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arXiv:2505.06353

Operator Spreading, Duality, and the Noisy Long-Range FKPP Equation

Tianci Zhou, Brunet, Éric, Xiaoguang Qi

Published May 9, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2505.06353

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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.
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A bstract The Ryu-Takayanagi (RT) formula and its interpretation in terms of quantum error correction (QEC) implies an emergent locality for the spread of quantum information in holographic CFTs, where information injected at a point in the boundary theory spreads within a sharp light-cone corresponding to the butterfly velocity. This emergent locality is a necessary condition for the existence of a geometric bulk dual with an RT-like formula for entanglement entropy. In this paper, we use tools from QEC to study the spread of quantum information and the emergence of a sharp light-cone in an analytically tractable model of chaotic dynamics, namely a one-dimensional Brownian SYK chain. We start with an infinite temperature state in this model and inject a qudit at time t = 0 at some point p on the chain. We then explicitly calculate the amount of information of the qudit contained in an interval of length 2 ℓ (centered around p ) at some later time t = T . We find that at strong coupling, this quantity shows a sharp transition as a function of ℓ from near zero to near maximal correlation. The transition occurs at ℓ ~ v B T , with v B being the butterfly velocity. Underlying the emergence of this sharp light-cone is a non-linear generalization of the diffusion equation called the FKPP equation, which admits sharp domain wall solutions at late times and strong coupling. These domain wall solutions can be understood on physical grounds from properties of operator growth in chaotic systems.
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.
We argue that chaotic power-law interacting systems have emergent limits on information propagation, analogous to relativistic light cones, which depend on the spatial dimension d and the exponent α governing the decay of interactions. Using the dephasing nature of quantum chaos, we map the problem to a stochastic model with a known phase diagram. A linear light cone results for α≥d+1/2. We also provide a Lévy flight (long-range random walk) interpretation of the results and show consistent numerical data for 1D long-range spin models with 200 sites.
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.