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Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws

Curt von Keyserlingk, Tibor Rakovszky, Frank Pollmann, Shivaji Sondhi

TL;DR

The paper addresses scrambling and thermalization in 1D quantum systems lacking local conservation laws by analyzing random local unitary circuits, a Floquet kicked Ising model, and Clifford circuits. It develops a hydrodynamic description of operator spreading, where a diffusing front propagates with a butterfly velocity $v_{ ext{B}}$ below the light-cone speed, and shows that entanglement growth is governed by a separate velocity $v_{ ext{E}}$ with $v_{ ext{E}}<v_{ ext{B}}$. It provides exact results for operator spreading in random circuits, including OTOC behavior across four regimes, and demonstrates that diffusively broadened fronts lead to reduced entanglement velocity and nontrivial scrambling, while Clifford circuits reveal that linear entanglement growth does not guarantee chaotic OTOC dynamics. The work argues for a universal hydrodynamic picture of operator spreading in generic 1D ergodic systems and suggests directions to extend these ideas beyond random circuits.

Abstract

Thermalization and scrambling are the subject of much recent study from the perspective of many-body quantum systems with locally bounded Hilbert spaces (`spin chains'), quantum field theory and holography. We tackle this problem in 1D spin-chains evolving under random local unitary circuits and prove a number of exact results on the behavior of out-of-time-ordered commutators (OTOCs), and entanglement growth in this setting. These results follow from the observation that the spreading of operators in random circuits is described by a `hydrodynamical' equation of motion, despite the fact that random unitary circuits do not have locally conserved quantities (e.g., no conserved energy). In this hydrodynamic picture quantum information travels in a front with a `butterfly velocity' $v_{\text{B}}$ that is smaller than the light cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do \emph{not} observe a prolonged exponential regime of the form $\sim e^{λ_\text{L}(t-x/v)}$ for a fixed Lyapunov exponent $λ_\text{L}$. We find that the diffusive broadening of the front has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description applies to more generic ergodic systems and support this by verifying numerically that the diffusive broadening of the operator wavefront also holds in a more traditional non-random Floquet spin-chain. We also compare our results to Clifford circuits, which have less rich hydrodynamics and consequently trivial OTOC behavior, but which can nevertheless exhibit linear entanglement growth and thermalization.

Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws

TL;DR

The paper addresses scrambling and thermalization in 1D quantum systems lacking local conservation laws by analyzing random local unitary circuits, a Floquet kicked Ising model, and Clifford circuits. It develops a hydrodynamic description of operator spreading, where a diffusing front propagates with a butterfly velocity below the light-cone speed, and shows that entanglement growth is governed by a separate velocity with . It provides exact results for operator spreading in random circuits, including OTOC behavior across four regimes, and demonstrates that diffusively broadened fronts lead to reduced entanglement velocity and nontrivial scrambling, while Clifford circuits reveal that linear entanglement growth does not guarantee chaotic OTOC dynamics. The work argues for a universal hydrodynamic picture of operator spreading in generic 1D ergodic systems and suggests directions to extend these ideas beyond random circuits.

Abstract

Thermalization and scrambling are the subject of much recent study from the perspective of many-body quantum systems with locally bounded Hilbert spaces (`spin chains'), quantum field theory and holography. We tackle this problem in 1D spin-chains evolving under random local unitary circuits and prove a number of exact results on the behavior of out-of-time-ordered commutators (OTOCs), and entanglement growth in this setting. These results follow from the observation that the spreading of operators in random circuits is described by a `hydrodynamical' equation of motion, despite the fact that random unitary circuits do not have locally conserved quantities (e.g., no conserved energy). In this hydrodynamic picture quantum information travels in a front with a `butterfly velocity' that is smaller than the light cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do \emph{not} observe a prolonged exponential regime of the form for a fixed Lyapunov exponent . We find that the diffusive broadening of the front has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description applies to more generic ergodic systems and support this by verifying numerically that the diffusive broadening of the operator wavefront also holds in a more traditional non-random Floquet spin-chain. We also compare our results to Clifford circuits, which have less rich hydrodynamics and consequently trivial OTOC behavior, but which can nevertheless exhibit linear entanglement growth and thermalization.

Paper Structure

This paper contains 18 sections, 2 theorems, 71 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Quantifying operator spreading
  3. Random circuit model
  4. Random walk dynamics of operator density
  5. Behavior of out-of-time-order commutators
  6. Fluctuations from circuit to circuit
  7. Relationship to entanglement spreading
  8. Comparison with the kicked Ising model
  9. Fractal Clifford circuits
  10. Conclusions
  11. Validity of Eq. \ref{['eq:Rapprox']}
  12. Application to OTOC
  13. Exact operator spread coefficients
  14. Useful limits
  15. Long time correlations
  16. ...and 3 more sections

Key Result

Theorem 1

Translation invariant centered CQCAs $U$ have linear in system size recurrence times (at most $t_{\text{rec}}=12L$), at least for system sizes $L=2^{n}$.

Figures (9)

  • Figure 1: Structure of the local unitary circuits studied in this paper. The on-site Hilbert space dimension is $q$. Each two-site gate is a $q^2 \times q^2$ unitary matrix. For the random circuit model of Eq. \ref{['eq:circuit_def']} each gate is randomly chosen from the Haar distribution. For the Floquet models considered in Sec. \ref{['s:kickedIsing']} and \ref{['s:Clifford']} the 2-site gates are defined by the Floquet unitaries in Eqs. \ref{['eq:Ising_def']} and \ref{['eq:fractalClifford']}, respectively.
  • Figure 2: Spreading of a one-site operator averaged over random unitary circuits. $\overline{\rho_R}(s,\tau)$ ($\overline{\rho_L}(s,\tau)$) is the total weight carried by Pauli strings with right (left) endpoint at site $s$ at time $\tau$. Figure (c) shows the sum of these two functions (multiplied by $\sqrt{\tau}$ to show the position of the front more clearly). Almost all the weight is carried by operators with endpoints at the two fronts propagating out from the initial site with speed $v_{\text{B}} = \frac{q^2-1}{q^2+1}$. These fronts in turn broaden diffusively in time as $~\sqrt{\tau}$. The two other velocity scales, the light cone velocity $v_\text{LC}$ and the entanglement velocity $v_\text{E}$ (see Eq. \ref{['eq:entanglement_velocity']}) are also indicated, satisfying $v_\text{E} < v_{\text{B}} < v_\text{LC}$. The values of $\overline{\rho_R}$ and $\overline{\rho_L}$ after 100 layers of the circuit are shown in Fig. (b). Fig. (a) shows the integrated operator weights $\overline{R}(s)$ ($\overline{L}(s)$), denoting the total weight left (right) of site $s$, along with the OTO commutator $\mathcal{C}(s,\tau)$. The OTOC saturates to 1 inside the front and has the value $1/2$ exactly at $\tau = s/v_{\text{B}}$
  • Figure 3: Time dependence of the average OTOC in the random circuit model. (a) Different time regimes for fixed separation $s=100$. The exact result for the OTOC follows Eq. \ref{['eq:lyaponov']} after the light cone hits site $s$. The behaviour than goes over to regime described by Eq \ref{['eq:otoc_intermediate']} after the front with speed $v_{\text{B}}$ arrives. The inset shows the exponential decay of the OTOC to its final value $1$, as described by Eq. \ref{['eq:otoc_late']}, for different separations. (b): scaling collapse of the OTOC at the front.
  • Figure 4: Average values and fluctuations of the (a) OTOC and (b) the total weight left of site $s$ for the time evolved operator $Z_0(\tau)$ after $\tau=12$ layers of the random circuit. Blue dots correspond to average values of 100 different random circuits while the error bars signify one standard deviation. Figure (c) shows the standard deviations of $R(s)$ for different times. The largest fluctuations decrease in time approximately as $\propto \tau^{-1/2}$, as shown by the inset.
  • Figure 5: Entanglement growth in the random circuit model. Left: comparing the exact formula, Eq. \ref{['eq:S_Renyi_exact']}, to matrix product state numerics shows that it captures both the typical value $S^{(2)}_\text{typ} = -\log{\overline{e^{-S^{(2)}}}}$, and the average $S^{(2)}_\text{avg} = \overline{S^{(2)}}$ of the second Rényi entropy. The main figure shows the time dependence for $L_\text{A} = 50$ sites, while the inset is for $L_\text{A} = 2$. Right: The entanglement velocity increases with $q$ according to Eq. \ref{['eq:entanglement_velocity']} while the saturation regime becomes smaller.
  • ...and 4 more figures

Theorems & Definitions (5)

  • proof
  • Theorem 1
  • proof
  • Corollary 1.1
  • proof