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

TL;DR

This work introduces a solvable extension of the SYK model where N interacting sites are coupled to M = pN quadratic peripheral sites, enabling a continuous dynamical quantum phase transition between a non-Fermi liquid with fast scrambling and a Fermi-liquid-like phase with slow scrambling. The transition is governed by the peripheral-to-core ratio p, with a density-dependent critical value p_c(n); the NFL phase exhibits conformal scaling, universal Lyapunov scrambling λ_L = 2πT, and a residual entropy S_0 that vanishes at the QCP. In the FL phase, scrambling becomes perturbative, λ_L ∝ T^2, and a finite low-energy quasiparticle description emerges. The NFL–FL transition thus marks a dynamical universality class change, with evidence from both analytic conformal solutions and numerical solutions of the self-consistency equations, and a holographic interpretation tied to the disappearance of the black-hole geometry at criticality.

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 phase is characterized by a universal Lyapunov exponent $λ_L\to 2πT$ 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 $T\to 0$, non zero 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 frequency limit. The interactions have a perturbative effect in this regime leading to scrambling with Lyapunov exponent $λ_L\propto T^2$.

Figures (15)

• Figure 1: The generalized SYK model. The SYK sites at the centre are coupled through random four fermion coupling $J_{ijkl}$. The sites at the periphery are connected to the SYK sites and to each other via random hoppings $V_{i{\alpha}}$ and $t_{{\alpha}{\beta}}$, respectively.
• Figure 2: Self-energy diagrams that contribute at leading order in $1/N$ for a fixed ratio $p=M/N$. The bold lines represent Green's function $G$ of the SYK sites and the double line the Green's function $\mathcal{G}$ for the peripheral sites. The dashed, dotted and dashed-double lines imply disorder averaging over $J_{ijkl}$, $V_{i{\alpha}}$ and $t_{{\alpha}{\beta}}$, respectively.
• Figure 3: (a) Phase diagram of the generalized SYK model in the plane of the average fermion filling $n$ and the ratio $p=M/N$. The dashed line indicates half filling. (b) Fermion density $n$, below half filling, as a function of the chemical potential $\mu$ for $p=0.1,\dots,0.9$ (bottom to top curve) at $T=0.025J$. The dashed lines indicate respective critical densities $n_c$'s at the lower phase boundary in panel (a). The plateaus for $p=0.1,0.3$ imply the presence of incompressible state at $n_c$.
• Figure 4: Numerical results for the single particle spectral function on the full frequency range taken on on the SYK sites [panels (a) to (c)] and the quadratic sites [panels (d) to (f)]. The spectral functions are computed at $T=0.025J$ and varying values of $p$ showing how the low temperature singularities vanish beyond the critical point.
• Figure 5: The zero temperature limit of the entropy as a function of $p$. The analytic result using the Luttinger theorem and an assumption $S(T\to 0)=0$ at the finite density phase transition is compared to a direct numerical evaluation of the low-temperature entropy and extrapolations to $T = 0$ for various values of $t$.