Solvable model for a dynamical quantum phase transition from fast to slow scrambling (1610.04619v2)

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 $\lambda_L\to 2\pi T$ 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 $T\to 0$, 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 $\lambda_L\propto T^2$.

Solvable Model for a Dynamical Quantum Phase Transition from Fast to Slow Scrambling

The paper "Solvable model for a dynamical quantum phase transition from fast to slow scrambling" by Banerjee and Altman presents an extension of the Sachdev-Ye-Kitaev (SYK) model, offering a significant contribution to understanding quantum phase transitions and quantum information scrambling in large $N$ limits. The authors introduce a model that transitions between a non-Fermi liquid (NFL) and a Fermi liquid (FL) phase, providing an analytically tractable platform for exploring the interplay between many-body quantum chaos and thermalization dynamics.

Model Description

The extended model builds upon the well-known SYK framework by coupling an $N$-site SYK system with $pN$ non-interacting peripheral sites. The coupling is introduced through quadratic hopping terms, maintaining solvability in the large $N$ limit. The model captures the transition from the strongly interacting SYK state to a screened, non-interacting state as the ratio $p$ varies. Importantly, the quantum phase transition occurs at a critical ratio, $p_c$, which is dependent solely on the fermion density $n$, but not on the coupling strength.

Analytical Findings

The paper extensively explores the model's phases using large $N$ techniques and conformal symmetry in the low-energy limit. In the NFL regime, characterized by $p < p_c$, the model exhibits a universal Lyapunov exponent $\lambda_L \to 2\pi T$, indicative of fast scrambling similar to black hole dynamics in AdS$_2$ spacetimes. The transition to the FL phase for $p > p_c$ results in a Lyapunov exponent $\lambda_L \propto T^2$, a haLLMark of slower scrambling dynamics indicative of a more conventional Fermi liquid.

Critically, the paper identifies a continuous vanishing of residual entropy at $T \to 0$ across the transition, with $S \to 0$ as $n \to n_c$. This behavior underscores the fundamental change in ground state degeneracy and marks the disappearance of the dense low-energy spectrum typical of SYK-like models.

Implications and Speculative Insights

This work enriches the theoretical landscape by methodically detailing a solvable transition between fast and slow information scrambling regimes. While the model's exact solutions in the NFL phase provide a concrete instance of an SYK-like chaotic phase, the complete resolution beyond the critical point remains an open problem, potentially marking a unique universality class of dynamical transitions. The elimination of the black hole analog in the dual gravity picture adds an intriguing holographic dimension to the problem.

Future Directions

The paper suggests several avenues for future exploration, particularly in understanding the quantum-critical dynamics at the transition point, which was left unaddressed in this analysis. Moreover, expanding the model to incorporate features such as disorder or external fields could yield applications to real-world systems exhibiting quantum chaos, such as cold atomic gases or certain solid-state systems.

This work stands as a vital step towards a classification of quantum matter based on its information scrambling properties, laying the groundwork for a deeper understanding of quantum chaos and its manifestations in condensed matter systems.