Competition between Chaotic and Non-Chaotic Phases in a Quadratically Coupled Sachdev-Ye-Kitaev Model

TL;DR

A generalization of the Sachdev-Ye-Kitaev model that contains two SYK models with a different number of Majorana modes coupled by quadratic terms is considered, and the solution shows a zero-temperature quantum phase transition between two non-Fermi liquid chaotic phases.

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a concrete solvable model to study non-Fermi liquid properties, holographic duality and maximally chaotic behavior. In this work, we consider a generalization of the SYK model that contains two SYK models with different number of Majorana modes coupled by quadratic terms. This model is also solvable, and the solution shows a zero-temperature quantum phase transition between two non-Fermi liquid chaotic phases. This phase transition is driven by tuning the ratio of two mode numbers, and a Fermi liquid non-chaotic phase sits at the critical point with equal mode number. At finite temperature, the Fermi liquid phase expands to a finite regime. More intriguingly, a different non-Fermi liquid phase emerges at finite temperature. We characterize the phase diagram in term of the spectral function, the Lyapunov exponent and the entropy. Our results illustrate a concrete example of quantum phase transition and critical regime between two non-Fermi liquid phases.

Figures (3)

• Figure 1: Schematic of the model that two SYK$_4$ models with different numbers of modes are coupled by quadratic couplings.
• Figure 2: (a) Schematic of a finite temperature phase diagram in terms of $p=N_2/N_1$ and temperature $k_\text{B}T$ in unit of $\sqrt{J^2+V^2}$, for a fixed $V/J=0.2$. At zero temperature, two non-Fermi liquid phases $(1/4,3/4)$ and $(3/4,1/4)$ are separated by a Fermi liquid point $(1/2,1/2)$ at $p=1$. At finite temperature, the color scheme and the dashed lines indicate the crossover between different phases. (b1-b4) The spectral functions $A(\omega)$ at four different representing points in the phase diagram as marked in (a). $A$ is in unit of $\beta$ and $\omega$ is in unit of $1/\beta$. The dashed lines (except for the fitted $\delta$-function) are results from the conformal limit analysis; while the solid lines are obtained from numerical solutions of the real time retarded Green's functions. In (b2-b4), two solid lines nearly coincide with each other and their difference is hard to see. (b1-b4) are computed at $\{p, 1/(\beta\sqrt{J^2+V^2})\}=\{0.1,0.2\}$, $\{1,0.005\}$, $\{0.9, 0.2\}$ and $\{0.9,0.9\}$, respectively.
• Figure 3: (a) A contour plot of the Lyapunov exponent $\lambda/(2\pi k_\text{B}T)$ in a three-dimensional parameter space in term of $p$, $V/J$ and $1/(\beta\sqrt{J^2+V^2})$. The regions of different phases are indicated by the red arrows. (b). The entropy $S/N_1$ is plotted as a function of $V/J$, with $p$ fixed at $0.25$ and $1/(\beta\sqrt{J^2+V^2})$ fixed at $0.02$. The values marked by dashed lines are zero-temperature entropy for the $(1/4,1/4)$ phase, the $(1/4,3/4)$ phase and the $V/J\rightarrow\infty$ limit, respectively.