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

We study a series of perturbations on the Sachdev-Ye-Kitaev (SYK) model. We show that the maximal chaotic non-Fermi-liquid phase described by the ordinary $q=4$ SYK model has marginally relevant or 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 ${\mathrm{SYK}}_q$ model with a $q$-fermion interaction, similar perturbations can lead to a series of new fixed points with continuously varying exponents.

Figure 1

The phase diagram of Eq. (2).

Figure 2

(a)–(c) Diagrams that we consider for the leading-order RG for the coupling constant $u$ in Eq. (2). Only the diagram in (a) contributes in the large-$N$ limit. (d) The leading-order RG for $u$ in Eq. (17), 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 (a) Eq. (2) and (b) Eq. (17). 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 (20, 21, 22). This confirms the scaling relation in Eq. (14).

Figure 5

The logarithmic static correlation $lnD\left(0\right)$ vs 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 $D\left(0\right)={lim}_{\omega \to 0}D\left(\omega \right)$ in these cases.

Figure 6

The logarithmic static correlation $lnD\left(0\right)$ vs the fermion number $N$ for the cases of $A=0.2$, $u<0$ (left) and $A=2$, $u>0$ (right). Neither case shows long-range correlation of the bosonic field $b^a$. Both $D(\omega =0)$ (red) and ${lim}_{\omega \to 0}D\left(\omega \right)$ (blue) are plotted.

Figure 7

The numerical solution of Eqs. (31) and (32) for $u=-1$, $J=1$, $\beta =300$ with different $M/N$, without assuming a conformal solution from the beginning. Both the boson and fermion Green's functions have nice power-law scaling with the frequency, whose scaling dimensions depend on $M/N$.

Figure 8

We numerically solve the Schwinger-Dyson equations (31) and (32) for $u=-1,J=1,\beta =300$ and fit the low-frequency part as a power law. The scaling dimensions are a continuous function of $M/N$, and for all the data points, the relation $2{\mathrm{\Delta }}_f+{\mathrm{\Delta }}_b=1$ is held. The solid curves plot the solution of the scaling dimensions based on Eq. (33). In particular, for $M/N=1$ (the dashed line), the scaling dimensions obtained from both the numerical and analytical solutions match the prediction from the SUSY SYK model. [24]