Complexity transitions in chaotic quantum systems: Nonstabilizerness, entanglement, and fractal dimension in SYK and random matrix models

TL;DR

This paper investigates how complexity measures like fractal dimension, entanglement entropy, and stabilizer Re9nyi entropy change during phase transitions in chaotic quantum systems, revealing that different markers detect different aspects of complexity.

Contribution

It provides a comparative analysis of multiple complexity markers across phase transitions in several quantum models, highlighting their differences and sensitivities.

Findings

Finite-size scaling reveals sharp transitions in complexity markers.

Markers diverge in the presence of an intermediate fractal phase.

Stabilizer Re9nyi entropy is more sensitive to symmetries.

Abstract

Complex quantum systems -- composed of many, interacting particles -- are intrinsically difficult to model. When a quantum many-body system is subject to disorder, it can undergo transitions to regimes with varying non-ergodic and localized behavior, which can significantly reduce the number of relevant basis states. It remains an open question whether such transitions are also directly related to an abrupt change in the system's complexity. In this work, we study the transition from chaotic to integrable phases in several paradigmatic models, the power-law random banded matrix model, the Rosenzweig--Porter model, and a hybrid SYK+Ising model, comparing three complementary complexity markers -- fractal dimension, von Neumann entanglement entropy, and stabilizer R\'enyi entropy. For all three markers, finite-size scaling reveals sharp transitions between high- and low-complexity regimes, which, however, can occur at different critical points. As a consequence, while in the ergodic and localized regimes the markers align, they diverge significantly in the presence of an intermediate fractal phase. Additionally, our analysis reveals that the stabilizer R\'enyi entropy is more sensitive to underlying many-body symmetries, such as fermion parity and time reversal, than the other markers. As our results show, different markers capture complementary facets of complexity, making it necessary to combine them to obtain a comprehensive diagnosis of phase transitions. The divergence between different complexity markers also has significant consequences for the classical simulability of chaotic many-body systems.