Pith. sign in

REVIEW 2 major objections 5 minor 58 references

Supersymmetry in the nonsupersymmetric Sachdev-Ye-Kitaev model

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The plain four-body SYK model is supersymmetric for six of its eight symmetry classes.

desk verdict The central claim—that the four-body SYK model has hidden supersymmetry for generic couplings—is sound and worth refereeing; the quantitative plateau prediction is a heuristic that the paper overstates. read the letter →

arxiv 1908.00995 v2 pith:26XL4WAW submitted 2019-08-02 cond-mat.str-el cond-mat.dis-nnhep-th

Jan Behrends , Benjamin Béri This is my paper · ORCID
classification cond-mat.str-elcond-mat.dis-nnhep-th MSC 81Q6082B4415B52
keywords Sachdev-Ye-KitaevmodelsupersymmetryMajoranafermionsAltland-Zirnbauerclassificationrandommatrixtheorymany-bodyzeromodesfermionparitycorrelationfunctions
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

This paper argues that the simplest Sachdev-Ye-Kitaev model, consisting of $k$ Majorana fermions with random four-body couplings, is exactly supersymmetric for all but two values of $k \bmod 8$, with no fine-tuning of the couplings. The supercharges square to the Hamiltonian, and their number and structure are fixed by $k \bmod 8$ and the model's Altland-Zirnbauer symmetry class. The discovery matters because supersymmetry in many-body physics is usually imposed by hand; here it emerges automatically from the model's particle-hole symmetry, with observable consequences for time-dependent correlation functions away from the ground state.

What carries the argument

The key machinery is the set of spectrally flattened supercharges $\Gamma_j = Q_j/\sqrt{H}$, which behave as many-body Majorana zero modes: they satisfy $\{\Gamma_j,\Gamma_k\}=2\delta_{jk}$, commute with $H$, and anticommute with fermion parity $P$. The existence of $N$ such operators gives a $2^{N/2}$-fold degeneracy for each energy level. The count $N$ and the number of local supercharges $N_{\text{loc}}$ are read off from the antiunitary symmetries $T_\pm$; in particular, the presence of particle-hole symmetry $T_-$ forces parity degeneracy and hence supersymmetry. Expanding the local $\Gamma_j$ in the basis of odd products of Majorana operators then determines which $q$-body observables couple to the long-time plateau of correlation functions.

What would settle it

Diagonalize $H$ for a specific supersymmetric instance, such as $k=6$ or $k=10$, and directly test $\{Q_a,Q_b\}=2H\delta_{ab}$ for supercharges built from parity-degenerate eigenstates; then compare the exact long-time plateau $C_{q,\infty}$ for $q=1,3,5$ against Eq. (13). A violation of the anticommutator would refute the existence of supersymmetry, while a plateau pattern that does not alternate with $q$ even at large $k$ would falsify the random-vector modeling assumption.

Watch

Extended reading notes

Core claim

The central claim is that the four-body SYK Hamiltonian with structureless Gaussian couplings $J_{qrst}$ admits Hermitian supercharges $Q_a$ satisfying $\{Q_a,Q_b\}=2H\delta_{ab}$ and $[H,Q_a]=0$ whenever $k \bmod 8$ is not $0$ or $4$. The number of independent supercharges is $N=2$ in the symmetry classes D, C, BDI, and CI, and $N=4$ in classes DIII and CII. In odd-$k$ classes, all but one supercharge are local to the $k$ Majorana modes, and the remaining supercharge is the auxiliary Majorana $\gamma_\infty$. The microscopic structure of the local supercharges is dictated by $k \bmod 8$: they expand only in products of $4n+1$ Majorana operators in classes with positive supercharge signature, and only in products of $4n+3$ operators when the signature is negative. The supersymmetry follows directly from particle-hole symmetry $T_-$, which pairs opposite fermion-parity sectors and forces an energy degeneracy that can be reorganized into supermultiplets.

Load-bearing premise

The plateau predictions assume that the coefficients expanding the projected supercharges into Majorana strings behave as random vectors subject only to normalization and symmetry constraints; if this fails, the plateau values would differ, although the existence of supersymmetry itself would not be affected.

Editorial extensions

If this is right

  • The long-time plateau of the infinite-temperature $q$-body correlation function alternates with $q$: it is nonzero only for $q=4n+1$ in classes with positive supercharge signature, with additional contributions in classes DIII and CII where the product supercharge $\Gamma_4$ enters for $q=4n+3$.
  • The ramp shape in $q$-body correlation functions is set by the Dyson index $\beta$, which equals 1, 2, or 4 and is tied to the number of local supercharges.
  • The plateau formula $C_{q,\infty}M/4 = N/(\beta N_{\text{loc}})$ for matching Majorana strings, with a $\beta=4$ correction from $\Gamma_4$, predicts the alternating plateau pattern observed numerically when $\binom{k}{q}/\binom{k}{\lfloor k/2\rfloor}$ is close to one.
  • The local supercharges can be viewed as emergent unpaired Majorana modes at the boundary of a one-dimensional topological phase, connecting supersymmetry to the $\mathbb{Z}_8$ classification of the SYK model.
  • Because these correlation functions are measurable in digital quantum simulation, the predicted plateau and ramp signatures provide dynamical, far-from-equilibrium tests of supersymmetry.

Reading between the lines

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

  • The argument that particle-hole symmetry generates supersymmetry is not specific to the four-body interaction: any parity-conserving Hamiltonian in an Altland-Zirnbauer class with $T_-$ should admit analogous supercharges, so extended SYK models with other interaction orders may also be supersymmetric without fine-tuning.
  • The random-vector assumption used for the plateau coefficients is likely exact only in the large-Hilbert-space limit; the finite-$k$ growth visible in the numerics suggests a systematic finite-size correction to the plateau formula that could be derived from the exact trace structure.
  • If the identification with a one-dimensional topological phase is taken literally, the emergent supersymmetry should be stable under perturbations that preserve $T_-$, which is a concrete stability prediction that could be tested numerically.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the four-body SYK model of k Majorana modes and claims that, for all k mod 8 except 0 and 4, the model is supersymmetric in the sense of supersymmetric quantum mechanics: there exist Hermitian supercharges satisfying {Q_a,Q_b}=2H δ_ab and [H,Q_a]=0. The proof is based on the antiunitary particle-hole symmetry T_-, which forces parity-degenerate spectra; the supercharges are explicitly built from pairs of degenerate eigenstates. The authors count the number of local supercharges Nloc using commutation with the chiral operator Z, classify their Majorana-operator content by k mod 8, and connect Nloc to the Dyson index and ramp shapes of q-body correlation functions. They further derive a formula (Eq. 13) for the long-time plateau C_{q,∞} under a random-vector assumption on the expansion coefficients, and compare it to exact-diagonalization data in Figs. 2 and 4.

Significance. If the claims hold, the paper identifies a universal, parameter-free supersymmetric structure in the standard SYK model and ties it to the Altland-Zirnbauer classification. The explicit construction of local supercharges and the connection between their number and the Dyson index/ramp shapes are valuable for the SYK and random-matrix communities. The manuscript is careful in its symmetry analysis, and the numerical data support the qualitative ramp classification and the existence of long-time plateaus in the supersymmetric classes. The mathematical core establishing parity-degenerate eigenstates and hence the superalgebra is straightforward and sound; the paper also gives an explicit, reproducible numerical study for a range of system sizes.

major comments (2)
  1. [Eqs. (12)-(13), Fig. 2] The derivation of the plateau formula (13) from Eq. (12) rests on the unproven assertion that the expansion coefficients v_{μj,a} of P_μ Γ_j in the Majorana basis behave as random vectors subject only to normalization and T± constraints. This assertion is not a consequence of supersymmetry, and it is not exact: the identity Σ_μ P_μ Γ_j = Γ_j imposes sum rules that correlate the v_{μj,a} across μ, and in classes BDI/CI the operator Γ_1=Z has a sparse expansion. The numerical data in Fig. 2 show that Eq. (13) fails when (k choose q)/(k choose ⌊k/2⌋) is not close to one, so the assumption is empirically false in those regimes. Because the abstract and conclusions present the plateau values as a consequence of SUSY, the quantitative connection is not secured; please either supply a controlled derivation or clearly mark Eq. (13) as a heuristic valid only in the large-(k choose q) regime.
  2. [Abstract and Conclusions] The abstract's statement that SUSY has consequences away from the ground state, including in q-body dynamical correlation functions, and the concluding claim that the plateau value is due to the imprint of how Γ_j transforms are stronger than what is established. The long-time plateau formula requires the additional random-vector hypothesis discussed above; the exact consequences of SUSY are the parity degeneracy, the algebraic structure of the supercharges, and the qualitative ramp shapes. Please distinguish these exact statements from the heuristic plateau estimate in the abstract and conclusions.
minor comments (5)
  1. [Supersymmetry from particle-hole symmetry] Please state explicitly in the abstract that the SUSY construction is equivalent to the parity degeneracy induced by T_-, so that the reader does not mistake the result for a new independent symmetry beyond particle-hole symmetry.
  2. [Eq. (13)] The first line of Eq. (13) is typographically ambiguous and appears to read N/(β Nloc), which is inconsistent with the numerical plateau values quoted in Appendix C (e.g., CII q=4n+3 plateau ≈3); the intended expression is presumably N Nloc/β. Please correct the typesetting.
  3. [Fig. 2 caption] The quantity c in panel (d) of Fig. 2 is used before it is defined; please define c as the random-matrix expectation from Eq. (13).
  4. [Appendix C b and Fig. 4] Please specify in the caption of Fig. 4 whether the plotted plateau is C_{q,∞} or C_{q,∞}M/4, so that the quoted values (e.g., "plateau at C_{q,∞} ≈ 2") match the main-text notation.
  5. [Abstract] The phrase that the structure of the supercharges is entirely set by the number of interacting Majorana modes should be qualified: the explicit supercharges in Eq. (4) depend on the disorder realization through the eigenstates, while what is fixed by k mod 8 is their symmetry type and Majorana-parity content.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SUSY existence is a genuine consequence of particle-hole symmetry T−, and the plateau estimate is an openly qualified random-vector ansatz, not a fitted prediction.

full rationale

The paper's central derivation is self-contained and non-circular. Supersymmetry is established by constructing supercharges Q_a from the parity-degenerate eigenstates forced by particle-hole symmetry T−; the input is the independently established antiunitary symmetry (Table I and Appendix A), not the target result. The construction in Eq. (4) is a standard witness argument: T− satisfies [T−,H]=0 and {T−,P}=0, hence ε_μ^+=ε_μ^− and the operators Q_1,Q_2 satisfy {Q_a,Q_b}=2Hδ_ab by explicit construction. This is a derivation rather than a restatement of the conclusion, because T− is defined independently of any supercharge. The locality count Nloc is checked by explicit (anti)commutation with Z, with the DIII/CII cases worked out in Appendix B. The eightfold classification, although citing the authors' Ref. 46, is rederived in Appendix A from Fidkowski-Kitaev Majorana constructions, so the self-citation is not load-bearing. The plateau estimate, Eq. (13), rests on an explicitly stated random-vector ansatz for the expansion coefficients v_{μj,a} of P_μΓ_j, subject only to normalization and symmetry constraints. This is an uncontrolled modeling assumption whose agreement with numerics is openly qualified in Fig. 2 ('the agreement is excellent when (k choose q)/(k choose floor(k/2)) is close to one'), but an unproven ansatz is a correctness risk, not circularity: Eq. (13) is not fitted to the data it predicts and does not reduce to any input by construction. No step in the claimed derivation chain is equivalent to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the parity degeneracy established by the particle-hole symmetry classification (from the authors' prior work [46]) and on standard supercharge construction. No free parameters are fitted; the random couplings are not parameters in the predictions. The only ad hoc ingredient is the random-vector assumption used to estimate the plateau heights.

assumptions (4)
  • domain assumption The Altland-Zirnbauer classification of the SYK model (Table I) is correct.
    The paper relies on the eightfold symmetry classification from Ref. [46] to identify which symmetry classes have particle-hole symmetry T- and thus parity degeneracy.
  • domain assumption For odd k, the Hilbert space is defined with an auxiliary Majorana gamma_infinity at infinity.
    The model with odd k is conventionally completed by a nonlocal Majorana; this is standard in the literature.
  • ad hoc to paper Random-vector assumption for the coefficients v_{mu j,a} in Eq. (12).
    Used to derive the plateau estimate Eq. (13); it is stated as an assumption in the text.
  • standard math An operator on the odd-k Hilbert space is local (built from gamma_{q neq infinity}) iff it commutes with Z.
    Clifford algebra structure; used to count Nloc.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Supersymmetry in the nonsupersymmetric Sachdev-Ye-Kitaev model." pith.science (2026). https://pith.science/paper/26XL4WAW

@misc{pith2026190800995,
  author       = {Pith},
  title        = {Pith review of: Supersymmetry in the nonsupersymmetric Sachdev-Ye-Kitaev model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/26XL4WAW}},
  note         = {Machine review of arXiv:1908.00995}
}
abstract

Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground state properties of complex quantum systems and gives relations between correlation functions. In this work, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes, and are thus fundamentally linked to the model's Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics, and has consequences away from the ground state, including in $q$-body dynamical correlation functions.

Figures

Figures reproduced from arXiv: 1908.00995 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Ensemble-averaged [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 49 canonical work pages

  1. [1]

    spectrally flattened

    The presence of particle-hole symmetry also guar- antees degeneracy between parity sectors, which as we now note, also implies SUSY. Parity degeneracy directly follows from particle-hole symmetry because |ψp µ⟩ and T−|ψp µ⟩ have the same energy εµ = εp µ = ε−p µ (since [T−,H ] = 0), but opposite parity ({T−,P} = 0) [45, 46]. Therefore, |ψp µ⟩⟨ψ−p µ | is a...

    work page

  2. [2]

    We refer to the first case as time- reversal symmetry: The level spacing statistics of each subblock are determined by the presence of T and by the signT 2 =±1

    Thus, when both operators commute, T is an antiu- nitary operator that maps each parity subblock of the Hamiltonian to itself, otherwise, the parity subblocks are exchanged by T . We refer to the first case as time- reversal symmetry: The level spacing statistics of each subblock are determined by the presence of T and by the signT 2 =±1. We refer to the s...

    work page

  3. [3]

    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)

    work page 1993

  4. [4]

    Kitaev, A simple model of quantum holography (2015), proceedings of the KITP Program: Entanglement in Strongly-Correlated Quantum Matter

    A. Kitaev, A simple model of quantum holography (2015), proceedings of the KITP Program: Entanglement in Strongly-Correlated Quantum Matter

    work page 2015

  5. [5]

    Maldacena, The Large-N Limit of Superconformal Field Theories and Supergravity, Int

    J. Maldacena, The Large-N Limit of Superconformal Field Theories and Supergravity, Int. J. Theor. Phys 38, 1113 (1999)

    work page 1999

  6. [6]

    Gubser, I

    S. Gubser, I. Klebanov, and A. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428, 105 (1998)

    work page 1998

  7. [7]

    Witten, Anti de Sitter Space and Holography, Adv

    E. Witten, Anti de Sitter Space and Holography, Adv. Theor. Math. Phys. 2, 253 (1998)

    work page 1998

  8. [8]

    S. H. Shenker and D. Stanford, Black holes and the but- terfly effect, J. High Energy Phys. 2014 (3), 67

    work page 2014

Show all 58 references
  1. [9]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 2016 (8), 106

    2016

  2. [10]

    A. M. Garc´ ıa-Garc´ ıa and J. J. M. Verbaarschot, Analyt- ical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N, Phys. Rev. D 96, 066012 (2017)

    2017

  3. [11]

    Altland and D

    A. Altland and D. Bagrets, Quantum ergodicity in the SYK model, Nucl. Phys. B 930, 45 (2018)

    2018

  4. [12]

    Iyoda, H

    E. Iyoda, H. Katsura, and T. Sagawa, Effective dimen- sion, level statistics, and integrability of Sachdev-Ye- Kitaev-like models, Phys. Rev. D 98, 086020 (2018)

    2018

  5. [13]

    Micklitz, F

    T. Micklitz, F. Monteiro, and A. Altland, Nonergodic Extended States in the Sachdev-Ye-Kitaev Model, Phys. Rev. Lett. 123, 125701 (2019)

    2019

  6. [14]

    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. B 95, 155131 (2017)

    2017

  7. [15]

    Gu, X.-L

    Y. Gu, X.-L. Qi, and D. Stanford, Local criticality, diffu- sion and chaos in generalized Sachdev-Ye-Kitaev models, J. High Energy Phys. 2017 (5), 125

    2017

  8. [16]

    Song, C.-M

    X.-Y. Song, C.-M. Jian, and L. Balents, Strongly Corre- lated Metal Built from Sachdev-Ye-Kitaev Models, Phys. Rev. Lett. 119, 216601 (2017)

    2017

  9. [17]

    Ben-Zion and J

    D. Ben-Zion and J. McGreevy, Strange metal from local quantum chaos, Phys. Rev. B 97, 155117 (2018)

    2018

  10. [18]

    Altland, D

    A. Altland, D. Bagrets, and A. Kamenev, Quantum Criti- cality of Granular Sachdev-Ye-Kitaev Matter, Phys. Rev. Lett. 123, 106601 (2019)

    2019

  11. [19]

    Sachdev, Bekenstein-hawking entropy and strange metals, Phys

    S. Sachdev, Bekenstein-hawking entropy and strange metals, Phys. Rev. X 5, 041025 (2015)

    2015

  12. [20]

    A. A. Patel and S. Sachdev, Critical strange metal from fluctuating gauge fields in a solvable random model, Phys. Rev. B 98, 125134 (2018)

    2018

  13. [21]

    J. D. Bekenstein, Black Holes and Entropy, Phys. Rev. D 7, 2333 (1973)

    1973

  14. [22]

    D. I. Pikulin and M. Franz, Black Hole on a Chip: Pro- posal for a Physical Realization of the Sachdev-Ye-Kitaev model in a Solid-State System, Phys. Rev. X 7, 031006 (2017)

    2017

  15. [23]

    A. Chew, A. Essin, and J. Alicea, Approximating the Sachdev-Ye-Kitaev model with Majorana wires, Phys. Rev. B 96, 121119(R) (2017)

    2017

  16. [24]

    Y.-Z. You, A. W. W. Ludwig, and C. Xu, Sachdev- Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states, Phys. Rev. B 95, 115150 (2017)

    2017

  17. [25]

    Maldacena and D

    J. Maldacena and D. Stanford, Remarks on the Sachdev- Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016)

    2016

  18. [26]

    D. J. Gross and V. Rosenhaus, A generalization of Sachdev-Ye-Kitaev, J. High Energy Phys. 2017 (2), 93. 9

    2017

  19. [27]

    W. Fu, D. Gaiotto, J. Maldacena, and S. Sachdev, Su- persymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95, 026009 (2017)

    2017

  20. [28]

    T. Li, J. Liu, Y. Xin, and Y. Zhou, Supersymmetric SYK model and random matrix theory, J. High Energy Phys. 2017 (6), 111

    2017

  21. [29]

    Kanazawa and T

    T. Kanazawa and T. Wettig, Complete random matrix classification of SYK models with N = 0, 1 and 2 super- symmetry, J. High Energy Phys. 2017 (9), 50

    2017

  22. [30]

    Hunter-Jones and J

    N. Hunter-Jones and J. Liu, Chaos and random matrices in supersymmetric SYK, J. High Energy Phys. 2018 (5), 202

    2018

  23. [31]

    A. M. Garc´ ıa-Garc´ ıa, Y. Jia, and J. J. M. Verbaarschot, Universality and Thouless energy in the supersymmet- ric Sachdev-Ye-Kitaev model, Phys. Rev. D 97, 106003 (2018)

    2018

  24. [32]

    Fendley, K

    P. Fendley, K. Schoutens, and J. de Boer, Lattice Models withN = 2 Supersymmetry, Phys. Rev. Lett. 90, 120402 (2003)

    2003

  25. [33]

    Fendley, B

    P. Fendley, B. Nienhuis, and K. Schoutens, Lattice fermion models with supersymmetry, J. Phys. A. Math. Gen. 36, 12399 (2003)

    2003

  26. [34]

    T. H. Hsieh, G. B. Hal´ asz, and T. Grover, All majorana models with translation symmetry are supersymmetric, Phys. Rev. Lett. 117, 166802 (2016)

    2016

  27. [35]

    Sannomiya, H

    N. Sannomiya, H. Katsura, and Y. Nakayama, Super- symmetry breaking and Nambu-Goldstone fermions with cubic dispersion, Phys. Rev. D 95, 065001 (2017)

    2017

  28. [36]

    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, J. High Energy Phys. 2017 (5), 118

    2017

  29. [37]

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

    1998

  30. [38]

    Witten, Constraints on supersymmetry breaking, Nucl

    E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202, 253 (1982)

    1982

  31. [39]

    Nicolai, Supersymmetry and spin systems, J

    H. Nicolai, Supersymmetry and spin systems, J. Phys. A. Math. Gen. 9, 1497 (1976)

    1976

  32. [40]

    C. V. Sukumar, Supersymmetric quantum mechanics of one-dimensional systems, J. Phys. A. Math. Gen. 18, 2917 (1985)

    1985

  33. [41]

    F¨ orster, Staggered spin and statistics in the super- symmetric t-J model, Phys

    D. F¨ orster, Staggered spin and statistics in the super- symmetric t-J model, Phys. Rev. Lett. 63, 2140 (1989)

    1989

  34. [42]

    Cooper, A

    F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rep. 251, 267 (1995)

    1995

  35. [43]

    Junker, Supersymmetric Methods in Quantum and Statistical Physics , Theoretical and Mathematical Physics (Springer, Berlin Heidelberg, 1996)

    G. Junker, Supersymmetric Methods in Quantum and Statistical Physics , Theoretical and Mathematical Physics (Springer, Berlin Heidelberg, 1996)

    1996

  36. [44]

    Correa and M

    F. Correa and M. S. Plyushchay, Hidden supersymmetry in quantum bosonic systems, Ann. Phys. (N. Y). 322, 2493 (2007)

    2007

  37. [45]

    Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi 44, 131 (2000)

    A. Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi 44, 131 (2000)

    2000

  38. [46]

    Each pair of Majoranas spans a two-dimensional Hilbert space; for an odd number of Majoranas, we insert an additional Majorana γ∞ to obtain a fermionic Hilbert space, giving M as above

  39. [47]

    Fidkowski and A

    L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B 83, 075103 (2011)

    2011

  40. [48]

    Behrends, J

    J. Behrends, J. H. Bardarson, and B. B´ eri, Tenfold way and many-body zero modes in the Sachdev-Ye-Kitaev model, Phys. Rev. B 99, 195123 (2019)

    2019

  41. [49]

    A. M. Garc´ ıa-Garc´ ıa and J. J. M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 126010 (2016)

    2016

  42. [50]

    Altland and M

    A. Altland and M. R. Zirnbauer, Nonstandard symme- try classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997)

    1997

  43. [51]

    Wess and J

    J. Wess and J. Bagger, Supersymmetry and Supergravity (Princeton University Press, Princeton, NJ, 1992)

    1992

  44. [52]

    We could have also chosen Γ j = diag{UµτjU† µ} with Uµ = exp (iαµτ3) having an arbitrary sequence of αµ. This does not mean that one can simultaneously choose a continuum different supercharges: Mutual anticommu- tation allows two supercharges from this continuum, uni- tary equ...

  45. [53]

    Goldstein and C

    G. Goldstein and C. Chamon, Exact zero modes in closed systems of interacting fermions, Phys. Rev. B 86, 115122 (2012)

    2012

  46. [54]

    Haake, Quantum Signatures of Chaos , Springer Se- ries in Synergetics, Vol

    F. Haake, Quantum Signatures of Chaos , Springer Se- ries in Synergetics, Vol. 54 (Springer, Berlin Heidelberg, 2010)

    2010

  47. [55]

    Garc´ ıa-´Alvarez, I

    L. Garc´ ıa-´Alvarez, I. L. Egusquiza, L. Lamata, A. del Campo, J. Sonner, and E. Solano, Digital Quantum Sim- ulation of Minimal AdS/CFT, Phys. Rev. Lett. 119, 040501 (2017)

    2017

  48. [56]

    Fendley and C

    P. Fendley and C. Hagendorf, Ground-state properties of a supersymmetric fermion chain, J. Stat. Mech. Theory Exp. 2011, P02014 (2011)

    2011

  49. [57]

    A. C. Cubero, G. Mussardo, and M. Panfil, Quench dy- namics in two-dimensional integrable SUSY models, J. Stat. Mech. Theory Exp. 2016, 033115 (2016)

    2016

  50. [58]

    de Wit and J

    B. de Wit and J. Smith, Field Theory in Particle Physics (North-Holland, Amsterdam, 1986)

    1986

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.