Velocity-dependent Lyapunov exponents in many-body quantum, semi-classical, and classical chaos

Abstract

The exponential growth or decay with time of the out-of-time-order commutator (OTOC) is one widely used diagnostic of many-body chaos in spatially-extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, $λ({\bf v})$, which is the growth or decay rate along "rays" at that velocity. We examine the behavior of $λ({\bf v})$ for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by $λ({\bf \hat n}v_B({\bf \hat n}))=0$, with a generally direction-dependent "butterfly speed" $v_B({\bf \hat n})$. In spatially local systems, $λ(v)$ is negative outside the light cone where it takes the form $λ(v) \sim -(v-v_B)^α$ near $v_b$, with the exponent $α$ taking on various values over the range of systems we examine. The regime inside the light cone with positive Lyapunov exponents may only exist for classical, semi-classical or large-$N$ systems, but not for "fully quantum" chaotic systems with strong short-range interactions and local Hilbert space dimensions of order one.

Figures (2)

• Figure 1: Schematic illustration of velocity-dependent Lyapunov exponents $\lambda(v)$ along rays $|x|=vt$ in a 1D system. The color scale encodes $\lambda(v)$ which is positive inside the light cone for $v < v_B$ in classical and semiclassical/large-$N$ quantum systems, but is ill-defined inside the cone in "fully" quantum thermalizing spin systems that do not see an extended period of exponential growth for $v < v_B$. All local systems have negative $\lambda(v) <0$ for $v > v_B$ following Lieb-Robinson, and $\lambda(v)$ smoothly interpolates from positive to negative in semiclassical/large $N$ systems, passing through zero at $v_B$. The OTOC $C(x_0,t)$ for a fixed position $x_0$ and increasing time (green arrow) cuts through rays with different $\lambda(v)$'s and thus will not show a simple-exponential-in-time growth unless $\lambda(v)$ scales linearly with $v$\ref{['eq:tdep']}.
• Figure 2: Schematic illustration of $\lambda(v)$ for the different models considered here. (a) In classical and semiclassical weak-scattering systems, $\lambda(v)$ smoothly interpolates from positive to negative, passing through zero at $v_B$. (b) In large $N$ holographic models and chains of coupled SYK dots, the OTOC has a simple exponential form $C \sim \epsilon e^{2\pi T(1-v/v_B)t}$ at low $T$ and leading order in $1/N$. For cases (a) and (b), $\lambda(v)$ scales linearly with $v$ near $v_B$: $\lambda(v) \sim (v-v_B)$, corresponding to $\alpha =1$ and a simple exponential in time growth for $C(x_0,t)$ near $v_B$. (c) $\lambda(v)$ in fully quantum thermalizing systems with $d<4$ and in integrable models. In these cases, a negative $\lambda(v)$ for $v>v_B$ smoothly approaches zero as $v\rightarrow v_B+$ with exponent $\alpha >1$ due to broadening of the operator front. As a result, these models do not show a simple exponential in time growth in $C(|{\bf x_0}|,t)$ outside the front. (d) In higher dimensions $(d>4)$, the KPZ model capturing the dynamics of operator spreading in chaotic quantum systems has a flat phase which may allow for $\alpha = 1$ and a simple exponential growth in $C(|{\bf x_0}|,t)$ outside the front. Cases (c) and (d) do not show an extended period of exponential growth inside the light cone due to the absence of additional small parameters $\epsilon$, and hence these do not have a well-defined $\lambda(v)$ for $v < v_B$.