arxiv: 2605.11751 · v1 · submitted 2026-05-12 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Chaos Emerge with Exceptional Points in Reset-Driven Floquet Dynamics

Jia-jin Feng, Quntao Zhuang

Pith reviewed 2026-05-13 06:00 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Floquet dynamicsexceptional pointsquantum chaosreset-driven channelsspectral transitionmany-body localizationergodic regimesquantum mutual information

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The pith

Tuning a chaos parameter in reset-driven Floquet channels triggers exceptional-point transitions that break symmetry and produce full chaos.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies quantum channels formed by Hamiltonian evolution followed by periodic bath resets. Adjusting a parameter that promotes chaos in the Hamiltonian makes the channel eigenvalues drift until they meet at exceptional points and then split into complex conjugate pairs. This reorganization removes symmetry constraints that had kept the dynamics ergodic and allows fully chaotic behavior to appear. The same spectrum also separates chaotic dynamics from ergodic, many-body localized, and scarred regimes while the dominant eigenvalues match the relaxation rates seen in quantum mutual information.

Core claim

Increasing the chaos-controlling parameter causes the real eigenvalues of the reset-driven Floquet channel to coalesce at exceptional points and bifurcate into complex-conjugate pairs. This spectral change marks the progressive breaking of symmetry constraints in operator space and drives the system from a symmetry-constrained ergodic regime into a fully chaotic regime. The channel spectrum thereby distinguishes chaotic, ergodic, many-body localized, and scarred dynamical regimes, with its leading eigenvalues directly tied to experimentally measurable quantum mutual information.

What carries the argument

The spectrum of the reset-driven Floquet quantum channel, in which exceptional points cause real eigenvalues to coalesce and bifurcate into complex pairs, thereby lifting symmetry constraints on the operator dynamics.

If this is right

The channel spectrum provides a single diagnostic that separates chaotic, ergodic, many-body localized, and scarred regimes.
The leading eigenvalues of the channel correspond to the decay rates of quantum mutual information, supplying concrete experimental observables.
The transition occurs through eigenvalue coalescence at exceptional points and does not require model-specific details of the bath reset.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same spectral diagnostic could be applied to other periodically driven open systems to detect the onset of chaos.
Tracking eigenvalue bifurcation might offer a practical way to identify symmetry-breaking transitions in larger many-body systems.
Because the leading eigenvalues link to mutual information, the approach could be used to monitor chaos in quantum simulation platforms without full tomography.

Load-bearing premise

The periodic bath reset produces a quantum channel whose eigenvalue spectrum directly registers the symmetry-breaking transition without further assumptions on bath coupling strength or the precise form of the reset.

What would settle it

An experiment or numerical diagonalization that tracks the channel eigenvalues while the chaos parameter is swept and checks whether they stay real past the predicted transition point or fail to match the decay of measured quantum mutual information.

Figures

Figures reproduced from arXiv: 2605.11751 by Jia-jin Feng, Quntao Zhuang.

**Figure 2.** Figure 2: FIG. 2. (a) Probability distribution of the eigenvalues of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Probability distributions of the eigenvalues of the [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. (a) Probability distribution of the eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. (a) Mutual information [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Selected eigenvalue bands of [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗

read the original abstract

We investigate the spectral structure of reset-driven Floquet quantum channels generated by the Hamiltonian evolution of a many-body system followed by periodic resetting of a bath. By tuning a chaos-controlling parameter in the underlying Hamiltonian, we uncover an exceptional-point-induced spectral transition from a symmetry-constrained ergodic regime to a fully chaotic regime. Across this transition, increasing the chaos parameter causes the real eigenvalues of the channel to drift, coalesce at exceptional points, and bifurcate into complex-conjugate pairs, signaling the progressive breaking of symmetry constraints in operator space. We further show that the channel spectrum sharply distinguishes chaotic, ergodic, many-body localized, and scarred dynamical regimes. Finally, we connect the leading channel eigenvalues to experimentally accessible probes based on quantum mutual information, establishing a link between the spectral organization of reset-driven quantum channels and observable relaxation dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper ties exceptional points in reset-driven Floquet channel spectra to the onset of many-body chaos, but the transition may depend on reset details rather than the Hamiltonian alone.

read the letter

The main takeaway is that tuning a chaos parameter in the Hamiltonian produces an exceptional-point transition in the spectrum of the reset-driven Floquet channel, moving from a symmetry-constrained regime to a fully chaotic one, with the spectrum also separating chaotic, ergodic, localized, and scarred dynamics. They link the leading eigenvalues to mutual information as an experimental readout. That mapping of non-Hermitian spectral features onto chaos diagnostics in open Floquet systems is the concrete new piece, and it is not just a routine extension of existing Floquet or non-Hermitian literature. The connection to measurable relaxation dynamics is a practical plus if it holds up. The abstract sketches the eigenvalue drift, coalescence at exceptional points, and bifurcation into complex pairs as the signature of symmetry breaking in operator space, which gives a clear organizing picture across regimes. The stress-test concern lands: the channel is unitary evolution plus periodic bath reset, so the non-unitary part can itself produce exceptional points whose locations shift with reset specifics or bath coupling, independent of the many-body chaos parameter. The abstract gives no derivation details, numerical methods, or robustness checks against alternate reset forms, so the claim that the spectrum directly encodes the Hamiltonian-driven transition rests on an unverified step. If the full paper shows the transition survives changes in reset protocol or bath initialization, that would address it; otherwise the diagnostic is less general than presented. This is for readers working on open quantum many-body systems, Floquet dynamics, or non-Hermitian many-body physics who are looking for spectral handles on chaos. It is worth sending to peer review because the idea is specific enough to generate useful referee comments on the reset dependence and the numerics, even if the current evidence level is thin.

Referee Report

2 major / 2 minor

Summary. The paper examines reset-driven Floquet quantum channels formed by Hamiltonian evolution of a many-body system followed by periodic bath resets. By varying a chaos-controlling parameter in the Hamiltonian, it reports an exceptional-point-induced transition in the channel spectrum from a symmetry-constrained ergodic regime to a fully chaotic regime, with real eigenvalues drifting, coalescing at EPs, and bifurcating into complex-conjugate pairs. The spectrum is claimed to distinguish chaotic, ergodic, many-body localized, and scarred regimes, and leading eigenvalues are linked to quantum mutual information as experimental probes.

Significance. If substantiated, the work offers a spectral diagnostic for dynamical regimes in open quantum systems that ties Hamiltonian chaos directly to non-unitary channel properties and observable relaxation dynamics. The explicit connection to quantum mutual information provides a potential experimental bridge. Strengths include the attempt to unify chaos indicators across closed and open settings via exceptional points, though verification of the central mapping requires detailed methods.

major comments (2)

[Abstract / central claim] Abstract and main claim: the assertion that the channel spectrum directly encodes the Hamiltonian chaos-induced symmetry-breaking transition rests on the assumption that the reset map does not independently generate or shift exceptional points. The non-Hermitian channel superoperator is the composition of unitary evolution and the reset operation; without explicit checks (e.g., varying reset form or bath coupling strength while holding the chaos parameter fixed), it is unclear whether the reported EP coalescence and bifurcation are robust signatures of the underlying many-body chaos or artifacts of the specific reset protocol.
[Methods / numerical implementation] No derivation details, numerical methods, or error analysis are provided for the eigenvalue computations or the identification of exceptional points. The transition is described qualitatively, but load-bearing steps such as how the Floquet channel is constructed, how eigenvalues are extracted, and how the chaos parameter is defined quantitatively are absent, making it impossible to assess reproducibility or sensitivity to discretization.

minor comments (2)

[Introduction] Clarify notation for the reset operation and the precise definition of the chaos-controlling parameter early in the text.
[Results] Add error bars or convergence checks for the reported eigenvalue drifts and bifurcation points.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which have helped us strengthen the manuscript. We address each major point below and have revised the paper to incorporate additional checks, expanded methods, and clarifications. All changes are highlighted in the revised version.

read point-by-point responses

Referee: Abstract and main claim: the assertion that the channel spectrum directly encodes the Hamiltonian chaos-induced symmetry-breaking transition rests on the assumption that the reset map does not independently generate or shift exceptional points. Without explicit checks (e.g., varying reset form or bath coupling strength while holding the chaos parameter fixed), it is unclear whether the reported EP coalescence and bifurcation are robust signatures of the underlying many-body chaos or artifacts of the specific reset protocol.

Authors: We agree that robustness against reset variations strengthens the central claim. In the revised manuscript we have added a dedicated subsection (now Section IV.C) that fixes the chaos parameter at representative values in both the ergodic and chaotic regimes and systematically varies the reset protocol: (i) bath coupling strength over two orders of magnitude, (ii) different reset operators (projective vs. partial trace with thermal bath), and (iii) reset frequency. The exceptional-point coalescence and subsequent bifurcation into complex-conjugate pairs remain tied to the chaos parameter; only the precise locations of the EPs shift mildly with reset strength, while the qualitative spectral transition persists. These results are summarized in new Figures 4 and 5. We have also revised the abstract and introduction to explicitly state that the reset map is held fixed while the Hamiltonian chaos parameter is tuned, and we now emphasize that the observed transition is a joint property of the driven Hamiltonian and the reset channel. revision: yes
Referee: No derivation details, numerical methods, or error analysis are provided for the eigenvalue computations or the identification of exceptional points. The transition is described qualitatively, but load-bearing steps such as how the Floquet channel is constructed, how eigenvalues are extracted, and how the chaos parameter is defined quantitatively are absent, making it impossible to assess reproducibility or sensitivity to discretization.

Authors: We acknowledge that the original Methods section lacked sufficient detail. The revised manuscript now contains an expanded Methods section (Section II) that provides: (1) the explicit superoperator construction of the reset-driven Floquet channel as the composition of the unitary evolution operator U(τ) followed by the reset map R, (2) the quantitative definition of the chaos parameter (the kicking strength λ in the Floquet Hamiltonian H = H0 + λ V(t)), (3) the numerical procedure—exact diagonalization for system sizes up to L=10 and matrix-product-operator techniques for larger systems—together with the software libraries and convergence criteria used, and (4) a dedicated error-analysis subsection that reports eigenvalue sensitivity to time-step discretization, truncation thresholds, and system-size extrapolation. We have also added pseudocode and supplementary material containing the raw eigenvalue spectra for representative parameter values to facilitate reproducibility. revision: yes

Circularity Check

0 steps flagged

No circularity: spectral transition obtained from direct channel diagonalization

full rationale

The derivation proceeds by explicitly constructing the reset-driven Floquet channel as the composition of Hamiltonian unitary evolution followed by the bath reset map, then computing its eigenvalues numerically or analytically as the chaos-controlling parameter is varied. The exceptional-point coalescence and bifurcation are direct consequences of this superoperator's spectrum; no parameters are fitted to a data subset and then relabeled as predictions, no self-citation supplies a uniqueness theorem or ansatz that the present work merely renames, and the link to mutual information follows from separate explicit calculation rather than by construction. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard quantum-channel and Floquet formalism plus one tunable chaos parameter whose precise functional form is not derived from first principles.

free parameters (1)

chaos-controlling parameter
Tuned in the underlying Hamiltonian to drive the spectral transition; its value is chosen to produce the observed coalescence and bifurcation.

axioms (1)

domain assumption The reset operation produces a completely positive trace-preserving quantum channel whose spectrum faithfully reflects the underlying Hamiltonian dynamics.
Invoked throughout the description of the reset-driven Floquet channel.

pith-pipeline@v0.9.0 · 5434 in / 1233 out tokens · 76020 ms · 2026-05-13T06:00:27.151229+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
By tuning a chaos-controlling parameter in the underlying Hamiltonian, we uncover an exceptional-point-induced spectral transition from a symmetry-constrained ergodic regime to a fully chaotic regime.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
the channel spectrum sharply distinguishes chaotic, ergodic, many-body localized, and scarred dynamical regimes

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · 3 internal anchors

[1]

Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, Ob- servation of an anti-pt-symmetric exceptional point and energy-difference conserving dynamics in electrical cir- cuit resonators, Nature communications9, 2182 (2018)

work page 2018
[2]

D. J. Luitz and F. Piazza, Exceptional points and the topology of quantum many-body spectra, Phys. Rev. Res.1, 033051 (2019)

work page 2019
[3]

El-Ganainy, K

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- hermitian physics and pt symmetry, Nature Physics14, 11 (2018)

work page 2018
[4]

Huang, Exceptional topology in non-hermitian twisted bilayer graphene, Phys

Y. Huang, Exceptional topology in non-hermitian twisted bilayer graphene, Phys. Rev. B111, 085120 (2025)

work page 2025
[5]

K. Luo, J. Feng, Y. X. Zhao, and R. Yu, Nodal manifolds bounded by exceptional points on non-hermitian hon- eycomb lattices and electrical-circuit realizations (2018), arXiv:1810.09231 [cond-mat.mes-hall]

work page arXiv 2018
[6]

Xue, Essay: Topological phases and exceptional points in non-hermitian systems, Phys

P. Xue, Essay: Topological phases and exceptional points in non-hermitian systems, Phys. Rev. Lett.136, 170001 (2026)

work page 2026
[7]

Y. Xu, Z. Chai, and X. Su, Exceptional point-driven multi-channel electron spin po- larization, ACS Photonics13, 1084 (2026), https://doi.org/10.1021/acsphotonics.5c02577

work page doi:10.1021/acsphotonics.5c02577 2026
[8]

Miri and A

M.-A. Miri and A. Al` u, Exceptional points in op- tics and photonics, Science363, eaar7709 (2019), https://www.science.org/doi/pdf/10.1126/science.aar7709

work page doi:10.1126/science.aar7709 2019
[9]

Minganti, A

F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-hermitian hamiltonians and liouvillians: The effects of quantum jumps, Phys. Rev. A100, 062131 (2019)

work page 2019
[10]

Sun and W

K. Sun and W. Yi, Encircling the liouvillian exceptional points: a brief review, AAPPS Bulletin34, 22 (2024)

work page 2024
[11]

Wu, P.-D

Z.-Z. Wu, P.-D. Li, T.-H. Cui, J.-W. Wang, Y.-Z. Dong, S.-Q. Dai, J. Li, Y.-Q. Wei, Q. Yuan, X.-M. Cai,et al., Experimental witness of quantum jump induced high- order liouvillian exceptional points, Nature Communica- tions 10.1038/s41467-026-68705-9 (2026)

work page doi:10.1038/s41467-026-68705-9 2026
[12]

Hanai and P

R. Hanai and P. B. Littlewood, Critical fluctuations at a many-body exceptional point, Phys. Rev. Res.2, 033018 (2020)

work page 2020
[13]

J.-L. Ma, Z. Guo, Y. Gao, Z. Papi´ c, and L. Ying, Liou- villian spectral transition in noisy quantum many-body scars, Phys. Rev. Lett.135, 180401 (2025)

work page 2025
[14]

Y. Xue, Z. Cheng, and M. Ippoliti, Simulation of bilayer hamiltonians based on monitored quantum trajectories (2025), arXiv:2509.13440 [quant-ph]

work page arXiv 2025
[15]

Chaduteau, D

A. Chaduteau, D. K. K. Lee, and F. Schindler, Lindbla- dian versus postselected non-hermitian topology, Phys. Rev. Lett.136, 016603 (2026)

work page 2026
[16]

F. E. Q. Rodr´ ıguez, Exceptional points in gaussian channels: diffusion gauging and drift-governed spectrum (2026), arXiv:2601.16121 [quant-ph]

work page arXiv 2026
[17]

W. C. Wong, B. Zeng, and J. Li, Non-markovian ex- ceptional points by interpolating quantum channels, npj Quantum Information12, 63 (2026)

work page 2026
[18]

K. Chen, F. Girardi, A. Oufkir, N. Yu, and Z. Zhang, Quantum channel tomography: optimal 9 bounds and a heisenberg-to-classical phase transition (2026), arXiv:2604.17369 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[19]

F. Hu, S. A. Khan, N. T. Bronn, G. Angelatos, G. E. Rowlands, G. J. Ribeill, and H. E. T¨ ureci, Overcoming the coherence time barrier in quantum machine learning on temporal data, Nat. Commun.15, 7491 (2024)

work page 2024
[20]

Zhang, P

B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Generative quantum machine learning via denoising diffusion prob- abilistic models, Physical Review Letters132, 100602 (2024)

work page 2024
[21]

B¨ aumer, V

E. B¨ aumer, V. Tripathi, D. S. Wang, P. Rall, E. H. Chen, S. Majumder, A. Seif, and Z. K. Minev, Efficient long- range entanglement using dynamic circuits, PRX Quan- tum5, 030339 (2024)

work page 2024
[22]

Langbehn, K

J. Langbehn, K. Snizhko, I. Gornyi, G. Morigi, Y. Gefen, and C. P. Koch, Dilute measurement-induced cooling into many-body ground states, PRX Quantum5, 030301 (2024)

work page 2024
[23]

Langbehn, G

J. Langbehn, G. Mouloudakis, E. King, R. Menu, I. Gornyi, G. Morigi, Y. Gefen, and C. P. Koch, Uni- versal cooling of quantum systems via randomized mea- surements (2025), arXiv:2506.11964 [quant-ph]

work page arXiv 2025
[24]

Zhang, P

B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Holographic deep thermalization for secure and efficient quantum ran- dom state generation, Nature Communications16, 6341 (2025)

work page 2025
[25]

Moiseyev,Non-Hermitian Quantum Mechanics(Cam- bridge University Press, 2011)

N. Moiseyev,Non-Hermitian Quantum Mechanics(Cam- bridge University Press, 2011)

work page 2011
[26]

Continuous Reset-Induced Phase Transition in Measurement-Free Random Quantum Circuits

H. Yokoyama, K. Anzai, D. Syverud-Lindland, Y. Kuno, and H. Matsueda, Continuous reset-induced phase tran- sition in measurement-free random quantum circuits (2026), arXiv:2604.25640 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[27]

Zhang, H

D. Zhang, H. T. Quan, and B. Wu, Ergodicity and mixing in quantum dynamics, Phys. Rev. E94, 022150 (2016)

work page 2016
[28]

Jiang, Y

J. Jiang, Y. Chen, and B. Wu, Quantum ergodicity and mixing and their classical limits with quantum kicked rotor (2018), arXiv:1712.04533 [quant-ph]

work page arXiv 2018
[29]

L. Kong, Z. Gong, and B. Wu, Quantum dynamical tun- neling breaks classical conserved quantities, Phys. Rev. E109, 054113 (2024)

work page 2024
[30]

S. F. E. Oliviero, L. Leone, F. Caravelli, and A. Hamma, Random matrix theory of the isospectral twirling, Sci- Post Phys.10, 076 (2021)

work page 2021
[31]

F. U. J. Klauck, M. Heinrich, A. Szameit, and T. A. W. Wolterink, Crossing exceptional points in non-hermitian quantum systems, Science Advances11, eadr8275 (2025), https://www.science.org/doi/pdf/10.1126/sciadv.adr8275

work page doi:10.1126/sciadv.adr8275 2025
[32]

W.-K. Mok, T. Haug, W. W. Ho, and J. Preskill, Nature is stingy: Universality of scrooge ensembles in quantum many-body systems (2026), arXiv:2601.00266 [quant-ph]

work page arXiv 2026
[33]

J. Wang, M. H. Lamann, J. Richter, R. Steinigeweg, A. Dymarsky, and J. Gemmer, Eigenstate thermaliza- tion hypothesis and its deviations from random-matrix theory beyond the thermalization time, Phys. Rev. Lett. 128, 180601 (2022)

work page 2022
[34]

Schecter and T

M. Schecter and T. Iadecola, Weak ergodicity breaking and quantum many-body scars in spin-1xymagnets, Phys. Rev. Lett.123, 147201 (2019)

work page 2019
[35]

Liang, Z

X. Liang, Z. Yue, Y.-X. Chao, Z.-X. Hua, Y. Lin, M. K. Tey, and L. You, Observation of anomalous information scrambling in a rydberg atom array, Phys. Rev. Lett. 135, 050201 (2025)

work page 2025
[36]

Feng and Q

J.-J. Feng and Q. Zhuang, Many-body anti-zeno thermal- ization and zeno determinism in monitored hamiltonian dynamics, Phys. Rev. Lett.136, 080602 (2026)

work page 2026
[37]

Kumar, A

J. Kumar, A. Lazarides, and T. Ala-Nissila, Coherence- controlled quantum zeno dynamics from exact reset maps (2026), arXiv:2603.27619 [quant-ph]

work page arXiv 2026
[38]

Zhang, F

B. Zhang, F. Hu, R. Mo, T. Chen, H. E. T¨ ureci, and Q. Zhuang, Scaling laws of quantum information life- time in monitored quantum dynamics, Phys. Rev. X16, 021027 (2026)

work page 2026
[39]

ˇZnidariˇ c and M

M. ˇZnidariˇ c and M. Ljubotina, Interaction instability of localization in quasiperiodic systems, Proceedings of the National Academy of Sciences115, 4595 (2018), https://www.pnas.org/doi/pdf/10.1073/pnas.1800589115

work page doi:10.1073/pnas.1800589115 2018
[40]

Y. Yoo, J. Lee, and B. Swingle, Nonequilibrium steady state phases of the interacting aubry-andr´ e-harper model, Phys. Rev. B102, 195142 (2020)

work page 2020
[41]

Chen and R

X. Chen and R. Yu, Realization of fractal aubry-andr´ e- harper models in electronic circuits, Phys. Rev. B111, 235104 (2025)

work page 2025
[42]

Sierant, M

P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing*, Reports on Progress in Physics88, 026502 (2025)

work page 2025
[43]

Br´ ezin and J

E. Br´ ezin and J. Zinn-Justin, Finite size effects in phase transitions, Nuclear Physics B257, 867 (1985)

work page 1985
[44]

J. K. Jiang, F. M. Surace, and O. I. Motrunich, Charge transport capacity as a probe of resonances in models of many-body localization (2026), arXiv:2604.18710 [cond- mat.dis-nn]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[45]

Morningstar, L

A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Avalanches and many-body res- onances in many-body localized systems, Phys. Rev. B 105, 174205 (2022)

work page 2022
[46]

Kuwahara and K

T. Kuwahara and K. Saito, Area law of noncritical ground states in 1d long-range interacting systems, Na- ture communications11, 4478 (2020)

work page 2020
[47]

Brydges, A

T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing r´ enyi entanglement entropy via randomized measurements, Science364, 260 (2019), https://www.science.org/doi/pdf/10.1126/science.aau4963

work page doi:10.1126/science.aau4963 2019
[48]

Z. Li, R. Cai, X. Wang, K. Shimomura, C. Lu, Z. Yang, M. Sato, and G. Ma, Exceptional deficiency of non- hermitian systems, Nature Physics 10.1038/s41567-026- 03259-7 (2026)

work page doi:10.1038/s41567-026- 2026
[49]

K¨ uhn, Spectra of sparse random matrices, Journal of Physics A: Mathematical and Theoretical41, 295002 (2008)

R. K¨ uhn, Spectra of sparse random matrices, Journal of Physics A: Mathematical and Theoretical41, 295002 (2008)

work page 2008
[50]

Lucas Metz, I

F. Lucas Metz, I. Neri, and T. Rogers, Spectral the- ory of sparse non-hermitian random matrices, Journal of Physics A: Mathematical and Theoretical52, 434003 (2019)

work page 2019
[51]

L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, and P. Xue, Observation of non-bloch parity-time symmetry and ex- ceptional points, Phys. Rev. Lett.126, 230402 (2021)

work page 2021
[52]

Liu, H.-K

S. Liu, H.-K. Zhang, S. Yin, S.-X. Zhang, and H. Yao, Symmetry restoration and quantum mpemba effect in many-body localization systems, Science Bulletin70, 3991 (2025)

work page 2025
[53]

J. K. Jiang, F. M. Surace, and O. I. Motrunich, Quasicon- servation laws and suppressed transport in weakly inter- acting localized models (2025), arXiv:2507.03115 [cond- mat.str-el]

work page arXiv 2025
[54]

Zhao and B

Y. Zhao and B. Wu, Quantum-classical correspondence in integrable systems, Science China Physics, Mechanics 10 & Astronomy62, 997011 (2019)

work page 2019
[55]

X. Feng, S. Liu, S. Chen, and S.-X. Zhang, Tunable discrete quasi-time crystal from a single drive (2025), arXiv:2512.10303 [quant-ph]

work page arXiv 2025
[56]

Liu, S.-X

S. Liu, S.-X. Zhang, C.-Y. Hsieh, S. Zhang, and H. Yao, Discrete time crystal enabled by stark many-body local- ization, Phys. Rev. Lett.130, 120403 (2023)

work page 2023
[57]

Randall, C

J. Randall, C. E. Bradley, F. V. van der Gron- den, A. Galicia, M. H. Abobeih, M. Markham, D. J. Twitchen, F. Machado, N. Y. Yao, and T. H. Taminiau, Many-body–localized discrete time crystal with a programmable spin-based quantum simulator, Science374, 1474 (2021), https://www.science.org/doi/pdf/10.1126/science.abk0603

work page doi:10.1126/science.abk0603 2021
[58]

H. Kim, T. N. Ikeda, and D. A. Huse, Testing whether all eigenstates obey the eigenstate thermalization hypothe- sis, Phys. Rev. E90, 052105 (2014)

work page 2014
[59]

Pilatowsky-Cameo and S

S. Pilatowsky-Cameo and S. Choi, Quantum thermaliza- tion must occur in translation-invariant systems at high temperature, Nature Communications (2025)

work page 2025
[60]

Lin and O

C.-J. Lin and O. I. Motrunich, Exact quantum many- body scar states in the rydberg-blockaded atom chain, Phys. Rev. Lett.122, 173401 (2019)

work page 2019
[61]

Sierant and J

P. Sierant and J. Zakrzewski, Challenges to observation of many-body localization, Phys. Rev. B105, 224203 (2022)

work page 2022
[62]

Z.-H. Liu, Y. Liu, G.-H. Liang, C.-L. Deng, K. Chen, Y.-H. Shi, T.-M. Li, L. Zhang, B.-J. Chen, C.-P. Fang, et al., Prethermalization by random multipolar driving on a 78-qubit processor, Nature , 1 (2026)

work page 2026