Anatomy of information scrambling and decoherence in the integrable Sachdev-Ye-Kitaev model

TL;DR

Problem: determine the full time dependence of the OTOC $C(t)$ in an integrable $q=2$ SYK model at finite temperature with Markovian dissipation. Method: analytically compute $C(t,\beta,\mu)$ for finite $N$ by diagonalizing $H=i\sum_k \lambda_k\widetilde{\chi}_{2k-1}\widetilde{\chi}_{2k}$, averaging over orthogonal rotations, and exploiting the factorization $F(t,\mu)=e^{-2\mu t}F(t,0)$; include leading $1/N$ and subleading $1/N^2$ corrections via the spectral form factor $K_c(t)$. Key findings: without dissipation, $C(t)$ shows initial polynomial growth, a power-law approach to saturation with oscillations, a linear-in-time decrease for $t\gtrsim \sqrt{N}$ driven by $K_c(t)$, and saturation near $t\sim 2N$; with dissipation, $C(t)$ decays exponentially as $e^{-2\mu t}$, ultimately dominating; numerics (ED and fermionic quantum trajectories) confirm the analytics. Significance: clarifies scrambling mechanisms in open, integrable many-body systems, demonstrating how spectral discreteness and the spectral form factor govern OTOC dynamics and providing benchmarks for future studies of more general Liouvillian dynamics.

Abstract

The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, $N$ Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For $t \lesssim \sqrt{N}$, after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At $t \sim \sqrt{N}$, the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading $1/N$ effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at $t \sim 2N$, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at $t \lesssim \sqrt{N}$ indicate lack of thermalization -- a desired feature for a better performance of quantum information devices.

Figures (3)

• Figure 1: Time evolution of the quantum uncertainty $C(t)$ with $t$ in units of $J = 1$ for $\mu=\beta=0$. Top: The blue ($N = 20$), red ($N = 200$), and green ($N = 2000$) curves correspond to $C(t)$ given in Eq. (\ref{['eq:ctsum']}) and (\ref{['eq:gtsum1']}) and $C_N = (N+1)/[(N-1)(N+2)]$. They converge to the $1/N$ prediction Eq. (\ref{['eq:ctsemibeta0']}), with $C_N = 1/N$, only for sufficiently large $N$. In both cases, $C_N$ is given by the time independent part of $C(t)$. Bottom: Comparison between the analytical exact $C(t)$, Eqs. (\ref{['eq:ctfull']}), (\ref{['eq:ctsum']}) and (\ref{['eq:gtsum1']}), in terms of the eigenvalues of $J_{ij}$ (purple circles) with the simple analytic $C(t)$, Eq. (\ref{['eq:ctn2']}), including up to $1/N^2$ corrections for $N=200$ on a linear scale. The black curve stands for the $1/N$ result, Eq. (\ref{['eq:ctsemibeta0']}), while the green curve stands for the $1/N+1/N^2$ result, Eqs. (\ref{['eq:ctn2']}) and (\ref{['eq:kct']}). The agreement between of analytical exact and the $1/N+1/N^2$ results is excellent for all timescales. Deviations from the $1/N$ expression occur at $t\sim \sqrt{N}$. The Heisenberg time, signaling saturation, is at $t \sim 2 N$.
• Figure 2: Top: $C(t)$ with $t$ in units of $J = 1$, for $\mu = \beta = 0$. $C(t)$ Eq. (\ref{['eq:ctfull']}) is computed using exact diagonalization for $N = 24$ (red) with $\sim 10^6$ disorder realizations. The exact diagonalization result shows almost no difference with respect to the exact analytic result (black), Eqs. (\ref{['eq:ctsum']}) and (\ref{['eq:gtsum1']}), and and also an excellent agreement with the $1/N+1/N^2$simple expression (blue) given in Eq. (\ref{['eq:ctn2']}). Substantial deviations are observed if only the leading $1/N$ correction (green) Eq. (\ref{['eq:ctsemibeta0']}) is kept. We are neglecting terms $1/N^2$ in $NC(t)$. Since $N = 24$ and, at least to lower order, prefactors are of the order one, differences between ED and the analytic $1/N+1/N^2$ expression (blue curve) should be $\sim 1/24^2 \approx 0.0017$, is fully consistent with the observed deviation. Bottom: $C(t)$ for $\mu = 0.15$, $N = 18$ and $\beta = 0$. We combine exact diagonalization with the quantum trajectory method molmer93dum1992. For each point in time, we use $200$ quantum trajectories, a time step of $dt = 0.01$, and at least $5\times10^5$ disorder realizations. The exponential approach to the steady state for $\mu\ne 0$ is correctly reproduced by the quantum trajectory method and we find good agreement with the $1/N$ semiclassical result.
• Figure 3: $C(t)$ with $t$ in units of $J = 1$ for $\beta=5$. Top: $\mu = 0$. The black curve stands for the semiclassical expression Eq. (\ref{['eq:ctsemifinitebeta']}) with $C_N$ given by the time independent part of the coefficient of $\exp[-\mu t]$ in Eq. (\ref{['eq:ctsemifinitebeta']}). The blue ($N = 20$), red ($N = 200$), and green ($N = 2000$) curves stand for the exact analytic $C(t)$, Eqs. (\ref{['eq:ctsum']}) and (\ref{['eq:gtsum1']}) with $C_N = 0.06, 0.044, 0.043$ for $N = 20, 200, 2000$, respectively. The initial approach to the steady state is still power-law, so thermal effects do not lead to qualitative changes. Bottom: $\mu = 0.15, \beta = 5$. For $t \gtrsim 1/2\mu$ the decay is exponential and fully controlled by the Markovian bath.