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Complexity transitions in chaotic quantum systems: Nonstabilizerness, entanglement, and fractal dimension in SYK and random matrix models
by Gopal Chandra Santra, Alex Windey, Soumik Bandyopadhyay, Andrea Legramandi, Philipp Hauke
Submission summary
| Authors (as registered SciPost users): | Gopal Chandra Santra |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2505.09707v3 (pdf) |
| Date submitted: | Nov. 27, 2025, 2:14 p.m. |
| Submitted by: | Gopal Chandra Santra |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
Generative AI has only been used to polish the text of the manuscript and debug the numerical code.
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 Rényi 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 Rényi 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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1) Timely topic 2) Scientifically sound and sufficient details
Weaknesses
1) The work is numerical and restricted to very small system sizes ( N = 12 )
Report
Using these markers, the authors analyze three paradigmatic models: (1) the Rosenzweig–Porter (RP) model, which features an intermediate non-ergodic fractal phase; (2) the power-law random banded matrix (PLRBM) model, which shows Anderson localization with multifractality at criticality; and (3) the SYK+Ising hybrid model, interpolating between chaotic SYK dynamics and an integrable Ising model.
The main finding of the paper is that all three markers detect sharp complexity transitions, but not necessarily at the same parameter values. In particular, in the RP model, the intermediate fractal phase reveals a splitting of transition points depending on the marker used. In contrast, the PLRBM model exhibits essentially a single transition where all markers agree. In the SYK+Ising model, the stabilizer Rényi entropy is shown to be sensitive to symmetries such as fermion parity and time-reversal symmetry.
I believe the work is suitable for publication in SciPost Physics, provided the authors can reasonably address the points raised below.
(1) The authors extract thermodynamic transition points from crossings of derivatives and extrapolate them using a specific finite-size scaling scheme based on the geometric mean of the system sizes ((N_1, N_2)). While such extrapolation procedures are employed in the literature, their reliability can be sensitive for small system sizes . Can the authors comment on the robustness of their extrapolation scheme? For example, do the inferred transition points remain stable if alternative fitting forms (e.g., simple linear extrapolation with system size ), different effective size definitions (such as arithmetic versus geometric mean), or restricted fitting windows (e.g, excluding the smallest system sizes) are used?
(2) The numerical analysis for the SYK+Ising model is limited to system sizes up to M=20 Majorana fermions (corresponding to N=M/2=10 spins). In the SYK+Ising results, the stabilizer Renyi entropy displays a non-monotonic behavior for certain symmetry classes.
Is this non-monotonicity of the stabilizer Renyi entropy a genuine feature, or is it instead a finite-size effect? It would be helpful to check this by extending the analysis to larger system sizes.
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