Pith. sign in

REVIEW 3 major objections 3 minor 89 references

Coexistence of ergodic and non-ergodic behavior and level spacing statistics in a one-dimensional model of a flat band superconductor

T0 review · 3 major / 3 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 claims that the level spacing distribution of a one-dimensional flat-band superconductor model stays Poissonian when a singlet-conversion term is added, signaling an extensive set of local integrals of motion.

desk verdict A useful new model with solid few-body analytics, but the headline claim that level statistics do not change with λ3 is contradicted by the paper's own Fig. 10, so the LIOM inference is not established. read the letter →

arxiv 2411.09196 v2 pith:SFES5K3M submitted 2024-11-14 cond-mat.supr-con cond-mat.dis-nncond-mat.stat-mechcond-mat.str-el

Meri Teeriaho , Ville-Vertti Linho , Koushik Swaminathan , Sebastiano Peotta This is my paper · ORCID
classification cond-mat.supr-concond-mat.dis-nncond-mat.stat-mechcond-mat.str-el
keywords flatbandsuperconductivitylevelspacingstatisticslocalintegralsofmotionergodicitybreakingexactdiagonalizationCreutzladderprojecteddicelatticequasiparticlelocalization
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

Flat-band superconductors are hard to analyze because projecting onto a flat band produces long-range, many-body interactions, and quasiparticle excitations may be localized even without disorder. This paper introduces the one-dimensional on-site/bond singlet (OBS) model as a tractable stand-in for the projected dice-lattice superconductor and uses exact diagonalization to study its spectrum and dynamics. Its central claim is that when the bond-singlet hopping term is absent, the level spacing statistics remain close to Poissonian as the on-site/bond singlet conversion term is varied, even though the explicitly known conserved quantities of the Creutz-ladder limit no longer commute. This is presented as evidence for an extensive number of local integrals of motion at $\lambda_2=0$ for all $\lambda_3$, making the OBS model a nontrivial integrable generalization of the projected Creutz ladder. The result matters because explicit local conserved quantities are rarely available for flat-band superconductors, and level spacing statistics may serve as a general diagnostic of quasiparticle localization.

What carries the argument

The central object is the three-term OBS Hamiltonian $\hat{\mathcal{H}}_{\lambda_2,\lambda_3}=\hat{\mathcal{H}}_1+\lambda_2\hat{\mathcal{H}}_2+\lambda_3\hat{\mathcal{H}}_3$, where $\hat{\mathcal{H}}_1$ is the projected Creutz-ladder term (on-site singlet hopping plus spin exchange), $\hat{\mathcal{H}}_2$ moves bond singlets along the chain, and $\hat{\mathcal{H}}_3$ converts on-site singlets into bond singlets and back. The diagnostic machinery is level spacing statistics: unfolded spacings $P(s)$ and the mean gap ratio $\langle r\rangle$, whose Poisson value signals integrability and whose Gaussian orthogonal ensemble value signals quantum chaos. The algebra carries the argument: $\hat{\mathcal{H}}_1$ and $\hat{\mathcal{H}}_3$ both satisfy a spectrum-generating relation with the total pseudospin raising operator, while $\hat{\mathcal{H}}_2$ does not, so only $\hat{\mathcal{H}}_2$ can destroy the integrable structure. In agreement with this, the numerical level statistics depend strongly on $\lambda_2$ but are almost independent of $\lambda_3$ at $\lambda_2=0$.

What would settle it

Compute the level spacing distribution for a model that has the same spectrum-generating algebra but is known to have no local integrals of motion; if it too stays sub-Poissonian, then the near-Poissonian signal at $\lambda_2=0$ does not by itself prove local integrals of motion. A positive check would be the explicit construction of a local operator commuting with $\hat{\mathcal{H}}_{0,\lambda_3}$ or a Bethe-ansatz solution for nonzero $\lambda_3$.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the level spacing distribution of the OBS Hamiltonian $\hat{\mathcal{H}}_{\lambda_2,\lambda_3}=\hat{\mathcal{H}}_1+\lambda_2\hat{\mathcal{H}}_2+\lambda_3\hat{\mathcal{H}}_3$ is essentially independent of $\lambda_3$ when $\lambda_2=0$. At $\lambda_2=\lambda_3=0$, the Hamiltonian $\hat{\mathcal{H}}_1$ is the projected Creutz-ladder Hamiltonian, known to be integrable with local integrals of motion given by the site spin parities; the authors show that at $\lambda_3=1$ the unfolded level spacing distribution and the mean gap ratio $\langle r\rangle$ remain sub-Poissonian and close to the $\lambda_3=0$ values, even though the site parities are no longer conserved. They interpret this as evidence that $\hat{\mathcal{H}}_{0,\lambda_3}$ has an extensive number of local integrals of motion for arbitrary $\lambda_3$, differing from the known Creutz-ladder operators, and that the model is an integrable generalization of the projected Creutz ladder. Including $\hat{\mathcal{H}}_2$ at $\lambda_2>0$ breaks this structure and drives the statistics toward the Gaussian orthogonal ensemble result, identifying $\hat{\mathcal{H}}_2$ as the term that restores ergodicity.

Load-bearing premise

The argument assumes that the near-Poissonian level spacing seen when the conversion term is switched on is caused by hidden local conservation laws, rather than by the special algebraic symmetry that both the integrable and non-integrable settings share.

Editorial extensions

If this is right

  • The full line $\lambda_2=0$ in the OBS model is integrable: an extensive number of local integrals of motion exists for every value of $\lambda_3$, even though the explicit Creutz-ladder operators no longer commute.
  • The near-Poissonian level spacing at $\lambda_2=0$ is robust to boundary conditions, appearing for both periodic and open chains, so the hidden conservation laws are not an artifact of one geometry.
  • Adding the bond-singlet hopping term $\lambda_2$ destroys the integrable structure and drives the gap ratio toward the Gaussian orthogonal ensemble value, identifying $\hat{\mathcal{H}}_2$ as the term responsible for ergodic behavior.
  • Level spacing statistics can serve as an unbiased probe of quasiparticle localization in flat-band superconductors, extending the dynamics-based evidence previously obtained for the projected dice lattice.

Reading between the lines

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

  • A natural next test is to construct the local integrals of motion for $\hat{\mathcal{H}}_{0,\lambda_3}$ perturbatively in small $\lambda_3$; the three-body scattering solution suggests they acquire a nonlocal component that moves an unpaired particle by two sites per collision, so candidate operators should involve bond-singlet strings rather than bare site parities.
  • The same level-spacing signature could also appear in models with a spectrum-generating algebra but no conserved charges, so checking the statistics against such a control model would calibrate the method before it is applied to the two-dimensional dice lattice.
  • If the interpretation is correct, the OBS model offers an efficient platform for studying quasiparticle transport in flat-band superconductors, since the integrable line reduces the Hilbert space and may enable a Bethe-ansatz treatment for nonzero $\lambda_3$.
  • A direct finite-size check on larger chains and other fillings would clarify whether the anomalously large small-spacing bin grows with system size, as expected if it is a genuine many-body degeneracy from local integrals of motion rather than a finite-size accident.
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

3 major / 3 minor

Summary. The paper introduces the on-site/bond singlet (OBS) model, a one-dimensional fermionic lattice model inspired by the projected dice lattice Hamiltonian, with three terms: H1 (the Creutz-ladder-type term with known local integrals of motion), H2 (bond singlet hopping), and H3 (on-site/bond singlet conversion). The authors use exact diagonalization to study the energy spectrum and time evolution, solve the two-body and three-body problems analytically for λ2=0, and analyze level spacing statistics as a function of λ2 and λ3. The central claim is that for λ2=0 the level spacing distribution is essentially independent of λ3 and remains close to Poisson, which is interpreted as evidence for an extensive number of LIOMs for H_{0,λ3}, establishing the OBS model as a nontrivial integrable generalization of the projected Creutz ladder.

Significance. If the LIOM claim were established, the paper would provide a useful one-dimensional platform for studying quasiparticle localization in flat-band superconductors and would demonstrate level spacing statistics as a practical probe of integrability in such systems. The analytic few-body results, particularly the three-body scattering solution and the transmission coefficient, are concrete and valuable contributions, and the presentation is generally honest about the indirect nature of level statistics. However, the headline claim is not quantitatively supported by the paper's own numerics, so the significance is currently prospective rather than established.

major comments (3)
  1. [Sec. V, Fig. 10] The claim that the level spacing distribution for λ2=0 is essentially independent of λ3 is not supported by the reported data. At λ2=0.0001, the mean gap ratio changes from ⟨r⟩=0.3559 (λ3=0) to ⟨r⟩=0.1863 (λ3=1), a shift comparable to the entire Poisson-to-GOE interval (0.386 to 0.536) and in the direction away from the Poisson value. The difference persists at λ2=0.02 (0.4047 vs 0.3616) and is also visible in the histograms of Fig. 8, where the low-s peak is much larger for λ3=1. Because the LIOM conclusion in the abstract and Sec. V rests on this invariance, the authors need to either quantify the similarity (with error bars and a stated criterion) or soften the central claim to a qualitative one.
  2. [Sec. V; Sec. II A] The sub-Poissonian level spacing for λ2=0 is also consistent with the spectrum generating algebra [H1,B+]=-(A+4)B+ and [H3,B+]=0, which holds for all λ3 and produces equally spaced towers of states; this mechanism is not exclusive to systems with LIOMs. The paper does not rule out the SGA as the source of the anomalous level clustering for H_{0,λ3}, so the inference of an extensive set of LIOMs from the level statistics alone is not secure. A control study (for example, a model with SGA but no LIOMs) or an explicit perturbative construction of the LIOMs is needed.
  3. [Sec. V, Figs. 8–11] The level statistics are computed only for L=10 and L=12 with no finite-size scaling; the paper itself notes, in the discussion of Fig. 10, that the finite-size effect on the degeneracies cannot be confirmed due to computational limits. For a claim of integrability, one should show that ⟨r⟩ and P(s) do not drift with system size, or at least provide a two-size scaling analysis. The histograms in Figs. 8 and 9 also lack error bars, so it is unclear whether the differences between the λ3=0 and λ3=1 rows are statistically significant.
minor comments (3)
  1. [Sec. IV] In the sentence introducing the three-body scattering problem, 'interacts with a a single unpaired particle' contains a duplicated article 'a'.
  2. [Fig. 7] The horizontal axis label appears to be missing; please label it explicitly (e.g., k a).
  3. [Sec. V] The sentence 'The level spacing distribution of a model that behaves chaotically will follow that of a Gaussian ensemble' should specify the Gaussian orthogonal ensemble for time-reversal-symmetric systems, as done later in the same paragraph.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the level-spacing results are direct numerical outputs, and cited integrability of H1 is independent background.

full rationale

The paper's central claim—that the level spacing distribution for λ2=0 is essentially independent of λ3 and indicates LIOMs—is supported by exact diagonalization data (Figs. 8–11) computed with QuSpin. No parameter is fitted to produce the reported ⟨r⟩ values; the comparison to Poisson and GOE values is an external benchmark. The integrability of H1 is cited to Refs. [24,61], which is independent support (Bethe-ansatz solvable Creutz ladder) and is used to interpret the λ3=0 case, not to define or force the λ3=1 result. The inference from sub-Poissonian statistics to LIOMs is an assumption (acknowledged in Secs. V and VI), but it is not circular because the statistics are computed directly from the Hamiltonian. Self-citations to Refs. [24,25] are present but not load-bearing: they provide context and known models, and the main numerical derivation does not rely on them.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The central claim rests on standard random matrix theory, an unfolding protocol, and an inductive inference from level spacing similarity to the existence of LIOMs. The free parameters are model couplings chosen by hand. No fitting to data is performed. The main invented entity is the set of unknown LIOMs, which is currently unfalsified.

free parameters (2)
  • A = 10
    Chemical potential / on-site pair energy in H1 (Eq. 9), fixed by hand to A = 10 in all numerical simulations. It is not fitted to data but affects the spectrum and is a free model parameter.
  • lambda3 = varied (0, 0.5, 1)
    Coupling of the on-site/bond singlet conversion term H3 (Eq. 11). The central claim concerns the independence of the level statistics from this parameter, so it is a hand-tuned parameter of the model.
assumptions (4)
  • domain assumption The unfolding procedure (discarding 2.5% of levels at each edge, splitting into 30 bins) yields unbiased level spacing statistics.
    Sec. V describes the specific unfolding recipe; it is a standard but non-unique procedure and could affect the resulting distributions.
  • domain assumption Near-Poisson (or sub-Poissonian) level spacing statistics imply the presence of an extensive number of local integrals of motion.
    The paper uses this to claim LIOMs for H_{0,lambda3}, while citing Caux and Mossel (Ref. [86]) noting that Poisson statistics do not automatically imply Bethe-ansatz integrability. The assumption is standard but not rigorous.
  • ad hoc to paper Similarity of level spacing distributions between lambda3 = 0 and lambda3 = 1 indicates that LIOMs survive for lambda3 nonzero.
    Sec. V interprets the numerical similarity as evidence for LIOMs, without excluding the alternative that the spectrum generating algebra (SGA, Sec. IIA) produces the same level clustering.
  • domain assumption The three-body scattering Ansatz in Sec. IVB spans the relevant invariant subspace and yields the full scattering solution.
    The authors state that S1 union S2 is invariant 'as one can verify directly from (39)' but do not show the verification; this completeness is assumed.
invented entities (1)
  • Unspecified local integrals of motion for H_{0,lambda3} when lambda3 != 0
    purpose: To explain the near-Poisson level spacing and the partially non-ergodic time dynamics observed for lambda2 = 0, lambda3 != 0
    The LIOMs are not constructed explicitly; their existence is inferred indirectly from level spacing statistics and spectral degeneracy patterns. No falsifiable prediction outside the paper is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Coexistence of ergodic and non-ergodic behavior and level spacing statistics in a one-dimensional model of a flat band superconductor." pith.science (2026). https://pith.science/paper/SFES5K3M

@misc{pith2026241109196,
  author       = {Pith},
  title        = {Pith review of: Coexistence of ergodic and non-ergodic behavior and level spacing statistics in a one-dimensional model of a flat band superconductor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SFES5K3M}},
  note         = {Machine review of arXiv:2411.09196}
}
read the original abstract

Motivated by recent studies of the projected dice lattice Hamiltonian [K. Swaminathan et al., Phys. Rev. Research 5, 043215 (2023)], we introduce the on-site/bond singlet (OBS) model, a one-dimensional model of a flat band superconductor, in order to better understand the quasiparticle localization and interesting coexistence of ergodic and non-ergodic behavior present in the former model. The OBS model is the sum of terms that have direct counterparts in the projected dice lattice Hamiltonian, each of which is parameterized by a coupling constant. Exact diagonalization reveals that the energy spectrum and non-equilibrium dynamics of the OBS model are essentially the same as that of the dice lattice for some values of the coupling constants. The quasiparticle localization and breaking of ergodicity manifest in a striking manner in the level spacing distribution. Its near Poissonian form provides evidence for the existence of local integrals of motion and establishes the OBS model as a non-trivial integrable generalization of the projected Creutz ladder Hamiltonian. These results show that level spacing statistics is a promising tool to study quasiparticle excitations in flat band superconductors.

Figures

Figures reproduced from arXiv: 2411.09196 by the authors.

Figure 1
Figure 1. FIG. 1. Dynamical regimes in the OBS model: (a) Ini [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of possible eigenstates of the integrable [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Energy spectrum of the Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Time-averaged particle and spin densities obtained [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Dispersion of the OBS model [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Transmission coefficient [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Level spacing distribution for the 1D OBS model subject to periodic boundary conditions. Level spacings were [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Level spacing distribution in the 1D OBS model for a chain of length [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Mean gap ratio [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

89 extracted references · 57 canonical work pages

  1. [1]

    Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E.Kaxiras,andP.Jarillo-Herrero,Unconventionalsuper- conductivity in magic-angle graphene superlattices, Na- ture 556, 43 (2018)

    2018

  2. [2]

    G. R. Stewart, Unconventional superconductivity, Ad- vances in Physics66, 75 (2017)

    2017

  3. [3]

    Keimer, S

    B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179 (2015)

    2015

  4. [4]

    G. R. Stewart, Superconductivity in iron compounds, Re- views of Modern Physics83, 1589 (2011)

    2011

  5. [5]

    D. J. Scalapino, A common thread: The pairing inter- action for unconventional superconductors, Reviews of Modern Physics 84, 1383 (2012)

    2012

  6. [6]

    J.P.Hong, T.Soejima,andM.P.Zaletel,DetectingSym- metry Breaking in Magic Angle Graphene Using Scan- ningTunnelingMicroscopy,PhysicalReviewLetters 129, 147001 (2022)

    2022

  7. [7]

    Astrakhantsev, G

    N. Astrakhantsev, G. Wagner, T. Westerhout, T. Neu- pert, and M. H. Fischer, Understanding symmetry break- ing in twisted bilayer graphene from cluster constraints, Physical Review Research5, 043214 (2023)

    2023

  8. [8]

    Bultinck, Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling, Physical Review X 10, 10.1103/PhysRevX.10.031034 (2020)

    N. Bultinck, Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling, Physical Review X 10, 10.1103/PhysRevX.10.031034 (2020)

    doi:10.1103/physrevx.10.031034 2020

Show all 89 references
  1. [9]

    J. G. Checkelsky, B. A. Bernevig, P. Coleman, Q. Si, and S. Paschen, Flat bands, strange metals and the Kondo effect, Nature Reviews Materials9, 509 (2024)

    2024

  2. [10]

    Törmä, S

    P. Törmä, S. Peotta, and B. A. Bernevig, Superconduc- tivity, superfluidity and quantum geometry in twisted multilayer systems, Nature Reviews Physics 4, 528 (2022)

    2022

  3. [11]

    Balents, C

    L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré flat bands, Nature Physics16, 725 (2020)

    2020

  4. [12]

    F. He, Y. Zhou, Z. Ye, S.-H. Cho, J. Jeong, X. Meng, and Y. Wang, Moiré Patterns in 2D Materials: A Review, ACS Nano 15, 5944 (2021)

    2021

  5. [13]

    K. P. Nuckolls and A. Yazdani, A microscopic perspec- tive on moiré materials, Nature Reviews Materials9, 460 (2024)

    2024

  6. [14]

    P. C. Adak, S. Sinha, A. Agarwal, and M. M. Deshmukh, Tunable moiré materials for probing Berry physics and topology, Nature Reviews Materials9, 481 (2024)

    2024

  7. [15]

    N. B. Kopnin, T. T. Heikkilä, and G. E. Volovik, High-temperature surface superconductivity in topolog- ical flat-band systems, Physical Review B 83, 220503 (2011)

    2011

  8. [16]

    T. T. Heikkilä, N. B. Kopnin, and G. E. Volovik, Flat bands in topological media, JETP Letters94, 233 (2011)

    2011

  9. [17]

    Leykam, A

    D. Leykam, A. Andreanov, and S. Flach, Artificial flat band systems: From lattice models to experiments, Ad- vances in Physics: X3, 1473052 (2018)

    2018

  10. [18]

    V. A. J. Pyykkönen, S. Peotta, P. Fabritius, J. Mohan, T. Esslinger, and P. Törmä, Flat band transport and Josephson effect through a finite-size sawtooth lattice, Physical Review B103, 144519 (2021)

    2021

  11. [19]

    Peotta and P

    S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nature Communications 6, 8944 (2015)

    2015

  12. [20]

    Törmä, L

    P. Törmä, L. Liang, and S. Peotta, Quantum metric and effective mass of a two-body bound state in a flat band, Physical Review B98, 220511 (2018). 17

    2018

  13. [21]

    Bravyi, D

    S. Bravyi, D. P. DiVincenzo, and D. Loss, Schrieffer– Wolff transformation for quantum many-body systems, Annals of Physics326, 2793 (2011)

    2011

  14. [22]

    S. D. Huber and E. Altman, Bose condensation in flat bands, Physical Review B82, 184502 (2010)

    2010

  15. [23]

    Tovmasyan, S

    M. Tovmasyan, S. Peotta, P. Törmä, and S. D. Huber, Effective theory and emergent SU ( 2 ) symmetry in the flatbandsofattractiveHubbardmodels,PhysicalReview B 94, 245149 (2016)

    2016

  16. [24]

    Tovmasyan, S

    M. Tovmasyan, S. Peotta, L. Liang, P. Törmä, and S. D. Huber, Preformed pairs in flat Bloch bands, Physical Re- view B 98, 134513 (2018)

    2018

  17. [25]

    Swaminathan, P

    K. Swaminathan, P. Tadros, and S. Peotta, Signatures of many-body localization of quasiparticles in a flat band superconductor, Physical Review Research 5, 043215 (2023)

    2023

  18. [26]

    Möller and N

    G. Möller and N. R. Cooper, Correlated Phases of Bosons in the Flat Lowest Band of the Dice Lattice, Physical Review Letters 108, 045306 (2012)

    2012

  19. [27]

    Pudleiner and A

    P. Pudleiner and A. Mielke, Interacting bosons in two- dimensional flat band systems, The European Physical Journal B 88, 207 (2015)

    2015

  20. [28]

    Danieli, A

    C. Danieli, A. Andreanov, and S. Flach, Many-body lo- calization transition from flat-band fine tuning, Physical Review B 105, L041113 (2022)

    2022

  21. [29]

    Danieli, A

    C. Danieli, A. Andreanov, D. Leykam, and S. Flach, Flat band fine-tuning and its photonic applications, Nanopho- tonics 13, 3925 (2024)

    2024

  22. [30]

    H. L. Stormer, D. C. Tsui, and A. C. Gossard, The frac- tional quantum Hall effect, Reviews of Modern Physics 71, S298 (1999)

    1999

  23. [31]

    Murthy and R

    G. Murthy and R. Shankar, Hamiltonian theories of the fractional quantum Hall effect, Reviews of Modern Physics 75, 1101 (2003)

    2003

  24. [32]

    J. K. Jain and R. K. Kamilla, Composite Fermions in the Hilbert Space of the Lowest Electronic Landau Level, In- ternationalJournal ofModernPhysicsB 11,2621 (1997)

    1997

  25. [33]

    Sodemann and A

    I. Sodemann and A. H. MacDonald, Landau level mixing and the fractional quantum Hall effect, Physical Review B 87, 245425 (2013)

    2013

  26. [34]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized gen- eralized Wannier functions for composite energy bands, Physical Review B56, 12847 (1997)

    1997

  27. [35]

    Vidal, R

    J. Vidal, R. Mosseri, and B. Douçot, Aharonov-Bohm Cages in Two-Dimensional Structures, Physical Review Letters 81, 5888 (1998)

    1998

  28. [36]

    Vidal, P

    J. Vidal, P. Butaud, B. Douçot, and R. Mosseri, Disor- der and interactions in Aharonov-Bohm cages, Physical Review B 64, 155306 (2001)

    2001

  29. [37]

    Herzog-Arbeitman, A

    J. Herzog-Arbeitman, A. Chew, K.-E. Huhtinen, P. Törmä, and B. A. Bernevig, Many-Body Su- perconductivity in Topological Flat Bands (2022), arXiv:2209.00007 [cond-mat]

    2022 arXiv

  30. [38]

    V. A. J. Pyykkönen, S. Peotta, and P. Törmä, Suppres- sion of Nonequilibrium Quasiparticle Transport in Flat- Band Superconductors, Physical Review Letters 130, 216003 (2023)

    2023

  31. [39]

    Ma and P

    M. Ma and P. A. Lee, Localized superconductors, Phys- ical Review B32, 5658 (1985)

    1985

  32. [40]

    M. Ma, B. I. Halperin, and P. A. Lee, Strongly disordered superfluids: Quantum fluctuations and critical behavior, Physical Review B34, 3136 (1986)

    1986

  33. [41]

    Kliczkowski, R

    M. Kliczkowski, R. Świętek, M. Hopjan, and L. Vidmar, Fadingergodicity,PhysicalReviewB 110,134206(2024)

    2024

  34. [42]

    Neill, P

    C. Neill, P. Roushan, M. Fang, Y. Chen, M. Kolodru- betz, Z. Chen, A. Megrant, R. Barends, B. Campbell, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, J. Mutus, P. J. J. O’Malley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. Polkovnikov, and J. M. Mar- tini...

    2016

  35. [43]

    Roy and A

    S. Roy and A. Lazarides, Strong ergodicity breaking due to local constraints in a quantum system, Physical Re- view Research 2, 023159 (2020)

    2020

  36. [44]

    Oganesyan and D

    V. Oganesyan and D. A. Huse, Localization of interact- ing fermions at high temperature, Physical Review B75, 155111 (2007)

    2007

  37. [45]

    Alet and N

    F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, Comptes Rendus Physique 19, 498 (2018)

    2018

  38. [46]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many-Body Localiza- tion and Thermalization in Quantum Statistical Mechan- ics, Annual Review of Condensed Matter Physics6, 15 (2015)

    2015

  39. [47]

    J. Z. Imbrie, V. Ros, and A. Scardicchio, Local integrals of motion in many-body localized systems, Annalen der Physik 529, 1600278 (2017)

    2017

  40. [48]

    D. A. Abanin, E. Altman, I. Bloch, and M. Ser- byn, Colloquium : Many-body localization, thermaliza- tion, and entanglement, Reviews of Modern Physics91, 021001 (2019)

    2019

  41. [49]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice, Science 349, 842 (2015)

    2015

  42. [50]

    J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science352, 1547 (2016)

    2016

  43. [51]

    S. S. Kondov, W. R. McGehee, W. Xu, and B. De- Marco, Disorder-Induced Localization in a Strongly Cor- related Atomic Hubbard Gas, Physical Review Letters 114, 083002 (2015)

    2015

  44. [52]

    Serbyn, Z

    M. Serbyn, Z. Papić, and D. A. Abanin, Local Conserva- tion Laws and the Structure of the Many-Body Localized States, Physical Review Letters111, 127201 (2013)

    2013

  45. [53]

    Serbyn, Z

    M. Serbyn, Z. Papić, and D. A. Abanin, Criterion for Many-Body Localization-Delocalization Phase Transi- tion, Physical Review X5, 041047 (2015)

    2015

  46. [54]

    Orito, Y

    T. Orito, Y. Kuno, and I. Ichinose, Nonthermalized dy- namics of flat-band many-body localization, Physical Re- view B 103, L060301 (2021)

    2021

  47. [55]

    S. J. Thomson and M. Schiró, Local Integrals of Motion in Quasiperiodic Many-Body Localized Systems, SciPost Physics 14, 125 (2023)

    2023

  48. [56]

    Danieli, A

    C. Danieli, A. Andreanov, and S. Flach, Many-body flat- bandlocalization,PhysicalReviewB 102,041116(2020)

    2020

  49. [57]

    Moudgalya, B

    S. Moudgalya, B. A. Bernevig, and N. Regnault, Quan- tum many-body scars and Hilbert space fragmentation: A review of exact results, Reports on Progress in Physics 85, 086501 (2022)

    2022

  50. [58]

    A.Chandran, T.Iadecola, V.Khemani,andR.Moessner, Quantum Many-Body Scars: A Quasiparticle Perspec- tive, Annual Review of Condensed Matter Physics14, 443 (2023), arXiv:2206.11528 [cond-mat, physics:quant- ph]. 18

    2023 arXiv

  51. [59]

    Chandran, I

    A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Constructing local integrals of motion in the many-body localized phase, Physical Review B91, 085425 (2015)

    2015

  52. [60]

    T. E. O’Brien, D. A. Abanin, G. Vidal, and Z. Papić, Ex- plicit construction of local conserved operators in disor- dered many-body systems, Physical Review B94, 144208 (2016)

    2016

  53. [61]

    Y. Kuno, T. Orito, and I. Ichinose, Flat-band many-body localization and ergodicity breaking in the Creutz ladder, New Journal of Physics22, 013032 (2020)

    2020

  54. [62]

    P.Wigner, On the statistical distribution of thewidths and spacings of nuclear resonance levels, Mathematical Proceedings of the Cambridge Philosophical Society47, 790 (1951)

    E. P.Wigner, On the statistical distribution of thewidths and spacings of nuclear resonance levels, Mathematical Proceedings of the Cambridge Philosophical Society47, 790 (1951)

    1951

  55. [63]

    F. J. Dyson, The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechan- ics, Journal of Mathematical Physics3, 1199 (1962)

    1962

  56. [64]

    T. Guhr, A. Müller–Groeling, and H. A. Weidenmüller, Random-matrix theories in quantum physics: Common concepts, Physics Reports299, 189 (1998)

    1998

  57. [65]

    L. F. Santos, Integrability of a disordered Heisenberg spin-1/2 chain, Journal of Physics A: Mathematical and General 37, 4723 (2004)

    2004

  58. [66]

    L. F. Santos and M. Rigol, Localization and the effects of symmetries in the thermalization properties of one- dimensional quantum systems, Physical Review E 82, 031130 (2010)

    2010

  59. [67]

    Creutz, End States, Ladder Compounds, and Domain-Wall Fermions, Physical Review Letters 83, 2636 (1999)

    M. Creutz, End States, Ladder Compounds, and Domain-Wall Fermions, Physical Review Letters 83, 2636 (1999)

    1999

  60. [68]

    Takayoshi, H

    S. Takayoshi, H. Katsura, N. Watanabe, and H. Aoki, Phase diagram and pair Tomonaga-Luttinger liquid in a Bose-Hubbard model with flat bands, Physical Review A 88, 063613 (2013)

    2013

  61. [69]

    Tovmasyan, E

    M. Tovmasyan, E. P. L. Van Nieuwenburg, and S. D. Hu- ber, Geometry-induced pair condensation, Physical Re- view B 88, 220510 (2013)

    2013

  62. [70]

    Jünemann, A

    J. Jünemann, A. Piga, S.-J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez, Exploring Interacting Topo- logical Insulators with Ultracold Atoms: The Synthetic Creutz-Hubbard Model, Physical Review X 7, 031057 (2017)

    2017

  63. [71]

    A. O. Barut and A. Böhm, Dynamical Groups and Mass Formula, Physical Review139, B1107 (1965)

    1965

  64. [72]

    B.Buča, J.Tindall,andD.Jaksch,Non-stationarycoher- ent quantum many-body dynamics through dissipation, Nature Communications 10, 1730 (2019)

    2019

  65. [73]

    Cao, W.-L

    J. Cao, W.-L. Yang, K. Shi, and Y. Wang, Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields, Nuclear Physics B877, 152 (2013)

    2013

  66. [74]

    Weinberg and M

    P. Weinberg and M. Bukov, QuSpin: A Python package for dynamics and exact diagonalisation of quantum many body systems part I: Spin chains, SciPost Physics2, 003 (2017)

    2017

  67. [75]

    Weinberg and M

    P. Weinberg and M. Bukov, QuSpin: A Python package for dynamics and exact diagonalisation of quantum many body systems. Part II: Bosons, fermions and higher spins, SciPost Physics 7, 020 (2019)

    2019

  68. [76]

    Poilblanc, T

    D. Poilblanc, T. Ziman, J. Bellissard, F. Mila, and G. Montambaux, Poisson vs. GOE Statistics in Inte- grable and Non-Integrable Quantum Hamiltonians, Eu- rophysics Letters 22, 537 (1993)

    1993

  69. [77]

    Giraud, N

    O. Giraud, N. Macé, É. Vernier, and F. Alet, Probing Symmetries of Quantum Many-Body Systems through Gap Ratio Statistics, Physical Review X 12, 011006 (2022)

    2022

  70. [78]

    M. L. Mehta,Random Matrices, 3rd ed., Pure and Ap- plied Mathematics, Vol. 142 (Academic Press, 2004)

    2004

  71. [79]

    M. R. Zirnbauer, Symmetry Classes in Random Ma- trix Theory, in Encyclopedia of Mathematical Physics, edited by J.-P. Françoise, G. L. Naber, and T. S. Tsun (Academic Press, Oxford, 2006) pp. 204–212

    2006

  72. [80]

    T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Random-matrix physics: Spectrum and strength fluctuations, Reviews of Modern Physics 53, 385 (1981)

    1981

  73. [81]

    Bruus and J.-C

    H. Bruus and J.-C. Angl‘es d’Auriac, Energy level statis- tics of the two-dimensional Hubbard model at low filling, Physical Review B55, 9142 (1997)

    1997

  74. [82]

    Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles, Physical Review Letters 110, 084101 (2013)

    2013

  75. [83]

    L. F. Santos and M. Rigol, Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization, Physical Review E81, 036206 (2010)

    2010

  76. [84]

    Pal and D

    A. Pal and D. A. Huse, Many-body localization phase transition, Physical Review B82, 174411 (2010)

    2010

  77. [85]

    Rigol and L

    M. Rigol and L. F. Santos, Quantum chaos and thermal- ization in gapped systems, Physical Review A82, 011604 (2010)

    2010

  78. [86]

    Caux and J

    J.-S. Caux and J. Mossel, Remarks on the notion of quan- tum integrability, Journal of Statistical Mechanics: The- ory and Experiment2011, P02023 (2011)

    2011

  79. [87]

    G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vish- wanath, and D. M. Stamper-Kurn, Ultracold Atoms in a Tunable Optical Kagome Lattice, Physical Review Let- ters 108, 045305 (2012)

    2012

  80. [88]

    Leung, M

    T.-H. Leung, M. N. Schwarz, S.-W. Chang, C. D. Brown, G. Unnikrishnan, and D. Stamper-Kurn, Interaction- Enhanced Group Velocity of Bosons in the Flat Band of an Optical Kagome Lattice, Physical Review Letters 125, 133001 (2020)

    2020

  81. [89]

    T. H. Barter, T.-H. Leung, M. Okano, M. Block, N. Y. Yao, and D. M. Stamper-Kurn, Spatial coherence of a strongly interacting Bose gas in the trimerized kagome lattice, Physical Review A101, 011601 (2020)

    2020

Pith tools

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