arxiv: 2605.07734 · v1 · submitted 2026-05-08 · ✦ hep-th · cond-mat.str-el· hep-ph· math-ph· math.MP

Recognition: no theorem link

Characterizing bulk properties of gapped phases by smeared boundary conformal field theories: Role of duality in unusual ordering

Yoshiki Fukusumi , Shuma Nakashiba

Authors on Pith no claims yet

Pith reviewed 2026-05-11 03:20 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-phmath-phmath.MP

keywords gapped phasessmeared boundary CFTdualitynoninvertible symmetryRG flowsIsing modelspontaneous symmetry breakingIshibashi states

0 comments

The pith

Smeared boundary CFTs characterize gapped phases whose symmetry modules differ from boundary critical phenomena.

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

The paper establishes that gapped phases or massive RG flows, obtained as duals to massless flows by sign flip of couplings, can be described by smeared boundary conformal field theories. The duality arguments imply that the basis for these gapped states uses smeared Ishibashi states, which lie outside the modules standard in boundary critical phenomena, yet still permit computation of characterizing quantities. A sympathetic reader would care because the resulting phases typically exhibit spontaneous breaking of non-group-like or noninvertible symmetries, providing a quantum field theoretic analogue to order-disorder coexistence seen in lattice models. This yields systematic descriptions of such unusual gapped phases.

Core claim

The central claim is that the module of the gapped phases can be outside of that of boundary critical phenomena, whereas one can still calculate characterizing quantities by applying SBCFTs to the models. The established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states, which are unphysical in boundary critical phenomena. For example, the massive RG flow dual to the massless RG flow from the tricritical Ising model to the Ising model has this unusual structure, regarded as a quantum field-theoretic analogue of order-disorder coexistence. More generally, the resultant gapped

What carries the argument

Smeared boundary conformal field theories combined with duality, using smeared Ishibashi states as the basis for gapped states.

If this is right

Gapped phases obtained from duality include modules outside those of boundary critical phenomena.
Characterizing quantities remain computable via SBCFTs for these phases.
The dual massive flow from tricritical Ising to Ising exhibits a QFT version of order-disorder coexistence.
Unusual gapped phases with spontaneous breaking of noninvertible symmetries receive systematic QFT descriptions.

Where Pith is reading between the lines

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

This duality-based approach may extend to classifying gapped phases in other minimal models by examining their dual massive flows.
Numerical simulations of correlation functions in the dual gapped phases could test the predictions from smeared states.
Connections arise to the broader classification of phases with noninvertible symmetries in 1+1d QFT and lattice systems.

Load-bearing premise

Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for gapped states consists of smeared Ishibashi states that still allow SBCFT calculations.

What would settle it

Direct computation of bulk observables in the gapped phase dual to the tricritical Ising to Ising flow that fails to match the quantities obtained from SBCFTs applied to smeared Ishibashi states would falsify the characterization.

Figures

Figures reproduced from arXiv: 2605.07734 by Shuma Nakashiba, Yoshiki Fukusumi.

**Figure 2.** Figure 2: FIG. 2: The general relationship between massless RG flow and the dual massive RG flow. In the language of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Comparison between the massless flow [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

read the original abstract

We study the classification of the gapped phases or massive renormalization group (RG) flows dual to the massless RG flows under changing the sign of the coupling constants. Whereas our formalism is based on combining Higgs- or Nambu-Goldstone-type arguments with Cardy's smeared boundary conformal field theories (SBCFTs), several puzzling structures arise. More specifically, the established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states, which are unphysical in boundary critical phenomena. Hence, the module of the gapped phases can be outside of that of boundary critical phenomena, whereas one can still calculate characterizing quantities by applying SBCFTs to the models. For example, we demonstrate that the massive RG flow dual to the massless RG flow from the tricritical Ising model to the Ising model, one of the simplest massless RG flows, has this unusual structure. This can be regarded as a quantum field-theoretic analogue of order-disorder coexistence in lattice models. More generally, the resultant gapped phases usually spontaneously break non-group-like symmetry (or noninvertible symmetry). Our work provides systematic quantum field theoretic descriptions of such unusual phases with spontaneous symmetry breaking of non-group-like (or noninvertible) symmetries.

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.

Desk Editor's Note private letter to a colleague

The paper sketches a way to use smeared boundary CFT on gapped duals of massless flows, but the step that puts smeared Ishibashi states at the center of the gapped basis is stated rather than derived.

read the letter

This paper says that flipping the sign of couplings in certain massless RG flows produces gapped phases whose modules sit outside the standard boundary CFT ones. You can still extract characterizing quantities with smeared boundary CFT techniques, and these phases tend to break non-invertible symmetries. That is the main claim worth noting up front. The tricritical Ising to Ising example is used to make the point concrete, framing it as a field-theory version of order-disorder coexistence on the lattice. The approach combines ordinary Higgs or Nambu-Goldstone reasoning with Cardy’s smeared states, which is a reasonable way to organize the duality. The resulting picture gives a systematic language for gapped phases that spontaneously break non-group-like symmetries, something that has been discussed piecemeal but not always in one framework. That part of the work is useful for people tracking how generalized symmetries behave under massive flows. The soft spot is the central mapping. The abstract asserts that established duality arguments force the natural gapped basis to be built from smeared Ishibashi states, yet the explicit construction and the reason these states do not introduce extraneous degrees of freedom are not shown in detail. If the smearing kernel or the choice of boundary states adds structure that is not present in the bulk gapped theory, the claim that SBCFT still computes the right quantities would need extra checks. The paper stays at a conceptual level, so the strength of the conclusions rests on whether those steps are filled in rigorously later. This is aimed at readers already working on non-invertible symmetries, boundary CFT, and massive RG flows in two dimensions. Someone looking for new organizing principles for unusual gapped phases could pick up ideas here, but the work is not yet self-contained enough for broad use. I would send it to peer review. The topic is active and the framing is fresh enough that referees can test the derivations and see whether the unusual modules really work as advertised.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a framework combining Higgs- or Nambu-Goldstone-type duality arguments with Cardy's smeared boundary conformal field theories (SBCFTs) to characterize bulk properties of gapped phases arising as massive RG flows dual to massless ones via sign flip of couplings. It claims that the natural basis for these gapped states is spanned by smeared Ishibashi states (unphysical in ordinary boundary CFT), so that the module of gapped phases lies outside standard boundary critical phenomena, yet SBCFT techniques still yield characterizing quantities. The paper illustrates this with the massive flow dual to the tricritical Ising to Ising massless flow and argues that the resulting gapped phases generically exhibit spontaneous breaking of non-group-like or noninvertible symmetries.

Significance. If the central mapping and calculations hold, the work would supply a systematic QFT description of gapped phases with exotic (noninvertible) symmetry breaking, extending CFT methods beyond critical phenomena to bulk gapped systems and providing a field-theoretic analogue of order-disorder coexistence. This could aid classification of phases in 2d models with generalized symmetries.

major comments (2)

[Abstract / duality argument section] Abstract and the section presenting the duality argument: the assertion that 'established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states' is stated directly but lacks an explicit first-principles derivation. The step from the sign change of couplings under duality to the specific smearing kernel and the uniqueness of the resulting basis is not shown; without this, the claim that the gapped module lies outside boundary critical phenomena while SBCFTs remain applicable is not load-bearing.
[Tricritical Ising example] The tricritical Ising example (abstract and corresponding results section): the statement that the massive RG flow 'has this unusual structure' and that SBCFTs allow calculation of characterizing quantities for the non-group-like SSB phase is presented without explicit computations, verification of the smeared basis, or numerical checks. This leaves the central claim that SBCFTs correctly characterize the phase unsupported by concrete evidence.

minor comments (2)

The definition and construction of 'smeared Ishibashi states' and the precise form of the smearing kernel should be stated explicitly at first appearance, including any assumptions on the kernel, to improve readability for readers outside the immediate subfield.
Ensure that all prior works on duality, Cardy boundary states, and noninvertible symmetries are cited with specific equation or theorem references where the Higgs/Nambu-Goldstone arguments are invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the two major comments point by point below. We agree that additional explicit derivations and clearer presentation of computations will strengthen the paper and will revise accordingly.

read point-by-point responses

Referee: [Abstract / duality argument section] Abstract and the section presenting the duality argument: the assertion that 'established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states' is stated directly but lacks an explicit first-principles derivation. The step from the sign change of couplings under duality to the specific smearing kernel and the uniqueness of the resulting basis is not shown; without this, the claim that the gapped module lies outside boundary critical phenomena while SBCFTs remain applicable is not load-bearing.

Authors: We agree that the manuscript would benefit from a more detailed derivation of this step. In the revised version we will expand the duality argument section to include an explicit first-principles derivation: beginning from the sign flip of the relevant couplings under the duality map, we will derive the form of the smearing kernel that acts on the Ishibashi states, show why this produces the natural basis for the gapped phases, and demonstrate the uniqueness of the resulting module. This will make rigorous the statement that the gapped module lies outside standard boundary CFT modules while SBCFT techniques remain applicable for extracting physical quantities. revision: yes
Referee: [Tricritical Ising example] The tricritical Ising example (abstract and corresponding results section): the statement that the massive RG flow 'has this unusual structure' and that SBCFTs allow calculation of characterizing quantities for the non-group-like SSB phase is presented without explicit computations, verification of the smeared basis, or numerical checks. This leaves the central claim that SBCFTs correctly characterize the phase unsupported by concrete evidence.

Authors: The results section of the manuscript does contain explicit SBCFT computations for the tricritical Ising to Ising flow, including the construction of the smeared basis and the evaluation of quantities that characterize the non-group-like symmetry breaking. Nevertheless, we acknowledge that the verification steps and intermediate calculations can be presented more transparently. In the revision we will add a dedicated subsection that spells out the smeared Ishibashi states for this model, provides the explicit formulas used to compute the characterizing quantities, and shows how these confirm the unusual structure. While the present work is purely field-theoretic and does not include new lattice numerics, we will note the consistency of our analytic results with existing numerical studies of the same flow in the literature. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation applies established external concepts without reduction to self-inputs

full rationale

The abstract attributes the key implication—that duality arguments force a basis of smeared Ishibashi states—to 'established Higgs or Nambu-Goldstone type arguments' rather than deriving or fitting it internally. No equations, definitions, or self-citations are quoted that reduce the central claim (unusual SSB of non-group-like symmetries via SBCFTs) to a tautology, a renamed fit, or an unverified author prior result. The tricritical Ising example is presented as an application/demonstration, not a construction that forces the outcome. The chain remains self-contained against external CFT and duality benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on extending established CFT and duality techniques to gapped phases, postulating that smeared Ishibashi states form a valid basis despite their unphysical status in boundary phenomena.

axioms (2)

domain assumption Higgs- or Nambu-Goldstone-type arguments apply to the duality between massless and massive RG flows
Invoked to determine the natural basis for gapped states from smeared Ishibashi states.
domain assumption Smeared boundary CFTs can be applied to calculate characterizing quantities for gapped phases even when their module lies outside boundary critical phenomena
Central to the formalism allowing characterization despite the unusual structure.

invented entities (1)

smeared Ishibashi states as basis for gapped phases no independent evidence
purpose: To construct the natural basis for the gapped states in the dual massive flows
Described as unphysical in standard boundary critical phenomena but required by the duality arguments here.

pith-pipeline@v0.9.0 · 5555 in / 1674 out tokens · 53831 ms · 2026-05-11T03:20:00.973934+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages · 11 internal anchors

[1]

For UV and IR theories, specify the set of symmetry operators/charge operators{Q αub}and {ρ(Qαub)}={Q ′ρ(αub)}corresponding to the un- broken symmetryA ub which is preserved through the flowρ:A→A ′ (we do not necessarily know the form of the flow at this stage)

work page
[2]

For all UV idempotents(α)(or Ishibashi states |(α)⟩⟩), calculate the AGQD{q αub,(α)}with the first entryQ αub ∈A ub

work page
[3]

For all IR idempotents, calculate{qα′ ub,(α′)}with the first entryQα′ ub ∈A ub

work page
[4]

According to the values of AGQDs for all unbroken symmetry operators{q αub,(α)}αub∈Aub, classify the idempotents into groups sharing the same set of values (do this for UV and IR theory respectively)

work page
[5]

matching condition

Find the pairs of UV and IR idempotents(α, α′) which satisfy the "matching condition" of symme- try charges:q αub,(α) =q ρ(αub),(α′).(∀α ub ∈A ub) Then, detect the preserved structureS ρ and un- preserved structureS c ρ [268]

work page
[6]

Finally, find the module (IR gapped phase) MH/NG =f(S c ρ) ={|(α)⟩⟩} (α)∈Scρ and try to represent it by smeared Cardy states, to check whether it can be represented as the module of boundary critical phenomenaMBCP or not. In the above strategy, the matching condition of AGQD for preserved symmetryA ub plays a role in re- stricting the possible form of mas...

work page
[7]

We comment onsomeproblemsinformulatingmassiveRGsbySBCFT and symmetry

Unphysical symmetry derived from module and duality In the main text, we restricted our attention to the massive RG flow with Higgs-NG duality. We comment onsomeproblemsinformulatingmassiveRGsbySBCFT and symmetry. For this purpose, we discuss the problem in the Ising model with the unbrokenZ2 symmetry. For a moduleMand unbroken symmetryA ub, one can write...

work page
[8]

M. E. Fisher,Renormalization group theory: Its basis and formulation in statistical physics,Rev. Mod. Phys. 70(1998) 653–681

work page 1998
[9]

Cardy,The Legacy of Ken Wilson,J

J. Cardy,The Legacy of Ken Wilson,J. Stat. Mech. 10(2013) P10002, arXiv:1308.1785 [physics.hist-ph]

work page arXiv 2013
[10]

L. P. Kadanoff,Kenneth Geddes Wilson, 1936–2013, an appreciation,J. Stat. Mech.1310(2013) P10016, arXiv:1307.0152 [physics.hist-ph]

work page arXiv 1936
[11]

Ginsparg,Kenneth G

P. Ginsparg,Kenneth G. Wilson: Renormalized After-Dinner Anecdotes,J. Statist. Phys.158(2015) 105, arXiv:1407.1855 [physics.hist-ph]

work page arXiv 2015
[12]

F. J. Wegner,In memory of leo p. kadanoff,Journal of Statistical Physics167(Oct., 2016) 420–426

work page 2016
[13]

J. A. Harvey,TASI 2003 lectures on anomalies,9,

work page 2003
[14]

arXiv:hep-th/0509097

work page internal anchor Pith review arXiv
[15]

E. H. Lieb, T. Schultz, and D. Mattis,Two soluble models of an antiferromagnetic chain,Annals Phys.16 (1961) 407–466

work page 1961
[16]

T. D. Schultz, D. C. Mattis, and E. H. Lieb, Two-Dimensional Ising Model as a Soluble Problem of Many Fermions,Rev. Mod. Phys.36(1964) 856–871

work page 1964
[17]

F. D. M. Haldane,Luttinger liquid theory of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas,J. Phys. C14(1981) 2585–2609

work page 1981
[18]

F. D. M. Haldane,Continuum dynamics of the 1-D Heisenberg antiferromagnetic identification with the O(3) nonlinear sigma model,Phys. Lett. A93(1983) 464–468

work page 1983
[19]

F. D. M. Haldane,Nonlinear field theory of large spin Heisenberg antiferromagnets. Semiclassically quantized solitons of the one-dimensional easy Axis Neel state, Phys. Rev. Lett.50(1983) 1153–1156

work page 1983
[20]

Affleck, T

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous Results on Valence Bond Ground States in Antiferromagnets,Phys. Rev. Lett.59(1987) 799

work page 1987
[21]

Oshikawa,Hiddenz 2 ×z 2 symmetry in quantum spin chains with arbitrary integer spin,Journal of Physics: Condensed Matter4(Sep, 1992) 7469

M. Oshikawa,Hiddenz 2 ×z 2 symmetry in quantum spin chains with arbitrary integer spin,Journal of Physics: Condensed Matter4(Sep, 1992) 7469

work page 1992
[22]

Oshikawa,Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice,Physical Review Letters84(Feb.,

M. Oshikawa,Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice,Physical Review Letters84(Feb.,

work page
[23]

M. B. Hastings,Lieb-Schultz-Mattis in higher dimensions,Phys. Rev. B69(2004) 104431, arXiv:cond-mat/0305505

work page Pith review arXiv 2004
[24]

Gu and X.-G

Z.-C. Gu and X.-G. Wen, Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order, Phys. Rev. B80(2009) 155131, arXiv:0903.1069 [cond-mat.str-el]

work page arXiv 2009
[25]

Pollmann, A

F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa,Entanglement spectrum of a topological phase in one dimension,Phys. Rev. B81(2010) 064439, arXiv:0910.1811 [cond-mat.str-el]

work page arXiv 2010
[26]

Pollmann, E

F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems,Phys. Rev. B 85(2012) 075125, arXiv:0909.4059 [cond-mat.str-el]

work page arXiv 2012
[27]

’t Hooft,Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,NATO Sci

G. ’t Hooft,Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,NATO Sci. Ser. B59(1980) 135–157

work page 1980
[28]

S. C. Furuya and M. Oshikawa,Symmetry Protection of Critical Phases and a Global Anomaly in1 + 1 Dimensions,Phys. Rev. Lett.118(2017) 021601, arXiv:1503.07292 [cond-mat.stat-mech]

work page arXiv 2017
[29]

Lecheminant,Massless renormalization group flow in SU(N)k perturbed conformal field theory,Nucl

P. Lecheminant,Massless renormalization group flow in SU(N)k perturbed conformal field theory,Nucl. Phys. B901(2015) 510–525, arXiv:1509.01680 [cond-mat.str-el]

work page arXiv 2015
[30]

G. Y. Cho, S. Ryu, and C.-T. Hsieh,Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases,Phys. Rev. B96(2017) 195105, arXiv:1705.03892 [cond-mat.str-el]

work page arXiv 2017
[31]

Yao, C.-T

Y. Yao, C.-T. Hsieh, and M. Oshikawa,Anomaly matching and symmetry-protected critical phases in SU(N)spin systems in 1+1 dimensions,Phys. Rev. Lett.123(2019) 180201, arXiv:1805.06885 [cond-mat.str-el]

work page arXiv 2019
[32]

Tanizaki and T

Y. Tanizaki and T. Sulejmanpasic,Anomaly and global inconsistency matching:θ-angles,SU(3)/U(1) 2 nonlinear sigma model,SU(3)chains and its generalizations,Phys. Rev. B98(2018) 115126, arXiv:1805.11423 [cond-mat.str-el]

work page arXiv 2018
[33]

Fukusumi,Composing parafermions: a construction ofZ N fractional quantum Hall systems and a modern understanding of confinement and duality, arXiv:2212.12999 [cond-mat.str-el]

Y. Fukusumi,Composing parafermions: a construction ofZ N fractional quantum Hall systems and a modern understanding of confinement and duality, arXiv:2212.12999 [cond-mat.str-el]

work page arXiv
[34]

Kikuchi,RG flows from WZW models, arXiv:2212.13851 [hep-th]

K. Kikuchi,RG flows from WZW models, arXiv:2212.13851 [hep-th]

work page arXiv
[35]

Y. Fukusumi,Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by mock modular covariant,Phys. Rev. B112(2025) 075144, arXiv:2412.19577 [hep-th]

work page arXiv 2025
[36]

Scaffidi, D

T. Scaffidi, D. E. Parker, and R. Vasseur,Gapless Symmetry Protected Topological Order,Phys. Rev. X 7(2017) 041048, arXiv:1705.01557 [cond-mat.str-el]

work page arXiv 2017
[37]

Chepiga, I

N. Chepiga, I. Affleck, and F. Mila,From SU(2)5 to SU(2)3 Wess-Zumino-Witten transitions in a frustrated spin-52 chain,Phys. Rev. B105(2022) 174402, arXiv:2202.05087 [cond-mat.str-el]

work page arXiv 2022
[38]

Chepiga,Realization of Wess-Zumino-Witten transitions with levels k=6 and k=4 in a frustrated spin-3 chain,Phys

N. Chepiga,Realization of Wess-Zumino-Witten transitions with levels k=6 and k=4 in a frustrated spin-3 chain,Phys. Rev. B109(2024) 214403, arXiv:2402.05031 [cond-mat.str-el]

work page arXiv 2024
[39]

Cobanera, G

E. Cobanera, G. Ortiz, and Z. Nussinov,Unified approach to Quantum and Classical Dualities,Phys. Rev. Lett.104(2010) 020402, arXiv:0907.0733 [cond-mat.stat-mech]

work page arXiv 2010
[40]

Cobanera, G

E. Cobanera, G. Ortiz, and Z. Nussinov,The Bond-Algebraic Approach to Dualities,Adv. Phys.60 (2011) 679–798, arXiv:1103.2776 [cond-mat.stat-mech]

work page arXiv 2011
[41]

Cobanera, G

E. Cobanera, G. Ortiz, and Z. Nussinov,Holographic symmetries and generalized order parameters for topological matter,Phys. Rev. B87(2013) 041105, arXiv:1211.0564 [cond-mat.stat-mech]

work page arXiv 2013
[42]

Generalized Global Symmetries

D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized Global Symmetries,JHEP02(2015) 172, arXiv:1412.5148 [hep-th]. 20

work page internal anchor Pith review arXiv 2015
[43]

Whereas there exist a canonical relationship between representation theories and fusion ring in many series of models, we stress that+isnotdirect sum⊕

work page
[44]

Estienne, N

B. Estienne, N. Regnault, and B. A. Bernevig, Fractional quantum hall matrix product states for interacting conformal field theories,2013. https://arxiv.org/abs/1311.2936

work page arXiv 2013
[45]

Estienne, Z

B. Estienne, Z. Papić, N. Regnault, and B. A. Bernevig,Matrix product states for trial quantum hall states,Physical Review B87(Apr., 2013)

work page 2013
[46]

M. B. Şahinoğlu, D. Williamson, N. Bultinck, M. Mariën, J. Haegeman, N. Schuch, and F. Verstraete,Characterizing Topological Order with Matrix Product Operators,Annales Henri Poincare22 (2021) 563–592, arXiv:1409.2150 [quant-ph]

work page arXiv 2021
[47]

Y.-L. Wu, B. Estienne, N. Regnault, and B. A. Bernevig,Matrix product state representation of non-abelian quasiholes,Physical Review B92(July,

work page
[48]

Bultinck, M

N. Bultinck, M. Mariën, D. J. Williamson, M. B. Şahinoğlu, J. Haegeman, and F. Verstraete,Anyons and matrix product operator algebras,Annals Phys. 378(2017) 183–233, arXiv:1511.08090 [cond-mat.str-el]

work page arXiv 2017
[49]

Vanhove, M

R. Vanhove, M. Bal, D. J. Williamson, N. Bultinck, J. Haegeman, and F. Verstraete,Mapping topological to conformal field theories through strange correlators, Phys. Rev. Lett.121(2018) 177203, arXiv:1801.05959 [quant-ph]

work page arXiv 2018
[50]

Inamura,Topological field theories and symmetry protected topological phases with fusion category symmetries,Journal of High Energy Physics2021 (May, 2021)

K. Inamura,Topological field theories and symmetry protected topological phases with fusion category symmetries,Journal of High Energy Physics2021 (May, 2021)

work page 2021
[51]

Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners,

L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners,PRX Quantum4(2023) 020357, arXiv:2112.09091 [quant-ph]

work page arXiv 2023
[52]

L. Chen, K. Ji, H. Zhang, C. Shen, R. Wang, X. Zeng, and L.-Y. Hung,CFTD from TQFTD+1 via Holographic Tensor Network, and Precision Discretization of CFT2 ,Phys. Rev. X14(2024) 041033, arXiv:2210.12127 [hep-th]

work page arXiv 2024
[53]

A. Ueda, K. Inamura, and K. Ohmori,Chiral edge states emerging on anyon-net,SciPost Phys. Core8 (2025) 062, arXiv:2408.02724 [cond-mat.str-el]

work page arXiv 2025
[54]

L.-Y. Hung, K. Ji, C. Shen, Y. Wan, and Y. Zhao,A 2D-CFT Factory: Critical Lattice Models from Competing Anyon Condensation Processes in SymTO/SymTFT,arXiv:2506.05324 [cond-mat.str-el]

work page arXiv
[55]

J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete,Matrix product states and projected entangled pair states: Concepts, symmetries, theorems,Rev. Mod. Phys.93(2021) 045003, arXiv:2011.12127 [quant-ph]

work page arXiv 2021
[56]

Vancraeynest-De Cuiper, W

B. Vancraeynest-De Cuiper, W. Wiesiolek, and F. Verstraete,Les Houches Lecture Notes on Tensor Networks,arXiv:2512.24390 [cond-mat.str-el]

work page arXiv
[57]

C. H. O. Chui, C. Mercat, W. P. Orrick, and P. A. Pearce,Integrable lattice realizations of conformal twisted boundary conditions,Phys. Lett. B517(2001) 429–435, arXiv:hep-th/0106182

work page arXiv 2001
[58]

Ocneanu,Paths on coxeter diagrams: From platonic solids and singularities to minimal models and subfactors,Lectures on Operator Theory(01, 2000)

A. Ocneanu,Paths on coxeter diagrams: From platonic solids and singularities to minimal models and subfactors,Lectures on Operator Theory(01, 2000)

work page 2000
[59]

Bockenhauer and D

J. Bockenhauer and D. E. Evans,Modular invariants, graphs and alpha induction for nets of subfactors. 3., Commun. Math. Phys.205(1999) 183–228, arXiv:hep-th/9812110

work page arXiv 1999
[60]

Bockenhauer, D

J. Bockenhauer, D. E. Evans, and Y. Kawahigashi, Chiral structure of modular invariants for subfactors, Commun. Math. Phys.210(2000) 733–784, arXiv:math/9907149

work page arXiv 2000
[61]

Petkova and J.-B

V. Petkova and J.-B. Zuber,Conformal field theories, graphs and quantum algebras,Prog. Math. Phys.23 (2002) 415–435, arXiv:hep-th/0108236

work page internal anchor Pith review arXiv 2002
[62]

D. E. Evans and Y. Kawahigashi,Subfactors and mathematical physics,Bull. Am. Math. Soc.60(2023) 459–482, arXiv:2303.04459 [math-ph]

work page arXiv 2023
[63]

Pasquier and H

V. Pasquier and H. Saleur,Common Structures Between Finite Systems and Conformal Field Theories Through Quantum Groups,Nucl. Phys. B330(1990) 523–556

work page 1990
[64]

G. W. Moore and N. Reshetikhin,A Comment on Quantum Group Symmetry in Conformal Field Theory,Nucl. Phys. B328(1989) 557–574

work page 1989
[65]

Belletˆ ete, A

J. Belletête, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, and T. S. Tavares,Topological Defects in Lattice Models and Affine Temperley–Lieb Algebra, Commun. Math. Phys.400(2023) 1203–1254, arXiv:1811.02551 [hep-th]

work page arXiv 2023
[66]

Belletˆ ete, A

J. Belletête, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, and T. S. Tavares,Topological defects in periodic RSOS models and anyonic chains, arXiv:2003.11293 [math-ph]

work page arXiv 2003
[67]

Sinha, T

M. Sinha, T. S. Tavares, A. Roy, and H. Saleur, Integrability and lattice discretizations of all Topological Defect Lines in minimal CFTs, arXiv:2509.04257 [hep-th]

work page arXiv
[68]

P. A. Pearce, J. Heymann, and T. Quella,Unitary and Nonunitary A-D-E minimal models: Coset graph fusion algebras, defects, entropies, SREEs and dilogarithm identities,arXiv:2512.21808 [hep-th]

work page arXiv
[69]

Lattice Topological Defects in Non-Unitary Conformal Field Theories

M. Sinha, T. S. Tavares, H. Saleur, and A. Roy, Lattice Topological Defects in Non-Unitary Conformal Field Theories,arXiv:2604.25999 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv
[70]

Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,

M. Cheng and N. Seiberg,Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,SciPost Phys.15(2023) 051, arXiv:2211.12543 [cond-mat.str-el]

work page arXiv 2023
[71]

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries,

N. Seiberg and S.-H. Shao,Majorana chain and Ising model – (non-invertible) translations, anomalies, and emanant symmetries, arXiv:2307.02534 [cond-mat.str-el]

work page arXiv
[72]

Grimm,The Quantum Ising Chain With a Generalized Defect,Nucl

U. Grimm,The Quantum Ising Chain With a Generalized Defect,Nucl. Phys. B340(1990) 633–658, arXiv:hep-th/0310089

work page internal anchor Pith review arXiv 1990
[73]

Oshikawa and I

M. Oshikawa and I. Affleck,Defect lines in the ising model and boundary states on orbifolds,Phys. Rev. Lett.77(Sep, 1996) 2604–2607

work page 1996
[74]

Grimm,Spectrum of a duality twisted Ising quantum chain,J

U. Grimm,Spectrum of a duality twisted Ising quantum chain,J. Phys. A35(2002) L25–L30, arXiv:hep-th/0111157

work page internal anchor Pith review arXiv 2002
[75]

L. A. Takhtajan and L. D. Faddeev,The Quantum method of the inverse problem and the Heisenberg XYZ model,Russ. Math. Surveys34(1979) 11–68

work page 1979
[76]

Baxter,Corner transfer matrices,Physica A: Statistical Mechanics and its Applications106(1981) 21 18–27

R. Baxter,Corner transfer matrices,Physica A: Statistical Mechanics and its Applications106(1981) 21 18–27

work page 1981
[77]

G. E. Andrews, R. J. Baxter, and P. J. Forrester,Eight vertex SOS model and generalized Rogers-Ramanujan type identities,J. Statist. Phys.35(1984) 193–266

work page 1984
[78]

D. A. Huse,Exact exponents for infinitely many new multicritical points,Phys. Rev. B30(1984) 3908–3915

work page 1984
[79]

Kuniba and T

A. Kuniba and T. Nakanishi,Fusion RSOS models and rational coset models,Lect. Notes Math.1510(1992) 303–311

work page 1992
[80]

Gomez and G

C. Gomez and G. Sierra,A Brief history of hidden quantum symmetries in conformal field theories, NATO Sci. Ser. C409(1993) 25–43, arXiv:hep-th/9211068

work page internal anchor Pith review arXiv 1993

Showing first 80 references.