Pith. sign in

REVIEW 1 major objections 6 minor 29 references

Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models

T0 review · 1 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper establishes that the entropic complexity C = S1 − S2 separates two kinds of physics in disordered quantum systems: a mid-phase eigenstate feature, reproducible across two random-matrix models and pinned to a fixed parameter value,

desk verdict A careful, honest numerical study showing an eigenstate-level entropic complexity peak in two random-matrix models that largely washes out under thermal averaging — worth engaging, but the Heisenberg leg rests on a self-cited result you should check. read the letter →

arxiv 2607.15798 v1 pith:R3JJCRGB submitted 2026-07-17 cond-mat.dis-nn

Imre Varga , Onur Aktan This is my paper · ORCID
classification cond-mat.dis-nn MSC 82B4415B5281Q50
keywords entropiccomplexityRosenzweig-Porterensemblepower-lawbandedrandommatricesmany-bodylocalizationthermalstatequantumcoherencedecoherencemultifractality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether thermal averaging destroys the fingerprints of individual eigenstate complexity in disordered quantum systems. It studies the entropic complexity C = S1 − S2 (Shannon minus second-order Rényi entropy) across the ergodic-to-localized crossover in three models: two random-matrix ensembles with an extended multifractal phase, and the random-field Heisenberg chain. Its central finding is that a sharp mid-phase maximum in the single-eigenstate complexity — located at γ* ≈ 1.5–1.6 in the Rosenzweig–Porter model and b* ≈ 0.31 in the power-law banded model, both at fractal dimension D2 ≈ 0.44 — is strongly but not completely suppressed by thermal mixing: a ~10–13% thermal shadow survives in one model, while a distinct low-temperature 'edge' feature has no eigenstate counterpart. The paper argues that entropic complexity thus cleanly separates thermal-state physics from eigenstate physics, and that care is needed in stating which quantity a measurement sees.

What carries the argument

The central object is C = S1 − S2, evaluated in three ways: on individual eigenstates (C_eig), on the thermal state's eigenvalue populations (C_tr), and on its dephased site-basis populations (C_diag). The coherence gap C_gap = C_diag − C_tr decomposes as C_rel − ΔS2, where C_rel is the relative entropy of coherence (a genuine monotone) and ΔS2 obeys ΔS2/ln N → D2 as T → 0, tying the thermal probe to the eigenstate fractal dimension. The washout mechanism is quantified by expanding C to quadratic order in site-intensity fluctuations, giving C ≈ v(γ)/(2k) for k-state mixtures, with a measured exponent α ≈ 0.85 rather than 1 due to eigenstate correlations.

What would settle it

Exact-diagonalize the random-field Heisenberg chain at L = 18–20 and scan C_eig(W) finely; if the maximum moves away from the ergodic-to-MBL crossover (W/J ≈ 4–5) into the ergodic phase as L grows, the paper's claim that the mid-phase feature is confined to random-matrix models fails. Alternatively, repeat the RP thermal scan near γ* with PLBRM-level statistics; a resolvable bump at γ* would show the RP 'no shadow' is a statistical limit rather than a structural difference.

Watch

Extended reading notes

Core claim

Entropic complexity C = S1 − S2, the difference between Shannon and second-order Rényi entropy of a distribution in a fixed site basis, is claimed to separate two structurally distinct features across the ergodic-to-localized crossover. In the Rosenzweig–Porter and power-law banded random-matrix ensembles, the single-eigenstate quantity C_eig peaks deep inside the fractal phase — γ* ≈ 1.561 ± 0.008 in RP and b* = 0.311 ± 0.008 in PLBRM, both size-independent — at fractal dimension D2 ≈ 0.44, invisible to D2 and to level statistics. Thermal averaging strongly suppresses the peak: no shadow is resolvable in RP, while a weak (~10–13%) bump survives in PLBRM’s temperature-maximized diagonal comp

Load-bearing premise

The classification of the Heisenberg chain as lacking a mid-phase eigenstate peak depends on a previously published, cited result that C_eig(W) is maximal at the ergodic-to-MBL crossover for L ≤ 16; that result is not recomputed in this paper, and if it drifts with system size the 'random-matrix-only' conclusion would need revision.

Editorial extensions

If this is right

  • Because C_eig peaks where D2 and level statistics are featureless, it offers a new, independent marker of deep-fractal-phase structure that could identify extended fractal phases in other disordered systems.
  • The strong but incomplete thermal suppression means energy-resolved, few-state observables, not full Gibbs states, are the right place to look for eigenstate-complexity signatures.
  • The scale-invariant log-ratio L = ln(T*_tr/T*_diag) vanishes on localization in all three models, giving a simple crossover indicator that is independent of the temperature rescaling.
  • In systems without an extended fractal phase, such as the Heisenberg chain, the eigenstate complexity peaks at the ergodic-to-MBL transition, so the mid-phase feature can serve to distinguish models with a genuine multifractal regime from those without.
  • The edge feature's N-dependent drift in PLBRM warns that thermal-edge locations extracted at finite size may not persist in the thermodynamic limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the mid-phase peak is driven by multifractal fluctuations rather than by the specific RP or PLBRM construction, a similar C_eig maximum should appear in any model with a broad fractal regime — e.g., quasiperiodic systems, Bethe-lattice Anderson models, or interacting models at criticality — a testable extension.
  • The RP/PLBRM difference in thermal shadow suggests a quantitative link: models with stronger energy-eigenstate correlations should show larger surviving shadows. Measuring the washout exponent α for other ensembles would predict where shadows appear.
  • The PLBRM edge feature receding with N hints that the RP edge at γ ≈ 0.8–0.9, reported as size-independent in the accessible range, may itself drift at larger N; checking this would test whether the two edges share a finite-size origin.
  • Because C_rel orders by density-of-states class rather than collapsing universally, a common scaling variable (e.g., an effective dimension) remains to be found; if found, it could collapse both edge and mid-phase features across models.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 6 minor

Summary. The paper studies the entropic complexity C = S_1 − S_2 (Shannon minus second-order Rényi entropy of a normalized distribution) across the ergodic-to-localized crossover of three disordered models: the Rosenzweig–Porter (RP) ensemble, the power-law banded random matrix (PLBRM) ensemble, and the random-field Heisenberg chain. It compares the single-eigenstate quantity C_eig with thermal-state quantities — the basis-independent trace complexity C_tr, the diagonal complexity C_diag of the dephased Gibbs state in the site (pointer) basis, the coherence gap C_gap = C_diag − C_tr, and the relative entropy of coherence C_rel. Principal findings: (i) C_eig develops a pronounced mid-phase maximum in RP at γ* ≈ 1.5–1.6 (1.561±0.008 at N=1600), located at D_2 ≈ 0.44, invisible to D_2(γ) and ⟨r⟩(γ); (ii) the same feature appears in PLBRM at b* = 0.31±0.01, N-independent over N=200–1600; (iii) thermal averaging strongly suppresses this feature — no resolvable shadow in RP at matched statistics, but a ~10–13% bump at b* in high-statistics PLBRM scans, attributed to a slower-than-1/k microcanonical washout (measured α≈0.85 vs. the α=1 independence prediction of Eq. (12)); (iv) a distinct thermal edge feature (γ≈0.8–0.9 in RP; W*/J≈0.5–0.7 in the Heisenberg chain) has no eigenstate-level counterpart and recedes with N in PLBRM (b*_edge ∝ N^{0.60±0.07}); (v) T*_tr and T*_diag merge upon localization, defining the scale-invariant indicator L = ln(T*_tr/T*_diag). The paper concludes that the mi

Significance. If the central taxonomy holds, the paper offers a useful organizing distinction: entropic complexity computed on individual eigenstates and on the thermal state tracks different physics, with a mid-phase wavefunction-level maximum reproduced in two structurally independent random-matrix ensembles at a comparable fractal dimension (D_2 ≈ 0.44) and cleanly separated from a thermal edge feature. The numerical discipline is a genuine strength: the Haar-random-basis rotation control flattens C_eig to the analytic GOE value 0.366±0.001; the D_2 = 1/2 hypothesis for the peak location is tested and ruled out at accessible sizes; the paper's analytic predictions are falsified rather than fitted — Eq. (10) predicts a flat plateau and Eq. (12) predicts α = 1, both contradicted by data, with the measured α ≈ 0.85 supplying the mechanism for the surviving PLBRM shadow; peak locations carry bootstrap errors; and the coherence-resource distinction (C_rel monotone under dephasing, C_gap not) is verified dynamically. The main risk to the significance of the classification is the Heisenberg-chain leg: the absence of a mid-phase eigenstate feature is imported from the self-cited Ref. [18] and not rec

major comments (1)
  1. [Sec. III E / III G; Abstract; Conclusion] The claim that the mid-phase eigenstate feature is 'confined to the two random-matrix models' (Abstract; Sec. III G; Conclusion) rests entirely on the assertion that Heisenberg-chain C_eig(W) peaks at the ETH–MBL transition (W≈4–5), not at an interior point. That assertion is not computed here: it is imported from Ref. [18], a prior paper by the first author, and the text marks it 'structural rather than numerical.' The gap is concrete: the paper already exact-diagonalizes the same chain at L=10–16 for the thermal diagnostics and ⟨r⟩ correlations of Sec. III E, so C_eig(W) on that ensemble is modest additional work. The same paper documents finite-size drift of the PLBRM edge feature (b*_edge ∝ N^{0.60±0.07}, Sec. III D), so an L≤16 null result cannot be assumed stable. Recommend either computing C_eig(W) directly, or re-scoping the abstract/conclusion to make the confinement claim condi
minor comments (6)
  1. [Sec. III B / Appendix A] Eq. (12) is labeled twice (Sec. III B and again in Appendix A). Renumber the second occurrence.
  2. [Sec. III A] At N=200 the RP peak height (C*_eig ≈ 0.70) is only ~1% above the flat mean-field value ln 2 ≈ 0.693; the phrase 'rises well above ln 2' is accurate only at larger N. State the N-dependence of the excess explicitly.
  3. [Sec. III A / Fig. 3] The 'essentially independent of N' statement for γ* rests on coarse production grids for N<1600; the high-resolution scan (Δγ≈0.03) is at N=1600 only. A high-resolution scan at one additional size, or a table of peak locations with bootstrap errors per size, would make the fixed-point claim quantitative.
  4. [Sec. III D / Sec. III B] The ~10–13% PLBRM thermal bump is described as 'right at b*' but no uncertainty or width is given for its location; a bootstrap estimate in the same style as the C_eig peak would strengthen the shadow identification. Relatedly, the matched-statistics RP null of Sec. III B is at N=200 only; state this explicitly wherever 'no thermal trace' is asserted.
  5. [Sec. III E] The per-sample Pearson correlations (|ρ|≲0.1) are quoted without standard errors; with 2000 samples the standard error of ρ is ≈0.022, so ρ≈0.06 is ~3σ from zero. 'Negligible' is defensible in variance-explained terms (~1%), but should be stated that way.
  6. [Sec. III E] The Heisenberg edge location W*/J ≈ 0.5–0.7 is 'set by adjacent points of the W-grid rather than a bootstrap-resolved location,' in contrast to the peak-location methodology used for RP and PLBRM. A parabolic interpolation on the log grid would improve cross-model comparability.

Circularity Check

1 steps flagged · score 4.0 of 10

Central taxonomy leans on un-recomputed self-cited Heisenberg C_eig null result; RP/PLBRM core is otherwise self-contained and benchmarked.

  1. self citation load bearing [Sec. III E (Heisenberg MBL); used again in Sec. III G, Abstract, and Conclusion]
    "The eigenstate-only entropic complexity C_eig(W) of this same chain – built from exactly the Shannon and Rényi-2 entropies used here (there termed the structural entropy) – has been characterized previously [18], where it is maximal on the non-ergodic but extended states in the critical region near the ETH–MBL transition (W≈4–5 for the lengths L≤16 accessible to exact diagonalization), rising monotonically out of the ergodic phase rather than turning over at an interior, mid-phase point."

    The manuscript's central structural claim — 'This mid-phase eigenstate feature is confined to the two random-matrix models' — depends on the Heisenberg chain having no mid-phase C_eig maximum. That leg is not computed here; the text imports it from [18], a self-cited paper by the first author, and even labels the move 'structural rather than numerical.' The assertion is load-bearing for the Abstract/Conclusion taxonomy. The same manuscript documents N-dependent drift of the PLBRM edge feature (b*_edge ∝ N^{0.60±0.07}), so finite-size drift is a live possibility for L≤16 Heisenberg results; without recomputation or an external check, the 'confined to random-matrix models' conclusion reduces, for one of its three model legs, to the author's own prior claim.

full rationale

Aside from the Heisenberg leg, the derivation chain is not circular. The RP and PLBRM observations are numerical measurements anchored to external benchmarks (analytic GOE plateau C_GOE ≈ 0.369, the D2 = 2−γ law, mean gap ratio), and the analytic 'predictions' in the paper are falsified rather than fitted: Eq. (10) predicts a flat C_eig ≈ ln 2 but the data show a γ-dependent maximum, and Eq. (12) predicts α = 1 in the washout while Appendix A measures α ≈ 0.85. The ∆S2 = D2 ln N relation is presented as an identity, not as a derived prediction. The only load-bearing step that reduces to a self-citation is the Heisenberg null result imported from Ref. [18]; because that null result is exactly what makes the feature 'confined to random-matrix models,' the score is 4 rather than 0–2. If Ref. [18] were independently reproduced, the score would drop to 0–2.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

Central measurements use standard exact diagonalization and textbook random-matrix ensembles; the only paper-specific premises are (a) the CLT-style independence/exchangeability assumption behind the washout law Eq. (12) (Appendix A, flagged by the authors as reasonable for RP but not guaranteed in general) and (b) the imported Heisenberg eigenstate-complexity result from self-cited [18]. Free parameters are the PLBRM edge exponent (four-size log-log fit, two degrees of freedom), the measured O(1) intensity variance v(γ), and the measured washout exponent α ≈ 0.85 — the last two are empirical inputs to the washout mechanism, not tuned knobs. No invented entities.

free parameters (3)
  • PLBRM edge-location scaling exponent = 0.60 ± 0.07 (b*_edge ∝ N^0.60)
    Weighted log-log regression over N = 200–1600 (four sizes, two degrees of freedom); quantifies the claim that the thermal edge recedes with N while the mid-phase peak does not. The authors admit this is a rough estimate.
  • v(γ): O(1) relative variance of single-eigenstate site intensities = O(1), N-independent (measured, not tuned)
    Enters the washout prediction Eq. (12), C(ε,k) ≈ v(γ)/(2k). Measured from eigenstates rather than fitted; the prediction built on it is only partially confirmed (washout exponent α ≈ 0.85 vs predicted 1), so it functions as an empirical input, not a fitted knob.
  • Washout exponent α (measured) = 0.85 ± 0.02 at b ∈ [0.2, 0.4], k = 64–256
    Measured from the microcanonical washout data (Fig. 8); the deviation of α from the independence value 1 is used as the mechanism explaining the survival of the PLBRM thermal shadow — an empirically determined quantity doing mechanistic work.
assumptions (7)
  • standard math Dephasing cannot increase purity: Tr ρ² ≥ Σ_i ρ_ii², hence ΔS_2 = S_2^diag − S_2^full ≥ 0
    Sec. II C, used to define the nonnegativity of the coherence-gap decomposition (Eq. 5).
  • standard math Second-order Taylor expansion of Shannon/Rényi entropies in the site-intensity fluctuations δ_i (Eq. A1) with the constraint Σ δ_i = 0
    Appendix A, the basis of the washout law Eq. (12); valid only for small fluctuations (large k).
  • domain assumption RP ensemble has exact transitions at γ = 1 and γ = 2 with mono-fractal dimension D_q = 2 − γ
    Sec. II D, from Kravtsov, Khaymovich, Cuevas, Amini [7]; used to place the measured fractal dimension and the mid-phase location in context.
  • domain assumption PLBRM at µ = 1 is critical for every b, with fractal dimension and level statistics varying continuously in b
    Sec. II E, from Mirlin et al. [19] and Varga–Braun [22]; grounds the interpretation that any C_eig(b) peak is a property of critical wavefunction statistics rather than proximity to a phase boundary.
  • domain assumption Random-field Heisenberg chain has its MBL crossover at W_c/J ≈ 3.5–4 for these sizes
    Sec. II F, from Luitz, Laflorencie, Alet [15]; the paper notes the sharpness/existence of the transition in the thermodynamic limit remains debated [3,9].
  • domain assumption Heisenberg-chain eigenstate complexity C_eig(W) peaks at the ETH–MBL crossover (W ≈ 4–5, L ≤ 16), not at an interior mid-phase point
    Sec. III E, imported from self-cited [18] (Varga, PRA 113, 022606) without recomputation; this is the load-bearing premise for the structural claim that the mid-phase feature is confined to random-matrix models. The weakest external input.
  • ad hoc to paper Approximate statistical independence of nearby-in-energy eigenstates and statistical exchangeability of sites (CLT for the microcanonical mixture)
    Appendix A, needed for the washout law Eq. (12); the paper itself states these are 'reasonable for RP but not guaranteed in general' and inapplicable to the Heisenberg chain.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models." pith.science (2026). https://pith.science/paper/R3JJCRGB

@misc{pith2026260715798,
  author       = {Pith},
  title        = {Pith review of: Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R3JJCRGB}},
  note         = {Machine review of arXiv:2607.15798}
}
read the original abstract

Does thermal averaging preserve signatures of eigenstate complexity? We study the entropic complexity C = S_1 - S_2 (Shannon minus second-order Renyi entropy) across the ergodic-to-localized crossover of three disordered models: the Rosenzweig-Porter (RP) ensemble, the power-law banded random matrix (PLBRM) ensemble, and the random-field Heisenberg chain. We compare a wavefunction-level quantity C_eig to the complexity of the thermal (Gibbs) state before and after pointer-basis dephasing, via the trace (C_tr) and diagonal (C_diag) complexities. The answer is mostly no: thermal averaging strongly suppresses, but does not eliminate, eigenstate-complexity signatures. C_eig develops a pronounced mid-phase maximum in RP and, at matching fractal dimension, in the structurally independent PLBRM model; a high-statistics scan resolves a weak (about 10%) but reproducible thermal shadow of this peak in the thermal diagonal complexity. This feature is confined to the two random-matrix models; in the Heisenberg chain the eigenstate complexity instead peaks at the many-body localization transition. A second, genuinely thermal feature, an edge just inside the ergodic phase, has no eigenstate counterpart and, in PLBRM, recedes with system size. Both features are tracked by scale-invariant crossover indicators: the log-ratio of the trace and diagonal crossover temperatures, which vanishes upon localization, and the relative entropy of coherence. Entropic complexity thus cleanly separates thermal-state and eigenstate physics: a sharp wavefunction-level feature, reproducible across unrelated random-matrix constructions, leaves only a faint, structurally distinct imprint on thermal observables.

Figures

Figures reproduced from arXiv: 2607.15798 by the authors.

Figure 1
Figure 1. FIG. 1. Temperature dependence of the trace ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Trace versus diagonal characteristic temperatures across the three models and their full range of system sizes: Heisenberg [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Rosenzweig–Porter model, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Power-law banded random matrix model, Eq. (8), [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. PLBRM thermal analysis, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Random-field Heisenberg chain, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Density-of-states [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Test of the self-averaging law Eq. (12) at the PLBRM [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 2 linked inside Pith

  1. [18]

    Complexity in multiqubit and many-body sys- tems,

    I. Varga, “Complexity in multiqubit and many-body sys- tems,” Phys. Rev. A113, 022606 (2026)

    2026

  2. [1]

    Disorder in quantum many-body systems,

    T. Vojta, “Disorder in quantum many-body systems,” Annu. Rev. Condens. Matter Phys.10, 233 (2019)

    2019

  3. [2]

    From quantum disor- der to quantum chaos,

    I. V. Gornyi and A. D. Mirlin, “From quantum disor- der to quantum chaos,” J. Low Temp. Phys.126, 1339 (2002)

    2002

  4. [3]

    Col- loquium: Many-body localization, thermalization, and entanglement,

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Col- loquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys.91, 021001 (2019)

    2019

  5. [4]

    Charac- terization of chaotic quantum spectra and universality of level fluctuation laws,

    O. Bohigas, M. J. Giannoni, and C. Schmit, “Charac- terization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett.52, 1 (1984)

    1984

  6. [5]

    Haake,Quantum Signatures of Chaos(Springer, 1991)

    F. Haake,Quantum Signatures of Chaos(Springer, 1991)

    1991

  7. [6]

    Repulsion of energy levels in complex atomic spectra,

    N. Rosenzweig and C. E. Porter, “Repulsion of energy levels in complex atomic spectra,” Phys. Rev.120, 1698 (1960)

    1960

  8. [7]

    A random matrix model with localization and ergodic transitions,

    V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, “A random matrix model with localization and ergodic transitions,” New J. Phys.17, 122002 (2015)

    2015

Show all 29 references
  1. [8]

    Metal- insulator transition in a weakly interacting many-electron system with localized single-particle states,

    D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metal- insulator transition in a weakly interacting many-electron system with localized single-particle states,” Ann. Phys. 321, 1126 (2006)

    2006

  2. [9]

    Quan- tum chaos challenges many-body localization,

    J. ˇSuntajs, J. Bonˇ ca, T. Prosen, and L. Vidmar, “Quan- tum chaos challenges many-body localization,” Phys. Rev. E102, 062144 (2020), arXiv:1905.06345

    2020 arXiv

  3. [10]

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

    G. C. Santra, A. Windey, S. Bandyopadhyay, A. Legra- mandi, and P. Hauke, “Complexity transitions in chaotic quantum systems: Nonstabilizerness, entanglement, and fractal dimension in SYK and random matrix models,” arXiv:2505.09707 (2025)

    2025

  4. [11]

    A sta- tistical measure of complexity,

    R. L´ opez-Ruiz, H. L. Mancini, and X. Calbet, “A sta- tistical measure of complexity,” Phys. Lett. A209, 321 (1995)

    1995

  5. [12]

    Features of the extension of a statistical measure of complexity to continuous systems,

    R. G. Catal´ an, J. Garay, and R. L´ opez-Ruiz, “Features of the extension of a statistical measure of complexity to continuous systems,” Phys. Rev. E66, 011102 (2002)

    2002

  6. [13]

    Quanti- fying coherence,

    T. Baumgratz, M. Cramer, and M. B. Plenio, “Quanti- fying coherence,” Phys. Rev. Lett.113, 140401 (2014)

    2014

  7. [14]

    Many-body localization phase transition,

    A. Pal and D. A. Huse, “Many-body localization phase transition,” Phys. Rev. B82, 174411 (2010)

    2010

  8. [15]

    Many-body localization edge in the random-field Heisenberg chain,

    D. J. Luitz, N. Laflorencie, and F. Alet, “Many-body localization edge in the random-field Heisenberg chain,” Phys. Rev. B91, 081103(R) (2015)

    2015

  9. [16]

    From non-ergodic eigenvectors to local resolvent statistics and back: A test of ergodicity for the Anderson model on the Bethe lat- tice,

    D. Facoetti, P. Vivo, and G. Biroli, “From non-ergodic eigenvectors to local resolvent statistics and back: A test of ergodicity for the Anderson model on the Bethe lat- tice,” Europhys. Lett.115, 47003 (2016)

    2016

  10. [17]

    Eigenfunction distribu- tion for the Rosenzweig-Porter model,

    E. Bogomolny and M. Sieber, “Eigenfunction distribu- tion for the Rosenzweig-Porter model,” Phys. Rev. E98, 032139 (2018)

    2018

  11. [19]

    Transition from localized to ex- tended eigenstates in the ensemble of power-law random banded matrices,

    A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, “Transition from localized to ex- tended eigenstates in the ensemble of power-law random banded matrices,” Phys. Rev. E54, 3221 (1996)

    1996

  12. [20]

    Power- law banded random matrix ensemble as a model for quantum many-body Hamiltonians,

    W. Buijsman, M. Haque, and I. M. Khaymovich, “Power- law banded random matrix ensemble as a model for quantum many-body Hamiltonians,” Phys. Rev. E113, 034116 (2026)

    2026

  13. [21]

    Complexity measure diagnostics of ergodic to many-body localization transi- tion,

    K. Cohen, Y. Oz, and D. Zhong, “Complexity measure diagnostics of ergodic to many-body localization transi- tion,” Phys. Rev. B110, L180101 (2024)

    2024

  14. [22]

    Critical statistics in a power- law random-banded matrix ensemble,

    I. Varga and D. Braun, “Critical statistics in a power- law random-banded matrix ensemble,” Phys. Rev. B61, R11859 (2000)

    2000

  15. [23]

    Fluctuation of correlation dimension and in- 14 verse participation number at the Anderson transition,

    I. Varga, “Fluctuation of correlation dimension and in- 14 verse participation number at the Anderson transition,” Phys. Rev. B66, 094201 (2002)

    2002

  16. [24]

    Entanglement in disordered systems at criticality,

    I. Varga and J. A. M´ endez-Berm´ udez, “Entanglement in disordered systems at criticality,” Phys. Status Solidi C 5, 867 (2008)

    2008

  17. [25]

    Scattering at the Anderson transition: Power-law banded random matrix model,

    J. A. M´ endez-Berm´ udez and I. Varga, “Scattering at the Anderson transition: Power-law banded random matrix model,” Phys. Rev. B74, 125114 (2006)

    2006

  18. [26]

    Conductance statistics for the power-law banded ran- dom matrix model,

    J. A. M´ endez-Berm´ udez, V. A. Gopar, and I. Varga, “Conductance statistics for the power-law banded ran- dom matrix model,” Phys. Rev. B82, 125106 (2010)

    2010

  19. [27]

    Scattering and transport statistics at the metal- insulator transition: A numerical study of the power-law banded random matrix model,

    J. A. M´ endez-Berm´ udez, V. A. Gopar, and I. Varga, “Scattering and transport statistics at the metal- insulator transition: A numerical study of the power-law banded random matrix model,” arXiv:1001.2568 (2010)

    2010 arXiv

  20. [28]

    Multifractal dimensions for critical random matrix en- sembles,

    J. A. M´ endez-Berm´ udez, A. Alc´ azar-L´ opez, and I. Varga, “Multifractal dimensions for critical random matrix en- sembles,” EPL98, 37006 (2012)

    2012

  21. [29]

    Strong eigenfunction correlations near the Anderson localization transition,

    Y. V. Fyodorov and A. D. Mirlin, “Strong eigenfunction correlations near the Anderson localization transition,” Phys. Rev. B55, R16001 (1997)

    1997

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.