Pith. sign in

REVIEW 4 major objections 5 minor 31 references

Analytical Theory for Anomalous Diffusion in the Anderson Model with Heavy Tails

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that a noninteracting Anderson model with heavy-tailed hopping amplitudes produces an intermediate subdiffusive regime whose spatial extent diverges as the localization transition is approached, while the crossover…

desk verdict Solid single-particle null model for Griffiths-like subdiffusion; the headline divergence is predicted outside the paper's own validity regime. read the letter →

arxiv 2608.10868 v1 pith:VQOFGM5X submitted 2026-08-11 cond-mat.dis-nn math-phmath.MPquant-ph

Elizaveta Safonova , Aleksey Lunkin , Mikhail Feigel'man This is my paper · ORCID
classification cond-mat.dis-nnmath-phmath.MPquant-ph
keywords heavy-tailedhoppingAndersonlocalizationanomalousdiffusionsubdiffusivecrossovermean-squaredisplacementrare-regiontransportmany-body
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 aims to establish that a single noninteracting particle moving on a disordered lattice with power-law-tailed hopping amplitudes can diffuse sublinearly—mean-square displacement growing with an effective exponent below 1—for a long intermediate time before ordinary diffusion takes over. The heavy-tail exponent $\mu$ controls the hopping statistics; as $\mu$ approaches $1$ from above, the spatial extent of this subdiffusive window grows as $R_{\rm sub}\sim r_0/\sqrt{\mu-1}$, while the crossover time measured in units of $\Gamma_0^{-1}$ stays finite. If true, this gives a minimal single-particle mechanism that reproduces the slow, rare-region-like transport often attributed to many-body effects near the many-body localization transition. The result matters because it provides an analytically controlled benchmark for separating disorder-driven dynamics from interaction-driven dynamics in numerical studies of disordered systems.

What carries the argument

The engine of the calculation is the saddle-point solution $g_\kappa(\varphi)\simeq(\beta\Gamma_\kappa \varphi^\dagger\varphi/4)^{\mu/2}$ for the supersymmetric effective action, which replaces the heavy-tailed hopping ensemble by an effective medium characterized by the self-consistent hybridization scale $\Gamma_\kappa$. This yields the correlation function $\hat K_q(\kappa)=c_0(\kappa)/(1-\tilde w(q)\lambda_0(\kappa))$, whose curvature in $q$ and dependence on $\kappa$ determine the conductivity and, through the exact relation to $\langle r^2(t)\rangle$, the mean-square displacement. The heavy-tail exponent $\mu$ enters through the stable distribution of hopping amplitudes and controls both the diffusion coefficient and the singular prefactor that produces the $(\mu-1)^{-1/2}$ growth of the subdiffusive window.

What would settle it

Directly simulate the three-dimensional Hamiltonian for a sequence of exponents approaching $\mu=1^+$ within the controlled parameter regime, and extract the distance at which the local exponent $\xi(t)$ reaches a fixed value such as 0.9. The central claim requires that distance to grow as $(\mu-1)^{-1/2}$ while the renormalized crossover time $\Gamma_0 t_\xi$ stays essentially constant; observing a saturation of the distance or a diverging crossover time would invalidate the claimed mechanism.

Watch

Extended reading notes

Core claim

The paper derives an analytical expression for the disorder-averaged retarded–advanced correlation function and from it the time-dependent mean-square displacement. The central quantitative result is that for $1<\mu<2$ the effective exponent $\xi(t)=d\log\langle r^2(t)\rangle/d\log t$ lies below 1 over an intermediate window, with the asymptotic form $\xi(t)\approx \Gamma_0 t/(\Gamma_0 t+(2-\mu)\Gamma(2-2/\mu)(\mu-1))$, and only reaches the diffusive value $\xi=1$ at long times. As $\mu\to 1^+$, the length scale $R_{\rm sub}$ over which the local exponent stays below a fixed value diverges as $r_0/\sqrt{\mu-1}$, whereas the corresponding crossover time $\Gamma_0 t_\xi$ approaches the finite constant $2\xi/(1-\xi)$. The paper therefore predicts a parametrically extended spatial crossover, not a true asymptotic subdiffusive phase.

Load-bearing premise

The derivation relies on a saddle-point approximation that is controlled only when the hopping range contains many sites, the dimensionless conductance is large, and the heavy-tail exponent stays above a threshold $\mu^*>1$; as $\mu\to 1^+$ the regime of validity shrinks, so the predicted divergence of the subdiffusive window is asserted in a limit the calculation does not fully control.

Editorial extensions

If this is right

  • For every $1<\mu<2$, transport is asymptotically diffusive, but the effective exponent $\xi(t)$ remains below 1 over an intermediate window that can extend over distances much larger than the microscopic hopping length $r_0$.
  • As the localization transition is approached, the distance over which subdiffusion appears grows without bound as $r_0/\sqrt{\mu-1}$, while the crossover time in units of $\Gamma_0^{-1}$ stays finite.
  • The frequency-dependent conductivity displays an apparent power-law $\sigma(\omega)\sim\omega^{\alpha(\omega)}$ with exponent below 1 at intermediate frequencies, before settling into the diffusive dc limit at low frequency.
  • A noninteracting disordered Hamiltonian can reproduce the slow-transport signatures often taken as evidence for many-body physics, so those signatures alone are not diagnostic of interactions.
  • The closed-form expressions provide a benchmark for numerical simulations of interacting disordered models, allowing disorder-driven subdiffusion to be identified and subtracted.

Reading between the lines

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

  • The same correlation function should produce a subdiffusive window in local observables such as the return probability or the participation entropy; the paper does not compute these, but the mechanism is directly carried by $\hat K_q(\kappa)$.
  • If the divergence is purely spatial and not temporal, then finite-time measurements at a fixed distance will misread the crossover as an asymptotic anomalous phase; checking system-size dependence of the effective exponent would expose the crossover.
  • Because the paper's own validity analysis leaves a threshold $\mu^*>1$, the most informative tests are those that push $\mu$ toward $1$ while scaling the model parameters to keep the saddle-point regime controlled—whether $R_{\rm sub}$ truly diverges or saturates remains an open quantitative question.
  • A natural next step is to add weak interactions to this single-particle model and ask whether the heavy-tail subdiffusive window survives or is cut off; the present paper stops at the noninteracting level.
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

4 major / 5 minor

Summary. The manuscript develops an analytical, supersymmetric field-theoretic treatment of transport in a noninteracting Anderson model with power-law (heavy-tailed) distributed hopping amplitudes. It derives a closed-form correlation function, Eq. (8), and from it the mean-square displacement and an effective dynamical exponent ξ(t). The central claim is that for 1<μ<2 the system exhibits a long intermediate subdiffusive regime before crossing over to asymptotic diffusion, and that as μ approaches the localization transition μ→1+, the spatial extent of the subdiffusive window diverges as R_sub ~ r0/√(μ−1), while the dimensionless crossover time stays finite. The analytical results are compared with a direct time-evolution simulation in three dimensions at μ=1.6.

Significance. If the central claim holds, the paper provides a single-particle, disorder-only mechanism for long-lived subdiffusive transport that qualitatively mimics Griffiths-type dynamics near the MBL transition, which would be a valuable null model for interaction-effect studies. The paper's strengths are its analytic closed-form expressions, the unified treatment of orthogonal and unitary symmetry classes, the explicit prediction for the frequency-dependent conductivity, and a direct numerical check. However, the headline μ→1+ divergence is obtained in a limit that the paper's own validity conditions exclude, and the numerical verification uses one parameter set with a fitted energy scale, so the claimed universality and the μ-dependence of the effect are not yet convincingly established.

major comments (4)
  1. [Physical discussion and regime of validity; Eq. (16)] The paper's central claim, Eq. (16) (R_sub ~ r0/√(μ−1) as μ→1+), is not controlled by the derivation. The section 'Physical discussion and regime of validity' states that a(μ) in Eq. (10) diverges as μ→1 and that 'there always remains a threshold μ* > 1 such that the theory is valid only for μ > μ*.' The saddle-point solution Eq. (33) and the delta-function approximation Eq. (38) are only justified in this valid regime, so the approach to μ=1 exits the controlled domain before the divergence can be established. To sustain the claim, the authors must analyze the double limit μ→1+ together with P_D(E)h0→0 and show that the scaling R_sub ~ 1/√(μ−1) is not altered; otherwise Eq. (16) is an extrapolation across an uncontrolled boundary.
  2. [Direct numerical simulations; Fig. 4] The numerical verification is too weak to support the μ-dependence of the theory. Only one value, μ=1.6, is simulated in 3D, and the energy scale Γ0 is not computed from Eq. (10) but is extracted by fitting the long-time tail of the same data to the predicted diffusive law, Eq. (12). The comparison with the full theory therefore does not independently test the predicted timescale or its μ-dependence. The figure also shows no error bars and only n=20–50 disorder realizations. Tests closer to μ=1, where the subdiffusive window is predicted to be large, or an independent determination of Γ0, are needed.
  3. [End Matter, Derivation of main results; Eq. (13)] Equation (13), the asymptotic form of the mean-square displacement, is load-bearing for the crossover time and length in Eqs. (14)–(16), but its derivation is not shown. The End Matter states that it follows from a higher-order small-κ expansion of Eq. (9), yet the expansion is not presented. Please include the expansion or an explicit reference to the derivation, together with the conditions under which the constant term is valid, because that constant term is what produces the preasymptotic enhancement over diffusion.
  4. [End Matter, Saddle-point equation; Eq. (38)] The delta-function approximation Eq. (38) and the separation of g(ϕ) into real- and imaginary-part contributions in Eq. (43) are imported from Ref. [27] and Ref. [29], respectively. The justification of Eq. (38) relies on the estimate ϕ†Lϕ ~ W/h0^2 in Eq. (44) from Ref. [27]. Since these steps are essential for the saddle-point solution Eq. (33) and hence for the main result Eq. (8), the manuscript should either reproduce these estimates in the present spatial setting or state explicitly that the zero-dimensional results are assumed to carry over and under what conditions.
minor comments (5)
  1. [Eq. (18)] The mean-square displacement is written as ⟨ψ(t)|Σ_n (r_n − r_n)^2|ψ(t)⟩; the second r_n should be the position of the initial state, e.g., r_0, otherwise the expression is identically zero.
  2. [Reference [14]] Reference [14] has a malformed author list; it should be corrected to 'Yan V. Fyodorov, Alexander D. Mirlin, and Hans-Jürgen Sommers.'
  3. [Fig. 4 caption] Fig. 4 reports no error bars; please add them or state that the scatter between realizations is smaller than the symbol size.
  4. [Eq. (9) and throughout] The notation Γκ should be defined as Γ_κ when first introduced, and the subscript should be used consistently instead of Γκ.
  5. [Table following Eq. (3)] For β=2, the distribution of hopping amplitudes should state explicitly whether the magnitude |t_nm| and the phase θ_nm are independent random variables.

Circularity Check

0 steps flagged · score 0.0 of 10

No equation-level circularity: the transport formulas are derived from the stated microscopic model; the fitted Gamma0 is a calibration of one time scale, and the mu -> 1 divergence is a regime-of-validity extrapolation rather than a definitional reduction.

full rationale

I walked the derivation chain: Eq. (2) -> supersymmetric action (24) -> saddle-point equation (32) -> solution (33) -> correlation function (8)-(9) -> mean-square displacement (11) -> asymptotic forms (13)-(14) -> crossover length (16). Each link is an explicit analytic manipulation of the stated heavy-tailed hopping model; none of the output formulas is identical by construction to an input parameter. The saddle-point solution (33) is quoted from the authors' earlier LRP papers (Refs. [27-29]), and the End Matter re-derives it, so the self-citations are derivation support rather than definitional substitution. The numerical check in Fig. 4 determines Gamma0 by fitting the same data's long-time diffusion constant through Eq. (12); this is a one-parameter calibration, and it does not by construction produce the subdiffusive shape, the finite crossover time in units of Gamma0^{-1}, or the R_sub ~ r0/sqrt(mu-1) divergence. The most serious weakness is the paper's own admission that below a threshold mu* > 1 the approximation fails ('there always remains a threshold mu* > 1 such that the theory is valid only for mu > mu*'), so Eq. (16) is an extrapolation across an uncontrolled boundary; that is a validity/correctness caveat, not a circular reduction. No step reduces to its input by definition, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The analytical result is not derived from first principles in this preprint: it imports exact zero-dimensional LRP results from Refs [27-29] and the supersymmetric formalism from Ref [14]. The only parameter fitted to data is the Gamma0 rescaling factor in the direct simulation comparison; h0, W, r0, p0, mu are model inputs. The calculation assumes broad on-site disorder, large connectivity, weak hybridization, and a saddle-point/delta-function approximation estimated using a self-cited result. No new particles, forces, dimensions, or conserved quantities are introduced.

free parameters (1)
  • Gamma0 rescaling factor in direct numerical simulations = not quoted in paper
    Gamma0 is extracted by fitting the numerical MSD to the long-time diffusive form 6Dt via Eq. (12), because Eq. (10) is only asymptotic in h0/W; this calibration absorbs finite-h0/W corrections into the time axis of Fig. 4.
assumptions (5)
  • domain assumption On-site disorder is much broader than the hopping scale, W/h0 >> 1, so averages over on-site and hopping disorder decouple.
    Invoked at the start of 'Averaging over off-diagonal disorder' in the End Matter; necessary to write the action in Eq. (24).
  • domain assumption The mean-field (saddle-point) approximation requires large connectivity w0 >> 1 and dimensionless conductance P_D(E)Gamma0 w0 >> 1, while weak hybridization requires P_D(E)Gamma0 << 1.
    Stated in 'Physical discussion and regime of validity'; these inequalities define the intermediate regime in which the theory lives.
  • domain assumption The real and imaginary parts of the self-energy fluctuate independently, and the ReSigma renormalization can be neglected.
    Used to replace the F integral by the delta function in Eq. (38) and to reduce g_kappa to Eq. (33); based on Refs [27,28,31].
  • domain assumption The estimate phi-dagger L phi ~ W/h0^2 >> 1/W, quoted from Ref. [27], justifies the delta-function approximation for on-site disorder.
    End Matter Eq. (44); this is a result from the authors' earlier paper, not re-derived in this preprint.
  • domain assumption Zero-dimensional LRP exact solutions from Refs [28,29] transfer to the spatially extended model, with only the nu=0 angular harmonic contributing to the correlation function.
    End Matter 'Derivation of main results', after Eq. (50); this is the bridge from 0D LRP to finite-range lattice transport.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Analytical Theory for Anomalous Diffusion in the Anderson Model with Heavy Tails." pith.science (2026). https://pith.science/paper/VQOFGM5X

@misc{pith2026260810868,
  author       = {Pith},
  title        = {Pith review of: Analytical Theory for Anomalous Diffusion in the Anderson Model with Heavy Tails},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQOFGM5X}},
  note         = {Machine review of arXiv:2608.10868}
}
abstract

We develop an analytical theory of anomalous transport in a noninteracting Anderson model with heavy-tailed hopping amplitudes. The broad distribution of hopping amplitudes gives rise to an extended intermediate-time regime with a subdiffusive effective exponent, despite the absence of interactions or genuine many-body effects. By solving the transport equations analytically, we derive the time dependence of the mean-square displacement and identify a continuous crossover from an intermediate anomalous regime to asymptotically diffusive transport. As the localization transition is approached, the spatial extent of the subdiffusive window diverges parametrically, while the crossover to conventional diffusion remains finite in units of $\Gamma_0^{-1}$. This produces an increasingly broad anomalous transport regime in space that can closely resemble Griffiths-type transport observed near the many-body localization transition. Our results demonstrate that rare hopping processes alone provide a microscopic single-particle mechanism for robust transport anomalies, establishing an analytical benchmark for distinguishing interaction-induced effects from disorder-driven dynamics.

Figures

Figures reproduced from arXiv: 2608.10868 by the authors.

Figure 1
Figure 1. FIG. 1. Color map of the normalized scaling exponent [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Top: Typical displacement [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Frequency-dependent conductivity scaling exponent [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

  1. [27]

    Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach.SciPost Phys., 18:010, 2025

    Elizaveta Safonova, Mikhail Feigel’man, and Vladimir Kravtsov. Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach.SciPost Phys., 18:010, 2025

    work page 2025

  2. [29]

    Local den- sity of states correlations in the L´ evy-Rosenzweig-Porter random matrix ensemble.SciPost Phys., 19:015, 2025

    Aleksey Lunkin and Konstantin Tikhonov. Local den- sity of states correlations in the L´ evy-Rosenzweig-Porter random matrix ensemble.SciPost Phys., 19:015, 2025

    work page 2025

  3. [1]

    P. W. Anderson. Absence of diffusion in certain random lattices.Phys. Rev., 109:1492–1505, Mar 1958

    1958

  4. [2]

    Fleishman and P

    L. Fleishman and P. W. Anderson. Interactions and the anderson transition.Phys. Rev. B, 21:2366–2377, Mar 1980

    work page 1980

  5. [3]

    Basko, I.L

    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.Annals of Physics, 321(5):1126–1205, 2006

    work page 2006

  6. [4]

    Rahul Nandkishore and David A. Huse. Many-body lo- calization and thermalization in quantum statistical me- chanics.Annual Review of Condensed Matter Physics, 6(1):15–38, 2015

    work page 2015

  7. [5]

    N. F. Mott. Conduction in non-crystalline materials.The Philosophical Magazine: A Journal of Theoretical Exper- imental and Applied Physics, 19(160):835–852, 1969

    work page 1969

  8. [6]

    Reichman

    Yevgeny Bar Lev and David R. Reichman. Dynamics of many-body localization.Phys. Rev. B, 89:220201(R), Jun 2014

    work page 2014

Show all 31 references
  1. [7]

    Reichman

    Yevgeny Bar Lev, Guy Cohen, and David R. Reichman. Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice.Phys. Rev. Lett., 114:100601, Mar 2015

    2015

  2. [8]

    Many-body local- ization in the age of classical computing.Reports on Progress in Physics, 88(2):026502, 2025

    Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio, Lev Vidmar, and Jakub Zakrzewski. Many-body local- ization in the age of classical computing.Reports on Progress in Physics, 88(2):026502, 2025

    2025

  3. [9]

    Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system.Phys

    Marko ˇZnidariˇ c, Antonello Scardicchio, and Vipin Kerala Varma. Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system.Phys. Rev. Lett., 117:040601, Jul 2016

    2016

  4. [10]

    Khaymovich

    Giuseppe De Tomasi, Soumya Bera, Antonello Scardic- chio, and Ivan M. Khaymovich. Subdiffusion in the an- derson model on the random regular graph.Phys. Rev. B, 101:100201(R), Mar 2020

    2020

  5. [11]

    Anomalous diffu- sion and griffiths effects near the many-body localization transition.Phys

    Kartiek Agarwal, Sarang Gopalakrishnan, Michael Knap, Markus M¨ uller, and Eugene Demler. Anomalous diffu- sion and griffiths effects near the many-body localization transition.Phys. Rev. Lett., 114:160401, Apr 2015

    2015

  6. [12]

    Huse, and Ehud Altman

    Ronen Vosk, David A. Huse, and Ehud Altman. The- ory of the many-body localization transition in one- dimensional systems.Phys. Rev. X, 5:031032, Sep 2015

    2015

  7. [13]

    Luitz, Nicolas Laflorencie, and Fabien Alet

    David J. Luitz, Nicolas Laflorencie, and Fabien Alet. Ex- tended slow dynamical regime close to the many-body localization transition.Phys. Rev. B, 93:060201(R), Feb 2016

    2016

  8. [14]

    A novel field theoretical approach to the An- derson lo- calization : sparse random hopping model.Journal de Physique I, 2 (8):1571–1605, 1992

    Hans-J¨ urgen Sommers Yan Fyodorov, Alexander Mirlin. A novel field theoretical approach to the An- derson lo- calization : sparse random hopping model.Journal de Physique I, 2 (8):1571–1605, 1992

    1992

  9. [15]

    Alexander D. Mirlin. Statistics of energy levels and eigenfunctions in disordered systems.Physics Reports, 326(5):259–382, 2000

    2000

  10. [16]

    Anoma- lous diffusion in disordered media: Statistical mecha- nisms, models and physical applications.Physics Re- ports, 195(4):127–293, 1990

    Jean-Philippe Bouchaud and Antoine Georges. Anoma- lous diffusion in disordered media: Statistical mecha- nisms, models and physical applications.Physics Re- ports, 195(4):127–293, 1990

    1990

  11. [17]

    Norbert Rosenzweig and Charles E. Porter. ”repulsion of energy levels” in complex atomic spectra.Phys. Rev., 120:1698–1714, Dec 1960

    1960

  12. [18]

    Ferdinand Evers and Alexander D. Mirlin. Anderson transitions.Rev. Mod. Phys., 80:1355–1417, Oct 2008

    2008

  13. [19]

    Tarquini, G

    E. Tarquini, G. Biroli, and M. Tarzia. Level statistics and localization transitions of l´ evy matrices.Phys. Rev. Lett., 116:010601, Jan 2016

    2016

  14. [20]

    Khaymovich

    Giuseppe De Tomasi and Ivan M. Khaymovich. Non- hermitian rosenzweig-porter random-matrix ensemble: Obstruction to the fractal phase.Phys. Rev. B, 106:094204, Sep 2022

    2022

  15. [21]

    V. E. Kravtsov and K. A. Muttalib. New class of random matrix ensembles with multifractal eigenvectors.Phys. Rev. Lett., 79:1913–1916, Sep 1997

    1913

  16. [22]

    I. M. Khaymovich, J. V. Koski, O.-P. Saira, V. E. Kravtsov, and J. P. Pekola. Multifractality of random eigenfunctions and generalization of jarzynski equality. Nature Communications, 6:7010, 2015

    2015

  17. [23]

    I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe. Fragile extended phases in the log- normal rosenzweig-porter model.Physical Review Re- search, 2:043346, 2020

    2020

  18. [24]

    multifractal

    I. M. Khaymovich and V. E. Kravtsov. Dynamical phases in a “multifractal” rosenzweig-porter model.SciPost Physics, 11:045, 2021

    2021

  19. [25]

    Cizeau and J

    P. Cizeau and J. P. Bouchaud. Theory of l´ evy matrices. Phys. Rev. E, 50:1810–1822, Sep 1994

    1994

  20. [26]

    Biroli and M

    G. Biroli and M. Tarzia. L´ evy-rosenzweig-porter random matrix ensemble.Phys. Rev. B, 103:104205, Mar 2021

    2021

  21. [28]

    Density of states correlations in L´ evy Rosenzweig-Porter model via supersymmetry approach

    Elizaveta Safonova, Aleksey Lunkin, and Mikhail Feigel’man. Density of states correlations in L´ evy Rosenzweig-Porter model via supersymmetry approach. SciPost Phys., 20:003, 2026

    2026

  22. [30]

    Disordered electronic system as a model of interacting matrices.Physics Reports, 67(1):15–24, 1980

    Franz Wegner. Disordered electronic system as a model of interacting matrices.Physics Reports, 67(1):15–24, 1980

    1980

  23. [31]

    M. A. Skvortsov, M. Amini, and V. E. Kravtsov. Sen- sitivity of (multi)fractal eigenstates to a perturbation of the hamiltonian.Phys. Rev. B, 106:054208, Aug 2022. End Matter 7 OUTLINING THE CALCULA TION Supersymmetric formalism.–In this End Matter, we briefly summarize the su...

    2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.