Quantum chaos, scrambling and operator growth in $T\bar{T}$ deformed SYK models

TL;DR

The paper probes how a 1D TTbar deformation alters quantum chaos, scrambling, and operator growth in finite-N SYK-family models. By analyzing spectral form factors, OTOCs, and Krylov complexity, it finds that TTbar acts mainly as a rescaling of the effective coupling J to J_eff for chaotic SYK4/SSYK4, leaving chaos diagnostics essentially unchanged, while TTbar-deformed SYK2 exhibits many-body localization-like behavior. The results articulate a universal structure: deformation preserves chaos signatures in the chaotic models and reshapes dynamics through time rescaling, with K-complexity offering nuanced insights at finite temperature. These findings contribute to understanding the robustness of quantum chaos under integrable deformations and highlight MBL-like phenomena in TTbar-deformed integrable systems.

Abstract

In this work, we investigate the quantum chaos in various $T\bar{T}$-deformed SYK models with finite $N$, including the SYK$_4$, the supersymmetric SYK$_4$, and the SYK$_2$ models. We numerically study the evolution of the spectral form factor (SFF), the out-of-time ordered correlator (OTOC), and the Krylov complexity. We find that the characteristic evolution of the SFF, OTOC and K-complexity of both the SYK$_4$ and SSYK$_4$ models remains unchanged under the deformation, which implies that the properties of quantum chaos is preserved. We also identify a many-body localization behavior in the deformed SYK$_2$ model.

Figures (13)

• Figure 1: Plot of the nearest neighbor energy level spacing distributions of the SYK$_4$, SSYK$_4$ and SYK$_2$ models with $\lambda=0$, $-0.001$, $-0.1$, $-1$, $-10$, $-100$, $-1000$. The curves with different $\lambda$ overlap each other exactly.
• Figure 2: Plot of SFF of the undeformed SYK$_4$ model with various values of $N$.
• Figure 3: Plot of SFF of the (a) deformed SYK$_4$ model and (b) deformed SSYK$_4$ model. We choose $\lambda=-1$, $\beta=0$ and a range of $N$ from 8 to 22.
• Figure 4: (a): SSF of the $T\bar{T}$-deformed SYK$_4$ model and (b): cSSF of the $T\bar{T}$-deformed SSYK$_4$ model with various value of $\lambda$. (c): SSF of the $T\bar{T}$-deformed SYK$_4$ model and (d): cSSF of the $T\bar{T}$-deformed Majorana SSYK$_4$ model with respect to $J_{\rm eff}t$ and various value of $\lambda$. We choose $N=22$, $\beta=0$ and $\lambda=0$, $-0.01$, $-0.1$, $-1$, $-10$, $-100$.
• Figure 5: (a): SSF of the $T\bar{T}$-deformed SYK$_2$ with various value of $N$ and $\lambda=-1$. (b): SSF of the $T\bar{T}$-deformed SYK$_2$ with various value of $\lambda$. (c): SSF of the $T\bar{T}$-deformed SYK$_2$ with respect to $J_{\rm eff}t$. We choose $N=22$ in (c) and (d).