Pith. sign in

REVIEW 3 major objections 5 minor 44 references

Twisting-operator approach to identifying gaplessness in SU($N$) fermionic systems

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read For any gapped SU(2)-symmetric fermion chain with finite degeneracy, ⟨U²⟩ = 1 + O(1/L); a thermodynamic-limit value not 1 certifies gaplessness, and the same holds for SU(N) with exponent M.

desk verdict Extends the twisting-operator gaplessness test to fermions and SU(N); the central O(1/L) estimate is inherited from earlier spin-chain work and needs a self-contained check. read the letter →

arxiv 2607.19694 v1 pith:S7J5L4MZ submitted 2026-07-22 cond-mat.str-el cond-mat.stat-mechhep-thmath-phmath.MP

Hao Cheng , Hang Su , Yuan Yao This is my paper · ORCID
classification cond-mat.str-elcond-mat.stat-mechhep-thmath-phmath.MP
keywords twistingoperatorgaplessnesscriterionfermionicchainSU(2)symmetrySU(N)HubbardmodelBCSHamiltonianground-statedegeneracy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to prove a symmetry-based necessary condition for a fermionic chain to be gapped: in any gapped SU(2)-symmetric fermion chain with finite ground-state degeneracy, the expectation value of a properly defined fermionic twisting operator must approach unity as 1 + O(1/L). If true, this turns the twisting operator into a practical gaplessness witness: computing ⟨U²⟩ and seeing it not tend to 1 rules out a conventional gapped phase. The authors verify the criterion on the solvable s-wave BCS Hamiltonian and on interacting Hubbard chains and ladders via determinant quantum Monte Carlo, and they extend it to SU(N)-symmetric fermionic systems, where the gapped ground-state sector must be an SU(N) singlet and ⟨U^M⟩ = 1 + O(1/L) for a suitable integer M. This matters because traditional symmetry arguments can distinguish gapped and gapless phases less directly and sometimes say nothing at all for integer or higher-symmetry settings.

What carries the argument

The central object is the fermionic twisting operator U = exp((4πi/L)Σ_{j=1}^L j S^z_{f,j}), built from the local spin operator S^z_{f,j} = (1/2)(c†_{j↑}c_{j↑} − c†_{j↓}c_{j↓}) on a four-dimensional local Hilbert space. The choice 4π, rather than 2π, makes the commutation of U with lattice translation a phase times a global rotation. The same structure is generalized to SU(N) using a Cartan element S^z and a rotation R whose conjugation property Σ_{k=0}^{n_R−1}(R†)^k S^z R^k = 0 guarantees that the combined operator V = R U has finite order n_R; the identity (R U)^{n_R} = 1 then forces the twisted ground-state expectation to equal 1 up to O(1/L).

What would settle it

A concrete check would be to compute |e^(n)⟩ for a small exactly solvable gapped SU(2)-symmetric fermionic chain (for example, a dimerized Hubbard chain at L = 8, 10, 12 under periodic boundary conditions) and verify that its norm stays bounded by C/L uniformly over all twisted states. If any state with empty or doubly occupied sites produces a decay slower than 1/L, the key bound in Proof of Lemma 5 fails and the theorems do not follow. At the level of the criterion itself, a counterexample would be an explicitly gapped SU(2)-symmetric fermionic chain with finite degeneracy whose extrapolated

Watch

Extended reading notes

Core claim

The paper's central statement is that, for any local, translation-invariant SU(2)-symmetric fermion chain with periodic boundary conditions and a spectral gap with finite ground-state degeneracy, ⟨gs|U²|gs⟩ = 1 + O(1/L) for the fermionic twisting operator U = exp((4πi/L)Σ_j j S^z_{f,j}). Hence a thermodynamic-limit value different from 1 is a sufficient criterion for gaplessness or infinite ground-state degeneracy. The authors also prove an SU(N) extension: the gapped ground-state sector must be an SU(N)-singlet, and ⟨gs|U^M|gs⟩ = 1 + O(1/L), where M is the gcd of the orders of a set of SU(N) rotations R satisfying Eq. (14). The BCS calculation and the Hubbard DQMC results are presented as n

Load-bearing premise

The load-bearing premise sits in the Supplemental Materials' Proof of Lemma 5: the excited-state component |e^(n)⟩ of the twisted ground state has norm O(1/L), which the paper carries over from the spin-chain proof rather than re-deriving for the fermionic local Hilbert space with its extra empty and doubly occupied singlet sectors; if that bound fails there, Eq. (17) and hence the SU(2) and SU(N) criteria no longer follow.

Editorial extensions

If this is right

  • A gapped SU(2)-symmetric fermion chain with finite ground-state degeneracy necessarily has ⟨U²⟩ = 1 + O(1/L); measuring any other thermodynamic value certifies gaplessness or infinite degeneracy.
  • The gapped ground-state sector of such a chain must be an SU(2) singlet, and likewise an SU(N) singlet in the SU(N) case, constraining candidate gapped phases before any microscopic calculation.
  • The BCS example shows that the 1/L correction has a nonzero coefficient controlled by the gap, so the criterion is visible at finite size and can be extrapolated numerically.
  • The interacting Hubbard versus Hubbard-ladder comparison gives a practical diagnostic: a gapless chain keeps ⟨U²⟩ near zero as L grows, while a gapped ladder pushes it toward unity.
  • Because the criterion is necessary but not sufficient, it complements rather than replaces existing symmetry-based gap arguments, and it extends to cases those arguments do not constrain.

Reading between the lines

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

  • A natural extension not spelled out in the paper is to use ⟨U²⟩ versus 1/L as a routine companion to level-spectroscopy in tensor-network or Monte Carlo studies of candidate gapped fermionic phases; the linear-in-1/L behavior seen in the paper's examples makes the intercept a cheap diagnostic.
  • If the SU(N) singlet statement is correct, many proposed gapped phases with ground states carrying nontrivial SU(N) representations are ruled out regardless of interaction strength, which could sharpen searches for gapped SU(N) spin liquids and symmetry-protected phases.
  • The paper leaves the inverse direction open—a chain with ⟨U²⟩ = 1 could still be gapless; a testable refinement would be to combine the twisting expectation with an independent probe, such as the entanglement gap, to seek an if-and-only-if gaplessness test.
  • Because the proof of Lemma 5 borrows the O(1/L) excited-state bound from the spin-chain setting, the most valuable follow-up would be a direct proof or numerical verification of that bound in the fermionic local Hilbert space with empty, singly, and doubly occupied sites.
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 / 5 minor

Summary. The paper proposes a twisting-operator gaplessness criterion for spinful SU(2) fermion chains and for SU(N)-symmetric fermion chains. It introduces a fermionic twisting operator \hat U = exp(4\pi i/L \sum_j j \hat S^z_{f,j}) and claims that for any gapped local Hamiltonian with finite ground-state degeneracy, \langle \hat U^2\rangle = 1 + O(1/L); therefore a thermodynamic-limit value different from unity is a sufficient condition for gaplessness (or infinite degeneracy). The claim is illustrated analytically by an s-wave BCS Hamiltonian and numerically by determinant quantum Monte Carlo for the half-filled Hubbard chain and the two-leg Hubbard ladder. The SU(N) generalization states that gapped ground states are SU(N) singlets and \langle \hat U^M\rangle = 1 + O(1/L) for a suitable integer M.

Significance. If the theorem holds, it provides a practical, symmetry-based numerical certificate of gaplessness that complements LSM-type constraints and applies directly to fermionic systems without requiring a particular filling or integer/half-integer spin per unit cell. The criterion involves no fitted parameters, and the BCS check and DQMC data are used as confirmations rather than inputs. The principal obstacle is that the proof of the central estimate is not self-contained and, as written, is in tension with the paper's own analytic BCS example. The contribution would be significant for the field if these gaps are closed.

major comments (3)
  1. [Theorem 2; Supplemental 'Proof of Lemma 5'] The central criterion is not proven in the manuscript. Theorem 2 is stated and left as 'It can be shown'; the actual argument is delegated to the Supplemental 'Proof of Lemma 5', which inherits the bound |e^(n)| = O(1/L) purely by citation to [14]. Since [14] is a spin-chain result, while the fermionic local Hilbert space here is four-dimensional with S^z_f = diag(0,1/2,-1/2,0), the transfer is not automatic: the two extra SU(2)-singlet sectors can in principle modify the relevant resolvent/Lieb-Robinson estimates. The sentence 'Since S^z and S^z_f follow the same commutation relation, we can directly replace S^z by S^z_f' (around Eq. (4)) is not a proof of this estimate. Because Eq. (17) is the load-bearing input for Theorems 2 and 6, this gap must be fixed by supplying the fermionic version of the O(1/L) estimate or by a careful reduction to [14] that addresses the zero modes of S^z_f.
  2. [Eq. (9) and Supplemental Eqs. (34)-(35); Lemma 5] The analytic BCS illustration is in internal tension with Lemma 5. For a unique gapped ground state, if \langle U^2\rangle = 1 - c/L (as in Eq. (34)), then unitarity gives ||P_exc U^2|gs>||^2 = 1 - |<gs|U^2|gs>|^2 = 2c/L + O(1/L^2), so the excited-state component has norm \Theta(L^{-1/2}), not O(1/L). Thus Lemma 5 as stated cannot hold for the BCS state, or the BCS expectation would need to be 1 + O(1/L^2). Additionally, Eq. (9) defines M = (2\pi/L)\Sigma_p ... and then uses M 8\pi/L, which is O(1/L^2), whereas the SM (Eqs. (34)-(35)) defines M = \Sigma_p ... and obtains 1 - 16\pi^2 M/L. These expressions need to be reconciled.
  3. [Theorem 6, Eqs. (20)-(22)] In Theorem 6, the step from Eq. (20) to Eq. (22) via Bezout's identity is not justified. From \langle U^{n_R}\rangle = 1 + O(1/L) for each R, it does not immediately follow that \langle U^M\rangle = 1 + O(1/L) with M = \sum a_R n_R unless one controls powers U^{a_R n_R} for the fixed integer coefficients a_R. This is plausibly true (using spectral concentration of the unitary operator), but it needs a proof; otherwise the SU(N) criterion is unsupported even if Lemma 5 is accepted.
minor comments (5)
  1. [Theorem 2] The condition 'lim_{L\to\infty} \langle U^2\rangle \neq 1 + O(1/L)' is imprecise, since 1 + O(1/L) denotes a family of sequences and any sequence whose limit is not 1 satisfies the condition. Rewrite as 'the limit is not 1' or 'liminf |\langle U^2\rangle - 1| > 0'.
  2. [Fig. 2(a)] Fig. 2(a) does not show statistical errors for the DQMC points; please add error bars or state that they are smaller than the marker size. Also clarify whether the twisting operator for the ladder is defined with L_x or with the total number of sites.
  3. [Table I] The matrix definitions of Z_m and X_m are garbled (e.g., X_m \equiv \binom{1}{I_{m-1}} is not written as a square matrix). Please redraw the table so that the permutation/reflection matrices are unambiguous.
  4. [Supplemental] Typos: 'Mine-filed approximation' should be 'mean-field'; 'instead Sz' should be 'replace Sz'; 'sufficient', 'recognization', and the reference '[44] ... p. 351 pages' need correction.
  5. [SU(N) generalizations] Theorem 6 refers to a 'spin-chain Hamiltonian', but the paper's declared topic is fermionic systems; clarify whether the SU(N) result is proved for local fermionic Hamiltonians or for abstract spin-chain Hamiltonians.

Circularity Check

1 steps flagged · score 4.0 of 10

Fermionic O(1/L) bound in Lemma 5 is imported from the authors' own spin-chain result [14] without proof in the fermionic Hilbert space; otherwise no fitted-input circularity.

  1. self citation load bearing [Eq. (4); Theorem 2; Supplemental Material, Proof of Lemma 5]
    "Since Sz and Sz f follow the same commutation relation, we can directly instead Sz by Sz f to obtain the corresponding twisting operator ... it has been shown that |e(n)⟩ has norm ∼ O(1/L) [14]."

    The O(1/L) bound on the excited-state component |e(n)⟩ is the load-bearing estimate in Lemma 5, and through Eq. (17) it is what supplies the O(1/L) correction in Theorem 2 and Theorem 6. The paper does not re-derive this bound for the four-dimensional fermionic local Hilbert space, which contains two extra SU(2)-singlet sectors with S^z_f = 0. Instead it imports the bound from [14], a spin-chain result by the same authors, and justifies the transfer only by the assertion that the commutation relations are the same. The central fermionic criterion therefore inherits its key finite-size control from a self-citation whose stated assumptions do not demonstrably cover the fermionic representation.

full rationale

The derivation is not circular in the fitted-input or definitional sense: the BCS analytic example and the DQMC simulations are independent checks, not fits used to manufacture the criterion, and Theorem 2 is not defined as its own conclusion. The main circularity-related concern is the O(1/L) excited-state norm bound. The proof of Lemma 5 says 'it has been shown that |e^(n)> has norm ~ O(1/L) [14]' and the main text says S^z can be 'directly' replaced by S^z_f because they follow the same commutation relation. Since [14] is a spin-chain result and the fermionic Hilbert space has additional singlet sectors, this transfer is asserted rather than proved, making the finite-size control in Theorems 2 and 6 rest on a same-author citation. That is a genuine load-bearing self-citation, though it is only one step and the paper does give independent analytic and numerical evidence for the criterion's behavior. A separate correctness aside: the BCS result <U^2>=1-O(1/L) may be in tension with an O(1/L) norm bound, which would naively give an O(1/L^2) deviation; this is a consistency question rather than a circularity. Overall, some self-citation but with independent content: score 4.

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

The paper introduces no fitted free parameters and no new physical entities. The central proof relies on prior twisting-operator theorems [13,14] (especially the O(1/L) excited-state norm bound) and standard group-theoretic facts.

assumptions (4)
  • domain assumption For a gapped, local, translation-invariant Hamiltonian, the component of U^n|gs⟩ outside the ground-state manifold has norm O(1/L).
    Invoked in the Supplemental Proof of Lemma 5 with citation [14]; it is the key step transferring the spin-chain result to fermionic and SU(N) chains.
  • domain assumption The local Hamiltonian decomposition can be chosen to commute with the on-site symmetry, [S^z_f, h_j]=[R,h_j]=0.
    Stated near Eq. (3) and justified by [13]; used to make the twisting argument local.
  • standard math For any traceless diagonal S^z, the permutation matrix R = exp(iα)X_N ∈ SU(N) has order N and satisfies Σ_{k=0}^{N-1}(R†)^k S^z R^k = 0.
    Lemma 3; the cyclic sum of permuted diagonal entries is (Tr S^z)I = 0.
  • standard math A state annihilated by the Cartan generators of all SU(2) subalgebras associated with simple roots is a trivial SU(N) representation.
    Used in the proof of Theorem 4; follows from the zero-weight highest-weight argument.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Twisting-operator approach to identifying gaplessness in SU($N$) fermionic systems." pith.science (2026). https://pith.science/paper/S7J5L4MZ

@misc{pith2026260719694,
  author       = {Pith},
  title        = {Pith review of: Twisting-operator approach to identifying gaplessness in SU($N$) fermionic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S7J5L4MZ}},
  note         = {Machine review of arXiv:2607.19694}
}
abstract

We propose a general necessary condition for a spinful fermion chain with SU(2) spin-rotation symmetry to be gapped. Specifically, we prove that the expectation value of a properly defined fermionic twisting operator asymptotically approaches unity in any gapped phase with finite ground-state degeneracy, with finite-size corrections bounded by $\mathscr{O}(1/L)$. Consequently, a non-unity value in the thermodynamic limit provides a sufficient criterion for identifying gapless fermion chains. We confirm this criterion using the $s$-wave Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and determinant quantum Monte Carlo (DQMC) simulations of interacting Hubbard models. We further extend the twisting-operator approach to SU($N$)-symmetric fermionic systems, where the gapped ground-state sector must be SU($N$)-singlet and $\langle \hat{\mathcal{U}}^M\rangle=1+\mathscr{O}(1/L)$ by a fermionic twisting operator $\hat{\mathcal{U}}$ with a suitably chosen integer $M$.

Figures

Figures reproduced from arXiv: 2607.19694 by the authors.

Figure 1
Figure 1. FIG. 1: Results for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (b) shows that ⟨Uˆ 2 ⟩ approaches unity as Lx in￾creases up to Lx = 100. A linear extrapolation in 1/Lx using the largest sizes gives the intercept 0.999(3), consis￾tent with the expected thermodynamic value for a gapped fermionic state. SU(N) generalizations.— We extend the Theorems 1 and 2 to the SU(N) case, where each site can host a fermion with arbitrary SU(N) “spin”. Its Hilbert space forms a linear representa… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references

  1. [14]

    H. Su, T. Zhang, Y. Yao, and A. Furusaki, Phys. Rev. Lett. 135, 146502 (2025)

    2025

  2. [1]

    L. D. Landau, Zh. Eksp. Teor. Fiz. 7, 19 (1937)

    1937

  3. [2]

    Zeng and X.-G

    B. Zeng and X.-G. Wen, Phys. Rev. B 91, 125121 (2015)

    2015

  4. [3]

    B. Zeng, X. Chen, D.-L. Zhou, X.-G. Wen, et al. , Quantum Information Meets Quantum Matter (Springer, 2019)

    2019

  5. [4]

    E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961)

    1961

  6. [5]

    Oshikawa, M

    M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev. Lett. 78, 1984 (1997)

    1984

  7. [6]

    Oshikawa, Phys

    M. Oshikawa, Phys. Rev. Lett. 84, 1535 (2000)

    2000

  8. [7]

    M. B. Hastings, Phys. Rev. B 69, 104431 (2004)

    2004

Show all 44 references
  1. [8]

    Nachtergaele and R

    B. Nachtergaele and R. Sims, Comm. Math. Phys. 276, 437 (2007)

    2007

  2. [9]

    Resta, Phys

    R. Resta, Phys. Rev. Lett. 80, 1800 (1998)

    1998

  3. [10]

    Resta and S

    R. Resta and S. Sorella, Phys. Rev. Lett. 82, 370 (1999)

    1999

  4. [11]

    A. A. Aligia and G. Ortiz, Phys. Rev. Lett. 82, 2560 (1999)

    1999

  5. [12]

    Tasaki, Phys

    H. Tasaki, Phys. Rev. Lett. 121, 140604 (2018)

    2018

  6. [13]

    H. Su, Y. Yao, and A. Furusaki, Phys. Rev. Lett. 133, 266705 (2024)

    2024

  7. [15]

    Wu, J.-P

    C. Wu, J.-P. Hu, and S.-C. Zhang, Phys. Rev. Lett. 91, 186402 (2003)

    2003

  8. [16]

    Honerkamp and W

    C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. 92, 170403 (2004)

    2004

  9. [17]

    M. A. Cazalilla, A. Ho, and M. Ueda, New J. Phys. 11 (2009)

    2009

  10. [18]

    A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nat. Phys. 6, 289 (2010)

    2010

  11. [19]

    S. Taie, R. Yamazaki, S. Sugawa, and Y. Takahashi, Nature Physics 8, 825 (2012)

    2012

  12. [20]

    Pagano, M

    G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J. Catani, C. Sias, M. In- guscio, and L. Fallani, Nature Physics 10, 198 (2014)

    2014

  13. [21]

    Scazza, C

    F. Scazza, C. Hofrichter, M. Höfer, P. C. De Groot, I. Bloch, and S. Fölling, Nat. Phys. 10, 779 (2014)

    2014

  14. [22]

    Zhang, M

    X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, and J. Ye, Science 345, 1467 (2014)

    2014

  15. [23]

    Hetényi, Phys

    B. Hetényi, Phys. Rev. Res. 2, 023277 (2020)

    2020

  16. [24]

    A. A. Aligia, Phys. Rev. B 107, 075153 (2023)

    2023

  17. [25]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, 162 (1957)

    1957

  18. [26]

    Altland and B

    A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, 2010)

    2010

  19. [27]

    Blankenbecler, D

    R. Blankenbecler, D. Scalapino, and R. Sugar, Phys. Rev. D 24, 2278 (1981)

    1981

  20. [28]

    Scalapino and R

    D. Scalapino and R. Sugar, Phys. Rev. B 24, 4295 (1981)

    1981

  21. [29]

    Assaad and H

    F. Assaad and H. Evertz, in Computational many-particle physics (Springer, 2008) pp. 277–356

    2008

  22. [30]

    Cohen-Stead, S

    B. Cohen-Stead, S. M. Costa, J. Neuhaus, A. T. Ly, Y. Zhang, R. Scalettar, K. Barros, and S. Johnston, SciPost Phys. Codebases , 29 (2024)

    2024

  23. [31]

    Duivenvoorden and T

    K. Duivenvoorden and T. Quella, Phys. Rev. B 86, 235142 (2012)

    2012

  24. [32]

    Greiter and S

    M. Greiter and S. Rachel, Phys. Rev. B 75, 184441 (2007)

    2007

  25. [33]

    Führinger, S

    M. Führinger, S. Rachel, R. Thomale, M. Greiter, and P. Schmitteckert, Annalen der Physik 520, 922 (2008)

    2008

  26. [34]

    Dufour, P

    J. Dufour, P. Nataf, and F. Mila, Phys. Rev. B 91, 174427 (2015)

    2015

  27. [35]

    Symmetry-protected topological order in su(n) heisenberg magnets –quantum entanglement and non-local order parameters,

    K. Tanimoto and K. Totsuka, “Symmetry-protected topological order in su(n) heisenberg magnets –quantum entanglement and non-local order parameters,” (2015)

    2015

  28. [36]

    Nataf and F

    P. Nataf and F. Mila, Phys. Rev. B 93, 155134 (2016)

    2016

  29. [37]

    Weichselbaum, S

    A. Weichselbaum, S. Capponi, P. Lecheminant, A. M. Tsvelik, and A. M. Läuchli, Phys. Rev. B 98, 085104 (2018)

    2018

  30. [38]

    Capponi, P

    S. Capponi, P. Fromholz, P. Lecheminant, and K. Tot- suka, Phys. Rev. B 101, 195121 (2020)

    2020

  31. [39]

    L. E. H. Affleck, Ian, Letters in Mathematical Physics 12, 57 (1986)

    1986

  32. [40]

    See Supplemental Materials

  33. [41]

    A. W. Sandvik, L. Balents, and D. K. Campbell, Phys. Rev. Lett. 92, 236401 (2004)

    2004

  34. [42]

    Ejima and S

    S. Ejima and S. Nishimoto, Phys. Rev. Lett. 99, 216403 (2007)

    2007

  35. [43]

    Noack, S

    R. Noack, S. White, and D. Scalapino, Physica C: Su- perconductivity 270, 281 (1996)

    1996

  36. [44]

    B. C. Hall, Lie groups, Lie algebras, and representations: an elementary introduction , Graduate Texts in Mathe- matics, Vol. 222 (Springer, New York, 2010) p. 351 pages. 6 Supplemental Materials Analytical results of ⟨G.S.| ˆU 2|G.S.⟩ for BCS Hamiltonian In this section, we g...

    2010

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.