Solvable model for a dynamical quantum phase transition from fast to slow scrambling

Abstract

We propose an extension of the Sachdev-Ye-Kitaev (SYK) model that exhibits a quantum phase transition from the previously identified non-Fermi-liquid fixed point to a Fermi-liquid-like state, while still allowing an exact solution in a suitable large-$N$ limit. The extended model involves coupling the interacting $N$-site SYK model to a new set of $pN$ peripheral sites with only quadratic hopping terms between them. The conformal fixed point of the SYK model remains a stable low-energy phase below a critical ratio of peripheral sites $p<{p}_{c}(n)$ that depends on the fermion filling $n$. The scrambling dynamics throughout the non-Fermi-liquid (NFL) phase is characterized by a universal Lyapunov exponent ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\rightarrow}2\ensuremath{\pi}T$ in the low-temperature limit; however, the temperature scale marking the crossover to the conformal regime vanishes continuously at the critical point ${p}_{c}$. The residual entropy at $T\ensuremath{\rightarrow}0$, nonzero in the NFL, also vanishes continuously at the critical point. For $p>{p}_{c}$ the quadratic sites effectively screen the SYK dynamics, leading to a quadratic fixed point in the low-temperature and low-frequency limit. The interactions have a perturbative effect in this regime leading to scrambling with Lyapunov exponent ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\propto}{T}^{2}$.