pith. sign in

arxiv: 2506.16176 · v2 · submitted 2025-06-19 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· quant-ph

Spectral statistics, non-equilibrium dynamics and thermalization in random matrices with global mathbb{Z}₂-symmetry

Adway Kumar Das This is my paper

Pith reviewed 2026-05-19 09:12 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnquant-ph
keywords Z2 symmetryrandom matrix theoryeigenstate thermalization hypothesisthermalizationnon-equilibrium dynamicscentrosymmetric matricesdisordered systemssurvival probability
0
0 comments X p. Extension

The pith

In random matrices with global Z2 symmetry, local observables still thermalize to the canonical ensemble even when the initial state and observable respect or violate the symmetry in different ways.

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

The paper models Hamiltonians conserving a global Z2 symmetry, such as exchange symmetry, by using random symmetric centrosymmetric matrices whose spectrum is a mixture of two independent subspaces. It computes the time evolution of survival probabilities for various initial states and finds that most states decay, though a measure-zero set of low-energy states can persist long enough to suggest spontaneous symmetry breaking in rare cases. For local observables like density-density correlations or kinetic energy, the long-time equilibrium value depends on whether the observable respects the symmetry and matches the initial state's sector, yet the diagonal fluctuations of these observables inside microcanonical energy windows shrink with increasing system size. This decay keeps the eigenstate thermalization hypothesis valid, and the equilibrium value matches the canonical ensemble average for every observable and initial state examined.

Core claim

When the observable violates the global Z2 symmetry of the Hamiltonian, its equilibrium value is independent of the symmetry of the initial state; when the observable respects the symmetry, the equilibrium value depends on the initial state's symmetry sector. Irrespective of these constraints, fluctuations of the diagonal matrix elements of observables within microcanonical shells decrease with system size, so the eigenstate thermalization hypothesis ansatz continues to hold. The long-time equilibrium converges to the canonical ensemble average for all observables and initial states, showing that thermalization occurs despite the presence of a global symmetry that decouples the Hilbert space

What carries the argument

Symmetric centrosymmetric random matrices whose spectrum is a superposition of two pure spectra from decoupled subspaces, allowing analytical calculation of survival probabilities and long-time averages for observables.

If this is right

  • Observables that respect the Z2 symmetry have equilibrium values that depend on whether the initial state lies in the even or odd symmetry sector.
  • Observables that violate the Z2 symmetry reach equilibrium values independent of the initial state's symmetry sector.
  • Fluctuations of observable diagonal elements inside microcanonical windows decay with system size, preserving the eigenstate thermalization hypothesis.
  • The long-time average of every examined local observable converges to the canonical ensemble value regardless of symmetry constraints.

Where Pith is reading between the lines

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

  • Global symmetries that split the spectrum into subspaces need not obstruct thermalization or ergodicity in disordered quantum systems.
  • The rare persistent states at low energy could be checked numerically in small Z2-symmetric spin chains to see whether they affect ground-state properties.
  • The independence of equilibrium values for symmetry-violating observables suggests a route to test thermalization by preparing states in one sector and measuring cross-sector operators.

Load-bearing premise

The analytically computed survival probabilities and long-time limits for the mixed spectrum of SC matrices correctly capture the non-equilibrium dynamics of generic disordered Hamiltonians with Z2 symmetry.

What would settle it

Numerical evolution of a concrete Z2-symmetric disordered spin chain or similar Hamiltonian that shows either persistent non-decay of survival probability in a finite fraction of realizations or failure of the equilibrium value to approach the canonical average.

Figures

Figures reproduced from arXiv: 2506.16176 by Adway Kumar Das.

Figure 1
Figure 1. Figure 1: FIG. 1. Energy correlation of random SC matrices. (a) den [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Ensemble averaged survival probability of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Survival probability of the superposition state [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Diagonal (diagonal ensemble), micro-canonical (micro-canonical ensemble), canonical (Gibbs ensemble) and generalized [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

$\mathbb{Z}_2$ symmetry is ubiquitous in quantum mechanics where it drives various phase transitions and emergent physics. The role of $\mathbb{Z}_2$ symmetry in the thermalization of a local observable in a disordered system can be understood using random matrix theory. To do so, we consider random symmetric centrosymmetric (SC) matrix as a toy model where a $\mathbb{Z}_2$ symmetry, namely, the exchange symmetry is conserved. Such a conservation law splits the Hilbert space into decoupled subspaces such that the energy spectrum of a SC matrix is a superposition of two pure spectra. After discussing the known results on the correlations of such mixed spectrum, we consider different initial states and analytically compute the time evolution of their survival probability and associated timescales. We show that there exist certain low-energy initial states which do not decay over a very long timescales such that a measure zero fraction of random SC matrices exhibit spontaneous symmetry breaking. Later, we look at the equilibrium values of local observables like the density-density correlation, kinetic energy operator and compare them against the average values from the microcanonical and canonical ensembles. We find that when the observable violates (respects) the global symmetry of the Hamiltonian, the equilibrium value is independent (dependent) of the symmetry of the initial state. However, irrespective of such symmetry constraints, the fluctuations of the diagonal terms of the observables within microcanonical shells decay with system size such that the ansatz of eigenstate thermalization hypothesis remains valid. We show that the equilibrium value converges to the canonical average for all the observables and initial states, indicating that thermalization occurs despite the presence of a global symmetry.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript employs symmetric centrosymmetric (SC) random matrices as a toy model for disordered quantum systems with global Z2 symmetry. It reviews spectral correlations in the resulting mixed spectrum from two decoupled subspaces, analytically computes survival probabilities and associated timescales for different initial states (including low-energy states that fail to decay), and compares long-time equilibrium values of local observables (density-density correlation, kinetic energy) against microcanonical and canonical ensemble averages. Central results are that equilibrium values are independent (dependent) of initial-state symmetry when the observable violates (respects) the Hamiltonian symmetry, that diagonal fluctuations of observables within microcanonical shells decay with system size (supporting ETH), and that thermalization occurs as equilibrium values converge to canonical averages for all cases examined.

Significance. If the SC ensemble faithfully captures the relevant spectral statistics, the work supplies useful analytical expressions for non-equilibrium dynamics and equilibrium observables in the presence of a global symmetry, together with explicit confirmation that ETH fluctuations decay and thermalization proceeds. Credit is due for the parameter-free derivations of survival probabilities from standard RMT assumptions on mixed spectra and for the clean separation of symmetry-respecting versus symmetry-violating observables.

major comments (2)
  1. [Section 3] Section 3 and the paragraphs on time evolution of survival probability: the central thermalization claim (equilibrium convergence to canonical averages and validity of ETH) rests on the intra- and inter-subspace level correlations of the SC ensemble matching those of generic Z2-symmetric disordered Hamiltonians. No explicit comparison is provided (e.g., of the two-point spectral form factor or the variance of diagonal matrix elements of local operators in microcanonical windows) to a physical model such as a disordered spin chain; this assumption is load-bearing for extending the analytical results beyond the toy ensemble.
  2. [time-evolution paragraphs] Abstract and time-evolution paragraphs: the reported long-time limits and the statement that 'thermalization occurs despite the presence of a global symmetry' for all observables and initial states rely on the analytically computed survival probabilities for the mixed spectrum. Without numerical verification on a concrete Z2-symmetric Hamiltonian, it remains unclear whether post-hoc choices in initial-state selection or spectrum superposition alter the claimed universality of the equilibrium values.
minor comments (2)
  1. [Abstract] The phrase 'a measure zero fraction of random SC matrices exhibit spontaneous symmetry breaking' in the abstract would benefit from an explicit cross-reference to the relevant equation or figure that quantifies this fraction.
  2. Notation for the two subspaces and the mixed spectrum should be introduced once with consistent symbols and then used uniformly; occasional redefinition of symbols for the same quantities reduces readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We are pleased that the referee recognizes the analytical contributions and the implications for thermalization in symmetric systems. We address each of the major comments in detail below, indicating the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Section 3] Section 3 and the paragraphs on time evolution of survival probability: the central thermalization claim (equilibrium convergence to canonical averages and validity of ETH) rests on the intra- and inter-subspace level correlations of the SC ensemble matching those of generic Z2-symmetric disordered Hamiltonians. No explicit comparison is provided (e.g., of the two-point spectral form factor or the variance of diagonal matrix elements of local operators in microcanonical windows) to a physical model such as a disordered spin chain; this assumption is load-bearing for extending the analytical results beyond the toy ensemble.

    Authors: We agree with the referee that the central claims regarding thermalization and ETH in the presence of Z2 symmetry rely on the SC ensemble capturing the spectral statistics of generic Z2-symmetric disordered Hamiltonians. The paper reviews the known results on the correlations in mixed spectra from two decoupled subspaces. To address this point, we will revise Section 3 to include an explicit numerical comparison of the spectral form factor and the variance of diagonal matrix elements for the SC ensemble versus a physical model, for example a disordered spin-1/2 chain with Z2 symmetry. This will provide direct evidence that the intra- and inter-subspace correlations match those expected in physical systems. revision: yes

  2. Referee: [time-evolution paragraphs] Abstract and time-evolution paragraphs: the reported long-time limits and the statement that 'thermalization occurs despite the presence of a global symmetry' for all observables and initial states rely on the analytically computed survival probabilities for the mixed spectrum. Without numerical verification on a concrete Z2-symmetric Hamiltonian, it remains unclear whether post-hoc choices in initial-state selection or spectrum superposition alter the claimed universality of the equilibrium values.

    Authors: The analytical computation of survival probabilities and the resulting long-time limits are derived rigorously from the standard assumptions of random matrix theory applied to the mixed spectrum of the SC ensemble. These results are independent of specific choices in initial states as long as they are within the ensemble framework. We acknowledge that numerical verification on a concrete Hamiltonian would be a valuable addition to demonstrate that the universality holds in physical systems. In the revised manuscript, we will add a paragraph in the time-evolution section discussing this point and include a small-scale numerical check on a Z2-symmetric disordered spin chain to confirm consistency with the analytical predictions. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation self-contained in SC ensemble properties and standard mixed-spectrum RMT

full rationale

The paper defines the SC matrix ensemble explicitly via its block-diagonal structure induced by Z2 symmetry, invokes established external results on correlations of mixed spectra, and derives survival probabilities, timescales, equilibrium values, and fluctuation decay analytically from those inputs. Equilibrium values are shown to match canonical averages and ETH fluctuations decay with size as direct consequences of the ensemble construction and standard RMT assumptions, without any parameter fitting renamed as prediction, self-definitional loops, or load-bearing self-citations. The chain remains independent of the target observables and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard assumption that the SC ensemble produces a superposition of two independent GOE-like spectra due to the conserved exchange symmetry; no additional free parameters or invented entities are introduced beyond the usual RMT level statistics.

axioms (1)
  • domain assumption Conservation of Z2 exchange symmetry splits the Hilbert space into two decoupled subspaces whose spectra are statistically independent.
    Stated in the abstract as the mechanism that makes the full spectrum a superposition of two pure spectra.

pith-pipeline@v0.9.0 · 5836 in / 1371 out tokens · 31994 ms · 2026-05-19T09:12:07.154757+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

134 extracted references · 134 canonical work pages

  1. [1]

    Since ⟨So-e⟩ is smaller than the global bandwidth Γ, comparing with Eq

    Such Gaussian decay relaxes at the time tR = 2 ⟨So-e⟩ q | ln ϵ| π given a tolerance value ϵ ≪ 1. Since ⟨So-e⟩ is smaller than the global bandwidth Γ, comparing with Eq. (18), we find that the average sur- vival probability of |Ψo-e⟩ has a much slower decay but a smaller relaxation time compared to a localized state with similar energy, as shown in Fig. 3(...

    work page

  2. [2]

    J. von Neumann, Proof of the ergodic theorem and the h-theorem in quantum mechanics: Translation of: Be- weis des ergodensatzes und des h-theorems in der neuen mechanik, The European Physical Journal H 35, 201 (2010)

    work page 2010

  3. [3]

    Goldstein, J

    S. Goldstein, J. L. Lebowitz, C. Mastrodonato, R. Tu- mulka, and N. Zangh` ı, Normal typicality and von neu- mann’s quantum ergodic theorem, Proc. R. Soc. A: Math. Phys. Eng. Sci. 466, 3203 (2010)

    work page 2010

  4. [4]

    Srednicki, Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994)

    work page 1994

  5. [5]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Ven- galattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011)

    work page 2011

  6. [6]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many-body localiza- tion and thermalization in quantum statistical mechan- ics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015)

    work page 2015

  7. [7]

    Borgonovi, F

    F. Borgonovi, F. Izrailev, L. Santos, and V. Zelevinsky, Quantum chaos and thermalization in isolated systems of interacting particles, Phys. Rep. 626, 1 (2016)

    work page 2016

  8. [8]

    J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018)

    work page 2018

  9. [9]

    E. J. Torres-Herrera and L. F. Santos, Extended noner- godic states in disordered many-body quantum systems, Ann. Phys. (Berlin) 529, 1600284 (2017)

    work page 2017

  10. [10]

    M. Pino, L. B. Ioffe, and B. L. Altshuler, Nonergodic metallic and insulating phases of josephson junction chains, Proceedings of the National Academy of Sciences 113, 536 (2016)

    work page 2016

  11. [11]

    Vidmar and M

    L. Vidmar and M. Rigol, Generalized gibbs ensemble in integrable lattice models, JSTAT 2016 (6), 064007. 9

    work page 2016

  12. [12]

    T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Ther- malization and prethermalization in isolated quantum systems: a theoretical overview, J. Phys. B: Atom. Mol. Opt. Phys 51, 112001 (2018)

    work page 2018

  13. [13]

    Sinha and S

    S. Sinha and S. Sinha, Chaos and quantum scars in bose- josephson junction coupled to a bosonic mode, Phys. Rev. Lett. 125, 134101 (2020)

    work page 2020

  14. [14]

    Dong, J.-Y

    H. Dong, J.-Y. Desaules, Y. Gao, N. Wang, Z. Guo, J. Chen, Y. Zou, F. Jin, X. Zhu, P. Zhang, H. Li, Z. Wang, Q. Guo, J. Zhang, L. Ying, and Z. Papi´ c, Disorder-tunable entanglement at infinite temperature, Science Advances 9, eadj3822 (2023)

    work page 2023

  15. [15]

    Roy and A

    S. Roy and A. Lazarides, Strong ergodicity breaking due to local constraints in a quantum system, Phys. Rev. Res. 2, 023159 (2020)

    work page 2020

  16. [16]

    Royen, S

    K. Royen, S. Mondal, F. Pollmann, and F. Heidrich- Meisner, Enhanced many-body localization in a kinet- ically constrained model, Phys. Rev. E 109, 024136 (2024)

    work page 2024

  17. [17]

    Serbyn, Z

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, Local conserva- tion laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013)

    work page 2013

  18. [18]

    Shtanko, D

    O. Shtanko, D. S. Wang, H. Zhang, N. Harle, A. Seif, R. Movassagh, and Z. Minev, Uncovering local integra- bility in quantum many-body dynamics, Nat. Commun. 16, 2552 (2025)

    work page 2025

  19. [19]

    A. K. Kulshreshtha, A. Pal, T. B. Wahl, and S. H. Si- mon, Behavior of l-bits near the many-body localization transition, Phys. Rev. B 98, 184201 (2018)

    work page 2018

  20. [20]

    Lyd˙ zba, Y

    P. Lyd˙ zba, Y. Zhang, M. Rigol, and L. Vidmar, Single- particle eigenstate thermalization in quantum-chaotic quadratic hamiltonians, Phys. Rev. B 104, 214203 (2021)

    work page 2021

  21. [21]

    Modak, S

    R. Modak, S. Mukerjee, E. A. Yuzbashyan, and B. S. Shastry, Integrals of motion for one-dimensional Ander- son localized systems, New J. Phys. 18, 033010 (2016)

    work page 2016

  22. [22]

    A. K. Das, A. Ghosh, and I. M. Khaymovich, Absence of mobility edge in short-range uncorrelated disordered model: Coexistence of localized and extended states, Phys. Rev. Lett. 131, 166401 (2023)

    work page 2023

  23. [23]

    A. K. Das and A. Ghosh, Nonergodic extended states in the β ensemble, Phys. Rev. E 105, 054121 (2022)

    work page 2022

  24. [24]

    A. K. Das, A. Ghosh, and I. M. Khaymovich, Robust nonergodicity of the ground states in the β ensemble, Phys. Rev. B 109, 064206 (2024)

    work page 2024

  25. [25]

    A. K. Das, A. Ghosh, and I. M. Khaymovich, Emer- gent multifractality in power-law decaying eigenstates (2025)

    work page 2025

  26. [26]

    A. K. Das and A. Ghosh, Eigenvalue statistics for gen- eralized symmetric and hermitian matrices, J. Phys. A: Math. Theor. 52, 395001 (2019)

    work page 2019

  27. [27]

    A. K. Das and A. Ghosh, Dynamical signatures of chaos to integrability crossover in 2 × 2 generalized random matrix ensembles, J. Phys. A: Mathematical and Theo- retical 10.1088/1751-8121/ad0b5a (2023)

    work page doi:10.1088/1751-8121/ad0b5a 2023

  28. [28]

    M. V. Berry and P. Shukla, Spacing distributions for real symmetric 2 × 2 generalized gaussian ensembles, J. Phys. A: Math. Theor. 42, 485102 (2009)

    work page 2009

  29. [29]

    A. K. Das and A. Ghosh, Transport in deformed cen- trosymmetric networks, Phys. Rev. E 106, 064112 (2022)

    work page 2022

  30. [30]

    A. K. Das and A. Ghosh, Chaos due to symmetry- breaking in deformed poisson ensemble, JSTAT 2022 (6), 063101

    work page 2022

  31. [31]

    Dutta, G

    A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, Quantum Phase Transi- tions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge Univer- sity Press, 2015)

    work page 2015

  32. [32]

    Mondal, S

    D. Mondal, S. Sinha, and S. Sinha, Chaos and quantum scars in a coupled top model, Phys. Rev. E 102, 020101 (2020)

    work page 2020

  33. [33]

    Wang and F

    Q. Wang and F. P´ erez-Bernal, Signatures of excited- state quantum phase transitions in quantum many- body systems: Phase space analysis, Phys. Rev. E 104, 034119 (2021)

    work page 2021

  34. [34]

    Casta˜ nos, R

    O. Casta˜ nos, R. L´ opez-Pe˜ na, J. G. Hirsch, and E. L´ opez-Moreno, Classical and quantum phase tran- sitions in the lipkin-meshkov-glick model, Phys. Rev. B 74, 104118 (2006)

    work page 2006

  35. [35]

    L. F. Santos, M. T´ avora, and F. P´ erez-Bernal, Excited- state quantum phase transitions in many-body systems with infinite-range interaction: Localization, dynamics, and bifurcation, Phys. Rev. A 94, 012113 (2016)

    work page 2016

  36. [36]

    P´ erez-Fern´ andez, A

    P. P´ erez-Fern´ andez, A. Rela˜ no, J. M. Arias, P. Cej- nar, J. Dukelsky, and J. E. Garc´ ıa-Ramos, Excited-state phase transition and onset of chaos in quantum optical models, Phys. Rev. E 83, 046208 (2011)

    work page 2011

  37. [37]

    M. Kloc, P. Str´ ansk´ y, and P. Cejnar, Quantum quench dynamics in dicke superradiance models, Phys. Rev. A 98, 013836 (2018)

    work page 2018

  38. [38]

    Hwang, R

    M.-J. Hwang, R. Puebla, and M. B. Plenio, Quantum phase transition and universal dynamics in the rabi model, Phys. Rev. Lett. 115, 180404 (2015)

    work page 2015

  39. [39]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum newton’s cradle, Nature 440, 900 (2006)

    work page 2006

  40. [40]

    Hofferberth, I

    S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Non-equilibrium coherence dy- namics in one-dimensional bose gases, Nature 449, 324 (2007)

    work page 2007

  41. [41]

    Gring, M

    M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Dem- ler, and J. Schmiedmayer, Relaxation and prethermal- ization in an isolated quantum system, Science 337, 1318 (2012)

    work page 2012

  42. [42]

    Langen, S

    T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized gibbs ensemble, Science 348, 207 (2015)

    work page 2015

  43. [43]

    Mallayya and M

    K. Mallayya and M. Rigol, Prethermalization, thermal- ization, and fermi’s golden rule in quantum many-body systems, Phys. Rev. B 104, 184302 (2021)

    work page 2021

  44. [44]

    Krupchitsky and V

    P. Krupchitsky and V. Kisin, Fundamental Research with Polarized Slow Neutrons (Springer Berlin Heidel- berg, 2014)

    work page 2014

  45. [45]

    G. E. Mitchell, E. G. Bilpuch, P. M. Endt, and J. F. Shriner, Broken symmetries and chaotic behavior in 26Al, Phys. Rev. Lett. 61, 1473 (1988)

    work page 1988

  46. [46]

    Guhr and H

    T. Guhr and H. Weidenm¨ uller, Isospin mixing and spec- tral fluctuation properties, Ann. Phys. 199, 412 (1990)

    work page 1990

  47. [47]

    Adams, G

    A. Adams, G. Mitchell, and J. Shriner Jr, Statistical distribution of reduced transition probabilities in 26al, Physics Letters B 422, 13 (1998)

    work page 1998

  48. [48]

    Scholak, F

    T. Scholak, F. de Melo, T. Wellens, F. Mintert, and A. Buchleitner, Efficient and coherent excitation trans- fer across disordered molecular networks, Phys. Rev. E 83, 021912 (2011)

    work page 2011

  49. [49]

    T. Zech, R. Mulet, T. Wellens, and A. Buchleitner, Cen- 10 trosymmetry enhances quantum transport in disordered molecular networks, New J. Phys. 16, 055002 (2014)

    work page 2014

  50. [50]

    Ortega, T

    A. Ortega, T. Stegmann, and L. Benet, Efficient quan- tum transport in disordered interacting many-body net- works, Phys. Rev. E 94, 042102 (2016)

    work page 2016

  51. [51]

    Ellegaard, T

    C. Ellegaard, T. Guhr, K. Lindemann, J. Nyg˚ ard, and M. Oxborrow, Symmetry breaking and spectral statis- tics of acoustic resonances in quartz blocks, Phys. Rev. Lett. 77, 4918 (1996)

    work page 1996

  52. [52]

    Schaadt, A

    K. Schaadt, A. P. B. Tufaile, and C. Ellegaard, Chaotic sound waves in a regular billiard, Phys. Rev. E 67, 026213 (2003)

    work page 2003

  53. [53]

    J. X. de Carvalho, M. S. Hussein, M. P. Pato, and A. J. Sargeant, Symmetry-breaking study with deformed en- sembles, Phys. Rev. E 76, 066212 (2007)

    work page 2007

  54. [54]

    J. X. de Carvalho, S. Jalan, and M. S. Hussein, Deformed gaussian-orthogonal-ensemble description of small-world networks, Phys. Rev. E 79, 056222 (2009)

    work page 2009

  55. [55]

    Jalan, Spectral analysis of deformed random net- works, Phys

    S. Jalan, Spectral analysis of deformed random net- works, Phys. Rev. E 80, 046101 (2009)

    work page 2009

  56. [56]

    Facchi and S

    P. Facchi and S. Pascazio, Quantum zeno subspaces, Phys. Rev. Lett. 89, 080401 (2002)

    work page 2002

  57. [57]

    L. F. Santos, F. Borgonovi, and G. L. Celardo, Coop- erative shielding in many-body systems with long-range interaction, Phys. Rev. Lett. 116, 250402 (2016)

    work page 2016

  58. [58]

    Gersho, Adaptive equalization of highly dispersive channels for data transmission, Bell System Technical Journal 48, 55 (1969)

    A. Gersho, Adaptive equalization of highly dispersive channels for data transmission, Bell System Technical Journal 48, 55 (1969)

    work page 1969

  59. [59]

    Magee and J

    F. Magee and J. Proakis, An estimate of the upper bound on error probability for maximum-likelihood se- quence estimation on channels having a finite-duration pulse response (corresp.), IEEE Transactions on Infor- mation Theory 19, 699 (1973)

    work page 1973

  60. [60]

    Datta and S

    L. Datta and S. D. Morgera, On the reducibility of centrosymmetric matrices—applications in engineering problems, Circuits, Systems and Signal Processing 8, 71 (1989)

    work page 1989

  61. [61]

    Gaudreau, Philippe and Safouhi, Hassan, Centrosym- metric matrices in the sinc collocation method for sturm-liouville problems, EPJ Web of Conferences 108, 01004 (2016)

    work page 2016

  62. [62]

    Christandl, N

    M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Perfect state transfer in quantum spin networks, Phys- ical review letters 92, 187902 (2004)

    work page 2004

  63. [63]

    Karbach and J

    P. Karbach and J. Stolze, Spin chains as perfect quan- tum state mirrors, Phys. Rev. A 72, 030301 (2005)

    work page 2005

  64. [64]

    T. Shi, Y. Li, Z. Song, and C.-P. Sun, Quantum-state transfer via the ferromagnetic chain in a spatially mod- ulated field, Physical Review A 71, 032309 (2005)

    work page 2005

  65. [65]

    Campos Venuti, C

    L. Campos Venuti, C. Degli Esposti Boschi, and M. Roncaglia, Qubit teleportation and transfer across antiferromagnetic spin chains, Phys. Rev. Lett. 99, 060401 (2007)

    work page 2007

  66. [66]

    G. M. Nikolopoulos, D. Petrosyan, and P. Lambropou- los, Electron wavepacket propagation in a chain of cou- pled quantum dots, Journal of Physics: Condensed Mat- ter 16, 4991 (2004)

    work page 2004

  67. [67]

    Rigol, V

    M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Re- laxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1d lattice hard-core bosons, Phys. Rev. Lett. 98, 050405 (2007)

    work page 2007

  68. [68]

    Gogolin, M

    C. Gogolin, M. P. M¨ uller, and J. Eisert, Absence of ther- malization in nonintegrable systems, Phys. Rev. Lett. 106, 040401 (2011)

    work page 2011

  69. [69]

    Caux and F

    J.-S. Caux and F. H. L. Essler, Time evolution of lo- cal observables after quenching to an integrable model, Phys. Rev. Lett. 110, 257203 (2013)

    work page 2013

  70. [70]

    Cantoni and P

    A. Cantoni and P. Butler, Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra and its Applications 13, 275 (1976)

    work page 1976

  71. [71]

    Pandey, A

    A. Pandey, A. Kumar, and S. Puri, Quantum chaotic systems and random matrix theory (2019)

    work page 2019

  72. [72]

    Mehta, Random Matrices, Pure and Applied Math- ematics (Elsevier Science, 2004)

    M. Mehta, Random Matrices, Pure and Applied Math- ematics (Elsevier Science, 2004)

    work page 2004

  73. [73]

    L. F. Santos and M. Rigol, Localization and the effects of symmetries in the thermalization properties of one- dimensional quantum systems, Phys. Rev. E 82, 031130 (2010)

    work page 2010

  74. [74]

    Bogomolny, U

    E. Bogomolny, U. Gerland, and C. Schmit, Short-range plasma model for intermediate spectral statistics, EPJ B 19, 121 (2001)

    work page 2001

  75. [75]

    T. Guhr, A. M¨ uller–Groeling, and H. A. Weidenm¨ uller, Random-matrix theories in quantum physics: common concepts, Phys. Rep. 299, 189 (1998)

    work page 1998

  76. [76]

    Buijsman, V

    W. Buijsman, V. Cheianov, and V. Gritsev, Sensitivity of the spectral form factor to short-range level statistics, Phys. Rev. E 102, 042216 (2020)

    work page 2020

  77. [77]

    Schiulaz, E

    M. Schiulaz, E. J. Torres-Herrera, and L. F. Santos, Thouless and relaxation time scales in many-body quan- tum systems, Phys. Rev. B 99, 174313 (2019)

    work page 2019

  78. [78]

    A. K. Das, A. Ghosh, and L. F. Santos, Spectral form factor and energy correlations in banded random matri- ces, Phys. Rev. B 111, 224202 (2025)

    work page 2025

  79. [79]

    Ourjoumtsev, R

    A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, Generating optical schr¨ odinger kittens for quantum information processing, Science 312, 83 (2006)

    work page 2006

  80. [80]

    Y. P. Huang and M. G. Moore, Creation, detection, and decoherence of macroscopic quantum superposition states in double-well bose-einstein condensates, Phys. Rev. A 73, 023606 (2006)

    work page 2006

Showing first 80 references.