Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Generalized Symmetries Phase Transitions with Local Quantum Fields

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper argues that ordinary local gauge theories can realize phase transitions between distinct SPT and SET phases of generalized one-form symmetries, including transitions between phases with the same topological order and the same…

desk verdict Useful toolkit with a flawed flagship example: the §5.1.1 claim of distinct SETs collapses because the two SO(3) Higgs routes end at conjugate Z2 subgroups, giving the same one-form fractionalization. read the letter →

arxiv 2506.07688 v1 pith:OPEZCXRJ submitted 2025-06-09 cond-mat.str-el hep-th

Po-Shen Hsin This is my paper · ORCID
classification cond-mat.str-elhep-th
keywords generalizedsymmetriesone-formsymmetryfractionalizationsymmetry-protectedtopologicalphasessymmetry-enrichedHiggstransitionsgaugetheorydeconfinedquantumcriticality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether ordinary quantum field theories built from local fields, like scalars or fermions coupled to gauge fields, can undergo phase transitions that are invisible to local operators: transitions between symmetry-protected (SPT) or symmetry-enriched (SET) phases whose labels live in generalized one-form symmetries, and symmetry-breaking transitions where the unbroken side is nontrivially SPT or SET. The answer it argues for is yes, in 3+1d and 2+1d gauge theories with bosons or fermions, driven by Higgs condensation or by flipping fermion masses. The core new result is that continuous gauge theories in 3+1d can sit in distinct phases that share the same unbroken one-form symmetry and the same topological order yet differ by how that symmetry fractionalizes on loop excitations. A sympathetic reader should care because these transitions are protected against arbitrary local-operator perturbations, making them robust signatures in systems where no local order parameter changes.

What carries the argument

The central machinery is fractionalization of one-form symmetries in the low-energy topological theory. When a gauge theory with gauge group G confines or is Higgsed, the infrared is described by a two-form gauge theory for the magnetic (or electric) center symmetry; the fractionalization is the statement that symmetry generators act on the deconfined loop excitations of that two-form theory by phases such as 1/N, visible as nontrivial junctions or as couplings like (1/2π)∫ da ∧ B in the low-energy action. The phase transitions are induced by Higgs condensation, which breaks the electric symmetry and confines monopoles, or by fermion mass sign flips, which shift θ by 2πk and thereby change which subgroup of the magnetic symmetry is unbroken and what SPT it carries.

What would settle it

Find, by lattice simulation or exact construction, the infrared theory of pure SO(2n+1) gauge theory in 3+1d at zero theta angle: if it is not the Z2 two-form gauge theory with broken magnetic one-form symmetry, the claimed SSB-SET transition in the SO(2n+1) Higgs model would not occur as described. Equivalently, check the negative-mass phase of SO(3) gauge theory with one adjoint fermion: the paper predicts a trivially gapped phase with unbroken Z2 magnetic symmetry and SPT -2π/4 ∫ P(B); observing deconfined anyonic excitations or a different SPT value there would falsify the central claim.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that standard local gauge theories realise the full menu of generalized-symmetry phase transitions. It shows that pure non-Abelian gauge theories confine to two-form gauge theories in which the generator of the magnetic one-form symmetry can carry fractional electric one-form charge, giving fractionalization tied to the mixed anomaly of the electric and magnetic symmetries; that Higgs phases can leave a magnetic one-form symmetry unbroken while fractionalizing it on vortex loops; and that fermion mass sign flips shift theta angles by integer multiples of 2π, converting a broken magnetic symmetry into an unbroken one equipped with a nontrivial SPT. In particular, it claims that a continuous gauge theory can have two phases with the same unbroken one-form symmetry and the same Z2 topological order but different fractionalizations of that symmetry, with the transition protected from local operator perturbations.

Load-bearing premise

The argument leans on the standard semiclassical infrared descriptions: that pure non-Abelian gauge theories confine into the stated two-form topological theories and that the Higgs phase is correctly described by the unbroken subgroup with the stated unbroken magnetic symmetry; if nonperturbative effects change those descriptions, the predicted fractionalizations and transitions could fail.

Editorial extensions

If this is right

  • If the claims hold, any local perturbation of these gauge theories cannot remove the phase transitions or change the asymptotic phases, so the distinctions are robust predictions for condensed-matter realizations.
  • In SO(2n) gauge theory with adjoint Higgs fields, the transition is a 3+1d analogue of deconfined quantum criticality: different subgroups of the Z2 × Z2 electric and magnetic one-form symmetry are spontaneously broken on the two sides.
  • In SO(3) gauge theory with an adjoint fermion, a sign flip of the fermion mass moves the system across a transition in which the Z2 magnetic one-form symmetry goes from broken to unbroken, with the unbroken phase carrying the SPT -2π/4 ∫ P(B).
  • Two Higgs phases of SO(3) can both be Z2 gauge theories with unbroken magnetic symmetry but differ in whether that symmetry fractionalizes, giving a genuine SET transition along a path in the phase diagram.

Reading between the lines

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

  • Going beyond the paper, a testable extension would be to put these models on the lattice: the fractionalization predictions imply quantized ground-state degeneracies or braiding signatures of the two-form gauge theory that could be measured in exactly solvable models or quantum simulators.
  • Going beyond the paper, the same mechanism likely produces analogous transitions for higher p-form symmetries in higher dimensions, by replacing the two-form gauge theory with the appropriate (p+1)-form gauge theory, though the paper does not work this out.
  • Going beyond the paper, because symmetries generated by condensation defects cannot be broken, the SPT and SET labels for one-form symmetries should transfer automatically to the condensation defects, suggesting that domain-wall or defect probes could observe the transitions even where local correlators are featureless.
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

2 major / 4 minor

Summary. The paper constructs continuum quantum field theory models with local fields—ordinary gauge theories in 3+1d and 2+1d—that exhibit phase transitions protected by generalized symmetries. Three classes are treated: SPT transitions, SET transitions with different one-form symmetry fractionalizations, and symmetry-breaking transitions in which the unbroken phase carries nontrivial SPT or SET data. The concrete mechanisms are Higgs transitions and fermion mass sign flips. The central novel claim is that continuous gauge theories in 3+1d can have different phases with the same unbroken one-form symmetry and the same topological order but different fractionalizations of that one-form symmetry, with the flagship example in Section 5.1.1.

Significance. If the central claim were correct, the paper would provide a useful systematic toolbox for studying phase transitions of generalized symmetries using local quantum fields, going beyond string-like order parameters and beyond the standard lore that higher-form symmetries cannot be used to distinguish RG flows. The paper contains many concrete, self-contained examples with explicit low-energy effective actions, anomaly-matching computations, and SPT jumps; these are strengths that make the framework easy to check and adapt. The arguments rely on standard semiclassical assumptions about confinement, monopole condensation, and the IR TQFTs of gauge theories, which is acceptable practice in this field even though those IR descriptions are not derived from first principles. The manuscript's main weakness is that its flagship SET example for different one-form fractionalizations is incorrect, as detailed below.

major comments (2)
  1. [Section 5.1.1, Eqs. (5.1)-(5.2)] The claimed SET transition between two Higgs phases with different fractionalizations of the Z2 magnetic one-form symmetry is not established. The two routes break SO(3) to K1 = ⟨R_z(π)⟩ (the Cartan Z2) and to K2 = ⟨r⟩, where r is an orientation-reversing element of O(2) such as r = R_x(π). In SO(3), these two subgroups are conjugate, because every element of order two is a π-rotation about some axis. The fractionalization of an exact one-form symmetry is a basis-independent property fixed by the pullback of w2^SO(3) along the embedding of the unbroken group into SO(3). For either π-rotation embedding, the 3d real representation decomposes as L⊕L⊕1, so i^* w2^SO(3) = a^2 with a the Z2 gauge field, a nontrivial class in H^2(BZ2;Z2). Consequently both routes give exactly the same low-energy coupling of the form Eq. (5.2), and the same SET data. The statement that "the magnetic flux of O(2) gauge field ... does not depend on the ZC2 gauge field" does not settle the physical coupling to the background B, which is determined by the pullback of w2. Thus the two Higgs phases are identical SETs, and this example does not support the abstract's central claim.
  2. [Abstract and Section 5] As a consequence of the failure of the Section 5.1.1 example, the paper does not contain an explicit model that realizes two phases with the same topological order and same unbroken one-form symmetry but different fractionalizations of that one-form symmetry. Section 5.1.2 describes a transition between a trivial SPT and a fractionalized SET, not between two different fractionalization patterns. Section 5.1.3 and Section 5.2.1 concern Lorentz symmetry fractionalization rather than a change of one-form symmetry fractionalization. The abstract's strongest claim is therefore not demonstrated by the examples in the manuscript; the authors should either correct the SET example with a genuinely different pair of embeddings (e.g., using a gauge group with non-conjugate finite subgroups that give different pullbacks of the relevant characteristic class) or explicitly restrict the abstract's claim to the cases that actually work.
minor comments (4)
  1. [Section 3.1.2] There is a typo in "magentic one-form symmetry" (should be "magnetic").
  2. [Section 3.1.3 heading] The heading reads "SU (N1N2)/ZN2 gauge theories"; the subscript notation for Z_{N_2} is easy to confuse with the product N1N2. Please use unambiguous typesetting, e.g., Z_{N_2}.
  3. [Section 5.1.1, Fig. 9] The reference to "a 1d path in the phase diagram" would be clearer if the text explained which tuning parameters realize the path (e.g., relative coefficients of the two Higgs potentials), since the figure itself is not part of the arXiv text.
  4. [Section 4.1.1] Equation (4.2) uses (kN1)^{-1} without specifying that the inverse is taken modulo N2; this should be stated explicitly to avoid ambiguity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the constructions are computed from explicit effective actions; self-citations support but do not by themselves force the central SET-transition claim.

full rationale

The paper's phase assignments are derived from explicit low-energy effective actions rather than from restating the desired conclusion. For example, the U(1) Higgs-phase fractionalization in §3.1.1 follows from dualizing the action to Eq. (3.3) and using the equation of motion, and the SPT responses in §3.2.1 and §4 are obtained by integrating out dynamical two-form gauge fields. The central SET example in §5.1.1 is presented as a two-route SO(3) construction: route 1 is reduced to the §3.1.1 computation, and route 2 is asserted to have no fractionalization because the O(2) magnetic flux 'does not depend on the ZC2 gauge field.' This latter step may be mathematically questionable, but it is not circular: the claimed difference in fractionalization is a computed/asserted output, not an input assumption, and it does not reduce by definition to the abstract's claim. The paper does cite the author's prior work extensively, but the load-bearing formulas are either re-derived or standard published results, and no uniqueness theorem imported from the author's own work is used to forbid alternatives. No fitted parameter is renamed as a prediction, and no equation is equivalent by construction to the claimed phase-transition result. The honest finding is therefore no significant circularity; score 1 reflects only the heavy self-citation pattern, which is not itself load-bearing.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard but unproven infrared lore about confinement, monopole condensation, and the Higgs mechanism. The model choices (gauge group, theta angle, fermion flavor number) are free parameters selected by hand to realize the desired phases. No new particles or forces are introduced.

free parameters (3)
  • Choice of gauge group G and Higgs representations = Various (e.g., U(1), SO(2n+1), SU(N1N2)/ZN2)
    Each example is engineered by selecting a gauge group and Higgs sector to produce the desired symmetry fractionalization and phase structure; these choices are not fixed by any data.
  • Theta angle θ = 2πk = Integer k (e.g., k=1 for SO(3) SPT transition)
    The theta angle is tuned to generate the discrete theta angles that yield the desired SPT phases; the SPT jump depends on k.
  • Number of Dirac fermion flavors Nf = Even integers
    Fermion mass transitions shift the theta angle by 2πx_R N_f; the paper restricts to even N_f to avoid time-reversal symmetry breaking, and chooses specific values to realize the desired k.
assumptions (4)
  • domain assumption Pure non-Abelian gauge theories with connected gauge group confine at zero theta angle and flow to the stated two-form C gauge theory.
    Invoked in Section 2.1 to identify the low-energy TQFT and the fractionalization of the electric one-form symmetry.
  • domain assumption The Higgs mechanism and monopole condensation determine the unbroken and broken one-form symmetries in the Higgs phase, with the unbroken magnetic symmetry given by Ker ι'* in Eq. (3.1).
    Used throughout Section 3 to assign the symmetry realization in each phase.
  • domain assumption Fermion mass transitions shift the theta angle by 2πx_R N_f via the APS index theorem.
    Used in Sections 3.2 and 4.2 to compute SPT jumps across the fermion mass transition.
  • standard math The discrete theta angle formulas for the listed gauge groups in Appendix A correctly capture the SPT phases.
    These formulas are taken from prior work by the author and others (e.g., Hsin-Lam-Seiberg [21], Hsin-Lam [25]) and used as inputs for the low-energy actions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalized Symmetries Phase Transitions with Local Quantum Fields." pith.science (2026). https://pith.science/paper/OPEZCXRJ

@misc{pith2026250607688,
  author       = {Pith},
  title        = {Pith review of: Generalized Symmetries Phase Transitions with Local Quantum Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OPEZCXRJ}},
  note         = {Machine review of arXiv:2506.07688}
}
read the original abstract

Symmetries are important guiding principle for phase transitions. We systematically construct field theory models with local quantum fields that exhibit the following phase transitions: (1) different symmetry protected topological (SPT) phases with generalized symmetries; (2) different symmetry enriched topological (SET) phases with generalized symmetries differ by symmetry fractionalizations; (3) spontaneously broken generalized symmetries, where the unbroken phases can have nontrivial SPT or SET. The models are ordinary gauge theories with bosons or fermions in 3+1d and 2+1d. We focus on one-form symmetries and symmetries generated by condensation defects, which do not act on local operators. The phase transitions are protected from local operator perturbations which do not change the asymptotic phases. In particular, we show that continuous gauge theories in 3+1d can have different phases distinguished by fractionalizations of unbroken one-form symmetries.

Figures

Figures reproduced from arXiv: 2506.07688 by the authors.

Figure 1
Figure 1. Scenarios for symmetry breaking transitions. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Phase diagram of the Higgs transition for [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Phase diagram of SO(2n + 1) gauge theory with Higgs fields that can break the gauge group to Z2 in 3+1d. Both phases are the same Z2 gauge theory topological order, but the Z2 magnetic one-form symmetry acts differently in the two phases. The magentic one-form symmetry fractionalizes in the unbroken phase, which is a nontrivial SET. Under the background gauge transformation B → B + dλ, B′ → B ′ + dλ′ , a → a − λ ′ ,… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Phase diagram for SO(2n) gauge theory with Higgs fields in 3+1d. This is an analogue of DQCP for Z e 2 × Z m 2 one-form symmetry in 3+1d. When n is odd, the two phases have nontrivial SETs. O(2) rotation symmetry (on the lattice, only a discrete rotation subgroup is ma…
Figure 5
Figure 5. Figure 5: Phase diagram of SU(N1N2)/ZN2 gauge theory with adjoint Higgs fields in 3+1d. This is an analogue of DQCP for Z e N1×Z m N2 one-form symmetry in 3+1d. When gcd(N1, N2) ̸= 1, the two phases have nontrivial SETs. SPT TO Unbroken Broken ⟨ϕ⟩ = 0 ⟨ϕ⟩ ̸= 0 Z e N ZN Z e N [P…
Figure 6
Figure 6. Figure 6: Phase diagram of SU(N) gauge theory with adjoint Higgs fields in 3+1d. This is a symmetry breaking transition for Z e N electric center one-form symmetry. SU(N1N2)/ZN2 gauge theory We can generalize the model to SU(N1N2)/ZN2 gauge theory with adjoint Higgs that has Z e…
Figure 7
Figure 7. Figure 7: Phase diagram for SPT transition. The symmetries are unbroken, the two phases [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Phase diagram for SET transitions. The symmetries are unbroken. We will focus [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: A 1d path in the SO(3) gauge theory phase diagrams between two Higgs phases with Z2 gauge groups in 3+1d. The path describes SET transition that separates Z2 gauge theories with different fractionalizations of the magnetic Z2 one-form symmetry of the SO(3) gauge theory…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Bockstein braiding statistics

    quant-ph 2026-07 unverdicted novelty 7.5 of 10

    Introduces Bockstein braiding statistics for invertible Z_N excitations in the adjacent dimension case via a field theory term with the Bockstein operation, with implications for condensation and symmetry fractionalization.

Reference graph

Works this paper leans on

54 extracted references · 8 canonical work pages · cited by 1 Pith paper

  1. [1]

    Time-reversal symmetric u(1) quantum spin liquids,

    C. Wang and T. Senthil, “Time-reversal symmetric u(1) quantum spin liquids,” Phys. Rev. X6 (Mar, 2016) 011034. https://link.aps.org/doi/10.1103/PhysRevX.6.011034

    work page doi:10.1103/physrevx.6.011034 2016

  2. [2]

    Symmetry-enriched quantum spin liquids in (3 + 1) d,

    P.-S. Hsin and A. Turzillo, “Symmetry-enriched quantum spin liquids in (3 + 1) d,” JHEP 09 (2020) 022, arXiv:1904.11550 [cond-mat.str-el]

    arXiv 2020

  3. [3]

    Intrinsically gapless topological phases,

    R. Thorngren, A. Vishwanath, and R. Verresen, “Intrinsically gapless topological phases,” Phys. Rev. B104 no. 7, (2021) 075132, arXiv:2008.06638 [cond-mat.str-el]

    arXiv 2021

  4. [4]

    Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases

    T. T. Dumitrescu and P.-S. Hsin, “Higgs-confinement transitions in QCD from symmetry protected topological phases,” SciPost Phys. 17 no. 3, (2024) 093, arXiv:2312.16898 [hep-th]

    work page Pith review arXiv 2024

  5. [5]

    Symmetry Fractionalization, Defects, and Gauging of Topological Phases,

    M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, “Symmetry Fractionalization, Defects, and Gauging of Topological Phases,” Phys. Rev. B100 no. 11, (2019) 115147, arXiv:1410.4540 [cond-mat.str-el]

    arXiv 2019

  6. [6]

    Theory of twist liquids: Gauging an anyonic symmetry,

    J. C. Teo, T. L. Hughes, and E. Fradkin, “Theory of twist liquids: Gauging an anyonic symmetry,” Annals of Physics360 (Sept., 2015) 349–445. http://dx.doi.org/10.1016/j.aop.2015.05.012

    work page doi:10.1016/j.aop.2015.05.012 2015

  7. [7]

    Symmetry fractionalization and twist defects,

    N. Tarantino, N. H. Lindner, and L. Fidkowski, “Symmetry fractionalization and twist defects,” New Journal of Physics18 no. 3, (Mar., 2016) 035006. http://dx.doi.org/10.1088/1367-2630/18/3/035006

    doi:10.1088/1367-2630/18/3/035006 2016

  8. [8]

    On 2-Group Global Symmetries and their Anomalies,

    F. Benini, C. C´ ordova, and P.-S. Hsin, “On 2-Group Global Symmetries and their Anomalies,” JHEP 03 (2019) 118, arXiv:1803.09336 [hep-th]

    arXiv 2019

Show all 54 references
  1. [9]

    Generalized charges, part I: Invertible symmetries and higher representations,

    L. Bhardwaj and S. Schafer-Nameki, “Generalized charges, part I: Invertible symmetries and higher representations,” SciPost Phys. 16 no. 4, (2024) 093, arXiv:2304.02660 [hep-th]

    2024 arXiv

  2. [10]

    Higher representations for extended operators,

    T. Bartsch, M. Bullimore, and A. Grigoletto, “Higher representations for extended operators,” arXiv:2304.03789 [hep-th]

    arXiv

  3. [11]

    Generalized Global Symmetries,

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, “Generalized Global Symmetries,” JHEP 02 (2015) 172, arXiv:1412.5148 [hep-th]

    2015 arXiv

  4. [12]

    Condensations in higher categories,

    D. Gaiotto and T. Johnson-Freyd, “Condensations in higher categories,” (5, 2019) , arXiv:1905.09566 [math.CT]. 31

    2019 arXiv

  5. [13]

    Higher Gauging and Non-invertible Condensation Defects,

    K. Roumpedakis, S. Seifnashri, and S.-H. Shao, “Higher Gauging and Non-invertible Condensation Defects,” Commun. Math. Phys.401 no. 3, (2023) 3043–3107, arXiv:2204.02407 [hep-th]

    2023 arXiv

  6. [14]

    Non-invertible condensation, duality, and triality defects in 3+1 dimensions,

    Y. Choi, C. C´ ordova, P.-S. Hsin, H. T. Lam, and S.-H. Shao, “Non-invertible condensation, duality, and triality defects in 3+1 dimensions,” Communications in Mathematical Physics402 no. 1, (May, 2023) 489–542, arXiv:2204.09025. http://dx.doi.org/10.1007/s00220-023-04727-4

    2023 arXiv

  7. [15]

    Non-Invertible Symmetries as Condensation Defects in Finite-Group Gauge Theories,

    C. Cordova, D. B. Costa, and P.-S. Hsin, “Non-Invertible Symmetries as Condensation Defects in Finite-Group Gauge Theories,” arXiv:2412.16681 [cond-mat.str-el]

    arXiv

  8. [16]

    Mean string field theory: Landau-Ginzburg theory for 1-form symmetries,

    N. Iqbal and J. McGreevy, “Mean string field theory: Landau-Ginzburg theory for 1-form symmetries,” SciPost Phys. 13 (2022) 114, arXiv:2106.12610 [hep-th]

    2022 arXiv

  9. [17]

    T. T. Dumitrescu and P.-S. Hsin. To appear, 2025

    2025

  10. [18]

    Anomalies in the Space of Coupling Constants and Their Dynamical Applications I,

    C. C´ ordova, D. S. Freed, H. T. Lam, and N. Seiberg, “Anomalies in the Space of Coupling Constants and Their Dynamical Applications I,” SciPost Phys. 8 no. 1, (2020) 001, arXiv:1905.09315 [hep-th]

    2020 arXiv

  11. [19]

    Anomalies in the Space of Coupling Constants and Their Dynamical Applications II,

    C. C´ ordova, D. S. Freed, H. T. Lam, and N. Seiberg, “Anomalies in the Space of Coupling Constants and Their Dynamical Applications II,” SciPost Phys. 8 no. 1, (2020) 002, arXiv:1905.13361 [hep-th]

    2020 arXiv

  12. [20]

    Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena,

    P.-S. Hsin, A. Kapustin, and R. Thorngren, “Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena,” Phys. Rev. B102 (2020) 245113, arXiv:2004.10758 [cond-mat.str-el]

    2020 arXiv

  13. [21]

    Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d,

    P.-S. Hsin, H. T. Lam, and N. Seiberg, “Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d,” SciPost Phys. 6 no. 3, (2019) 039, arXiv:1812.04716 [hep-th]

    2019 arXiv

  14. [22]

    Lattice models that realize Zn-1 symmetry-protected topological states for even n,

    L. Tsui and X.-G. Wen, “Lattice models that realize Zn-1 symmetry-protected topological states for even n,” Phys. Rev. B101 no. 3, (2020) 035101, arXiv:1908.02613 [cond-mat.str-el]

    2020 arXiv

  15. [23]

    Generalized Hall Conductivities in Local Commuting Projector Models: Generalized Symmetries and Protected Surface Modes,

    P.-S. Hsin and R. Kobayashi, “Generalized Hall Conductivities in Local Commuting Projector Models: Generalized Symmetries and Protected Surface Modes,” arXiv:2505.20384 [cond-mat.str-el]

  16. [24]

    Gauge theories and magnetic charge,

    P. Goddard, J. Nuyts, and D. Olive, “Gauge theories and magnetic charge,” Nuclear Physics B 125 no. 1, (1977) 1–28. https://www.sciencedirect.com/science/article/pii/0550321377902218. 32

    1977

  17. [25]

    Discrete theta angles, symmetries and anomalies,

    P.-S. Hsin and H. T. Lam, “Discrete theta angles, symmetries and anomalies,” SciPost Phys. 10 no. 2, (2021) 032, arXiv:2007.05915 [hep-th]

    2021 arXiv

  18. [26]

    Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups,

    C. Cordova, P.-S. Hsin, and N. Seiberg, “Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups,” SciPost Phys. 4 no. 4, (2018) 021, arXiv:1711.10008 [hep-th]

    2018 arXiv

  19. [27]

    Line Operators of Gauge Theories on Non-Spin Manifolds,

    J. P. Ang, K. Roumpedakis, and S. Seifnashri, “Line Operators of Gauge Theories on Non-Spin Manifolds,” JHEP 04 (2020) 087, arXiv:1911.00589 [hep-th]

    2020 arXiv

  20. [28]

    Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d,

    R. Thorngren, “Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d,” JHEP 02 (2015) 152, arXiv:1404.4385 [hep-th]

    2015 arXiv

  21. [29]

    Gravitational anomaly of (3+1)-dimensional Z2 toric code with fermionic charges and fermionic loop self-statistics,

    L. Fidkowski, J. Haah, and M. B. Hastings, “Gravitational anomaly of (3+1)-dimensional Z2 toric code with fermionic charges and fermionic loop self-statistics,” Phys. Rev. B106 no. 16, (2022) 165135, arXiv:2110.14654 [cond-mat.str-el]

    2022 arXiv

  22. [30]

    Exactly solvable lattice Hamiltonians and gravitational anomalies,

    Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies,” SciPost Phys. 14 no. 5, (2023) 089, arXiv:2110.14644 [cond-mat.str-el]

    2023 arXiv

  23. [31]

    Generalized Statistics on Lattice,

    R. Kobayashi, Y. Li, H. Xue, P.-S. Hsin, and Y.-A. Chen, “Generalized Statistics on Lattice,” arXiv:2412.01886 [quant-ph]

  24. [32]

    Lorentz Symmetry Fractionalization and Dualities in (2+1)d,

    P.-S. Hsin and S.-H. Shao, “Lorentz Symmetry Fractionalization and Dualities in (2+1)d,” SciPost Phys. 8 (2020) 018, arXiv:1909.07383 [cond-mat.str-el]

    2020 arXiv

  25. [33]

    Radiative corrections as the origin of spontaneous symmetry breaking,

    S. Coleman and E. Weinberg, “Radiative corrections as the origin of spontaneous symmetry breaking,” Phys. Rev. D7 (Mar, 1973) 1888–1910. https://link.aps.org/doi/10.1103/PhysRevD.7.1888

    1973 doi

  26. [34]

    Deconfined quantum critical points,

    T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, “Deconfined quantum critical points,” Science 303 no. 5663, (Mar., 2004) 1490–1494. http://dx.doi.org/10.1126/science.1091806

    2004 doi

  27. [35]

    Exactly solvable model for a deconfined quantum critical point in 1d,

    C. Zhang and M. Levin, “Exactly solvable model for a deconfined quantum critical point in 1d,” Physical Review Letters130 no. 2, (Jan., 2023) . http://dx.doi.org/10.1103/PhysRevLett.130.026801

    2023 doi

  28. [36]

    Deconfined quantum critical points: a review,

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

    arXiv

  29. [37]

    Dyons of Charge e theta/2 pi,

    E. Witten, “Dyons of Charge e theta/2 pi,” Phys. Lett. B86 (1979) 283–287. 33

    1979

  30. [38]

    Reading between the lines of four-dimensional gauge theories,

    O. Aharony, N. Seiberg, and Y. Tachikawa, “Reading between the lines of four-dimensional gauge theories,” JHEP 08 (2013) 115, arXiv:1305.0318 [hep-th]

    2013 arXiv

  31. [39]

    Higher-group symmetry in finite gauge theory and stabilizer codes,

    M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry in finite gauge theory and stabilizer codes,” SciPost Phys. 16 no. 4, (2024) 089, arXiv:2211.11764 [cond-mat.str-el]

    2024 arXiv

  32. [40]

    Spectral asymmetry and Riemannian Geometry 1,

    M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and Riemannian Geometry 1,” Math. Proc. Cambridge Phil. Soc.77 (1975) 43

    1975

  33. [41]

    Fermion Path Integrals And Topological Phases,

    E. Witten, “Fermion Path Integrals And Topological Phases,” Rev. Mod. Phys.88 no. 3, (2016) 035001, arXiv:1508.04715 [cond-mat.mes-hall]

    2016 arXiv

  34. [42]

    Theta, Time Reversal, and Temperature,

    D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, “Theta, Time Reversal, and Temperature,” JHEP 05 (2017) 091, arXiv:1703.00501 [hep-th]

    2017 arXiv

  35. [43]

    Milnor, J

    J. Milnor, J. Stasheff, J. Stasheff, and P. University, Characteristic Classes. Annals of mathematics studies. Princeton University Press, 1974. https://books.google.com/books?id=5zQ9AFk1i4EC

    1974

  36. [44]

    Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,

    J. Kaidi, K. Ohmori, and Y. Zheng, “Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories,” Phys. Rev. Lett.128 no. 11, (2022) 111601, arXiv:2111.01141 [hep-th]

    2022 arXiv

  37. [45]

    Exotic invertible phases with higher-group symmetries,

    P.-S. Hsin, W. Ji, and C.-M. Jian, “Exotic invertible phases with higher-group symmetries,” SciPost Phys. 12 no. 2, (2022) 052, arXiv:2105.09454 [cond-mat.str-el]

    2022 arXiv

  38. [46]

    What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,

    S.-H. Shao, “What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries,” arXiv:2308.00747 [hep-th]

    arXiv

  39. [47]

    ICTP lectures on (non-)invertible generalized symmetries,

    S. Schafer-Nameki, “ICTP lectures on (non-)invertible generalized symmetries,” Phys. Rept. 1063 (2024) 1–55, arXiv:2305.18296 [hep-th]

    2024 arXiv

  40. [48]

    Higgsing Transitions from Topological Field Theory & Non-Invertible Symmetry in Chern-Simons Matter Theories,

    C. Cordova, D. Garc ´ ıa-Sep´ ulveda, and K. Ohmori, “Higgsing Transitions from Topological Field Theory & Non-Invertible Symmetry in Chern-Simons Matter Theories,” arXiv:2504.03614 [hep-th]

    arXiv

  41. [49]

    Non-Invertible Defects in Nonlinear Sigma Models and Coupling to Topological Orders,

    P.-S. Hsin, “Non-Invertible Defects in Nonlinear Sigma Models and Coupling to Topological Orders,” (12, 2022) , arXiv:2212.08608 [cond-mat.str-el]

    2022 arXiv

  42. [50]

    Solitonic Symmetry beyond Homotopy: Invertibility from Bordism and Noninvertibility from Topological Quantum Field Theory,

    S. Chen and Y. Tanizaki, “Solitonic Symmetry beyond Homotopy: Invertibility from Bordism and Noninvertibility from Topological Quantum Field Theory,” Phys. Rev. Lett. 131 no. 1, (2023) 011602, arXiv:2210.13780 [hep-th]. 34

    2023 arXiv

  43. [51]

    On topology of the moduli space of gapped Hamiltonians for topological phases,

    P.-S. Hsin and Z. Wang, “On topology of the moduli space of gapped Hamiltonians for topological phases,” J. Math. Phys.64 no. 4, (2023) 041901, arXiv:2211.16535 [cond-mat.str-el]

    2023 arXiv

  44. [52]

    Emergent generalized symmetries in ordered phases and applications to quantum disordering,

    S. D. Pace, “Emergent generalized symmetries in ordered phases and applications to quantum disordering,” SciPost Phys. 17 no. 3, (2024) 080, arXiv:2308.05730 [cond-mat.str-el]

    2024 arXiv

  45. [53]

    Generalized symmetries in singularity-free nonlinear σ models and their disordered phases,

    S. D. Pace, C. Zhu, A. Beaudry, and X.-G. Wen, “Generalized symmetries in singularity-free nonlinear σ models and their disordered phases,” Phys. Rev. B110 no. 19, (2024) 195149, arXiv:2310.08554 [cond-mat.str-el]

    2024 arXiv

  46. [54]

    Emergent Non-Invertible Symmetries Bridging UV and IR Phases – The Adjoint QCD Example,

    M. Del Zotto, S. N. Meynet, D. Migliorati, and K. Ohmori, “Emergent Non-Invertible Symmetries Bridging UV and IR Phases – The Adjoint QCD Example,” arXiv:2408.07123 [hep-th]. 35

    arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.