Parity-dependent double degeneracy and spectral statistics in the projected dice lattice

Anouar Moustaj; Jose L. Lado; Koushik Swaminathan; Sebastiano Peotta

REVIEW 4 major objections 5 minor 97 references

Parity-dependent double degeneracy and spectral statistics in the projected dice lattice

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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

Pith's one-line read For an odd particle number, every eigenstate of the projected dice-lattice Hubbard model is exactly doubly degenerate even after resolving all known symmetries, and the spacings between doublets follow the Gaussian Unitary Ensemble, while e

desk verdict Solid numerical evidence for parity-dependent exact degeneracy in a projected flat-band model, but the 'unprecedented' GOE/GUE claim likely reduces to a hidden Kramers degeneracy — the mechanism is the real story. read the letter →

arxiv 2602.11844 v2 pith:G3ACWG2H submitted 2026-02-12 cond-mat.str-el math-phmath.MPquant-ph

Koushik Swaminathan , Anouar Moustaj , Jose L. Lado , Sebastiano Peotta This is my paper

classification cond-mat.str-elmath-phmath.MPquant-ph

keywords flatbandsdicelatticeprojectedHubbardmodelspectralstatisticsrandommatrixensembleslevelspacinggapratiosdoubledegeneracy

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

An interacting fermion model, built by projecting the Hubbard interaction onto the two lowest flat bands of the π-flux dice lattice, is shown to have spectral statistics that depend sharply on the parity of the particle number. For even particle number the spectrum is nondegenerate and its level-spacing and gap-ratio distributions match a superposition of several independent Gaussian Orthogonal Ensemble (GOE) blocks, with the block count varying with system size. For odd particle number every eigenstate is exactly doubly degenerate even after all known conserved quantities (particle numbers, momentum, spin-flip parity, total spin, pseudospin) are resolved, and the spacings between the doublets follow the Gaussian Unitary Ensemble (GUE). The coexistence of GOE and GUE statistics in one physical system is presented as an unprecedented finding, and the paper argues the GUE is not a disguised two-GOE-block superposition because the third-order statistics fail to match that description; the mechanism behind the double degeneracy remains open.

What carries the argument

The argument runs on two tools: fully symmetry-resolved diagonalization, where total spin S and pseudospin B are fixed by a penalty-term construction on top of the particle numbers, momentum, and spin-flip parity; and higher-order spectral statistics, namely the k-th neighbor spacings s_n^k = E_{n+k}-E_n with the associated non-overlapping gap ratios, compared against Wigner-surmise curves and Monte-Carlo-generated superpositions of m independent GOE/GUE blocks. The decisive identity is the equivalence P_{2k*}(s/r,1,2)=P_{k*}(s/r,2) between two superposed GOE blocks and a single GUE block; the 3rd-order statistics, which fit neither, are used to reject that equivalence as an explanation for

What would settle it

Find the explicit antiunitary operator that exchanges the two Wannier bands and commutes with the Hamiltonian in the odd-N sector; if such an operator exists with square -1, the double degeneracy is Kramers-type and the central claim fails. A cheaper numerical test: add a small translation-invariant perturbation that mixes the two Wannier orbitals while preserving all resolved quantum numbers; if the doublets split, the degeneracy is symmetry-protected and the generator can be extracted. Alternatively, sparse diagonalization of the unprojected Hubbard model on larger clusters can test whether

Watch

Extended reading notes

Core claim

On the projected dice-lattice Hamiltonian with interaction parameters λ2=λ3=1, the paper's central claim is a strict parity dichotomy in the fully symmetry-resolved spectrum. In every even-N sector the eigenvalues are nondegenerate, and the k-th-neighbor spacing and gap-ratio distributions, k=1,...,4, are those of a superposition of m independent GOE spectra, with m taking the values 4, 6, and 2 for the three accessible cluster sizes—behaviour attributed to an extensive set of local integrals of motion. In every odd-N sector, the full spectrum consists of exact doublets that persist after resolving N↑, N↓, kx, ky, f, S, and B; the 2nd- and 4th-order statistics between doublets match the GUE

Load-bearing premise

The load-bearing premise is that the list of resolved commuting quantum numbers—N↑, N↓, kx, ky, f, S, B—is complete; if a further discrete symmetry (such as the suspected band-index antiunitary operation) exists but was not resolved, the double degeneracy becomes a known Kramers-type effect and the GUE statistics follow from an unresolved symmetry sector, undercutting the claim that the coexisting ensembles are a new phenomenon.

Editorial extensions

If this is right

Any exact-diagonalization study of the projected dice lattice must treat odd-N sectors as split into doublets; nearest-neighbor statistics are blind to the physically meaningful correlations, which appear only at orders k≥2.
The GUE matching for odd N shows unitary-class spectral correlations can arise in a real, time-reversal-symmetric Hamiltonian without magnetic fields, once a double degeneracy is present and properly factored out.
The even-N multi-block GOE with system-size-dependent m supports an extensive set of (possibly deformed) local integrals of motion, giving a concrete microscopic route to the nonergodic dynamics reported earlier for this model.
Adding or removing a single particle changes the ergodicity class, so finite-size analyses of level statistics in flat-band models must respect particle-number parity.

Reading between the lines

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

A natural next step is to search directly for the hypothesized antiunitary band-exchange operator; if it exists with square -1 only in the odd-N sector, the degeneracy is Kramers-type and the GUE statistics follow from an unresolved symmetry, making the 'unprecedented' claim weaker—the paper itself flags this as an open possibility.
If the degeneracy is instead topological, the doublet degeneracy should depend on the spatial topology (for instance, acquire a multiplicity related to the genus of the cluster), which could be tested in future exact-diagonalization studies with different boundary conditions.
The same parity effect may appear in other flat-band lattices with two inequivalent Wannier orbitals per unit cell, such as kagome or Lieb under π flux; a quick check of their projected spectra would tell whether this is a general flat-band phenomenon or peculiar to the dice lattice.
Because the doublet states differ by exchanging the two Wannier bands, a site-resolved probe of local density in a cold-atom realization could observe the doublet structure directly; particle-counting statistics, mentioned in the paper, are a promising observable.

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.

Desk Editor's Note

read the letter

Read the paper. The core observation is real and reproducible: in the projected dice lattice, odd particle-number sectors have an exactly doubly degenerate spectrum even after resolving spin, pseudospin, momentum, and spin-flip, and the k=2 and k=4 spacing distributions between doublets match GUE for the 3x2 and 4x2 clusters. Even parity shows multi-block GOE. The authors handle the higher-order spacing toolkit with care — Monte Carlo m-block references, KS distances, explicit discussion of the unfolding. They also show the two states in each doublet are related by a Wannier-index swap, which is direct evidence for a hidden antiunitary symmetry.

Now the soft spots, in proportion. The 'unprecedented coexistence of GOE and GUE' hinges on the assumption that all antiunitary symmetries have been resolved. The paper itself floats a hidden antiunitary symmetry in the Discussion, and the supplemental density plots suggest its action. If such an operator exists with square -1 in odd sectors, the degeneracy is textbook Kramers degeneracy and the GUE-looking spacing between doublets is a known property of doublet spectra — not a new random-matrix phenomenon. Nothing in the paper constructs or rules out that operator, so the novelty claim is not established. Second, the largest odd sector, 3x3, has a strong peak at s=0; the automatic fitting is abandoned and the GUE match is hand-picked. The supplement says this, but it means the cleanest GUE statement rests on the two smaller clusters. Third, the even-sector block count is nonmonotonic (4,6,2), so the extensive-LIOM interpretation is speculative, as the authors concede.

What the paper does well is the careful symmetry resolution and the honest framing of the open mechanism. The phenomenon — parity-controlled double degeneracy — is concrete and likely to interest the flat-band and RMT communities, and the observational part deserves a serious referee. The referee's main task should be to force the authors to either identify the hidden antiunitary symmetry or defend the completeness of their resolution. If the symmetry exists, the paper becomes a clean example of Kramers degeneracy with a band-index time-reversal, which is worth publishing but not a paradigm shift. If it does not, the degeneracy is a real puzzle. The math and numerics are competent; the claim just overreaches.

Referee Report

4 major / 5 minor

Summary. The paper studies the spectral statistics of an interacting fermionic model obtained by projecting the Hubbard interaction onto the two lowest flat bands of the π-flux dice lattice. Using exact diagonalization in sectors resolved by particle numbers, momentum, spin-flip parity, total spin, and total pseudospin, the authors report a parity dichotomy: for even total particle number the spectrum is nondegenerate and its level-spacing and gap-ratio statistics match a superposition of several GOE blocks, whereas for odd particle number the spectrum is exactly doubly degenerate even after resolving all listed symmetries, and the spacing between doublets (2NN and 4NN statistics) matches the GUE. They interpret this as the simultaneous emergence of GOE and GUE within a single physical system, call it unprecedented, and discuss possible mechanisms (hidden antiunitary symmetry, topological order), leaving the mechanism open.

Significance. If the central observation holds—exact odd-parity double degeneracy beyond all resolved symmetries, with GUE statistics between doublets—this is a genuinely interesting result for both random-matrix theory and flat-band physics. The paper makes good use of higher-order (kNN) spectral statistics, which is an appropriate tool for degenerate spectra, and the comparison to Monte Carlo multi-block reference distributions is carefully described. The exact-diagonalization results are reproducible in principle, and the authors are honest about where the automatic fitting procedure was abandoned and where the mechanism is unknown. However, the 'unprecedented' claim goes beyond what is demonstrated, because the completeness of the symmetry resolution is not proven and one proposed mechanism (hidden antiunitary symmetry) appears inconsistent with the observed GUE statistics.

major comments (4)

[Supplemental Material IIID, Fig. 8] For the 3×3 odd-N sector, the largest accessible size, the automatic model-selection procedure is explicitly not used; instead the GUE distributions are hand-picked 'if the peak at s/r = 0 is ignored'. This conflicts with the main-text statement that the GUE behavior is 'present in the odd-N sector for all system sizes'. The peak indicates a large number of near-degenerate states at very low filling, and ignoring it by eye is not a quantitative test. The authors should either perform the fit on the collapsed spectrum, exclude the low-energy window using a stated criterion, or report KS distances with and without the peak. As it stands, the largest finite-size evidence is weaker than the other system sizes.
[Discussion] The proposed 'hidden antiunitary symmetry involving the band degree of freedom' is incompatible with the observed GUE statistics. An antiunitary symmetry squaring to −1 would place the sector in the symplectic class (GSE), for which the spacing between doublets has β=4. The data in Fig. 3 give α_fit≈1.94 for k=2, matched to β=2 (GUE). Thus the Kramers mechanism, as stated, would predict GSE, not GUE, and cannot explain the central observation. This tension should be explicitly addressed, either by showing a different antiunitary structure that yields GUE or by removing/rewording the Kramers hint. As written, the proposed mechanism undermines the 'unprecedented' interpretation.
[Abstract / Results] The claim that the coexistence of GOE and GUE in one physical system is 'unprecedented' is load-bearing but not established. The paper resolves all *known* mutually commuting charges, but there is no proof or exhaustive search showing that no additional unitary or antiunitary symmetry or hidden block structure exists. The Discussion explicitly leaves the mechanism open. Until a mechanism is identified or a no-hidden-symmetry argument is supplied, the wording should be softened to something like 'unexpected' or 'not explainable by any symmetry resolved here'. The empirical findings are interesting even without the 'unprecedented' label.
[Results, Fig. 3, odd-sector k=3] For the odd sector the paper argues that k=3 rules out the superposition of two GOE blocks, but it never compares k=3 to the prediction obtained by collapsing the doublets of a GUE spectrum. Since the doublets are exact, the 3NN statistics of the full spectrum should be a mixture of 1NN and 2NN spacings of the collapsed GUE spectrum. Showing that this prediction agrees with the data would be a direct positive test of the GUE interpretation. The current analysis only excludes one alternative, rather than confirming the proposed one.

minor comments (5)

[Discussion] The statement that the double degeneracy is 'present in the full dice lattice with a Hubbard interaction, even before the projection' is unsupported; no data or reference is given. If this is an unpublished numerical observation, it should appear in the Supplemental Material or be explicitly marked as a conjecture.
[Table I caption] The notation '(k; β, m)' and the equality signs between distributions are not defined in the caption. Readers without the Supplemental Material may find the table cryptic; a one-sentence explanation in the caption would help.
[Fig. 3(c)] The curve P_3(s,1,5) is shown as the closest m-block match, but the text states it 'does not resemble any m-block distribution'. The figure would benefit from also showing the GUE prediction for the collapsed doublet spectrum, or from stating explicitly that no m-block curve fits and the shown curve is merely the least bad within the tested set.
[Supplemental Eq. (25)] The variance formula uses k both as the order and in the logarithm, which is confusing. Please rename one of the variables or add a clarifying sentence.
[Results, even-N block count] The inferred block number m is non-monotonic in system size (4, 6, 2 for 3×2, 4×2, and 3×3). The claim that this 'is consistent with the presence of an extensive number of LIOMs' is not supported by only three sizes with this non-monotonic trend; a more explicit caveat or finite-size scaling discussion would be appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral-statistics claims are benchmarked against externally defined RMT distributions, and the exact double degeneracy is a direct exact-diagonalization observation.

full rationale

The paper's derivation chain is: (i) define the projected dice-lattice Hamiltonian using explicit expressions in the Supplemental Material (Eqs. (5)-(8)), with Ref. [20] used only for notation and prior LIOM analysis; (ii) perform exact diagonalization in sectors labeled by N_up, N_down, k, f, S, B; (iii) compare level-spacing and gap-ratio distributions to externally defined references—the generalized Wigner surmise and Monte Carlo beta-ensemble spectra generated with the Dumitriu-Edelman construction; (iv) infer the best-matching (k*, beta*) and block number m by KS/MSD. No parameter is fitted to the target ensemble and then renamed as a prediction; the RMT benchmarks do not depend on the paper's data. The odd-sector double degeneracy is observed directly in the ED spectrum, not imposed by a fitting procedure. The paper explicitly leaves the degeneracy mechanism open and even proposes a hidden antiunitary symmetry, so its 'unprecedented' interpretation is a stated interpretation, not a self-referential proof. The only self-citations (Refs. [20,39]) supply the model, LIOMs, and numerical methodology; they do not carry the novel spectral-statistics conclusion. The manual GUE assignment in Supplemental Fig. 8 for the largest odd sector is a transparent limitation rather than a circular reduction, since the reference distribution is still externally defined. Thus no circular step meeting the quoted-evidence threshold was found.

Assumptions & free parameters 5 free parameters · 6 assumptions · 3 invented entities

The paper's central numerical claims rest on the completeness of the symmetry resolution and on the RMT fitting procedure. The free parameters are mostly analysis choices. The main invented entities are the conjectured hidden symmetry and LIOMs, neither of which is independently evidenced.

free parameters (5)

A (on-site pair binding energy) = 10
Set to A=10 for all calculations; authors state it has no qualitative effect, but it is an arbitrary choice setting the pair binding scale.
penalty coefficients α, β for S² and B² resolution = not specified ('large')
Used in the penalty-term ED method to resolve total spin S and pseudospin B; values are not given, hampering exact reproduction and potentially allowing sector mixing if not large enough.
fedge (edge-eigenvalue discard fraction) = 0.025
Hand-chosen fraction of levels discarded at each spectrum edge before unfolding.
q (number of unfolding blocks) = 30
Hand-chosen; authors state varying q in [20,100] did not affect results, but the value is still a choice.
NRMT, Nsamp (Monte Carlo reference sizes) = 1000, 1000
Matrix dimension and sample count for generating multi-block RMT reference distributions.

assumptions (6)

domain assumption Projection of the Hubbard interaction onto the two lowest flat bands is a valid low-energy reduction; the effective Hamiltonian is Eq. (1) with λ2=λ3=1.
The entire study operates in the projected subspace; corrections from higher bands are neglected.
domain assumption The Wannier functions are compactly localized, allowing the projected Hamiltonian to be written as a finite collection of short-range terms (Eqs. 6–8).
This locality underpins the explicit form of H and the ED tractability.
standard math Time-reversal symmetry (T²=-1) plus full SU(2) spin symmetry fixes the Dyson class of a fully symmetry-resolved sector to GOE in the even-N case.
Standard RMT symmetry classification used to interpret the even-N statistics.
standard math The generalized Wigner surmise (Eqs. 4,5) and the variance-corrected exponent (Eq. 25) accurately approximate the exact kNN spacing distributions of the Gaussian ensembles for k≤4.
The fitting procedure and the extraction of β*,k* rely on these approximation formulas.
domain assumption The reference m-block distributions P_k(s/r, β, m) are generated assuming each block contributes an equal number of levels.
The physical sectors may have unequal dimensions; this could bias the inferred block number m and the 'not two GOE blocks' argument. Location: Supplement Sec. II D.
domain assumption The statistics are computed in the largest symmetry-resolved sector; smaller sectors are ignored.
The choice of the largest (S,B) sector as representative could bias the spectral statistics if different sectors have different symmetry classes.

invented entities (3)

Hidden antiunitary symmetry involving the Wannier/band index n=1↔2
purpose: To explain the exact double degeneracy of all odd-N eigenstates via a Kramers-like theorem.
Proposed in the Discussion; no explicit symmetry operator is constructed and no falsifiable prediction beyond the degeneracy is given.
Deformed local integrals of motion (LIOMs) in the even-N sector
purpose: To explain the multi-block GOE structure and the nonmonotonic block count m.
No explicit deformed LIOM operators are constructed; m varies nonmonotonically with system size.
Parity-dependent topological order (analogous to toric code)
purpose: Alternative explanation for the odd-N double degeneracy.
Mentioned as an 'intriguing possibility' in the Discussion; no diagnostic (e.g., topological entanglement entropy) is computed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Parity-dependent double degeneracy and spectral statistics in the projected dice lattice." pith.science (2026). https://pith.science/paper/G3ACWG2H

@misc{pith2026260211844,
  author       = {Pith},
  title        = {Pith review of: Parity-dependent double degeneracy and spectral statistics in the projected dice lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3ACWG2H}},
  note         = {Machine review of arXiv:2602.11844}
}

abstract

We investigate the spectral statistics of an interacting fermionic system derived by projecting the Hubbard interaction onto the two lowest-energy, degenerate flat bands of the dice lattice subjected to a $\pi$-flux. Surprisingly, the distributions of level spacings and gap ratios correspond to distinct Gaussian ensembles, depending on the parity of the particle number. For an even number of particles, the spectra conform to the Gaussian Orthogonal Ensemble, as expected for a time-reversal-symmetric Hamiltonian. In stark contrast, the odd-parity sector exhibits exact double degeneracy of all eigenstates even after resolving all known symmetries, and the Gaussian Unitary Ensemble accurately describes the spacing distribution between these doublets. The simultaneous emergence of two different random-matrix ensembles within a single physical system constitutes an unprecedented finding, opening new avenues for both random matrix theory and flat-band physics.

Figures

Figures reproduced from arXiv: 2602.11844 by the authors.

**Figure 2.** FIG. 2. Level spacing (top) and gap ratio (bottom) statistics of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 1.** Figure 1: FIG. 1. (left) Schematic of the dice lattice. The magnetic unit cell is highlighted in the rectangular box with orbital indices [PITH_FULL_IMAGE:figures/full_fig_p009_1.png]

**Figure 2.** Figure 2: FIG. 2. Level spacing statistics for [PITH_FULL_IMAGE:figures/full_fig_p014_2.png]

**Figure 3.** Figure 3: FIG. 3. (left): Mean [PITH_FULL_IMAGE:figures/full_fig_p015_3.png]

**Figure 4.** Figure 4: FIG. 4. Site-resolved total particle density [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: FIG. 5. Level spacing (top) and gap ratio (bottom) statistics of [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: FIG. 7. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

97 extracted references · 1 linked inside Pith

[1]

Diﬀerent colors indicate diﬀerent pairs (S, B) of total spin and pseudospin quantum numbers

for system size Nx, Ny = 3 , 2, momentum k = 0 , and particle numbers N↑, N↓ shown in each panel. Diﬀerent colors indicate diﬀerent pairs (S, B) of total spin and pseudospin quantum numbers. For even par- ticle numbers (b), the eigenvalues Ei are nondegenerate. For odd particle numbers (c), the spectrum consists of degenerate doublets. When λ 2 = λ 3 = 0 ...

2024
[2]

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

2018
[3]

Balents, C

L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré ﬂat bands, Nature Physics 16, 725 (2020)

2020
[4]

E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nature Reviews Materials 6, 201 (2021)

2021
[5]

Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Pauli-limit violation and re-entrant su- perconductivity in moiré graphene, Nature 595, 526–531 (2021)

2021
[6]

J. M. Park, S. Sun, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Experimental evidence for nodal su- perconducting gap in moiré graphene, Science 391, 79–83 (2026)

2026
[7]

H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F. Young, Superconductivity in rhombohedral trilayer graphene, Nature 598, 434–438 (2021)

2021
[8]

T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. Shariﬁ Sedeh, H. Weldeyesus, J. Yang, J. Seo, S. Ye, M. Zhou, H. Liu, G. Shi, Z. Hua, K. Watan- abe, T. Taniguchi, P. Xiong, D. M. Zumbühl, L. Fu, and L. Ju, Signatures of chiral superconductivity in rhombo- hedral graphene, Nature 643, 654–661 (2025)

2025

Show all 97 references

[9]

L. Wang, Y. Gao, B. Wen, Z. Han, T. Taniguchi, K. Watanabe, M. Koshino, J. Hone, and C. R. Dean, Evidence for a fractional fractal quantum Hall eﬀect in graphene superlattices, Science 350, 1231 (2015)

2015
[10]

Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous Hall eﬀect in multilayer graphene, Nature 626, 759 (2024)

2024
[11]

D. Kim, S. Jin, T. Taniguchi, K. Watanabe, J. H. Smet, G. Y. Cho, and Y. Kim, Observation of 1/3 fractional quantum Hall physics in balanced large angle twisted bi- layer graphene, Nature Communications 16, 179 (2025)

2025
[12]

J. Zhao, L. Liu, Y. Zhang, H. Zhang, Z. Feng, C. Wang, S. Lai, G. Chang, B. Yang, and W. Gao, Exploring the Fractional Quantum Anomalous Hall Eﬀect in Moiré Ma- terials: Advances and Future Perspectives, ACS Nano 19, 19509 (2025)

2025
[13]

S. D. Huber and E. Altman, Bose condensation in ﬂat bands, Physical Review B 82, 184502 (2010)

2010
[14]

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
[15]

Derzhko, J

O. Derzhko, J. Richter, and M. Maksymenko, Strongly correlated ﬂat-band systems: The route from Heisen- berg spins to Hubbard electrons, International Journal of Modern Physics B 29, 1530007 (2015)

2015
[16]

Tovmasyan, S

M. Tovmasyan, S. Peotta, P. Törmä, and S. D. Huber, Eﬀective theory and emergent SU ( 2 ) symmetry in the ﬂat bands of attractive Hubbard models, Physical Review B 94, 245149 (2016)

2016
[17]

Leykam, A

D. Leykam, A. Andreanov, and S. Flach, Artiﬁcial ﬂat band systems: From lattice models to experiments, Ad- vances in Physics: X 3, 1473052 (2018)

2018
[18]

Tovmasyan, S

M. Tovmasyan, S. Peotta, L. Liang, P. Törmä, and S. D. 6 Huber, Preformed pairs in ﬂat Bloch bands, Physical Re- view B 98, 134513 (2018)

2018
[19]

Y. Kuno, T. Orito, and I. Ichinose, Flat-band many-body localization and ergodicity breaking in the Creutz ladder, New Journal of Physics 22, 013032 (2020)

2020
[20]

Nicolau, A

E. Nicolau, A. M. Marques, R. G. Dias, and V. Ahuﬁn- ger, Flat band induced local Hilbert space fragmentation, Physical Review B 108, 205104 (2023)

2023
[21]

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
[22]

Vidal, R

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

1998
[23]

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
[25]

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, Berlin, Heidelberg, 2010)

2010
[26]

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

2004
[27]

P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltoni- ans, Physical Review X 10, 011047 (2020)

2020
[28]

Moudgalya and O

S. Moudgalya and O. I. Motrunich, Hilbert Space Frag- mentation and Commutant Algebras, Physical Review X 12, 011050 (2022)

2022
[29]

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
[33]

See Supplemental Material [URL] for the explicit repre- sentation of the Hamiltonian, details on the unfolding procedure and spectral statistics analysis, and additional supporting results. Refs. [ 69–74] are included in this Sup- plementary Material
[35]

A. O. Barut and A. Böhm, Dynamical Groups and Mass Formula, Physical Review 139, B1107 (1965)

1965
[36]

B. Buča, J. Tindall, and D. Jaksch, Non-stationary coher- ent quantum many-body dynamics through dissipation, Nature Communications 10, 1730 (2019)

2019
[39]

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
[40]

Teeriaho, V.-V

M. Teeriaho, V.-V. Linho, K. Swaminathan, and S. Peotta, Coexistence of ergodic and nonergodic behav- ior and level spacing statistics in a one-dimensional model of a ﬂat band superconductor, Physical Review Research 7, 013318 (2025)

2025
[42]

Oganesyan and D

V. Oganesyan and D. A. Huse, Localization of interact- ing fermions at high temperature, Physical Review B 75, 155111 (2007)

2007
[43]

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
[44]

R. Shir, P. Martinez-Azcona, and A. Chenu, Surmise for random matrices’ level spacing distributions beyond nearest-neighbors, Journal of Physics A: Mathematical and Theoretical 58, 445206 (2025)

2025
[46]

Bleher and A

P. Bleher and A. R. Its, Random Matrix Models and Their Applications , Mathematical Sciences Research In- stitute Publications No. 40 (Cambridge university press, Cambridge, 2001)

2001
[47]

G. W. Anderson, A. Guionnet, and O. Zeitouni, An In- troduction to Random Matrices , 1st ed. (Cambridge Uni- versity Press, 2009)

2009
[48]

Deift and P

P. Deift and P. J. Forrester, eds., Random Matrix The- ory, Interacting Particle Systems, and Integrable Sys- tems , Mathematical Sciences Research Institute Publi- cations No. 65 (Cambridge University Press, Cambridge, 2014)

2014
[50]

F. J. Massey, The Kolmogorov-Smirnov Test for Good- ness of Fit, Journal of the American Statistical Associa- tion 46, 68 (1951)

1951
[52]

P. J. Forrester and E. M. Rains, Correlations for superpo- sitions and decimations of Laguerre and Jacobi orthogo- nal matrix ensembles with a parameter, Probability The- ory and Related Fields 130, 518 (2004)

2004
[53]

P. J. Forrester, A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1), Communications in Mathematical Physics 285, 653 (2009)

2009
[54]

S. H. Tekur and M. S. Santhanam, Symmetry deduction from spectral ﬂuctuations in complex quantum systems, 7 Physical Review Research 2, 032063 (2020)

2020
[55]

U. T. Bhosale, Superposition and higher-order spacing ratios in random matrix theory with application to com- plex systems, Physical Review B 104, 054204 (2021)

2021
[56]

J. J. Sakurai and J. Napolitano, Modern Quantum Me- chanics, 3rd ed. (Cambridge University Press, 2020)

2020
[57]

Pietracaprina, N

F. Pietracaprina, N. Macé, D. J. Luitz, and F. Alet, Shift- invert diagonalization of large many-body localizing spin chains, SciPost Physics 5, 045 (2018)

2018
[58]

Sierant, M

P. Sierant, M. Lewenstein, and J. Zakrzewski, Polynomi- ally Filtered Exact Diagonalization Approach to Many- Body Localization, Physical Review Letters 125, 156601 (2020)

2020
[59]

Guan and W

H. Guan and W. Zhang, Dual applications of Cheby- shev polynomials method: Eﬃciently ﬁnding thousands of central eigenvalues for many-spin systems, SciPost Physics 11, 103 (2021)

2021
[60]

A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003)

2003
[61]

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
[62]

C. K. Thomas, T. H. Barter, T.-H. Leung, M. Okano, G.- B. Jo, J. Guzman, I. Kimchi, A. Vishwanath, and D. M. Stamper-Kurn, Mean-Field Scaling of the Superﬂuid to Mott Insulator Transition in a 2D Optical Superlattice, Physical Review Letters 119, 100402 (2017)

2017
[63]

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
[64]

S. Taie, H. Ozawa, T. Ichinose, T. Nishio, S. Naka- jima, and Y. Takahashi, Coherent driving and freezing of bosonic matter wave in an optical Lieb lattice, Science Advances 1, e1500854 (2015)

2015
[65]

S. Taie, T. Ichinose, H. Ozawa, and Y. Takahashi, Spa- tial adiabatic passage of massive quantum particles in an optical Lieb lattice, Nature Communications 11, 257 (2020)

2020
[66]

Andrijauskas, E

T. Andrijauskas, E. Anisimovas, M. Rači¯ unas, A. Mekys, V. Kudriašov, I. B. Spielman, and G. Juzeli¯ unas, Three- level Haldane-like model on a dice optical lattice, Physi- cal Review A 92, 033617 (2015)

2015
[67]

Möller and N

G. Möller and N. R. Cooper, Synthetic gauge ﬁelds for lattices with multi-orbital unit cells: Routes towards a π - ﬂux dice lattice with ﬂat bands, New Journal of Physics 20, 073025 (2018)

2018
[68]

Tassi and D

C. Tassi and D. Bercioux, Implementation and Char- acterization of the Dice Lattice in the Electron Quan- tum Simulator, Advanced Physics Research 3, 2400038 (2024)

2024
[69]

Dixmerias, G

M. Dixmerias, G. D. V. D. Vecchio, C. Daix, J. Ver- straten, T. de Jongh, B. Peaudecerf, P. L. Dous- sal, G. Schehr, and T. Yefsah, Universal Random Ma- trix Behavior of a Fermionic Quantum Gas (2025), arXiv:2510.25735 [cond-mat]

2025
[70]

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
[71]

F. J. Dyson and M. L. Mehta, Statistical Theory of the Energy Levels of Complex Systems. IV, Journal of Math- ematical Physics 4, 701 (1963)

1963
[74]

Kapustin and L

A. Kapustin and L. Fidkowski, Local Commuting Pro- jector Hamiltonians and the Quantum Hall Eﬀect, Com- munications in Mathematical Physics 373, 763 (2020)

2020
[76]

collapsed

is that the total pseudospin operator ^B 2 = ∑ α =x,y,z ( ^Bα )2, with ^Bα = ∑ n,l ^Bα nl, is also a conserved quantity. 4 e. Local integrals of motion of ^H0,0. The triangular Hamiltonian alone ( ^Htri. = ^H0,0) possesses an extensive set of strictly local conserved quantitie...
[77]

We therefore generate reference distributions by Monte Carlo sampling

Monte Carlo generation of P k(s/r;β,m ) Closed-form expressions for P k(s/r,β,m ) are generally not available for arbitrary (k,β,m ). We therefore generate reference distributions by Monte Carlo sampling. For each β ∈ { 1, 2}, we generate independent RMT spectra using the Dumi...
[78]

We use two complementary metrics:

Choosing the closest (β,m ) reference To determine the best matching (β,m ) for a given set of eigenvalues obtained from exact diagonalization, we compare the empirical distributions to the Monte Carlo reference distributions and select the closest match. We use two complement...
[79]

The Kolmogorov-Smirnov (KS) distance between empirical and reference cumulative distribution functions (CDFs) [ 23, 24]
[80]

stick statistics

A mean-square deviation (MSD) between empirical and reference distributions, evaluated on the histogram bins. Let {x(i)}N i=1 denote the sorted data values (either spacings or ratios), and deﬁne the empirical CDF evaluated at these points as Femp(x(i)) = i N . (26) For an anal...
[81]

Swaminathan, P

K. Swaminathan, P. Tadros, and S. Peotta, Signatures of many-body localization of quasiparticles in a ﬂat band super- conductor, Physical Review Research 5, 043215 (2023) . 10 0 1 2 3s 0.0 0.2 0.4 0.6 0.8 1.0 (a) k = 1 α fit = 0. 001 GOE k∗ = 1 Pois1(s) P 1(s, 1, 6) 0 1 2 3s (...

2023
[82]

Tovmasyan, S

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

2018
[83]

S. M. Zhang and L. Jin, Compact localized states and localization dynamics in the dice lattice, Physical Review B 102, 054301 (2020)

2020
[84]

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

2004
[85]

Poilblanc, T

D. Poilblanc, T. Ziman, J. Bellissard, F. Mila, and G. Montambaux, Poisson vs. GOE Statistics in Integrable and Non- Integrable Quantum Hamiltonians, Europhysics Letters 22, 537 (1993) . 11 0 1 2 3s 0.0 0.5 1.0 1.5 (a) k = 1 α fit = 0. 177 GOE k∗ = 1 Pois1(s) P 1(s, 1, 2) 0 1 ...

1993
[86]

Teeriaho, V.-V

M. Teeriaho, V.-V. Linho, K. Swaminathan, and S. Peotta, Coexistence of ergodic and nonergodic behavior and level spacing statistics in a one-dimensional model of a ﬂat band superconductor, Physical Review Research 7, 013318 (2025)

2025
[87]

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 Physics 2, 003 (2017)

2017
[88]

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
[89]

Oganesyan and D

V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Physical Review B 75, 155111 (2007)

2007
[90]

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) . 12

2013
[91]

S. H. Tekur, U. T. Bhosale, and M. S. Santhanam, Higher-order spacing ratios in random matrix theory and complex quantum systems, Physical Review B 98, 104305 (2018)

2018
[92]

Rao, Higher-order level spacings in random matrix theory based on Wigner’s conjecture, Physical Review B 102, 054202 (2020)

W.-J. Rao, Higher-order level spacings in random matrix theory based on Wigner’s conjecture, Physical Review B 102, 054202 (2020)

2020
[93]

R. Shir, P. Martinez-Azcona, and A. Chenu, Surmise for random matrices’ level spacing distributions beyond nearest- neighbors, Journal of Physics A: Mathematical and Theoretical 58, 445206 (2025)

2025
[94]

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

2004
[95]

Y. Y. Atas, E. Bogomolny, O. Giraud, P. Vivo, and E. Vivo, Joint probability densities of level spacing ratios in random matrices, Journal of Physics A: Mathematical and Theoretical 46, 355204 (2013)

2013
[96]

Bleher and A

P. Bleher and A. R. Its, Random Matrix Models and Their Applications , Mathematical Sciences Research Institute Publi- cations No. 40 (Cambridge university press, Cambridge, 2001)

2001
[97]

G. W. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices , 1st ed. (Cambridge University Press, 2009)

2009
[98]

Deift and P

P. Deift and P. J. Forrester, eds., Random Matrix Theory, Interacting Particle Systems, and Integrable Systems , Mathe- matical Sciences Research Institute Publications No. 65 (Cambridge University Press, Cambridge, 2014)

2014
[99]

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
[100]

S. H. Tekur and M. S. Santhanam, Symmetry deduction from spectral ﬂuctuations in complex quantum systems, Physical Review Research 2, 032063 (2020)

2020
[101]

U. T. Bhosale, Superposition and higher-order spacing ratios in random matrix theory with application to complex systems, Physical Review B 104, 054204 (2021)

2021
[102]

Dumitriu and A

I. Dumitriu and A. Edelman, Matrix models for beta ensembles, Journal of Mathematical Physics 43, 5830 (2002)

2002
[103]

F. J. Massey, The Kolmogorov-Smirnov Test for Goodness of Fit, Journal of the American Statistical Association 46, 68 (1951)

1951
[104]

M. A. Stephens, EDF Statistics for Goodness of Fit and Some Comparisons, Journal of the American Statistical Association 69, 730 (1974)

1974
[105]

Kapustin and L

A. Kapustin and L. Fidkowski, Local Commuting Projector Hamiltonians and the Quantum Hall Eﬀect, Communications in Mathematical Physics 373, 763 (2020)

2020
[106]

M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 356, 375 (1977)

1977
[107]

F. He, A. Hutsalyuk, G. Mussardo, and A. Stampiggi, Statistical Signatures of Integrable and Non-Integrable Quantum Hamiltonians (2025), arXiv:2510.02440 [cond-mat]

2025
[108]

Chakrabarti, A

B. Chakrabarti, A. Biswas, V. K. B. Kota, K. Roy, and S. K. Haldar, Energy-level statistics of interacting trapped bosons, Physical Review A 86, 013637 (2012)

2012
[109]

A. D. Kerin, B. Dietz, and J. Brand, Quartic level repulsion in a quantum chaotic three-body system without symplectic symmetry (2025), arXiv:2510.06772 [cond-mat]

2025 arXiv
[110]

Haake, Quantum Signatures of Chaos , Springer Series in Synergetics, Vol

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

2010
[111]

T. Guhr, A. Müller–Groeling, and H. A. Weidenmüller, Random-matrix theories in quantum physics: Common concepts, Physics Reports 299, 189 (1998)

1998

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Diﬀerent colors indicate diﬀerent pairs (S, B) of total spin and pseudospin quantum numbers

for system size Nx, Ny = 3 , 2, momentum k = 0 , and particle numbers N↑, N↓ shown in each panel. Diﬀerent colors indicate diﬀerent pairs (S, B) of total spin and pseudospin quantum numbers. For even par- ticle numbers (b), the eigenvalues Ei are nondegenerate. For odd particle numbers (c), the spectrum consists of degenerate doublets. When λ 2 = λ 3 = 0 ...

2024

[2] [2]

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

2018

[3] [3]

Balents, C

L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré ﬂat bands, Nature Physics 16, 725 (2020)

2020

[4] [4]

E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nature Reviews Materials 6, 201 (2021)

2021

[5] [5]

Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Pauli-limit violation and re-entrant su- perconductivity in moiré graphene, Nature 595, 526–531 (2021)

2021

[6] [6]

J. M. Park, S. Sun, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Experimental evidence for nodal su- perconducting gap in moiré graphene, Science 391, 79–83 (2026)

2026

[7] [7]

H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F. Young, Superconductivity in rhombohedral trilayer graphene, Nature 598, 434–438 (2021)

2021

[8] [8]

T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. Shariﬁ Sedeh, H. Weldeyesus, J. Yang, J. Seo, S. Ye, M. Zhou, H. Liu, G. Shi, Z. Hua, K. Watan- abe, T. Taniguchi, P. Xiong, D. M. Zumbühl, L. Fu, and L. Ju, Signatures of chiral superconductivity in rhombo- hedral graphene, Nature 643, 654–661 (2025)

2025