system of $L=30$ evolving from the separating state with ${\theta }_i=\pi /2$ and ${\varphi }_i=0$ for different values of the longitudinal magnetic field.

Figure 10

The second Rényi entropy for a kicked Ising system evolving from “tilted” initial states (12) in the thermodynamic limit. The colored lines correspond to nonseparating initial states (we take $\varphi =\theta$, with $\theta$ and $h$ specified in the legend) and are determined numerically evaluating Eq. (98), while the gray dashed lines report, for comparison, the result from separating states. The inset shows the instantaneous slope $\mathrm{\Delta }{S}_A^{(2)}(t)$ [cf. Eq. (100)] as a function of $1/t$. The points are computed by evaluating Eq. (98), while the lines are a linear fit of the last two points.

Figure 11

Time evolution of different Rényi entropies in a kicked Ising system with longitudinal magnetic field $h=-0.6$ in the thermodynamic limit. The initial state is of the form (12) with $\theta =\varphi =1$. The colored lines correspond to Renyi indices $n=1$, 2, 3, 4 and are determined numerically evaluating Eq. (98), while the gray dashed line reports, for comparison, the result from separating states. The inset shows the instantaneous slope $\mathrm{\Delta }{S}_A^{(n)}(t)$ [cf. Eq. (100)] as a function of $1/t$. The points are computed by evaluating Eq. (98), while the lines are linear extrapolations from the last two data points.