Tunable Quantum Chaos in the Sachdev-Ye-Kitaev Model Coupled to a Thermal Bath

TL;DR

The paper addresses how quantum chaos in SYK models can be tuned by coupling to a thermal bath. It develops exactly solvable (0+1)-d and (1+1)-d SYK constructions with a small system interfacing a much larger bath, analyzed via Schwinger-Dyson equations, ladder kernels for four-point functions, and a Schwarzian-like effective action. The key findings are that the Lyapunov exponent can be continuously reduced from $2\pi/\\beta$ to zero by increasing bath coupling, and that the butterfly velocity in a SYK chain exhibits a crossover from $v_B\propto\sqrt{T}$ at high temperature to $v_B\propto T$ at low temperature, with spatial variations when the bath is local. These results provide canonical, solvable models for studying thermalization and chaos in chaotic quantum systems and offer pathways to explore possible holographic interpretations.

Abstract

The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with $N$ Majorana fermion modes coupled to another SYK model with $N^2$ Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from $2π/β$ ($β=1/(k_BT)$, $T$ is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a $\sqrt{T}$-dependence at relatively high temperature to a linear $T$-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.

Figures (9)

• Figure 1: (a). Left: The Green's function $G_{\chi}(\tau)$ of SYK$_{\chi}$; Right: The Green's function $G_{\psi}(\tau)$ of SYK$_{\psi}$. (b). Order of $N$ for self energy diagram of $\chi$: Left: $\sim\frac{J^2}{N^3}\times N^3 \sim \mathcal{O}(N^{0})$; Right: $\sim \frac{u^2}{N^{5}}\times N\times (N^2)^2 \sim \mathcal{O}(N^{0})$. The dashed line means the two vertices should have same coupling. (c). Order of $N$ for self energy diagram of $\psi$: Left: $\sim\frac{J^2}{N^6}\times (N^2)^3 \sim \mathcal{O}(N^{0})$; Right: $\sim \frac{u^2}{N^5}\times N^2\times N^2 \sim \mathcal{O}(\frac{1}{N})$.
• Figure 2: (a). Three contributions of $K_{R,\chi\chi}(t_1 ... t_4)$. The first two diagrams are of order $\frac{1}{N}$ while the third one is of order $\frac{1}{N^2}$. (b). Diagrammatic representation of the self consistency equation Eq. (\ref{['eq10']}).
• Figure 3: The dependence of Lyapunov exponent of $F_{\chi\chi}$ on $u^2/J^2$.
• Figure 4: (a). Diagrammatic representation of the self consistency equation (\ref{['eq18']}). (b). At order $\mathcal{O}(\frac{1}{N^2})$, the OTOC $F_{\chi\chi}$ contains $F_{\psi\psi}$ as an inner propagator.
• Figure 5: A pictorial representation of the model discussed in sec. \ref{['chain1']}: An SYK chain with $N$ fermions each site in the bath of an $N^2$ fermions SYK model.