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arxiv: 2604.01320 · v2 · submitted 2026-04-01 · ✦ hep-th · cond-mat.str-el· quant-ph

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Solving L\'{e}vy Sachdev-Ye-Kitaev Model

Budhaditya Bhattacharjee , William. E. Salazar , Alexei Andreanov , Dario Rosa
Authors on Pith no claims yet

Pith reviewed 2026-05-13 21:31 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords Lévy SYK modellarge-N limitSchwinger-Dyson equationsquantum chaosLyapunov exponentholographynon-Fermi liquids
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The pith

The Lévy SYK model admits an exact large-N solution for any tail exponent mu that tunes chaos continuously from free to maximal.

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

The paper derives and solves the large-N Schwinger-Dyson equations for the Lévy Sachdev-Ye-Kitaev model whose random couplings follow a stable distribution with tail index mu between 0 and 2. Starting from a bosonic oscillator representation of the partition function, the equations are treated analytically in the large-q and infrared limits to obtain the Green's function. From this the Krylov and Lyapunov exponents are computed, showing that chaos is non-maximal for all intermediate mu and reaches the bound only at mu=2. Thermodynamic quantities including entropy and specific heat are evaluated and compared to the Gaussian SYK case, with discussion of their meaning for holographic duals and non-Fermi liquids.

Core claim

In the large-N limit the LSYK model is solved by Schwinger-Dyson equations obtained from the bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-q and infrared regimes, yielding the two-point function from which the Krylov exponent and the Lyapunov exponent are extracted. The tail exponent mu provides a continuous interpolation: the model is free at mu=0 and recovers the maximally chaotic Gaussian SYK model at mu=2, with non-maximal chaos for all intermediate values. Thermodynamic observables are computed explicitly and interpreted in the context of holography and non-Fermi liquid physics.

What carries the argument

The bosonic oscillator representation of the action that converts the Lévy-stable couplings into large-N Schwinger-Dyson equations solvable in the infrared and large-q limits.

If this is right

  • The Lyapunov exponent remains strictly below the chaos bound for every mu in (0,2) and saturates only at mu=2.
  • Entropy, free energy, and specific heat vary continuously with mu and differ from the corresponding Gaussian SYK quantities at each intermediate value.
  • The infrared Green's function admits closed-form expressions in the large-q limit that recover the free-fermion result at mu=0.
  • An alternative decomposition of the Lévy distribution establishes an explicit non-trivial mapping onto the Gaussian SYK model.

Where Pith is reading between the lines

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

  • The mu-tuned family could serve as a controlled interpolation between integrable and chaotic regimes for studying the onset of scrambling in other disordered systems.
  • Thermodynamic scaling with mu may be tested in cold-atom or trapped-ion realizations that can realize power-law disorder distributions.
  • The same bosonic representation technique might be applied to other heavy-tailed interaction distributions to obtain solvable large-N limits.

Load-bearing premise

The bosonic oscillator representation of the action correctly encodes the effects of the Lévy-stable distributed couplings when deriving the large-N Schwinger-Dyson equations.

What would settle it

Finite-N exact diagonalization or Monte Carlo sampling of the Hamiltonian for N around 20-50 that yields a Lyapunov exponent versus mu curve differing from the analytic large-N prediction.

Figures

Figures reproduced from arXiv: 2604.01320 by Alexei Andreanov, Budhaditya Bhattacharjee, Dario Rosa, William. E. Salazar.

Figure 1
Figure 1. Figure 1: Phase diagram of the L´evy Sachdev-Ye-Kitaev model. Top: for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the Bosonic oscillator approach. Each [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical solution of the Schwinger-Dyson equations (52-53): the [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Solution ν (63) as a function of Ja and Jb. The non-trivial dependence of ν on µ present on the plot vs. Ja weakens, e.g. different curves almost collapse, on the plot vs. Ja. Using the boundary conditions on the thermal circle G∗(0) = G∗(β) and introducing the parameter ν defined via c = πν/β and τ0 = β(1 − ν)/2ν, we obtain the following constraint equations for ν and K in terms of β and J µq  β 2J 2 2 q… view at source ↗
Figure 5
Figure 5. Figure 5: The scaling of ν (63) with β (left) and β µ/2 (right) vs β and Jb ∼ β µ/2 . A nearly perfect collapse is observed for the plot of ν/βν/2 vs Jb. The solid black line J is the limit limβ→0 ν/β for the Gaussian SYK. (βJµ) µ/2 = Jµ/2 a for different µ. Using this, we find the following expression for e g(τ) e g(τ) =   cos πν 2  cos  πν  1 2 − |τ| β    2 , (65) where the parameter ν is given by the sol… view at source ↗
Figure 5
Figure 5. Figure 5: 4.2 The Infrared Regime The SYK model is known to feature emergent symmetries in the infrared regime, which is achieved at strong interaction strength or large time-scales [15]. In this section, we investigate the properties of LSYK in the same regime and discuss the related symmetries and spontaneous symmetry breaking. Let us recall that the large−N effective action for LSYK is given by Sb,on-shell N = − … view at source ↗
Figure 6
Figure 6. Figure 6: Schematic representation of the shrinking of the conformal window in [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Dependence of ν(β, µ) on parameters β and µ . Left, (a): magnitude of ν as a function of µ and β. Right, (b): the sign of the rate ∂ν/∂µ for fixed β) fixed β and µ Yellow/bright and violed/dark colors indicate regions with increase and decrease of ∂ν/∂µ respectively. Recall that ν ∈ [0, 1], corresponding to β ∈ [0, ∞) for any µ. 5 Chaos exponents at large-q The 2− point Greens’ function GR(t) encodes infor… view at source ↗
Figure 8
Figure 8. Figure 8: Dependence of γ(β, µ) and λLβ/2π on parameters β and µ. Left (a): the ratio λL/2α – the non-saturation of the Q−complexity bound. Right (b): the OTOC exponent λL in units of 2π/β. The Gaussian SYK result corresponds to the vertical line at µ = 2.0 (dark red). In the last equality for s1, we use Eqn. (63). Note that for µ = 2, s2 = 0 and s1 = 2, which makes the above differential equation the P¨oschl-Teller… view at source ↗
Figure 9
Figure 9. Figure 9: (Left) Thermodynamic entropy per fermion as a function of [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The average energy per fermion ⟨H⟩/N (left) and log of partition function log Z/N (right) for different values of µ vs β −1 (top) and on T 2µ µ+2 = β − 2µ µ+2 (bottom). We use q = 4 for all the plots. This agrees with the scaling obtained from the reparameterization action in Eqn (87). Here we added a tilde to the coefficients by ˜e0, s˜0, c . . . ˜ to distinguish them from the coefficients in Eqn. (115).… view at source ↗
Figure 11
Figure 11. Figure 11: Thermodynamic coefficients ˜e0, s˜0, c˜ and ˜c2 from the log of partition function log Z/N (118) for different values of µ. Left: dependence on µ for fixed values of q. Right: dependence on q for fixed values of µ [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Heat capacity per unit temperature C(T)/T . Left: scaling with temperature T. Right: The numerical scaling coefficient for small T, Λ, extracted ˜ and compared with the analytical expectation of −1 − 2µ µ+2 = −3µ−2 µ+2 (120). sub-linear entropy growth (S ∼ T ν , ν = 2µ µ+1 < 1). The effective low-energy description via the Schwarzian mode becomes increasingly rigid (indicated by the divergence of the coef… view at source ↗
Figure 13
Figure 13. Figure 13: Schematic representation of the shrinking dual black hole picture of [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Stochastic representation of L´evy SYK model in terms a sum of in [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison between the numerical and the analytical large- [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Numerical scaling of 1 − ν for large β (low temperatures). (Left) scaling of log(1 − ν) with respect to log β indicates power-law scaling of 1 − ν with a single dominant exponent. (Right) The slope of linear fit of log(1 − ν) with log β is compared with the analytical expectation − 2µ µ+2 . expected. Surprisingly, the agreement also becomes better for small µ. The scaling of ν at low temperatures has also… view at source ↗
Figure 17
Figure 17. Figure 17: The lowest eigenvalue of the Schr¨odinger operator (133): numerical [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
read the original abstract

We present an exact solution in the large-$N$ limit of the L\'{e}vy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a L\'{e}vy Stable distribution parameterized by a tail exponent $\mu \in [0, 2]$. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-$N$ Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-$q$ and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-$q$ Green's function and extracting the Lyapunov exponent from the $4$-point function. The parameter $\mu$ continuously interpolates between a free theory at $\mu = 0$ and the conventional, maximally chaotic Gaussian SYK model at $\mu = 2$, with non-maximal chaos persisting throughout the intermediate regime $0 < \mu < 2$. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the L\'{e}vy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.

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

1 major / 2 minor

Summary. The manuscript claims an exact large-N solution to the Lévy Sachdev-Ye-Kitaev (LSYK) model with couplings drawn from a Lévy stable distribution of tail exponent μ ∈ [0,2]. Starting from the Hamiltonian, it derives the Schwinger-Dyson equations via a bosonic oscillator representation of the action, solves them numerically and analytically in the large-q and infrared limits, extracts the Krylov exponent from the large-q Green's function and the Lyapunov exponent from the four-point function, computes thermodynamic quantities (entropy, free energy, energy, specific heat), and compares them to Gaussian SYK. An alternative decomposition of the Lévy distribution that connects to Gaussian SYK is presented in the appendices.

Significance. If the central derivation is valid, the work supplies a continuous family of models interpolating between a free theory at μ=0 and the maximally chaotic Gaussian SYK at μ=2, with non-maximal chaos for 0<μ<2. The combination of numerical SD solutions, analytic large-q/IR limits, explicit Lyapunov extraction, and thermodynamic comparisons provides concrete, falsifiable predictions relevant to non-Fermi liquids and holographic duality. The alternative representation in the appendices is a useful cross-check.

major comments (1)
  1. [Derivation of SD equations via bosonic oscillator representation] The derivation of the large-N Schwinger-Dyson equations rests on a bosonic oscillator representation of the action to perform the disorder average over Lévy-stable couplings. For μ<2 the characteristic function is exp(−|k|^μ) rather than Gaussian; it is not obvious that a standard oscillator (quadratic) representation reproduces the correct infinite-variance cumulant-generating functional. This step is load-bearing for all subsequent SD equations, large-q solutions, and chaos exponents. A explicit verification that the representation generates the proper stable-distribution moments (or a demonstration that it reduces to the known Gaussian case only at μ=2) is required.
minor comments (2)
  1. The abstract cites Ref. [1] for the original LSYK definition; the bibliography entry should be expanded with full bibliographic details.
  2. In the thermodynamic analysis, the specific-heat curves for intermediate μ would benefit from an explicit statement of the temperature range over which the IR solution is used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work's significance, and constructive feedback. We address the single major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Derivation of SD equations via bosonic oscillator representation] The derivation of the large-N Schwinger-Dyson equations rests on a bosonic oscillator representation of the action to perform the disorder average over Lévy-stable couplings. For μ<2 the characteristic function is exp(−|k|^μ) rather than Gaussian; it is not obvious that a standard oscillator (quadratic) representation reproduces the correct infinite-variance cumulant-generating functional. This step is load-bearing for all subsequent SD equations, large-q solutions, and chaos exponents. A explicit verification that the representation generates the proper stable-distribution moments (or a demonstration that it reduces to the known Gaussian case only at μ=2) is required.

    Authors: We thank the referee for identifying this key point. The bosonic oscillator representation is constructed so that the Gaussian integral over the auxiliary fields exactly yields the characteristic function exp(−|k|^μ) of the Lévy stable distribution for general μ ∈ [0,2]. This is achieved by a μ-dependent rescaling of the oscillator frequencies and couplings that reproduces the correct cumulant-generating functional, including the non-Gaussian higher cumulants and infinite variance for μ < 2. The μ = 2 limit recovers the standard Gaussian SYK averaging procedure. In the revised manuscript we will add an explicit verification subsection (new Appendix C) that computes the first several moments generated by the representation and demonstrates the reduction to the known Gaussian case. This check confirms the validity of the Schwinger-Dyson equations and all downstream results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper starts from the Hamiltonian and its partition function, highlights differences from Gaussian SYK, and derives the large-N Schwinger-Dyson equations via a bosonic oscillator representation of the action. This representation is introduced as a technical step to handle the disorder average, after which the equations are solved numerically and analytically in large-q and IR limits to obtain Green's functions, chaos exponents, and thermodynamics. No quoted step reduces a claimed prediction or result to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation whose content is unverified within the paper. The model is referenced to Ref. [1], but the present derivation chain is self-contained and proceeds from the stated Hamiltonian without redefining its own outputs as inputs. The alternative representation discussed in the appendices is presented as an additional connection rather than a foundational assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard large-N saddle-point approximation applied to a new coupling distribution; no additional free parameters beyond the model parameter mu are introduced.

axioms (1)
  • domain assumption Large-N limit permits saddle-point evaluation of the partition function and derivation of closed Schwinger-Dyson equations
    Invoked immediately after stating the Hamiltonian and partition function.

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Bhattacharjee, W

    B. Bhattacharjee, W. E. Salazar, D. Rosa and A. Andreanov,L´ evy sachdev-ye-kitaev model(2025),2506.04343

    work page arXiv 2025

  2. [2]

    Bohigas, M

    O. Bohigas, M. J. Giannoni and C. Schmit,Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett.52, 1 (1984), doi:10.1103/PhysRevLett.52.1

    work page doi:10.1103/physrevlett.52.1 1984

  3. [3]

    Haake,Quantum signatures of chaos, InQuantum Coherence in Mesoscopic Systems

    F. Haake,Quantum signatures of chaos, InQuantum Coherence in Mesoscopic Systems. Springer (1991)

    work page 1991

  4. [4]

    M. L. Mehta,Random Matrices, Pure and applied mathematics: v. 142. Else- vier/Academic Press, Amsterdam, 3rd ed. edn., ISBN 0120884097 (2004)

    work page 2004

  5. [5]

    T. Guhr, A. M¨ uller–Groeling and H. A. Weidenm¨ uller,Random-matrix theo- ries in quantum physics: common concepts, Physics Reports299(4), 189 (1998), doi:https://doi.org/10.1016/S0370-1573(97)00088-4

    work page doi:10.1016/s0370-1573(97)00088-4 1998

  6. [6]

    Srednicki,Chaos and quantum thermalization, Phys

    M. Srednicki,Chaos and quantum thermalization, Phys. Rev. E50, 888 (1994), doi:10.1103/PhysRevE.50.888

    work page doi:10.1103/physreve.50.888 1994

  7. [7]

    J. M. Deutsch,Eigenstate thermalization hypothesis, Reports on Progress in Physics 81(8), 082001 (2018), doi:10.1088/1361-6633/aac9f1

    work page doi:10.1088/1361-6633/aac9f1 2018

  8. [8]

    M. Fava, J. Kurchan and S. Pappalardi,Designs via free probability, Phys. Rev. X 15, 011031 (2025), doi:10.1103/PhysRevX.15.011031

    work page doi:10.1103/physrevx.15.011031 2025

  9. [9]

    X. Wang, S. Ghose, B. C. Sanders and B. Hu,Entanglement as a signature of quantum chaos, Phys. Rev. E70, 016217 (2004), doi:10.1103/PhysRevE.70.016217

    work page doi:10.1103/physreve.70.016217 2004

  10. [10]

    Xu and B

    S. Xu and B. Swingle,Scrambling dynamics and out-of-time-ordered cor- relators in quantum many-body systems, PRX Quantum5, 010201 (2024), doi:10.1103/PRXQuantum.5.010201

    work page doi:10.1103/prxquantum.5.010201 2024

  11. [11]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman,A universal operator growth hypothesis, Phys. Rev. X9, 041017 (2019), doi:10.1103/PhysRevX.9.041017

    work page doi:10.1103/physrevx.9.041017 2019

  12. [12]

    Sachdev and J

    S. Sachdev and J. Ye,Gapless spin-fluid ground state in a random quantum heisenberg magnet, Phys. Rev. Lett.70, 3339 (1993), doi:10.1103/PhysRevLett.70.3339

    work page doi:10.1103/physrevlett.70.3339 1993

  13. [13]

    Kitaev,A simple model of quantum holography

    A. Kitaev,A simple model of quantum holography. 36 SciPost Physics Submission

    work page

  14. [14]

    Polchinski and V

    J. Polchinski and V. Rosenhaus,The spectrum in the sachdev-ye-kitaev model, Journal of High Energy Physics2016(4), 1–25 (2016), doi:10.1007/jhep04(2016)001

    work page doi:10.1007/jhep04(2016)001 2016

  15. [15]

    Maldacena and D

    J. Maldacena and D. Stanford,Remarks on the sachdev-ye-kitaev model, Phys. Rev. D94, 106002 (2016), doi:10.1103/PhysRevD.94.106002

    work page doi:10.1103/physrevd.94.106002 2016

  16. [16]

    J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stan- ford, A. Streicher and M. Tezuka,Black Holes and Random Matrices, JHEP 05, 118 (2017), doi:10.1007/JHEP05(2017)118, [Erratum: JHEP 09, 002 (2018)], 1611.04650

    work page doi:10.1007/jhep05(2017)118 2017

  17. [17]

    A. M. Garc´ ıa-Garc´ ıa and J. J. M. Verbaarschot,Spectral and thermodynamic properties of the sachdev-ye-kitaev model, Phys. Rev. D94, 126010 (2016), doi:10.1103/PhysRevD.94.126010

    work page doi:10.1103/physrevd.94.126010 2016

  18. [18]

    Sekino and L

    Y. Sekino and L. Susskind,Fast scramblers, Journal of High Energy Physics2008(10), 065 (2008), doi:10.1088/1126-6708/2008/10/065

    work page doi:10.1088/1126-6708/2008/10/065 2008

  19. [19]

    S. H. Shenker and D. Stanford,Black holes and the butterfly effect, Journal of High Energy Physics2014(3), 67 (2014), doi:10.1007/JHEP03(2014)067

    work page doi:10.1007/jhep03(2014)067 2014

  20. [20]

    T. G. Mertens and G. J. Turiaci,Solvable models of quantum black holes: a re- view on jackiw–teitelboim gravity, Living Reviews in Relativity26(1), 4 (2023), doi:10.1007/s41114-023-00046-1

    work page doi:10.1007/s41114-023-00046-1 2023

  21. [21]

    Sarosi,AdS 2 holography and the SYK model, InProceedings of XIII Modave Summer School in Mathematical Physics — PoS(Modave2017), vol

    G. Sarosi,AdS 2 holography and the SYK model, InProceedings of XIII Modave Summer School in Mathematical Physics — PoS(Modave2017), vol. 323, p. 001, doi:10.22323/1.323.0001 (2018)

    work page doi:10.22323/1.323.0001 2018

  22. [22]

    Rosenhaus,An introduction to the syk model, Journal of Physics A: Mathematical and Theoretical52(32), 323001 (2019), doi:10.1088/1751-8121/ab2ce1

    V. Rosenhaus,An introduction to the syk model, Journal of Physics A: Mathematical and Theoretical52(32), 323001 (2019), doi:10.1088/1751-8121/ab2ce1

    work page doi:10.1088/1751-8121/ab2ce1 2019

  23. [23]

    Chowdhury, A

    D. Chowdhury, A. Georges, O. Parcollet and S. Sachdev,Sachdev-ye-kitaev models and beyond: Window into non-fermi liquids, Rev. Mod. Phys.94, 035004 (2022), doi:10.1103/RevModPhys.94.035004

    work page doi:10.1103/revmodphys.94.035004 2022

  24. [24]

    Jha, Introduction to Sachdev-Ye-Kitaev Model: A Strongly Correlated System Perspective (2025), arXiv:2507.07195 [hep-th]

    R. Jha,Introduction to Sachdev-Ye-Kitaev Model: A Strongly Correlated System Perspective(2025),2507.07195

    work page arXiv 2025

  25. [25]

    Tezuka, O

    M. Tezuka, O. Oktay, E. Rinaldi, M. Hanada and F. Nori,Binary-coupling sparse sachdev-ye-kitaev model: An improved model of quantum chaos and holography, Phys. Rev. B107, L081103 (2023), doi:10.1103/PhysRevB.107.L081103

    work page doi:10.1103/physrevb.107.l081103 2023

  26. [26]

    A. M. Garc´ ıa-Garc´ ıa, B. Loureiro, A. Romero-Berm´ udez and M. Tezuka,Chaotic- integrable transition in the sachdev-ye-kitaev model, Phys. Rev. Lett.120, 241603 (2018), doi:10.1103/PhysRevLett.120.241603

    work page doi:10.1103/physrevlett.120.241603 2018

  27. [27]

    A. M. Garc´ ıa-Garc´ ıa, Y. Jia, D. Rosa and J. J. M. Verbaarschot,Sparse sachdev-ye- kitaev model, quantum chaos, and gravity duals, Phys. Rev. D103, 106002 (2021), doi:10.1103/PhysRevD.103.106002

    work page doi:10.1103/physrevd.103.106002 2021

  28. [28]

    Andreanov, M

    A. Andreanov, M. Carrega, J. Murugan, J. Olle, D. Rosa and R. Shir,From dyson models to many-body quantum chaos, Phys. Rev. B111, 035147 (2025), doi:10.1103/PhysRevB.111.035147. 37 SciPost Physics Submission

    work page doi:10.1103/physrevb.111.035147 2025

  29. [29]

    C. Monthus,Localization transition in random l´ evy matrices: multifractality of eigenvectors in the localized phase and at criticality, Journal of Statistical Me- chanics: Theory and Experiment2016(9), 093304 (2016), doi:10.1088/1742- 5468/2016/09/093304

    work page doi:10.1088/1742- 2016

  30. [30]

    Kutlin and I

    A. Kutlin and I. M. Khaymovich,Anatomy of the eigenstates distribu- tion: A quest for a genuine multifractality, SciPost Phys.16, 008 (2024), doi:10.21468/SciPostPhys.16.1.008,2309.06468

    work page doi:10.21468/scipostphys.16.1.008 2024

  31. [31]

    Janzen, A

    K. Janzen, A. K. Hartmann and A. Engel,Replica theory for levy spin glasses*, Journal of Statistical Mechanics: Theory and Experiment2008(04), P04006 (2008), doi:10.1088/1742-5468/2008/04/P04006

    work page doi:10.1088/1742-5468/2008/04/p04006 2008

  32. [32]

    Janzen, A

    K. Janzen, A. Engel and M. M´ ezard,The l´ evy spin glass transition, Europhysics Letters89(6), 67002 (2010), doi:10.1209/0295-5075/89/67002

    work page doi:10.1209/0295-5075/89/67002 2010

  33. [33]

    Janzen, A

    K. Janzen, A. Engel and M. M´ ezard,Thermodynamics of the l´ evy spin glass, Phys. Rev. E82, 021127 (2010), doi:10.1103/PhysRevE.82.021127

    work page doi:10.1103/physreve.82.021127 2010

  34. [34]

    Maldacena, S

    J. Maldacena, S. H. Shenker and D. Stanford,A bound on chaos, Journal of High Energy Physics2016(8), 106 (2016), doi:10.1007/JHEP08(2016)106

    work page doi:10.1007/jhep08(2016)106 2016

  35. [35]

    J. P. Nolan,Univariate Stable Distributions, Springer Cham, doi:https://doi.org/10.1007/978-3-030-52915-4 (2020)

    work page doi:10.1007/978-3-030-52915-4 2020

  36. [36]

    I. Neri, F. L. Metz and D. Boll´ e,The phase diagram of l´ evy spin glasses, Jour- nal of Statistical Mechanics: Theory and Experiment2010(01), P01010 (2010), doi:10.1088/1742-5468/2010/01/P01010

    work page doi:10.1088/1742-5468/2010/01/p01010 2010

  37. [37]

    Lee,Recent developments in non-fermi liquid theory, Annu

    S.-S. Lee,Recent developments in non-fermi liquid theory, Annu. Rev. Condens. Matter Phys.9(1), 227 (2018)

    work page 2018

  38. [38]

    Luo, Y.-Z

    Z. Luo, Y.-Z. You, J. Li, C.-M. Jian, D. Lu, C. Xu, B. Zeng and R. Laflamme, Quantum simulation of the non-fermi-liquid state of sachdev-ye-kitaev model, npj Quantum Information5(1), 53 (2019), doi:10.1038/s41534-019-0166-7

    work page doi:10.1038/s41534-019-0166-7 2019

  39. [39]

    A. E. Davis,Non-Fermi Liquid Physics and Chaos in Yukawa-SYK Models, Ph.D. thesis, Florida U. (2023)

    work page 2023

  40. [40]

    Maldacena, D

    J. Maldacena, D. Stanford and Z. Yang,Conformal symmetry and its breaking in two-dimensional nearly anti-de sitter space, Progress of Theoretical and Experimental Physics2016(12), 12C104 (2016), doi:10.1093/ptep/ptw124

    work page doi:10.1093/ptep/ptw124 2016

  41. [41]

    X. Dong, S. Harrison, S. Kachru, G. Torroba and H. Wang,Aspects of holography for theories with hyperscaling violation, Journal of High Energy Physics2012(6), 41 (2012), doi:10.1007/JHEP06(2012)041

    work page doi:10.1007/jhep06(2012)041 2012

  42. [42]

    Huijse, S

    L. Huijse, S. Sachdev and B. Swingle,Hidden fermi surfaces in com- pressible states of gauge-gravity duality, Phys. Rev. B85, 035121 (2012), doi:10.1103/PhysRevB.85.035121

    work page doi:10.1103/physrevb.85.035121 2012

  43. [43]

    Ryu and T

    S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence, Phys. Rev. Lett.96, 181602 (2006), doi:10.1103/PhysRevLett.96.181602. [44]Wave equation in oblate spheroidal coordinates, doi:DLMF-S30.14. 38

    work page doi:10.1103/physrevlett.96.181602 2006