Pith. sign in

REVIEW 3 major objections 6 minor 3 cited by

Landau-Ginzburg Paradigm of Topological Phases

T0 review · 3 major / 6 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 Topological phase transitions admit a Landau-Ginzburg-Higgs description: in an enlarged string-net gauge theory, anyon condensation breaks the gauge category and produces order parameters, Goldstone modes, and gapped gauge fields.

desk verdict A useful gauge-theoretic reformulation of anyon condensation with real worked examples, but the universal Landau-Ginzburg-Higgs claim rests on an unproven reduction to fluxon condensation. read the letter →

arxiv 2506.05319 v2 pith:573VAIBB submitted 2025-06-05 cond-mat.str-el hep-thmath-phmath.MP

Yu Zhao , Yidun Wan This is my paper
classification cond-mat.str-elhep-thmath-phmath.MP
keywords topologicalphasesanyoncondensationHiggsmechanismstring-netmodellatticegaugetheoryLandau-GinzburgparadigmunitaryfusioncategoryGoldstonemodes
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 argues that topological phases are not beyond the Landau-Ginzburg symmetry-breaking paradigm after all. The authors enlarge the HGW string-net model, a variant of the Levin-Wen model, until it is a genuine lattice gauge theory: anyons become matter fields with explicit internal degrees of freedom, coupled to a gauge field whose variables are the simple objects of a unitary fusion category, a finite algebraic structure generalizing a gauge group. In this language, anyon condensation, the standard mechanism for topological phase transitions, is shown to be a Higgs mechanism: condensation projectors restrict the gauge degrees of freedom to a subcategory, some gauge fields become gapped, and order parameters and Goldstone modes appear exactly as in Landau's effective theory. If the claim holds, topological phase transitions and conventional spontaneous gauge symmetry breaking are the same phenomenon, with the electroweak Higgs boson and the Cooper pair as direct analogues.

What carries the argument

The load-bearing object is the enlarged HGW string-net model, the Hu-Geer-Wu extension of the Levin-Wen model further modified so that anyons carry explicit internal Hilbert spaces, one per flux type and charge sector, and are treated as matter fields rather than as punctures in lattice-field states. The argument is carried by the condensation projectors P_E of Eq. (3.2): sums of creation operators for the condensed anyons, chosen so that applying them forces every edge label into the simple objects of a preselected full subcategory S of the input unitary fusion category F. These projectors convert condensation into a gauge-symmetry-breaking step, and the half-braiding tensor components [z^J_{xy}] turn the hopping operators into covariant gauge connections, which is what earns the model the name 'genuine lattice gauge theory.'

What would settle it

Take a concrete input category and list every topological phase reachable by condensing bosonic dyons, then check whether each reachable child phase is the HGW model of a full subcategory S of the parent, equivalently whether the condensation-projector linear system has a solution for that S; one reachable child phase whose input category is not a full subcategory would refute the claim that condensation is exactly this Higgs-type restriction.

Watch

Extended reading notes

Core claim

The central claim is that the enlarged HGW string-net model is a genuine Hamiltonian lattice gauge theory rather than a model of structureless punctures. Gauge variables, the simple objects of a unitary fusion category F, reside on lattice edges and tails, while anyonic matter with internal flux and charge degrees of freedom (dyonic sectors) resides in plaquettes, and a magnetic Gauss law ties an anyon's flux to the gauge-field curvature at its tail. Hopping operators built from half-braiding tensors act as the gauge connection, and tube-algebra plaquette operators act as gauge transformations. Anyon condensation is implemented by adding commuting projector terms that restrict every gauge degree of freedom to a full subcategory S of F, and the paper claims that all resulting phenomena, namely splitting, confinement, condensation, and identification of anyon types, together with order parameters, Goldstone modes, and gapped gauge degrees of freedom that absorb the Goldstone modes, are exactly the Landau-Ginzburg-Higgs effective theory. The child phase is the enlarged HGW model with input S, and the broken gauge invariance survives as a global symmetry that enriches the child topological phase.

Load-bearing premise

Every anyon condensation that ends in a topological child phase can be produced by adding projector terms that merely restrict the allowed gauge degrees of freedom to a preselected full subcategory of the parent category; the paper states this as natural rather than deriving it, so if some topological transitions cannot be written this way, the framework covers only a special class of transitions.

Editorial extensions

If this is right

  • After condensation the child phase is the enlarged HGW model with input S, and the broken gauge invariance persists as a global symmetry of the child phase, making it a symmetry-enriched topological phase.
  • Each element of the electroweak Higgs mechanism has a counterpart: dyonic sectors split like isospin doublets, identified anyons form like the Dirac electron, gapped contractible loops play the massive gauge bosons, and the Goldstone modes labeling domains are absorbed by the gapped degrees of freedom.
  • Confinement of certain anyons in the child phase follows from the discrete moduli space of gauge degrees of freedom, which contrasts with the simply connected moduli space that prevents flux confinement in the electroweak transition.
  • Local excitations such as the ψ̄ψ particle after ψ̄ψ condensation carry fractional global symmetry charges, 1/2 for ψ̄ψ and 1/4 for the identified anyon ε under the e↔m exchange symmetry of the toric code, phenomena the paper calls fragmentation.
  • Explicit order-parameter operators are supplied whose expectation value jumps from 0 to 1 at a critical value of the condensation scale, making the predicted first-order transitions directly accessible to numerical simulation.

Reading between the lines

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

  • If the projector construction is as general as claimed, the framework is also an algorithm: for any input category F and any full subcategory S, the linear system (3.6) yields the condensation projectors, so the paper implicitly provides a recipe to enumerate the topological child phases of a given parent.
  • The paper leaves the F-gauge transformations incomplete; if they cannot be consistently incorporated, the 'genuine lattice gauge theory' status and therefore the strongest reading of the Landau-Ginzburg claim would be restricted to the T-gauge sector.
  • A testable extension suggested by the dictionary is to compute the analogue of the Weinberg angle, the ratio of the gapped gauge-field mass scale to the anyon mass in concrete models such as Vec(S3) or Ising, and to look for universal ratios across Morita-equivalent descriptions.
  • The electroweak analogy also predicts topological analogues of Higgs-boson production: local excitations that braid trivially with all anyons but transform nontrivially across gapped loops, which numerical studies of the finite-coupling Hamiltonian (4.1) could detect.
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 / 6 minor

Summary. The paper constructs an "enlarged HGW string-net model" in which the gauge field and anyonic matter are treated as separate, covariantly coupled sectors, and argues that anyon condensation in this model is a genuine lattice realization of the Landau-Ginzburg-Higgs mechanism. The main technical device is a set of commuting projector terms, Eq. (3.2), that restrict the gauge degrees of freedom to a fusion-closed subset L_S of the input UFC F, thereby producing a child HGW model with input UFC S. From this projector the paper derives the standard anyon-condensation phenomena — splitting, confinement, condensation, and identification — and introduces analogues of order parameters, Goldstone modes, and gapped gauge fields. Detailed data and worked examples are given for Vec(Z2), Vec(Z3), Vec(S3), Ising, and Fibonacci input categories.

Significance. If the central claim is correct, the paper offers a conceptual unification: topological phase transitions induced by anyon condensation become examples of spontaneous gauge-symmetry breaking, with explicit lattice operators playing the roles of order parameters, Goldstone modes, and massive gauge fields. The paper is valuable for its detailed microscopic construction of the enlarged Hilbert space, the T-gauge transformations, the dyonic S matrix, and the extensive appendices with explicit data for several concrete models. The worked fluxon-condensation examples (Ising to Z2 toric code, Vec(S3) to Vec(Z3), Fibonacci to trivial) are internally consistent and provide a useful testing ground. However, the universality of the claimed Landau-Ginzburg-Higgs description rests on the assertion, not derived in this manuscript, that every anyon condensation can be represented by such projectors after passing to an equivalent model; this limitation is load-bearing.

major comments (3)
  1. [Section 3.1, Eqs. (3.2)-(3.6)] The projector construction is explicitly solved only for fluxon condensation, i.e., for dyons with trivial flux type 1, where Eq. (3.4) makes the creation operators diagonal in the gauge-field basis. The statement that "arbitrary bosonic dyon condensations are equivalent to certain fluxon condensation in certain equivalent HGW model" is attributed to Ref. [8] but no derivation or independent verification is provided. Since the concluding claim of Section 4 that topological phase transitions admit a fully fledged Landau-Ginzburg-Higgs description depends on every relevant condensation being realizable in this way, this missing step is a load-bearing gap. The authors should either include the reduction argument or explicitly restrict the universality claim to fluxon condensations and their Morita-equivalent images.
  2. [Section 3.1, Eq. (3.6)] The claim that the number of creation operators W^{J;11}_E equals the number of simple objects of F, and hence that Eq. (3.6) has a unique solution for every subset L_S, is asserted without proof. The matrix [z^J_{jj}]^{1β}_{1α} indexed by j ∈ L_F and (J,α,β) must be square and invertible for the statement to hold; neither property is demonstrated in the manuscript. Relatedly, the normalization π^1_1 = 1/Σ_J n^J_Cond d_J stated below Eq. (3.9) is also asserted without derivation. These are not merely technicalities: the existence and uniqueness of the projector is what turns anyon condensation into a well-defined Hamiltonian perturbation. A proof or a precise reference to a theorem establishing these linear-algebraic facts is needed.
  3. [Section 3.1 and Appendix D.3] The mechanism as presented applies directly only when the child input UFC S is a full subcategory of the parent F. Standard anyon condensation can produce child categories that are bimodule categories over a Frobenius algebra, which are not generally full subcategories. The paper's bridge to the general case is the cited reduction to fluxon condensation in a Morita-equivalent model, but no concrete non-fluxon example is worked out. A natural test case is the pure-flux dyon D in the Vec(S3) model (Appendix D.3), whose flux set is {s,sr,rs} and whose topological spin is 1, making it a candidate condensate outside the diagonal projector construction. Until such a condensation is explicitly realized by Eq. (3.2) (or shown to reduce to one in an equivalent model), the claim that the framework covers all topological transitions remains unsupported.
minor comments (6)
  1. [Appendix D.2] The text says "Let Z2 = {e,r,r^2}" but the section is about Vec(Z3); this should be Z3.
  2. [Appendix D.3] The sentence "Anyon F, G, and H both have three flux type L_F = L_G = L_H = {r,r^2}" is internally inconsistent: the displayed set has two elements. It should read "two flux types".
  3. [Appendix D.6] In the Vec(S3) to Vec(Z3) table, the row for parent anyon B lists the parent dyon as (A,e); this should presumably be (B,e).
  4. [Various] There are several typographical errors: "Fobonacci" in the heading of Appendix D.5, "Golden ration" in Section 3.6.3, "Arharonov-Bohm" in Section 2.7, "tuebalgebra" in the discussion after Eq. (2.23), and "a very example" in Section 3.6.1.
  5. [Section 3.1, Eq. (3.5)] The notation δ_{j∈L_S}|ψ⟩ in Eq. (3.5) is ambiguous: the Kronecker delta should act on the edge label j_E carried by the state, not on the whole state. Writing δ_{j_E∈L_S}|ψ⟩ would be clearer.
  6. [Section 2.8] The assumption that δ_{J1J2K*} ≤ 1 for all fusion triples excludes fusion multiplicities greater than one. This is a restriction on the class of input categories, and it should be stated as a limitation in the main text rather than only in the middle of Section 2.8.

Circularity Check

2 steps flagged · score 4.0 of 10

The general reduction of arbitrary dyon condensations to fluxon condensations is imported from the authors' own Ref. [8], and the child category S is an input chosen before the projector is solved; the central LG-Higgs claim is therefore partially supported by self-citation and by construction rather than independent derivation.

  1. self citation load bearing [Section 3.1, 'Coherent States and Anyon Condensation', immediately before Eq. (3.4)]
    "In Ref. [8], we devised a more efficient algorithm for determining these projectors. We showed that arbitrary bosonic dyon condensations are equivalent to certain fluxon condensation in certain equivalent HGW model. So it is sufficient to study the condensations of dyon sectors with trivial flux type 1."

    The paper's central claim, stated in Section 4, is that topological phase transitions admit a fully fledged Landau-Ginzburg-Higgs description. The projector construction (Eqs. 3.4-3.6) is justified only for fluxon condensations, i.e. dyons with trivial flux. The assertion that every arbitrary bosonic dyon condensation reduces to a fluxon condensation in an equivalent HGW model is attributed to Ref. [8], a prior paper by the same authors, and is not derived or independently checked in the present text. All worked examples in the paper (Ising psi-psi-bar, Vec(S3) B/C, Fibonacci tau-tau-bar) are fluxon condensations, so they do not test the general reduction.

  2. self definitional [Section 3.1, Eqs. (3.5)-(3.6), with child model defined in Eq. (3.3)]
    "We first select a subset L_S ⊂ L_F of simple objects that is closed under fusion... The transition from the parent model to the child model involves gapping out those dofs not in L_S, such that the condensation projector P_E (Eq. (3.2)) becomes P^{S|F}_E |ψ⟩ = δ_{j∈L_S}|ψ⟩... the system of linear equations... ∑_J ∑_{α,β} π^{(1,α)(1,β)}_J [z^J_{jj}]^{1β}_{1α} = δ_{j∈L_S} always has a unique solution for any subset L_S."

    The child input UFC S is preselected before the projector is constructed; Eq. (3.6) is solved to enforce the condition δ_{j∈L_S}. Equation (3.3) then defines the child Hilbert space and Hamiltonian by applying the projector ∏_E P_E to the parent model. Consequently, the statement that the child phase is the enlarged HGW model with input S is true by construction: S is an input, not a predicted outcome. The subsequent splitting, confinement, and identification rules are derived consequences of that chosen S, so the circularity is partial and is analogous to choosing an unbroken subgroup in ordinary Landau theory; however, if the formalism is presented as predicting the child phase from condensation dynamics, the child phase has already been inserted by hand.

full rationale

The bulk of the paper is a self-contained and mathematically grounded reformulation: the enlarged HGW lattice model, the half-braiding tensors, the tube-algebra representations, and the dyon braiding/fusion data are built from standard tensor-category mathematics and are checked on explicit examples. The condensation projector solution (Eq. 3.6) is a genuine linear-algebra computation once the child subcategory S is chosen. The main circularity concern is twofold. First, the claim that arbitrary bosonic dyon condensations reduce to fluxon condensations is imported from the authors' prior Ref. [8] without proof in the present text; because this reduction is needed for the general LG-Higgs claim, the universality of the framework rests on a self-citation. Second, the child category S is an input chosen before solving the projector equations, so the resulting child phase is constructed rather than independently predicted; this is a definitional feature of the Landau-Ginzburg approach but weakens any reading of the child phase as a derived prediction. The paper also states, rather than derives, that the child input must be a full subcategory of the parent UFC, which limits the scope of the claimed paradigm. These issues justify a moderate circularity score; the worked examples and the underlying categorical computations are not circular and retain independent content.

Assumptions & free parameters 4 free parameters · 6 assumptions · 3 invented entities

The construction contributes the projector mechanism and the Higgs dictionary, but pulls the input UFC, the Drinfeld center/tube algebra mathematics, and the fluxon-condensation algorithm from prior literature (much of it by the same authors). The free parameters are mostly undetermined scales and hand-chosen condensation coefficients.

free parameters (4)
  • condensation coefficients π_J^(pα)(qβ) = example-dependent, e.g. 1/2, 1/3, 1/(1+φ^2)
    Chosen by hand to make PE a projector onto a selected child subcategory LS; they determine which anyons condense (Section 3.1, Eqs. (3.2)-(3.6)).
  • coupling constant g = not fixed
    Introduced in the hopping term Hhop (Eq. 2.39); not determined by any principle, and the paper notes it breaks exact solvability.
  • mass M = not fixed
    Energy scale of the mass term Hmass (Eq. 2.26); arbitrary.
  • condensation strength Λ = ∞ in the child limit, finite above critical point
    Parameter controlling the projector term in Eq. (3.1); the phase transition is asserted to be first-order at finite critical Λ (Eq. 3.12).
assumptions (6)
  • domain assumption Input data are a unitary fusion category F with multiplicity-free fusion rules (δabc ∈ {0,1}) and, in the main text, commutative fusion rules.
    Restricts the HGW models covered; the paper says the construction extends to noncommutative categories via appendices of [9], but the detailed claims are only made in the multiplicity-free commutative setting (Appendix A).
  • ad hoc to paper Anyon condensation that preserves topological order is realized by adding commuting projectors PE that project the parent gauge degrees of freedom onto a full subcategory S of the input UFC F, with the child model being the HGW model with input S.
    This is the mechanism that makes the Higgs analogy work; it is asserted rather than derived in Section 3.1 (Eqs. 3.2-3.5).
  • domain assumption During condensation, the parent gauge invariance F is spontaneously broken to the child gauge structure S; the broken part becomes a global symmetry of the child phase.
    This is the central physical interpretation; it borrows from gauge theory but its validity for discrete fusion categories is assumed (Sections 3.3, 3.4).
  • domain assumption At most one anyon per plaquette and well-separated anyons in physical states are assumed (topological superselection/separation rule).
    Used to define single-anyon Hilbert spaces and hopping operators (Section 2.3, Eq. 2.11).
  • standard math Peter-Weyl decomposition of the tube algebra and the Verlinde formula for the Drinfeld center Z(F) are taken as standard.
    Used to define projectors ΠJ in Eq. (2.25) and fusion coefficients in Eq. (2.34).
  • domain assumption For any triple (J1,J2,K), the fusion multiplicity δJ1J2K* ≤ 1.
    Assumed in Section 2.8 and used in defining Clebsch-Gordan coefficients; excludes multi-channel fusion.
invented entities (3)
  • F-gauge space (F-charge dofs)
    purpose: Enlargement of the anyon Hilbert space to represent F-gauge transformations built from Frobenius algebra bimodules.
    The paper states in Appendix C.3 that 'the precise structure of this enlargement is still unclear and requires further investigation', so it is a postulated entity without a completed definition or independent verification.
  • Goldstone modes as discrete domain labels
    purpose: Stand in for Goldstone bosons after condensation, labeling domains separated by gapped loops.
    The gauge structures are discrete, so there are no gapless modes; these labels are a redefinition introduced by the paper, and the authors concede the analogy requires modifications.
  • Gapped loops (gapped gauge-field configurations) independent evidence
    purpose: Contractible loops of gapped gauge dofs that separate domains and absorb the discrete Goldstone labels, analogous to massive gauge bosons.
    In the Ising example, the gapped σ-loops implement the known e↔m exchange symmetry of the toric code, which is independently established, so there is some external support.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Landau-Ginzburg Paradigm of Topological Phases." pith.science (2026). https://pith.science/paper/573VAIBB

@misc{pith2026250605319,
  author       = {Pith},
  title        = {Pith review of: Landau-Ginzburg Paradigm of Topological Phases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/573VAIBB}},
  note         = {Machine review of arXiv:2506.05319}
}
read the original abstract

Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.

Discussion (0). Sign in to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries

    hep-th 2026-05 unverdicted novelty 7.0 of 10

    Gapped phases dual to massless RG flows in 2D CFTs exhibit unusual ordering via spontaneous breaking of non-group-like symmetries and are characterized using smeared boundary CFTs applied to smeared Ishibashi states.

  2. A 2D-CFT Factory: Critical Lattice Models from Competing Anyon Condensation Processes in SymTO/SymTFT

    cond-mat.str-el 2025-06 conditional novelty 7.0 of 10

    Competing anyon condensates in string-net models generate critical lattice models, including new Haagerup-symmetric CFT candidates with central charges near 1.3, 1.8 and 2.5.

  3. Anyon Condensation In Symmetry-Enriched Topological Phases: $G$-Grading of Multifusion Categories

    cond-mat.str-el 2026-07 conditional novelty 6.0 of 10

    G-preserving anyon condensation in SET string-net models is equivalent to a compatible N-grading of the input multifusion category, which constructs the child SET input and works even with symmetry fractionalization.

Reference graph

Works this paper leans on

75 extracted references · 56 canonical work pages · cited by 3 Pith papers

  1. [8]

    Y. Zhao, H. Wang, Y. Hu, and Y. Wan, Journal of High Energy Physics2024, 1 (2024)

    work page 2024

  2. [1]

    V. G. Turaev and O. Y. Viro, Topology31, 865 (1992). – 67 –

    work page 1992

  3. [2]

    M. A. Levin and X.-G. Wen, Physical Review B—Condensed Matter and Materials Physics 71, 045110 (2005)

    work page 2005

  4. [3]

    Y. Hu, N. Geer, and Y.-S. Wu, Physical Review B97, 195154 (2018)

    work page 2018

  5. [4]

    Y. Zhao, S. Huang, H. Wang, Y. Hu, and Y. Wan, SciPost Physics Core6, 076 (2023)

    work page 2023

  6. [5]

    Zhao and Y

    Y. Zhao and Y. Wan, arXiv preprint arXiv:2409.05852 (2024)

    arXiv 2024

  7. [6]

    Hu and Y

    Y. Hu and Y. Wan, Journal of High Energy Physics2020, 170 (2020)

    work page 2020

  8. [7]

    H. Wang, Y. Li, Y. Hu, and Y. Wan, Journal of High Energy Physics2020, 30 (2020)

    work page 2020

Show all 75 references
  1. [9]

    Zhao and Y

    Y. Zhao and Y. Wan, arXiv preprint arXiv:2408.02664 (2024)

    2024

  2. [10]

    Z. Jia, S. Tan, and D. Kaszlikowski, Journal of High Energy Physics2024, 1 (2024)

    2024

  3. [11]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, Physical Review Letters133, 161601 (2024)

    2024

  4. [12]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, S. Schäfer-Nameki, and A. Tiwari, Physical Review B111, 054432 (2025)

    2025

  5. [13]

    Kogut and L

    J. Kogut and L. Susskind, Physical Review D11, 395 (1975)

    1975

  6. [14]

    Susskind, Physical Review D16, 3031 (1977)

    L. Susskind, Physical Review D16, 3031 (1977)

    1977

  7. [15]

    Hung and Y

    L.-Y. Hung and Y. Wan, International Journal of Modern Physics B28, 1450172 (2014)

    2014

  8. [16]

    Symmetry-enriched topological orders and their gauging: A string-net model realization,

    N. Fu, Y. Zhao, and Y. Wan, “Symmetry-enriched topological orders and their gauging: A string-net model realization,” (2025), companion paper, to be submitted to arXiv soon

    2025

  9. [17]

    Nonlinear symmetry-fragmentation of nonabelian anyons in symmetry-enriched topological phases: A string-net model realization,

    N. Fu, Y. Zhao, S. Wang, and Y. Wan, “Nonlinear symmetry-fragmentation of nonabelian anyons in symmetry-enriched topological phases: A string-net model realization,” (2025), companion paper, to be submitted to arXiv soon

    2025

  10. [18]

    Bhardwaj and Y

    L. Bhardwaj and Y. Tachikawa, Journal of High Energy Physics2018, 1 (2018)

    2018

  11. [19]

    D. S. Freed and C. Teleman, Geom. Topol.26, 1907 (2022), arXiv:1806.00008 [math.AT]

    2022 arXiv

  12. [21]

    Kong and H

    L. Kong and H. Zheng, JHEP02, 150 (2020), arXiv:1905.04924 [cond-mat.str-el]

    2020 arXiv

  13. [22]

    L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, Physical Review Research2, 043086 (2020)

    2020

  14. [23]

    L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng, Journal of High Energy Physics 2020, 1 (2020)

    2020

  15. [24]

    Gaiotto and J

    D. Gaiotto and J. Kulp, Journal of High Energy Physics2021, 1 (2021)

    2021

  16. [25]

    Apruzzi, F

    F. Apruzzi, F. Bonetti, I. n. García Etxebarria, S. S. Hosseini, and S. Schafer-Nameki, Commun. Math. Phys.402, 895 (2023), arXiv:2112.02092 [hep-th]

    2023 arXiv

  17. [26]

    D. S. Freed, G. W. Moore, and C. Teleman, Quantum Topology15, 779 (2024)

    2024

  18. [27]

    Chatterjee and X.-G

    A. Chatterjee and X.-G. Wen, Phys. Rev. B107, 155136 (2023), arXiv:2203.03596 [cond-mat.str-el]

    2023 arXiv

  19. [28]

    K. Ji, C. Shen, Y. W. Wan, and Y. Zhao, arXiv preprint arXiv:2506.05324 (2025). – 68 –

    2025 arXiv

  20. [29]

    Lan and X

    T. Lan and X. G. Wen, Physical Review B - Condensed Matter and Materials Physics90, 115119 (2014), arXiv:1311.1784

    2014 arXiv

  21. [30]

    Kitaev, Annals of Physics303, 2 (2003)

    A. Kitaev, Annals of Physics303, 2 (2003)

    2003

  22. [31]

    J. K. Pachos,Introduction to topological quantum computation(Cambridge University Press, 2012)

    2012

  23. [32]

    Kitaev, Annals of Physics321, 2 (2006)

    A. Kitaev, Annals of Physics321, 2 (2006)

    2006

  24. [33]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. DasSarma, Reviews of Modern Physics 80, 1083 (2008)

    2008

  25. [34]

    Wen, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons (2004)

    X.-G. Wen, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons (2004)

    2004

  26. [35]

    Müger, inProceedings of the London Mathematical Society, Vol

    M. Müger, inProceedings of the London Mathematical Society, Vol. 87 (Oxford University Press (OUP), 2003) pp. 291–308, arXiv:0201017 [math]

    2003

  27. [36]

    Müger, Journal of Pure and Applied Algebra180, 81 (2003), arXiv:0111204 [math]

    M. Müger, Journal of Pure and Applied Algebra180, 81 (2003), arXiv:0111204 [math]

    2003

  28. [37]

    Muger, Journal of Pure and Applied Algebra180, 159 (2003)

    M. Muger, Journal of Pure and Applied Algebra180, 159 (2003)

    2003

  29. [38]

    Hung and Y

    L.-Y. Hung and Y. Wan, Physical Review B86, 235132 (2012), arXiv:1207.6169

    2012 arXiv

  30. [39]

    M. H. Freedman, M. Larsen, and Z. Wang, Communications in Mathematical Physics227, 605 (2002)

    2002

  31. [40]

    N. E. Bonesteel, L. Hormozi, G. Zikos, and S. H. Simon, Physical review letters95, 140503 (2005)

    2005

  32. [41]

    Hormozi, G

    L. Hormozi, G. Zikos, N. E. Bonesteel, and S. H. Simon, Physical Review B75, 165310 (2007)

    2007

  33. [42]

    Y.-a. Fan, Y. Li, Y. Hu, Y. Li, X. Long, H. Liu, X. Yang, X. Nie, J. Li, T. Xin,et al., The Innovation4 (2023)

    2023

  34. [43]

    Wen, Int

    X. Wen, Int. J. Mod. Phys. B239 (1990)

    1990

  35. [44]

    Rowell, R

    E. Rowell, R. Stong, and Z. Wang, Communications in Mathematical Physics292, 343 (2009)

    2009

  36. [45]

    Verlinde, Nuclear Physics B300, 360 (1988)

    E. Verlinde, Nuclear Physics B300, 360 (1988)

    1988

  37. [46]

    F. D. M. Haldane, Physical review letters61, 2015 (1988)

    1988

  38. [47]

    Senthil, A

    T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. Fisher, Science303, 1490 (2004)

    2004

  39. [48]

    Ran and X.-G

    Y. Ran and X.-G. Wen, arXiv preprint cond-mat/0609620 (2006)

    2006 arXiv

  40. [49]

    Wen, Reviews of Modern Physics89, 041004 (2017)

    X.-G. Wen, Reviews of Modern Physics89, 041004 (2017)

    2017

  41. [50]

    F. A. Bais and J. K. Slingerland, Physical Review B79, 045316 (2009)

    2009

  42. [51]

    Kitaev and L

    A. Kitaev and L. Kong, Communications in Mathematical Physics313, 351 (2012)

    2012

  43. [52]

    I. S. Eliëns, J. C. Romers, and F. A. Bais, Physical Review B90, 195130 (2014), arXiv:1310.6001

    2014 arXiv

  44. [53]

    Kong, Nuclear Physics B886, 436 (2014), arXiv:1307.8244

    L. Kong, Nuclear Physics B886, 436 (2014), arXiv:1307.8244

    2014 arXiv

  45. [54]

    Gu, L.-Y

    Y. Gu, L.-Y. Hung, and Y. Wan, Phys. Rev. B90, 245125 (2014), arXiv:1402.3356

    2014 arXiv

  46. [55]

    Burnell, Annual Review of Condensed Matter Physics9, 307 (2018)

    F. Burnell, Annual Review of Condensed Matter Physics9, 307 (2018). – 69 –

    2018

  47. [56]

    Y. Hu, Z. Huang, L.-y. Hung, and Y. Wan, Journal of High Energy Physics2022, 26 (2022), arXiv:2109.06145v1

    2022 arXiv

  48. [57]

    M. D. Schulz, S. Dusuel, K. P. Schmidt, and J. Vidal, Physical review letters110, 147203 (2013)

    2013

  49. [58]

    M. D. Schulz and F. J. Burnell, Physical Review B94, 165110 (2016)

    2016

  50. [59]

    Z. D. Shi and A. Chatterjee, arXiv preprint arXiv:2407.07941 (2024)

    2024 arXiv

  51. [60]

    Lin and F

    C.-H. Lin and F. J. Burnell, arXiv preprint arXiv:2303.07291 (2023)

    2023 arXiv

  52. [61]

    Christian, D

    J. Christian, D. Green, P. Huston, and D. Penneys, Journal of High Energy Physics2023, 1 (2023)

    2023

  53. [62]

    Delcamp and A

    C. Delcamp and A. Tiwari, SciPost Physics16, 110 (2024)

    2024

  54. [63]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik,Tensor categories, Vol. 205 (American Mathematical Soc., 2016)

    2016

  55. [64]

    Kawagoe, C

    K. Kawagoe, C. Jones, S. Sanford, D. Green, and D. Penneys, Communications in Mathematical Physics405, 266 (2024)

    2024

  56. [65]

    Polishchuk and E

    A. Polishchuk and E. Zaslow, arXiv preprint math/9801119 (1998)

    1998 arXiv

  57. [66]

    Hung and Y

    L.-Y. Hung and Y. Wan, Physical Review B87, 195103 (2013)

    2013

  58. [67]

    Baratin and L

    A. Baratin and L. Freidel, Journal of Mathematical Physics56 (2015)

    2015

  59. [68]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Journal of High Energy Physics2015, 172 (2015), arXiv:1412.5148v2

    2015 arXiv

  60. [69]

    Ji and X.-G

    W. Ji and X.-G. Wen, Physical Review Research2, 033417 (2020)

    2020

  61. [70]

    Bartsch, M

    T. Bartsch, M. Bullimore, and A. Grigoletto, arXiv preprint arXiv:2305.17165 (2023)

    2023 arXiv

  62. [71]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, S. Schäfer-Nameki, and A. Tiwari, SciPost Physics14, 007 (2023)

    2023

  63. [72]

    Bhardwaj, S

    L. Bhardwaj, S. Schäfer-Nameki, and J. Wu, Fortschritte der Physik70, 2200143 (2022)

    2022

  64. [73]

    Y. Choi, H. T. Lam, and S.-H. Shao, Physical Review Letters129, 161601 (2022)

    2022

  65. [74]

    Y. Choi, H. T. Lam, and S.-H. Shao, Physical Review Letters130, 131602 (2023)

    2023

  66. [75]

    Y. Choi, M. Forslund, H. T. Lam, and S.-H. Shao, Physical Review Letters132, 121601 (2024)

    2024

  67. [76]

    Bartsch, M

    T. Bartsch, M. Bullimore, A. E. Ferrari, and J. Pearson, SciPost Physics17, 015 (2024). – 70 –

    2024

Pith tools

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