Instability of the non-Fermi liquid state of the Sachdev-Ye-Kitaev Model

TL;DR

This work analyzes the stability of the chaotic non-Fermi liquid state of the SYK4 model against symmetry-preserving four-fermion perturbations. Through RG, mean-field theory, and exact diagonalization, it shows that these perturbations are marginally relevant or irrelevant depending on sign, driving a KT-like transition to a time-reversal symmetry-breaking, non-chaotic phase. The study further generalizes to SYK_q with multiple perturbations, revealing a family of interacting conformal fixed points with continuously tunable scaling dimensions, including a SUSY-SYK point at a specific limit. Overall, the results connect chaos, symmetry breaking, and conformal fixed points, offering insights into the stability and richness of SYK-like quantum critical states and their potential holographic interpretations.

Abstract

We study a series of perturbations on the Sachdev-Ye-Kitaev (SYK) model. We show that the chaotic non-Fermi liquid phase described by the ordinary $q = 4$ SYK model has marginally relevant/irrelevant (depending on the sign of the coupling constants) four-fermion perturbations allowed by symmetry. Changing the sign of one of these four-fermion perturbations leads to a continuous chaotic-nonchaotic quantum phase transition of the system accompanied by a spontaneous time-reversal symmetry breaking. Starting with the SYK$_q$ model with a $q-$fermion interaction, similar perturbations can lead to a series of new interacting conformal field theory fixed points.

Figures (8)

• Figure 1: The phase diagram of Eq. \ref{['cluster']}.
• Figure 2: ($a$), ($b$), ($c$), the diagrams that we consider for the leading order RG for the coupling constant $u$ in Eq. \ref{['cluster']}. Only diagram ($a$) contributes in the large$-N$ limit. ($d$), the leading order RG for $u$ in Eq. \ref{['replica']}, which is equivalent to ($a$), the solid and dashed lines are fermion and boson Green's functions.
• Figure 3: The fermion wave function renormalization based on Eq. \ref{['cluster']} and Eq. \ref{['replica']} respectively. These diagrams correspond to a $u^3$ term in the beta function, and it carries a factor of $1/N$.
• Figure 4: Transition temperature $T_c$ as a function of $u$ by numerically solving the mean field equations (\ref{['green']}-\ref{['wsaddle']}). This confirms the scaling relation in Eq. \ref{['scale']}.
• Figure 5: The logarithmic static correlation $\ln D(0)$ v.s. the fermion number $N$ for the case of $u > 0$ and $A = 0.2$. The error bar shows the statistical deviation over different random realizations of the coefficient $C_{ij}^a$. When $N\text{ mod }8=0$, $D(\omega=0)$ vanishes exactly, so we use the finite frequency extrapolation to obtain the static correlation $D(0)=\lim_{\omega\to 0}D(\omega)$ in these cases.