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arXiv:2208.01649 · Physical review. B./Physical review. B

Hydrodynamic theory of scrambling in chaotic long-range interacting systems

Tianci Zhou, Andrew Y. Guo, Shenglong Xu, Xiao Chen, Brian G. Swingle

Published Jan 3, 2023• 10 citations
arXivHTMLPDF

Abstract

The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a mean-field theory of out-of-time-ordered commutators in locally interacting quantum chaotic systems at high energy density. In systems with power-law interactions, the corresponding fractional-derivative FKPP equation provides an analogous mean-field theory. However, the fractional FKPP description is potentially subject to strong quantum fluctuation effects, so it is not clear a priori if it provides a suitable effective description for generic chaotic systems with power-law interactions. Here we study this problem using a model of coupled quantum dots with interactions decaying as $1/{r}^{\ensuremath{\alpha}}$, where each dot hosts $N$ degrees of freedom. The large-$N$ limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by $1/N$ corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for $N=1$ and generic finite $N$. In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.

DOI
10.1103/physrevb.107.014201

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Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.
Abstract Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit calculations in specific models. In this paper we propose a unified description of the exponential growth and ballistic butterfly spreading of OTOCs across different systems using a newly formulated “quantum hydrodynamics,” which is valid at finite ℏ and to all orders in derivatives. The scrambling of a generic few-body operator in a chaotic system is described as building up a “hydrodynamic cloud,” and the exponential growth of the cloud arises from a shift symmetry of the hydrodynamic action. The shift symmetry also shields correlation functions of the energy density and flux, and time ordered correlation functions of generic operators from exponential growth, while leads to chaotic behavior in OTOCs. The theory also predicts an interesting phenomenon of the skipping of a pole at special values of complex frequency and momentum in two-point functions of energy density and flux. This pole-skipping phenomenon may be considered as a “smoking gun” for the hydrodynamic origin of the chaotic mode. We also discuss the possibility that such a hydrodynamic description could be a hallmark of maximally chaotic systems.
Operator spreading provides a new characterization of quantum chaos beyond the semi-classical limit. There are two complementary views of how the characteristic size of an operator, also known as the butterfly light cone, grows under chaotic quantum time evolution: A discrete stochastic population dynamics or a stochastic reaction-diffusion equation in the continuum. When the interaction decays as a power function of distance, the discrete population dynamics model features superlinear butterfly light cones with stretched exponential or power-law scaling. Its continuum counterpart, a noisy long-range Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation, remains less understood. We use a mathematical duality to demonstrate their equivalence through an intermediate model, which replaces the hard local population limit by an equilibrium population. Through an algorithm with no finite size effect, we demonstrate numerically remarkable agreements in their light cone scalings.
We use an out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one-dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: The number of qubits NNN on each site (the on-site Hilbert space dimension) and the power law exponent $\ensuremath{\alpha}$. Three light cone structures of OTOC appear at N=1N=1N=1: (1) logarithmic when $0.5<\ensuremath{\alpha}\ensuremath{\lesssim}0.8$, (2) sublinear power law when $0.8\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\lesssim}1.5$, and (3) linear when $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$. The OTOC scales as $exp(\ensuremath{\lambda}t)/{x}^{2\ensuremath{\alpha}}$ and ${t}^{2\ensuremath{\alpha}/\ensuremath{\zeta}}/{x}^{2\ensuremath{\alpha}}$, respectively, beyond the light cones in the first two cases. When $\ensuremath{\alpha}\ensuremath{\ge}2$, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large NNN limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$. This implies the locality is never fully recovered for finite $\ensuremath{\alpha}$. Our result provides a unified physical picture for the chaos dynamics in a long-range power law interaction system.
Operator scrambling is commonly diagnosed by the growth of out-of-time-ordered correlators (OTOCs), yet a general symmetry principle underlying their effective dynamics has remained elusive. For Brownian or short-time-correlated large-NNN Majorana systems, we develop a symmetry-based effective field theory for operator scrambling, organized by a strong-to-weak U(1) symmetry breaking in operator space. The key observation is that, in the noninteracting fermion limit, the four-fold Keldysh contour representation of an OTOC admits an emergent strong U(1) symmetry in a doubled Hilbert-space description, even when the original system has no ordinary conserved quantity. The associated slow mode is the phase of the strong-charge creation operator, whose conjugate density is identified with the local operator size. Generic interactions explicitly break the strong symmetry and generate a mass term at lowest order for the would-be Goldstone mode, thereby converting diffusive operator spreading into chaotic growth. We further show that higher-order symmetry breaking terms are tightly constrained by an emergent duality that combines time reversal with contour permutation. This duality fixes the effective action up to quadratic order in the response field, relates the multiplicative noise strength directly to the Lyapunov exponent, and makes the positivity of the Lyapunov exponent a consequence of convergence of the real-time path integral. The resulting OTOC dynamics is governed by a noisy FKPP equation, which captures within a unified framework the early-time exponential growth, ballistic propagation, nonlinear saturation, and stochastic front broadening of operator scrambling. We verify this construction in a Brownian SYK chain, where a direct saddle-point expansion reproduces the symmetry-based effective action. Our results reveal a symmetry origin of operator-size hydrodynamics and scrambling.
Hydrodynamics as an effective field theory (EFT) has been immensely successful for predicting the behaviour of time-ordered correlation functions in the long-time, large-distance regime. In this paper, we analyse the predictions of hydrodynamics for scrambling as probed by out-of-time order correlators (OTOCs). To do so, we consider an all-derivative order generalisation of the Schwinger-Keldysh EFT of hydrodynamics extended to a four-fold time contour required to capture OTOCs. A crucial ingredient of the construction is a local SL(2,R) symmetry, which has previously been suggested as a mechanism for scrambling. We also derive this exact EFT from the microscopic formulation of the maximally chaotic low-temperature limit of the SYK lattice, thereby determining all transport coefficients for this model. As a novel application, we compute the OTOC of energy currents, find that its exponential growth saturates the chaos bound, and match it to expectations based on a shockwave (or scramblon) effective description. Our work shows that the universality of hydrodynamics degrees of freedom extends beyond traditional time-ordered correlators to more complex observables such as OTOCs.