Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

Grand Canonical vs Canonical Krylov Complexity in Double-Scaled Complex SYK Model

T0 review · 2 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The paper establishes that in the double-scaled complex SYK model the grand canonical Krylov complexity is exactly the chemical-potential-weighted sum of the Krylov complexities in each U(1) charge sector, because the grand canonical transf

desk verdict Solid cSYK result with a fixable normalization bug in the central formula. read the letter →

arxiv 2512.07715 v2 pith:YJBHPGUY submitted 2025-12-08 hep-th cond-mat.str-elquant-ph

Stefan Forste , Yannic Kruse , Saurabh Natu This is my paper
classification hep-thcond-mat.str-elquant-ph
keywords complexSYKmodeldoublescalinglimitKrylovcomplexitygrandcanonicalensembleorientedchorddiagramstransfermatrixq-deformedoscillator
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 show that two standard ways of defining a thermal ensemble—fixed U(1) charge (canonical) and variable charge with chemical potential (grand canonical)—produce Krylov complexities that are related in the simplest possible way in the double-scaled complex SYK model: the grand canonical complexity is a weighted sum of the canonical sector complexities, with weights set by the grand canonical probability. The key is a symmetrized oriented-chord transfer matrix that turns out to be block diagonal in charge, each block being the canonical transfer matrix of that sector. This realizes the equality case of a previously conjectured inequality between the two complexities at all times. The authors also provide analytic early- and late-time expressions for the sector complexity and verify the weighted-sum relation numerically.

What carries the argument

The load-bearing object is the symmetrized grand canonical transfer matrix T_µ acting on oriented chord states, strings of X and O symbols marking endpoints of left- and right-pointing chords. It is built from q-deformed creation and annihilation operators—combined bosonic operators à and Æ satisfying [Ã,Æ]_{q(µ)} = 1—and matched to the H-chord transfer matrix T_H(Q), where an H-chord is a pair of oppositely oriented chords corresponding to one ψ/ψ̄ pair contraction. Promoting the charge Q to an operator gives a Hilbert space F = ⊕_Q H_Q, and the block diagonality of T in Q is what forces the weighted-sum identity.

What would settle it

Compute the grand canonical moments m_k(µ) while keeping all m sectors, or directly evaluate off-diagonal matrix elements ⟨0,Q|T^ℓ|0,Q'⟩ for Q≠Q'; if any off-diagonal entry is nonvanishing, or if m≠0 contributions change the moments, the weighted-sum formula (3.29) fails and the conjectured inequality becomes strict.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is equation (3.29): C_K(µ,t) = ∫ dQ e^{-βNΩ(µ,Q)} C_K^{(Q)}(t). The grand canonical transfer matrix T in the (n,Q) chord basis is block diagonal, with each block the canonical transfer matrix T_H(Q) of a fixed charge sector, so time evolution does not mix sectors. Consequently the grand canonical state Krylov complexity—the spread complexity of the chord number operator—is the expectation value of that operator against block-diagonal evolution, and the conjectured inequality that grand canonical complexity minus the weighted sector average is non-negative, with equality iff the transfer matrix does not mix sectors, is saturated: equality holds

Load-bearing premise

Everything rests on the claim that only chord diagrams with equal numbers of left- and right-pointing chords (the m=0 sector) contribute to the moments; if m≠0 sectors contribute, the grand canonical transfer matrix can mix charge sectors and the equality becomes a strict inequality.

Editorial extensions

If this is right

  • For any operator commuting with U(1) charge in double-scaled cSYK, the grand canonical Krylov complexity is completely determined by the canonical sector complexities and the grand canonical weights.
  • The conjectured inequality between grand canonical and charge-weighted canonical Krylov complexity is saturated with equality at all times, not merely asymptotically.
  • The critical chord number n* = (1−4Q²)/(4λ) and the early-to-late time crossover t* depend on charge, so different charge sectors scramble at different rates and the grand canonical curve is their probability-weighted mixture.
  • The same q-deformed oscillator algebra appears in every charge sector with a Q-dependent deformation parameter q = exp(−4λ/(1−4Q²)), extending the known single-sector structure to the full grand canonical Hilbert space.
  • The block-diagonal form provides a concrete target for a bulk interpretation: the chord-number expectation value in the grand canonical state decomposes into a sum over charge-resolved contributions.
  • The grand canonical Lanczos coefficients b_n(µ) directly generate the grand canonical Krylov complexity, and the analytic asymptotic formulas for C_K^{(Q)}(t) reproduce the numerical curves in the early and late time regimes.

Reading between the lines

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

  • If the block-diagonal structure survives outside the double-scaling limit, a similar reconstruction of grand canonical complexity from sector data may hold in other U(1)-symmetric chaotic models; the present derivation does not establish that.
  • The m=0 projection is the quiet hinge: a direct finite-N computation of moments with unequal left/right chord counts would test whether the equality is exact or an artifact of the double-scaling/H-chord approximation.
  • The charge-dependent crossover time t*(Q) suggests a practical diagnostic: time-resolved spread complexity in charge-resolved measurements could reveal the charge dependence of scrambling dynamics in a thermal system.
  • The authors note that the N=2 supersymmetric SYK transfer matrix may mix charge sectors; if that mixing is real, the same methods would predict a strict inequality there, making a testable contrast with the cSYK equality.
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

2 major / 6 minor

Summary. The paper studies the double-scaled complex SYK model and constructs a symmetrized transfer matrix in an oriented-chord basis. Its central claim is that the grand canonical transfer matrix is block diagonal in U(1) charge, with each block equal to the canonical transfer matrix of the corresponding charge sector, and hence that the grand canonical Krylov complexity is a μ-dependent weighted sum of the canonical Krylov complexities, saturating the inequality (1.1) conjectured in [34]. The authors derive analytic early- and late-time asymptotics in fixed charge sectors and compare them with numerical Lanczos solutions for both canonical and grand canonical ensembles.

Significance. If the result stands, it provides the first explicit example of equality in the symmetry-resolved Krylov complexity inequality of [34] in a holographic SYK-type model, and it supplies a symmetrized oriented-chord transfer matrix that may be useful for future bulk reconstructions. The paper is largely self-contained, re-derives known double-scaled cSYK results from the chord combinatorics, has no fitted parameters, and the numerics match the analytic asymptotes in Figs. 3.2–3.4. The two main caveats are a missing normalization in the central weighted-sum formula and an insufficiently justified restriction to the m=0 sector; both are technical but affect the derivation of the main result.

major comments (2)
  1. [§3.2, Eqs. (3.12), (3.13), (3.27), (3.29); App. B (B.15)] The grand canonical state is correctly defined with normalization in App. B, Eq. (B.15): |0,μ⟩ = (1/√𝒩)∫dQ e^{-βNΩ/2}|0,Q⟩ with 𝒩=∫dQ e^{-βNΩ}. However this normalization is dropped in Eqs. (3.12), (3.27), and consequently in Eq. (3.29). For a block-diagonal observable, ⟨0,μ|O|0,μ⟩ = (1/𝒩)∫dQ e^{-βNΩ} o(Q). As written, (3.29) misses the 1/𝒩 factor, so the weights e^{-βNΩ} are not a probability function and the equality is false unless 𝒩=1. This is load-bearing because the paper's central claim is exactly this weighted-sum formula. Please correct the equations and define p(Q)=e^{-βNΩ}/𝒩; also state explicitly which normalization was used in the numerical comparison of Fig. 3.4.
  2. [§2.5, Eq. (2.46), and App. A (A.12)] The restriction to the m=0 sector is asserted rather than proved: 'we only point out that this implies an equal number of left and right pointing chords, i.e. m=0.' This restriction is used to reduce the moments to mk;0 and is essential for identifying the oriented-chord transfer matrix with the H-chord transfer matrix in §3.2. If m≠0 terms contributed, the local transfer matrix construction and the block-diagonal form could be altered. Please provide an explicit derivation from the fermionic contraction rules (or charge conservation) showing that mk;m vanishes for m≠0, and reconcile this with the sum over m that still appears in the saddle-point expression (A.12).
minor comments (6)
  1. [Abstract, §3.3.1] The text calls e^{-βNΩ} a probability function. After the normalization fix, it becomes a probability; please add the normalization factor explicitly in the abstract and in the discussion of Eq. (3.29).
  2. [Eq. (3.30)] The notation 'cosh p(˜µ)' is ambiguous; it should be cosh^p(˜µ) or an explicit power. Please define the rescaled variable ˜µ consistently with App. A as well.
  3. [Fig. 3.2 caption] The caption uses 'f (Q = 0.1) ∝ t²' without defining f. This should be the Krylov complexity C_K^{(Q)} or the appropriate analytic expression.
  4. [App. A, Eq. (A.2)] There is a typo: 'cos h(μ/2)' should be 'cosh(μ/2)'.
  5. [Eq. (2.20)] The summation range is typeset as 'X_{m=-k,···,k}' with missing summation symbols. Please fix the typesetting.
  6. [§2.6, after Eq. (2.51)] The statement that a† is the adjoint of a (and b† of b) after modding out null vectors is left to the reader. A short verification or a reference would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central weighted-sum relation is derived from U(1) charge conservation and standard ensemble definitions, not assumed.

full rationale

The paper's central result, Eq. (3.29), follows from the block-diagonality of the transfer matrix in the U(1) charge basis. That block-diagonality is derived rather than imposed: the complex SYK Hamiltonian conserves charge, and each Hamiltonian insertion produces one ψ and one ψ-bar insertion, so the oriented-chord argument in Sec. 2.5 (equal numbers of left- and right-pointing chords, m = 0) is a consequence of the model's structure. The grand canonical state |0, μ⟩ is defined in App. B.15 as a standard superposition of canonical states with weight exp(−βNΩ/2), and the weighted average of canonical Krylov complexities is then a direct expectation-value consequence once block diagonality is established. No parameter is fitted to data, and no 'prediction' is a renamed input. The paper relies on prior chord-diagram literature ([22], [25], [27]) for tools, but it re-derives the transfer matrices and operator algebras needed for its conclusion; these are not self-citations of the present authors, and the cited results are external published work. The only substantive issue is a correctness problem, not a circularity: Eq. (3.29) omits the normalization factor 𝒩 = ∫ dQ exp(−βNΩ) that App. B.15 explicitly introduces, so the weights are not normalized as a probability distribution. Once the 1/𝒩 factor is restored, the relation is exactly what block diagonality implies. This is an error in presentation, not a circular derivation, and it does not reduce the result to its own assumptions.

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

The central claim depends on the standard double-scaling/chord-diagram framework and on a small number of structural assumptions, of which the m=0 sector restriction is the most fragile. No free parameters are fit, and no new physical entities are introduced.

assumptions (5)
  • domain assumption Double-scaling limit N,p→∞ with λ=p^2/N finite
    All chord-diagram and saddle-point manipulations in Sections 2-3 and Appendix A assume this limit.
  • domain assumption Gaussian disorder average is captured by summing over oriented chord diagrams with multiplicative weights
    The moment expansion (2.10)-(2.21) and the penalty factors (2.27)-(2.28) rely on this standard SYK technique.
  • ad hoc to paper Only m=0 (equal numbers of left/right chords) configurations contribute
    Asserted in Section 2.5 from the form of the cSYK Hamiltonian without proof; drives block diagonality.
  • standard math Saddle-point (Laplace) approximation is valid for the Fourier integrals defining canonical moments
    Used in Appendix A; valid at large N but makes the equality in (3.29) approximate at finite N.
  • ad hoc to paper The q-deformed oscillator algebra (2.70)-(2.72) uniquely fixes the transfer-matrix action on all chord states
    Used in Section 3.2 to identify T^2 with \tilde A+\tilde A† by matching commutators.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Grand Canonical vs Canonical Krylov Complexity in Double-Scaled Complex SYK Model." pith.science (2026). https://pith.science/paper/YJBHPGUY

@misc{pith2026251207715,
  author       = {Pith},
  title        = {Pith review of: Grand Canonical vs Canonical Krylov Complexity in Double-Scaled Complex SYK Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJBHPGUY}},
  note         = {Machine review of arXiv:2512.07715}
}
read the original abstract

We consider the complex SYK model in the double-scaling limit. We obtain the transfer matrix for the grand canonical ensemble and symmetrize it. In the (n,Q)- basis of chord states, the grand canonical transfer matrix is block diagonal, where each block is the canonical transfer matrix for the respective charge sector. We therefore conclude that the Krylov complexity for the grand canonical ensemble is given by the sum of the complexities in the charge sectors weighted by a probability function that depends on the chemical potential. Finally, we compute the Krylov complexity analytically in the limit of early and late time in the charge sector and numerically for both canonical and grand canonical ensemble.

Figures

Figures reproduced from arXiv: 2512.07715 by the authors.

Figure 2.1
Figure 2.1. Part of a cut-open moment diagram expressed in terms of string of [PITH_FULL_IMAGE:figures/full_fig_p009_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. States in the physical Hilbert space that contribute to the [PITH_FULL_IMAGE:figures/full_fig_p012_2_2.png] view at source ↗
Figure 3.1
Figure 3.1. Numerical results for C (Q) K (t) for different values of Q at p = 10, λ = 0.001 and J = 1. 3.3.1 Krylov Complexity in the Grand Canonical Ensemble A state |Ψ(µ, t)⟩ in the grand canonical ensemble at arbitrary time t can be defined as, |Ψ(µ, t)⟩ = e −itT |0, µ⟩. (3.28) 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3_1.png] view at source ↗
Figures from the paper (3 more)
Figure 3.2
Figure 3.2. Figure 3.2: We compare the linear and quadratic asymptotes (dashed lines) for [PITH_FULL_IMAGE:figures/full_fig_p022_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Numerical results for CK(t) calculated via the µ dependent Lanczos coeffi￾cients (3.30) for different values of µ at p = 10, λ = 0.001 and J = 1. The number operator in the grand canonical ensemble is given by (3.23). Using this, the grand canonical state Krylov comp…
Figure 3.4
Figure 3.4. Figure 3.4: We compare the numerical results for CK(t) to analytic results obtained by integrating eq. (3.41) weighted with exp [−NβΩ] as explained in section 3.3.1. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_3_4.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantum gravity around ultracold black holes from DSSYK

    hep-th 2026-08 conditional novelty 6.0 of 10

    Ultracold Reissner-Nordström de Sitter black hole fluctuations are proposed to be described by a gauged near-flat dilaton gravity model with a Gaussian spectral density, yielding a finite partition function and dynami...

Reference graph

Works this paper leans on

40 extracted references · 33 linked inside Pith · cited by 1 Pith paper

  1. [34]

    Growth of block diagonal operators and symmetry-resolved Krylov complexity,

    P. Caputa, G. Di Giulio, and T. Q. Loc, “Growth of block diagonal operators and symmetry-resolved Krylov complexity,”arXiv:2507.02033 [hep-th]

    arXiv

  2. [35]

    Scaling limits of complex Sachdev-Ye-Kitaev models and holographic geometry,

    E. Gubankova, S. Sachdev, and G. Tarnopolsky, “Scaling limits of complex Sachdev-Ye-Kitaev models and holographic geometry,”arXiv:2512.05294 [hep-th]

    arXiv

  3. [1]

    Dimensional reduction in quantum gravity,

    G. ’t Hooft, “Dimensional reduction in quantum gravity,”Conf. Proc. C930308 (1993) 284–296,arXiv:gr-qc/9310026

    arXiv 1993

  4. [2]

    The World as a hologram,

    L. Susskind, “The World as a hologram,”J. Math. Phys.36(1995) 6377–6396, arXiv:hep-th/9409089

    arXiv 1995

  5. [3]

    The LargeNlimit of superconformal field theories and supergravity,

    J. M. Maldacena, “The LargeNlimit of superconformal field theories and supergravity,”Adv. Theor. Math. Phys.2(1998) 231–252, arXiv:hep-th/9711200

    arXiv 1998

  6. [4]

    Complexity, action, and black holes,

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, “Complexity, action, and black holes,”Phys. Rev. D93no. 8, (2016) 086006, arXiv:1512.04993 [hep-th]

    arXiv 2016

  7. [5]

    A Universal Operator Growth Hypothesis,

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, “A Universal Operator Growth Hypothesis,”Phys. Rev. X9no. 4, (2019) 041017, arXiv:1812.08657 [cond-mat.stat-mech]

    arXiv 2019

  8. [6]

    On The Evolution Of Operator Complexity Beyond Scrambling,

    J. L. F. Barbón, E. Rabinovici, R. Shir, and R. Sinha, “On The Evolution Of Operator Complexity Beyond Scrambling,”JHEP10(2019) 264, arXiv:1907.05393 [hep-th]

    arXiv 2019

Show all 40 references
  1. [7]

    Operator complexity: a journey to the edge of Krylov space,

    E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, “Operator complexity: a journey to the edge of Krylov space,”JHEP06(2021) 062,arXiv:2009.01862 [hep-th]

    2021 arXiv

  2. [8]

    Quantum chaos and the complexity of spread of states,

    V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, “Quantum chaos and the complexity of spread of states,”Phys. Rev. D106no. 4, (2022) 046007, arXiv:2202.06957 [hep-th]

    2022 arXiv

  3. [9]

    Computational complexity and black hole horizons,

    L. Susskind, “Computational complexity and black hole horizons,”Fortschritte der Physik64no. 1, (2016) 24–43, https://onlinelibrary.wiley.com/doi/pdf/10.1002/prop.201500092. https://onlinelibrary.wiley.com/doi/abs/10.1002/prop.201500092

    2016 doi

  4. [10]

    Computational Complexity and Black Hole Horizons,

    L. Susskind, “Computational Complexity and Black Hole Horizons,”Fortsch. Phys.64(2016) 24–43,arXiv:1403.5695 [hep-th]. [Addendum: Fortsch.Phys. 64, 44–48 (2016)]

    2016 arXiv

  5. [11]

    Krylov Complexity,

    E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, “Krylov Complexity,” arXiv:2507.06286 [hep-th]

    arXiv

  6. [12]

    Lower Dimensional Gravity,

    R. Jackiw, “Lower Dimensional Gravity,”Nucl. Phys. B252(1985) 343–356

    1985

  7. [13]

    Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,

    C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,”Phys. Lett. B126(1983) 41–45. 34

    1983

  8. [14]

    The Factorization Problem in Jackiw-Teitelboim Gravity,

    D. Harlow and D. Jafferis, “The Factorization Problem in Jackiw-Teitelboim Gravity,”JHEP02(2020) 177,arXiv:1804.01081 [hep-th]

    2020 arXiv

  9. [15]

    Gapless spin fluid ground state in a random, quantum Heisenberg magnet,

    S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet,”Phys. Rev. Lett.70(1993) 3339,arXiv:cond-mat/9212030

    1993 arXiv

  10. [16]

    Entangled 2015: Kitaev lecture

    A. Kitaev, “Entangled 2015: Kitaev lecture.” https://online.kitp.ucsb.edu/online/entangled15/kitaev/, 2015.KITP UCSB Online

    2015

  11. [17]

    Remarks on the Sachdev-Ye-Kitaev model,

    J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,”Phys. Rev. D94no. 10, (2016) 106002,arXiv:1604.07818 [hep-th]

    2016 arXiv

  12. [18]

    The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,

    A. Kitaev and S. J. Suh, “The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,”JHEP05(2018) 183,arXiv:1711.08467 [hep-th]

    2018 arXiv

  13. [19]

    Black Holes and Random Matrices,

    J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, “Black Holes and Random Matrices,” JHEP05(2017) 118,arXiv:1611.04650 [hep-th]. [Erratum: JHEP 09, 002 (2018)]

    2017 arXiv

  14. [20]

    Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction,

    M. Berkooz, P. Narayan, and J. Simon, “Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction,”JHEP08(2018) 192, arXiv:1806.04380 [hep-th]

    2018 arXiv

  15. [21]

    Towards a full solution of the large N double-scaled SYK model,

    M. Berkooz, M. Isachenkov, V. Narovlansky, and G. Torrents, “Towards a full solution of the large N double-scaled SYK model,”JHEP03(2019) 079, arXiv:1811.02584 [hep-th]

    2019 arXiv

  16. [22]

    Complex Sachdev-Ye-Kitaev model in the double scaling limit,

    M. Berkooz, V. Narovlansky, and H. Raj, “Complex Sachdev-Ye-Kitaev model in the double scaling limit,”JHEP02(2021) 113,arXiv:2006.13983 [hep-th]

    2021 arXiv

  17. [23]

    SYK Model with global symmetries in the double scaling limit,

    P. Narayan and S. T. S, “SYK Model with global symmetries in the double scaling limit,”JHEP05(2023) 083,arXiv:2302.11882 [hep-th]

    2023 arXiv

  18. [24]

    Supersymmetric Sachdev-Ye-Kitaev models,

    W. Fu, D. Gaiotto, J. Maldacena, and S. Sachdev, “Supersymmetric Sachdev-Ye-Kitaev models,”Phys. Rev. D95no. 2, (2017) 026009, arXiv:1610.08917 [hep-th]. [Addendum: Phys.Rev.D 95, 069904 (2017)]

    2017 arXiv

  19. [25]

    The double scaled limit of Super–Symmetric SYK models,

    M. Berkooz, N. Brukner, V. Narovlansky, and A. Raz, “The double scaled limit of Super–Symmetric SYK models,”JHEP12(2020) 110,arXiv:2003.04405 [hep-th]

    2020 arXiv

  20. [26]

    Looking at supersymmetric black holes for a very long time,

    H. W. Lin, J. Maldacena, L. Rozenberg, and J. Shan, “Looking at supersymmetric black holes for a very long time,”SciPost Phys.14no. 5, (2023) 128, arXiv:2207.00408 [hep-th]. 35

    2023 arXiv

  21. [27]

    The bulk Hilbert space of double scaled SYK,

    H. W. Lin, “The bulk Hilbert space of double scaled SYK,”JHEP11(2022) 060, arXiv:2208.07032 [hep-th]

    2022 arXiv

  22. [28]

    Sachdev–Ye–Kitaev model as Liouville quantum mechanics,

    D. Bagrets, A. Altland, and A. Kamenev, “Sachdev–Ye–Kitaev model as Liouville quantum mechanics,”Nucl. Phys. B911(2016) 191–205,arXiv:1607.00694 [cond-mat.str-el]

    2016 arXiv

  23. [29]

    Hilbert Spaces of Analytic Functions and Generalized Coherent States,

    M. Arik and D. D. Coon, “Hilbert Spaces of Analytic Functions and Generalized Coherent States,”J. Math. Phys.17(1976) 524

    1976

  24. [30]

    A symmetry algebra in double-scaled SYK,

    H. W. Lin and D. Stanford, “A symmetry algebra in double-scaled SYK,”SciPost Phys.15no. 6, (2023) 234,arXiv:2307.15725 [hep-th]

    2023 arXiv

  25. [31]

    Operator K-complexity in DSSYK: Krylov complexity equals bulk length,

    M. Ambrosini, E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, “Operator K-complexity in DSSYK: Krylov complexity equals bulk length,” arXiv:2412.15318 [hep-th]

    arXiv

  26. [32]

    Krylov complexity and chaos in deformed Sachdev-Ye-Kitaev models,

    S. Chapman, S. Demulder, D. A. Galante, S. U. Sheorey, and O. Shoval, “Krylov complexity and chaos in deformed Sachdev-Ye-Kitaev models,”Phys. Rev. B111 no. 3, (2025) 035141,arXiv:2407.09604 [hep-th]

    2025 arXiv

  27. [33]

    Krylov Complexity of Supersymmetric SYK Models,

    J. Chryssanthacopoulos and D. Vegh, “Krylov Complexity of Supersymmetric SYK Models,”arXiv:2511.20769 [hep-th]

    arXiv

  28. [36]

    Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography,

    R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen, and S. Sachdev, “Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography,”Phys. Rev. B95no. 15, (2017) 155131,arXiv:1612.00849 [cond-mat.str-el]

    2017 arXiv

  29. [37]

    SYK Model, Chaos and Conserved Charge,

    R. Bhattacharya, S. Chakrabarti, D. P. Jatkar, and A. Kundu, “SYK Model, Chaos and Conserved Charge,”JHEP11(2017) 180,arXiv:1709.07613 [hep-th]

    2017 arXiv

  30. [38]

    Notes on the complex Sachdev-Ye-Kitaev model,

    Y. Gu, A. Kitaev, S. Sachdev, and G. Tarnopolsky, “Notes on the complex Sachdev-Ye-Kitaev model,”JHEP02(2020) 157,arXiv:1910.14099 [hep-th]

    2020 arXiv

  31. [39]

    A bulk manifestation of Krylov complexity,

    E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, “A bulk manifestation of Krylov complexity,”JHEP08(2023) 213,arXiv:2305.04355 [hep-th]

    2023 arXiv

  32. [40]

    Geometry of Krylov complexity,

    P. Caputa, J. M. Magan, and D. Patramanis, “Geometry of Krylov complexity,” Phys. Rev. Res.4no. 1, (2022) 013041,arXiv:2109.03824 [hep-th]. 36

    2022 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.