pith. sign in

arxiv: 2605.13606 · v2 · pith:35WJKJJ5new · submitted 2026-05-13 · ❄️ cond-mat.str-el · cond-mat.stat-mech

Lieb-Schultz-Mattis theorem from gauge constraints

Bhandaru Phani Parasar This is my paper

Pith reviewed 2026-05-20 20:56 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mech
keywords Lieb-Schultz-Mattis theoremgauge theoryGauss law subspaceone-dimensional chainemergent U(1) symmetrygapless phaseDirac fermionscorrelation function
0
0 comments X

The pith

Gauge constraints on a one-dimensional chain produce a U(1) symmetry that yields a Lieb-Schultz-Mattis theorem ruling out trivial gapped states.

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

The paper builds a Z2 times Z2 gauge theory coupled to matter on a one-dimensional chain and restricts attention to the subspace obeying the Gauss law. Within that subspace an emergent U(1) symmetry appears whose generator commutes with translations yet anticommutes with reflections. The resulting Lieb-Schultz-Mattis argument shows that any Hamiltonian invariant under both symmetries cannot support a trivial gapped ground state; every point in parameter space must realize either spontaneous symmetry breaking or a gapless phase. At one particular gapless point the low-energy excitations are free Dirac fermions subject to a total-number constraint, and the simplest gauge-invariant two-point function decays as cos of pi r times r to the minus two-ninths. The construction illustrates how a kinematic constraint of a gauge theory can itself supply the symmetry needed for an LSM theorem.

Core claim

The theory in the Gauss law subspace has a U(1) symmetry whose generator commutes with lattice translations but anticommutes with the lattice reflection operator. This leads to a Lieb-Schultz-Mattis theorem that always rules out a trivial gapped ground state in the Gauss law subspace if the Hamiltonian is invariant under translations and reflection. Any point in the parameter space must realize either a spontaneously symmetry broken ground state or a gapless ground state. Imposing the Gauss law is pivotal for the existence of the U(1) symmetry and hence of the LSM theorem. At the gapless point the excitations admit a description in terms of free Dirac fermions with a constraint on the total

What carries the argument

The emergent U(1) symmetry in the Gauss law subspace, generated such that its operator commutes with translations and anticommutes with reflections.

If this is right

  • No trivial gapped ground state that respects both translations and reflections can exist in the physical subspace.
  • The ground state at every point in parameter space is either spontaneously symmetry broken or gapless.
  • At the identified gapless point the excitations are free Dirac fermions with a fixed total fermion number.
  • The two-point correlation function of the simplest local gauge-invariant operator decays proportionally to cos(pi r) times r to the power minus two-ninths.
  • The model provides a natural setting in which to examine topological defects that connect different families of symmetry-broken phases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same constraint-driven mechanism could be tested in two-dimensional gauge theories to see whether it likewise prohibits trivial gapped states.
  • The specific decay exponent of minus two-ninths may correspond to a known constrained-fermion conformal theory whose central charge or operator content can be checked independently.
  • Varying the gauge coupling or adding longer-range interactions while remaining in the Gauss law subspace could map how the gapless point moves or splits into multiple critical lines.
  • One could search for analogous emergent symmetries generated by other local constraints, such as those appearing in lattice gauge theories with different groups.

Load-bearing premise

The Hamiltonian is invariant under translations and reflection while the analysis is performed strictly inside the Gauss law subspace.

What would settle it

Numerical or exact diagonalization evidence of a unique, gapped, translation- and reflection-symmetric ground state inside the Gauss law subspace for some set of Hamiltonian parameters would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.13606 by Bhandaru Phani Parasar.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) - The one-dimensional chain on which the model is [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Conservation of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
read the original abstract

We construct a $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory coupled to matter on a one-dimensional chain, aiming to study the ground-state physics in the Gauss law subspace. We show that the theory in the Gauss law subspace has a U$(1)$ symmetry whose generator commutes with lattice translations, but anticommutes with the lattice reflection operator. This leads to a Lieb-Schultz-Mattis (LSM) theorem that always rules out a trivial gapped ground state in the Gauss law subspace, if the hamiltonian is invariant under translations and reflection. Any point in the parameter space must realize either a spontaneously symmetry broken (SSB) ground state, or a gapless ground state. Imposing the Gauss law is pivotal for the existence of the U$(1)$ symmetry, and hence of the LSM theorem. We thus demonstrate a novel mechanism to obtain an LSM-type theorem, wherein the symmetry responsible for the theorem originates from the kinematic constraints of a gauge theory. We identify a point in the parameter space at which the system is gapless. At the gapless point, the excitations admit a description in terms of free Dirac fermions with a constraint on the total fermion number. The asymptotic behavior of the two-point correlation function of the simplest local gauge-invariant quantity at the gapless point is found to be $ \propto \cos{(\pi r)}\,r^{-2/9}$, where $r$ is the lattice separation between the two points. This model is also a natural platform to study phase diagram topological defects residing in families of SSB phases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs a ℤ₂ × ℤ₂ gauge theory coupled to matter on a one-dimensional chain. It claims that, strictly inside the Gauss-law subspace, a U(1) symmetry emerges whose generator commutes with lattice translations but anticommutes with reflection. This symmetry is used to derive an LSM theorem that prohibits a trivial gapped ground state for any Hamiltonian invariant under translations and reflection. The authors identify a gapless point whose excitations are described by free Dirac fermions with a total-fermion-number constraint and report that the two-point function of the simplest local gauge-invariant operator decays as cos(πr) r^{-2/9}. The model is also proposed as a platform for studying topological defects in families of SSB phases.

Significance. If the central construction holds, the work supplies a novel kinematic mechanism for an LSM obstruction that originates from gauge constraints rather than from an explicitly imposed global symmetry. This could be useful for analyzing constrained Hilbert spaces and for constructing models with enforced gaplessness or SSB. The explicit correlation exponent at the gapless point and the suggestion for studying defects in SSB phases add concrete, falsifiable content.

major comments (2)
  1. [Derivation of the U(1) symmetry (following the model definition)] The abstract states that the U(1) generator 'originates from the kinematic constraints' and that imposing the Gauss law is 'pivotal' for the LSM theorem. However, the manuscript does not supply an explicit operator expression for this generator nor a direct verification that it commutes with every local, translation- and reflection-invariant term that preserves the Gauss law. Without such a demonstration, the claimed conservation [G, H] = 0 for arbitrary symmetry-allowed H inside the subspace remains unproven, which is load-bearing for the universality of the LSM conclusion.
  2. [LSM theorem statement and proof] The LSM theorem is asserted to rule out a trivial gapped state whenever the Hamiltonian is invariant under translations and reflection. The proof sketch relies on the anticommutation {G, R} = 0 together with [G, T] = 0. A concrete check is needed that no local gauge-invariant operator allowed by these symmetries can violate the commutation relations with G while still preserving the Gauss law; otherwise the obstruction applies only to a restricted subclass of models.
minor comments (2)
  1. [Gapless-point analysis] The correlation-function exponent -2/9 is reported at the gapless point; the derivation of this specific value (presumably from the free-Dirac description with the fermion-number constraint) should be expanded with intermediate steps so that the result can be reproduced independently.
  2. [Model construction] Notation for the gauge-invariant operators and the precise definition of the Gauss-law subspace should be introduced earlier and used consistently to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of our results. We address each major comment below by providing the requested explicit constructions and verifications. These will be incorporated into the revised manuscript.

read point-by-point responses

  1. Referee: The abstract states that the U(1) generator 'originates from the kinematic constraints' and that imposing the Gauss law is 'pivotal' for the LSM theorem. However, the manuscript does not supply an explicit operator expression for this generator nor a direct verification that it commutes with every local, translation- and reflection-invariant term that preserves the Gauss law. Without such a demonstration, the claimed conservation [G, H] = 0 for arbitrary symmetry-allowed H inside the subspace remains unproven, which is load-bearing for the universality of the LSM conclusion.

    Authors: We agree that an explicit operator expression and direct verification will strengthen the manuscript. The U(1) generator G is the global charge operator that follows directly from summing the local Gauss-law constraints over the chain; its explicit form is G = (1/2) ∑_i (Z_i^gauge * fermion parity terms) projected onto the physical subspace. In the revised version we will insert this definition early in the model section and prove [G, H_local] = 0 for every local, gauge-invariant, translation- and reflection-invariant term by showing that each such term is built from operators whose individual commutators with G cancel or vanish inside the Gauss-law subspace. This establishes [G, H] = 0 for arbitrary Hamiltonians obeying the stated symmetries. revision: yes

  2. Referee: The LSM theorem is asserted to rule out a trivial gapped state whenever the Hamiltonian is invariant under translations and reflection. The proof sketch relies on the anticommutation {G, R} = 0 together with [G, T] = 0. A concrete check is needed that no local gauge-invariant operator allowed by these symmetries can violate the commutation relations with G while still preserving the Gauss law; otherwise the obstruction applies only to a restricted subclass of models.

    Authors: We thank the referee for highlighting the need for this explicit check. In the revision we will add a dedicated paragraph demonstrating that any local gauge-invariant operator O that (i) preserves the Gauss law, (ii) commutes with translations, and (iii) commutes with reflection must satisfy [G, O] = 0 (or the appropriate relation compatible with {G, R} = 0). The argument proceeds by expressing O in the gauge-invariant basis and showing that its action cannot change the eigenvalue of G while remaining local and symmetry-allowed; consequently the anticommutation {G, R} = 0 is preserved for the full Hamiltonian and the LSM obstruction applies to the entire class of models considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity; U(1) symmetry derived from external Gauss-law constraint

full rationale

The derivation begins with a standard Z2 x Z2 gauge theory on a 1D chain and restricts to the Gauss-law subspace, an external kinematic constraint independent of the target LSM result. The claimed U(1) generator is constructed to commute with translations and anticommute with reflection inside this subspace, yielding the LSM obstruction for any translation- and reflection-invariant Hamiltonian. This follows directly from the imposed constraint plus lattice symmetries rather than from a self-referential definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The gapless-point correlation function is computed as an explicit result, not forced by construction. The paper is self-contained against external benchmarks with no reduction of the central claim to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard lattice gauge theory definitions and the existence of the U(1) symmetry inside the constrained subspace; no free parameters, additional axioms beyond domain assumptions, or invented entities are introduced.

axioms (1)
  • domain assumption Standard formulation and properties of Z2 x Z2 lattice gauge theory coupled to matter, including the definition of the Gauss law constraint.
    The paper invokes the usual definitions of gauge theories and local constraints without deriving them from more basic principles.

pith-pipeline@v0.9.0 · 5817 in / 1384 out tokens · 78132 ms · 2026-05-20T20:56:59.125041+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Imposing the Gauss law is pivotal for the existence of the U(1) symmetry, and hence of the LSM theorem. We thus demonstrate a novel mechanism to obtain an LSM-type theorem, wherein the symmetry responsible for the theorem originates from the kinematic constraints of a gauge theory.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the theory in the Gauss law subspace has a U(1) symmetry whose generator commutes with lattice translations, but anticommutes with the lattice reflection operator

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages · 1 internal anchor

  1. [1]

    hopping term

    as twisted gauging [38] for the spin-1 XYZ chain. Our construction described above corresponds to an ordinary gauging of the spin-1 XYZ chain. Let us discuss spatial symmetries of the model. Let Lbe even,Tbe the lattice translation operator (that translates by one unit cell), and letRbe the operator implementing the reflection of the periodic chain about ...

    work page

  2. [2]

    E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics16, 407 (1961)

    work page 1961

  3. [3]

    Affleck and E

    I. Affleck and E. H. Lieb, A proof of part of haldane’s con- jecture on spin chains, Letters in Mathematical Physics 12, 57 (1986)

    work page 1986

  4. [4]

    haldane gap

    M. Oshikawa, M. Yamanaka, and I. Affleck, Magneti- zation plateaus in spin chains: “haldane gap” for half- integer spins, Phys. Rev. Lett.78, 1984 (1997)

    work page 1984

  5. [5]

    Oshikawa, Commensurability, excitation gap, and topology in quantum many-particle systems on a peri- odic lattice, Phys

    M. Oshikawa, Commensurability, excitation gap, and topology in quantum many-particle systems on a peri- odic lattice, Phys. Rev. Lett.84, 1535 (2000)

    work page 2000

  6. [6]

    M. B. Hastings, Lieb-schultz-mattis in higher dimensions, Phys. Rev. B69, 104431 (2004)

    work page 2004

  7. [7]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Classification of gapped symmetric phases in one-dimensional spin sys- tems, Phys. Rev. B83, 035107 (2011)

    work page 2011

  8. [8]

    S. A. Parameswaran, A. M. Turner, D. P. Arovas, and A. Vishwanath, Topological order and absence of band insulators at integer filling in non-symmorphic crystals, Nature Physics9, 299 (2013)

    work page 2013

  9. [9]

    Watanabe, H

    H. Watanabe, H. C. Po, A. Vishwanath, and M. Zaletel, Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals, Proceedings of the National Academy of Sciences112, 14551 (2015), https://www.pnas.org/doi/pdf/10.1073/pnas.1514665112

    work page doi:10.1073/pnas.1514665112 2015

  10. [10]

    Watanabe, Lieb-schultz-mattis-type filling constraints in the 1651 magnetic space groups, Phys

    H. Watanabe, Lieb-schultz-mattis-type filling constraints in the 1651 magnetic space groups, Phys. Rev. B97, 165117 (2018)

    work page 2018

  11. [11]

    Tasaki, Lieb–schultz–mattis theorem with a local twist for general one-dimensional quantum systems, Journal of Statistical Physics170, 653 (2018)

    H. Tasaki, Lieb–schultz–mattis theorem with a local twist for general one-dimensional quantum systems, Journal of Statistical Physics170, 653 (2018)

    work page 2018

  12. [12]

    Ogata, Y

    Y. Ogata, Y. Tachikawa, and H. Tasaki, General lieb–schultz–mattis type theorems for quantum spin chains, Communications in Mathematical Physics385, 79 (2021)

    work page 2021

  13. [13]

    Kobayashi, K

    R. Kobayashi, K. Shiozaki, Y. Kikuchi, and S. Ryu, Lieb- schultz-mattis type theorem with higher-form symmetry and the quantum dimer models, Phys. Rev. B99, 014402 (2019)

    work page 2019

  14. [14]

    D. V. Else and R. Thorngren, Topological theory of lieb- schultz-mattis theorems in quantum spin systems, Phys. Rev. B101, 224437 (2020)

    work page 2020

  15. [15]

    Dubinkin, J

    O. Dubinkin, J. May-Mann, and T. L. Hughes, Lieb- schultz-mattis-type theorems and other nonperturbative results for strongly correlated systems with conserved dipole moments, Phys. Rev. B103, 125133 (2021)

    work page 2021

  16. [16]

    F. J. Burnell, S. Moudgalya, and A. Prem, Filling con- straints on translation invariant dipole conserving sys- tems, Phys. Rev. B110, L121113 (2024)

    work page 2024

  17. [17]

    Yao and A

    Y. Yao and A. Furusaki, Geometric approach to lieb- schultz-mattis theorem without translation symmetry under inversion or rotation symmetry, Phys. Rev. B106, 045125 (2022)

    work page 2022

  18. [18]

    Y. Yao, L. Li, M. Oshikawa, and C.-T. Hsieh, Lieb- schultz-mattis theorem for 1d quantum magnets with antiunitary translation and inversion symmetries, Phys. Rev. Lett.133, 136705 (2024)

    work page 2024

  19. [19]

    Cheng, M

    M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, Translational symmetry and microscopic constraints on symmetry-enriched topological phases: A view from the surface, Phys. Rev. X6, 041068 (2016)

    work page 2016

  20. [20]

    G. Y. Cho, C.-T. Hsieh, and S. Ryu, Anomaly mani- festation of lieb-schultz-mattis theorem and topological phases, Phys. Rev. B96, 195105 (2017)

    work page 2017

  21. [21]

    C.-M. Jian, Z. Bi, and C. Xu, Lieb-schultz-mattis theo- rem and its generalizations from the perspective of the symmetry-protected topological phase, Phys. Rev. B97, 054412 (2018)

    work page 2018

  22. [22]

    Cheng and N

    M. Cheng and N. Seiberg, Lieb-Schultz-Mattis, Lut- tinger, and ’t Hooft - anomaly matching in lattice sys- tems, SciPost Phys.15, 051 (2023)

    work page 2023

  23. [23]

    Seifnashri, Lieb-Schultz-Mattis anomalies as obstruc- tions to gauging (non-on-site) symmetries, SciPost Phys

    S. Seifnashri, Lieb-Schultz-Mattis anomalies as obstruc- tions to gauging (non-on-site) symmetries, SciPost Phys. 16, 098 (2024)

    work page 2024

  24. [24]

    Mudry, A

    ¨Omer Mert Aksoy, C. Mudry, A. Furusaki, and A. Tiwari, Lieb-Schultz-Mattis anomalies and web of dualities in- duced by gauging in quantum spin chains, SciPost Phys. 16, 022 (2024)

    work page 2024

  25. [25]

    Ebisu, B

    H. Ebisu, B. Han, and W. Cao, Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies, SciPost Phys.20, 117 (2026)

    work page 2026

  26. [26]

    J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys.51, 659 (1979)

    work page 1979

  27. [27]

    Senthil and M

    T. Senthil and M. P. A. Fisher,Z 2 gauge theory of electron fractionalization in strongly correlated systems, Phys. Rev. B62, 7850 (2000)

    work page 2000

  28. [28]

    Moessner, S

    R. Moessner, S. L. Sondhi, and E. Fradkin, Short-ranged resonating valence bond physics, quantum dimer models, and ising gauge theories, Phys. Rev. B65, 024504 (2001)

    work page 2001

  29. [29]

    Wen, Quantum orders and symmetric spin liquids, Phys

    X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B65, 165113 (2002)

    work page 2002

  30. [30]

    Senthil, (2023), arXiv:2306.12638 [cond-mat.str-el]

    T. Senthil, Deconfined quantum critical points: a review (2023), arXiv:2306.12638 [cond-mat.str-el]

    work page arXiv 2023

  31. [31]

    Nandkishore, M

    R. Nandkishore, M. A. Metlitski, and T. Senthil, Orthog- onal metals: The simplest non-fermi liquids, Phys. Rev. B86, 045128 (2012)

    work page 2012

  32. [32]

    Banerjee, M

    D. Banerjee, M. B¨ ogli, C. P. Hofmann, F.-J. Jiang, P. Widmer, and U.-J. Wiese, Interfaces, strings, and a soft mode in the square lattice quantum dimer model, Phys. Rev. B90, 245143 (2014)

    work page 2014

  33. [33]

    Gazit, M

    S. Gazit, M. Randeria, and A. Vishwanath, Emergent dirac fermions and broken symmetries in confined and deconfined phases ofZ 2 gauge theories, Nature Physics 13, 484 (2017)

    work page 2017

  34. [34]

    Gazit, F

    S. Gazit, F. F. Assaad, S. Sachdev, A. Vishwanath, and 7 C. Wang, Confinement transition ofZ 2 gauge theories coupled to massless fermions: Emergent quantum chromodynamics and SO(5) symmetry, Proceedings of the National Academy of Sciences115, E6987 (2018), https://www.pnas.org/doi/pdf/10.1073/pnas.1806338115

    work page doi:10.1073/pnas.1806338115 2018

  35. [35]

    Borla, B

    U. Borla, B. Jeevanesan, F. Pollmann, and S. Moroz, Quantum phases of two-dimensionalZ 2 gauge theory coupled to single-component fermion matter, Phys. Rev. B105, 075132 (2022)

    work page 2022

  36. [36]

    H. Das, N. P. Nayak, S. Bera, and V. B. Shenoy, Critical states of fermions withZ 2 flux disorder, Phys. Rev. Lett. 135, 256305 (2025)

    work page 2025

  37. [37]

    Without loss of generality,K z may be set to zero because of the identity P α(Sα)2 = 2

    work page

  38. [38]

    Haegeman, K

    J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, Gauging quantum states: From global to local symmetries in many-body systems, Phys. Rev. X 5, 011024 (2015)

    work page 2015

  39. [39]

    Blanik, J

    D. Blanik, J. Garre-Rubio, and N. Schuch, Gauging quan- tum phases: A matrix product state approach, Phys. Rev. B112, 115110 (2025)

    work page 2025

  40. [40]

    Kennedy and H

    T. Kennedy and H. Tasaki, Hidden Z 2×Z2 symmetry breaking in haldane-gap antiferromagnets, Phys. Rev. B 45, 304 (1992)

    work page 1992

  41. [41]

    L. Li, M. Oshikawa, and Y. Zheng, Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases, Phys. Rev. B108, 214429 (2023)

    work page 2023

  42. [42]

    Supplemental Material

    work page

  43. [43]

    D. Sen, R. Shankar, D. Dhar, and K. Ramola, Spin-1 kitaev model in one dimension, Phys. Rev. B82, 195435 (2010)

    work page 2010

  44. [44]

    F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, Lattice gauge theories and string dynamics in rydberg atom quantum simula- tors, Phys. Rev. X10, 021041 (2020)

    work page 2020

  45. [45]

    Mukherjee, D

    B. Mukherjee, D. Banerjee, K. Sengupta, and A. Sen, Minimal model for hilbert space fragmentation with local constraints, Phys. Rev. B104, 155117 (2021)

    work page 2021

  46. [46]

    Zhang, Y.-N

    W.-Y. Zhang, Y.-N. Wang, D. Liu, J. Ren, J. Li, N. Wu, A. M. Ole´ s, and W.-L. You, Quantum many-body scars in spin-1 kitaev chain with uniaxial single-ion anisotropy, Phys. Rev. B108, 104411 (2023)

    work page 2023

  47. [47]

    Mohapatra and A

    S. Mohapatra and A. C. Balram, Pronounced quantum many-body scars in the one-dimensional spin-1 kitaev model, Phys. Rev. B107, 235121 (2023)

    work page 2023

  48. [48]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Experiment2004, P06002 (2004)

    work page 2004

  49. [49]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, Code- base release 0.3 for ITensor, SciPost Phys. Codebases , 4 (2022)

    work page 2022

  50. [50]

    Jin and V

    B.-Q. Jin and V. E. Korepin, Quantum spin chain, toeplitz determinants and the fisher—hartwig conjecture, Journal of Statistical Physics116, 79 (2004)

    work page 2004

  51. [51]

    Deift, A

    P. Deift, A. Its, and I. Krasovsky, Toeplitz matrices and toeplitz determinants under the impetus of the ising model: Some history and some recent results, Commu- nications on Pure and Applied Mathematics66, 1360 (2013)

    work page 2013

  52. [52]

    Pujari and H

    S. Pujari and H. Nigam, AnticommutingZ 2 quantum spin liquids, Phys. Rev. B113, 064402 (2026)

    work page 2026

  53. [53]

    Prakash, M

    A. Prakash, M. Fava, and S. A. Parameswaran, Multiver- sality and unnecessary criticality in one dimension, Phys. Rev. Lett.130, 256401 (2023)

    work page 2023

  54. [54]

    P.-S. Hsin, A. Kapustin, and R. Thorngren, Berry phase in quantum field theory: Diabolical points and boundary phenomena, Phys. Rev. B102, 245113 (2020)

    work page 2020

  55. [55]

    In search of diabolical critical points

    N. Manjunath and D. V. Else, In search of diabolical critical points (2026), arXiv:2601.10783 [cond-mat.str-el]. 1 Supplemental Material for Lieb-Schultz-Mattis theorem from gauge constraints by Bhandaru Phani Parasar Recap of some notation from the main text— On each site and link, the operatorsZandXare defined asZ:=− P α ξα (Sα)2 ,X:=−(S xSy +S ySz +S z...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  56. [56]

    cyclic raising operator

    Hence,P α j−1τ +P α jσ +P α jτ −1 = 2P α j−1τ P α jτ = 2P α j−1τ P α jσ = 2P α jσ P α jτ . Now, using P α P α = 1, we see thatP α j−1τ P α jτ =P α j−1τ P α jσ =P α jσ P α jτ = 0 for alljand α=x, y, z. i.e., for eachj, the states on the sitejσ, and the linksj−1τandjτmust all be different from one another. B: Dimension ofV G In this section, we calculate th...

    work page

  57. [57]

    Clearly, D c† jcj E = 1/2

    Correlation function as a Toeplitz determinant In the ground state, consider the correlation functionD c† jck E = 1 L P q e−iq(j−k) ⟨nq⟩. Clearly, D c† jcj E = 1/2. ForLa multiple of 4, D c† jck E = 1 L PL/4−1 m=−L/4 e−i 2πm L (j−k) . Forj̸=k, D c† jck E = sin π 2 (j−k) Lsin π L(j−k) eiπ(j−k)/L D c† jck E − D c† kcj E = 2i L sin π 2 (j−k) (F.1) IfLis even...

    work page

  58. [58]

    We now state the Fisher-Hartwig conjecture (see Ref

    Asymptotic limit To understand the asymptotic behavior of the corre- lation function when the separation between the lattice pointsr→ ∞(note we have already taken the limit L→ ∞), we need to know the asymptotic behavior of det(F) as the dimension of the matrixr→ ∞. We now state the Fisher-Hartwig conjecture (see Ref. [50] for a review), which can be used ...

    work page

  59. [59]

    Then the Schur multiplier H2 (G,U(1)) ∼= ZN(N−1)/2 2

    Dimension of irrep LetG ∼= ZN 2 for someN. Then the Schur multiplier H2 (G,U(1)) ∼= ZN(N−1)/2 2 . LetUbe a projective repre- sentation ofGon a Hilbert space with Ω as the associated multiplier system. Also, let Ω(g,h) 2 = 1∀g,h∈G. We want to find the irreducible representations contained in the direct sum decomposition ofU. Now, the projective representat...

    work page