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Integrability and lattice discretizations of all Topological Defect Lines in minimal CFTs

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Every topological defect line of the diagonal minimal models M(p+1,p) admits an integrable RSOS lattice discretization with one modified spectral parameter, and its full spectrum follows analytically from the Bethe ansatz.

desk verdict A serious, mostly solid paper on lattice realizations of all TDLs in diagonal minimal models; the analytic core is convincing, but the claim that (r>=2,1) lines become exactly topological in the continuum is numerically weaker than the text admits. read the letter →

arxiv 2509.04257 v1 pith:E3BKAK23 submitted 2025-09-04 hep-th cond-mat.stat-mechmath-phmath.MP

Madhav Sinha , Thiago Silva Tavares , Ananda Roy , Hubert Saleur This is my paper
classification hep-thcond-mat.stat-mechmath-phmath.MP MSC 81R1282B2381T40 PACS 11.25.Hf05.50.+q
keywords topologicaldefectlinesminimalmodelsRSOSintegrablelatticeaffineTemperley-LiebalgebraBetheansatzVerlindefusionhierarchy
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 claims that every topological defect line (TDL) of the diagonal minimal models M(p+1,p) has a concrete lattice realization: an integrable RSOS model with a single row or column of faces whose spectral parameter has been shifted. For the (1,s) family the lattice defect operators sit in the center of the affine Temperley-Lieb algebra, making them exactly topological already on the lattice; for the (r,1) family the operators are built from fused transfer matrices at spectral parameter ±π/2 and become topological only in the continuum limit. In both cases the paper derives all eigenvalues and degeneracy factors analytically from the Bethe ansatz and the fusion hierarchy, and verifies them numerically for several minimal models. The payoff of the dictionary is that any Verlinde line can be simulated with exactly known spectra, opening numerical access to quantities such as entanglement cuts in the presence of defects.

What carries the argument

The machinery is the fused RSOS Boltzmann weights of spin J, organized into transfer matrices T^{(J)}(u), with one inhomogeneous row or column carrying an extra spectral parameter ũ. In the crossed channel the (1,s) lines are τ^{1-s} times the ũ → i∞ limit of T^{(s-1)}: the resulting Y-operators lie in the center of the affine Temperley-Lieb algebra, which is exactly why these lines can be deformed freely on the lattice. The (r,1) lines are τ^{1-r} T^{(r-1)}(±π/2); they are not central, hence only continuum-topological. The Bethe-ansatz treatment — the NLIE equations together with the generalized T-system relations (7.37)-(7.38) — supplies every eigenvalue, with the functions e^{(J)}_0 isola

What would settle it

Two concrete checks. (1) Diagonalize the ũ = π/2 defect Hamiltonian at sizes beyond the paper's L ≲ 40 (DMRG tables already reach 128 sites): if the low-lying levels extrapolate to the (2,1) defect Hilbert space including the O(1/L) momentum corrections of Tables 9/16/22, the fixed-point identification survives; a level crossing or a state flowing to a different primary would mean a relevant perturbation is generated and the identification fails. (2) Compute the normalized commutator [D^{(latt)}_{(2,1)}, e_i] / ||D^{(latt)}_{(2,1)} e_i|| at increasing sizes with higher-order fits: the paper's

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Extended reading notes

Core claim

The central claim is a complete identification, in the scaling limit, of every Verlinde line D(r,s) of the diagonal minimal models with a lattice operator built from fused RSOS transfer matrices with one modified spectral parameter. For (1,s), the operator is proportional to τ^{1-s} times the ũ → i∞ limit of the fused transfer matrix T^{(s-1)}(ũ); these operators generate the center of the affine Temperley-Lieb algebra and are topological on the lattice. For (r,1), the operator is proportional to τ^{1-r} T^{(r-1)}(±π/2) and is topological only in the continuum. The quantitative payload is Eq. (7.59): after stripping non-universal bulk factors, τ^{-J} T^{(J)} converges to (-1)^{R(J+1)} D_{(J,

Load-bearing premise

The load-bearing premise is that an impurity column or row with fixed finite spectral parameter (ũ = ±π/2 for the (2,1) line, and the fused analogues for (r≥3,1) lines) flows, in the large-lattice limit, to exactly the intended (r,1) defect fixed point — with no extra relevant perturbation generated — a point the paper explicitly sets aside in the introduction and supports only indirectly through Bethe-ansatz eigenvalue computations and commutator extrapolations that do not c

Editorial extensions

If this is right

  • Every Verlinde line of an A-type minimal model acquires an integrable lattice approximation with analytically known eigenvalues, so defect spectra, including O(1/R) corrections, can be computed numerically without ambiguity about which defect one is studying.
  • The (1,s) operators being central elements of the affine Temperley-Lieb algebra means their action and fusion relations hold exactly at finite system size, not merely in the scaling limit.
  • Because the same family T^{(J)}(u) realizes different defects in different regions of the spectral-parameter plane, one family interpolates between fixed points, giving a lattice handle on defect renormalization-group flows when parameters are scaled with system size.
  • The direct-channel defect Hamiltonians are matched to the conformal defect partition functions of Eq. (1.6), so entanglement entropies and other defect observables along the line become accessible to simulation with known finite-size behavior.
  • The construction is stated to carry over to D- and E-type RSOS models and to suggest a route for other coset diagonal minimal models, since it relies on the centralizer and the fusion hierarchy rather than the specific A_p weights.

Reading between the lines

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

  • The paper's own commutator fits for the (2,1) operator level off near 0.021 (maximum) and 0.038 (average) in A4 rather than cleanly at zero; a sharper check of the weakest assumption would be to push these commutators to much larger sizes and determine whether the extrapolated constant is genuinely zero or a small nonzero number.
  • The boundary between the two regions in Eq. (7.59) is where convergence must be slowest; if the identification is right, numerical studies there should see the slowest approach to the conformal eigenvalues, which could serve as a diagnostic for where the lattice approximation breaks down.
  • A testable corollary of the full dictionary: products of lattice defect operators with parameters chosen in different regions should reproduce the direct sums (e.g., (2,1)×(2,1) = (1,1)+(3,1)) already at finite size up to known power-law corrections, providing a purely numerical way to read off fusion rules of non-integrable deformations.
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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 manuscript proposes a lattice discretization of every Verlinde topological defect line (TDL) in the diagonal unitary minimal models M(p+1,p), using A_p RSOS models / anyonic chains. For (1,s) defects it constructs exact lattice TDLs (lTDLs) as central elements of the affine Temperley-Lieb algebra, realized as i∞ limits of fused transfer matrices (Eq. (6.14)); for (r,1) defects it constructs continuum TDLs (dTDLs) from fused transfer matrices at fixed spectral parameter ±π/2 (Eq. (6.15)), which are claimed to be topological only in the scaling limit. The quantitative core is Eq. (7.59): after subtracting bulk non-universal terms, τ^{-J}T^{(J)} flows to D_{(J,1)} or D_{(J+1,1)} depending on the spectral-parameter region. The support is a fused-weight Yang-Baxter construction, a generalized T-system and NLIE eigenvalue calculation, and numerical checks for A3, A4, A5, and A10 models, including direct-channel defect spectra and crossed-channel expectation values.

Significance. If valid, the paper gives a uniform integrable lattice realization of all Verlinde lines in diagonal minimal models, with analytically computed eigenvalues and degeneracy factors, and clarifies the relation between lattice topological invariance and degenerate Yang-Baxter moves. The strengths are substantial: the NLIE eigenvalue computation is parameter-free and reproduces the Verlinde S-matrix ratios against an external CFT benchmark; the fusion hierarchy is explicit; and the numerical tests cover several models, including a high-level A10 case. The paper also discloses overlap with its own precursors ([9], [17], [19], [20]) and with [11]. The main caveat is that the exactness of the dTDL identification relies on a fixed-point assumption that is not resolved in this manuscript, and the direct commutator extrapolations give small nonzero intercepts rather than cleanly zero.

major comments (3)
  1. [§8.3.1, §8.4.1 (Figs. 36, 39)] The article states that normalized commutators [D^(latt)_{(2,1)}, e_i] 'converge to 0.02088502 and 0.03777982' for A4 and 'converge to 0.01243544 and 0.01180966' for A5, and then says this 'confirms the topological nature... in the scaling limit.' A nonzero extrapolated intercept does not confirm exact topological behavior; it is evidence against it. This is the direct numerical support for the defining property of dTDLs. Please either show that these intercepts are finite-size artifacts (e.g., due to truncation to the lowest eight states, normalization, or an incorrect extrapolant) and provide a corrected extrapolation, or weaken the claim to 'approximately topological' with a quantified accuracy.
  2. [Introduction, §7.4.2, §9] The identification of all (r≥2,1) lines as dTDLs assumes that the impurity models with fixed finite defect spectral parameter (notably ũ=±π/2) flow to the exact (r,1) defect fixed point. The paper explicitly brackets this: 'We are not concerned by this aspect here, as we keep the spectral parameter fixed, and simply worry about identifying the corresponding defect in the scaling limit' (Introduction, citing [20]). Since all (r,1) lines are obtained by fusing (2,1), a relevant perturbation generated by the spectral-parameter impurity would propagate to every non-(1,s) defect. The eigenvalue and commutator checks are suggestive, but they do not rule out a different IR fixed point with an additional local perturbation. A concrete test would be to compare the full low-lying defect-Hilbert-space tower with Eq. (1.6) at larger L, and to study the defect RG flow of [20] at the fixed point.
  3. [§7.3.2 and §7.4.2] The derivation of the main eigenvalue identifications (7.55)–(7.59) relies on analyticity/zero-structure assumptions in the NLIE treatment. The paper states in §7.3.2 that completeness of the spectrum is not addressed ('It is not our objective to dig into these matters further') and notes that the generalized T-system relations are inferred from eigenvalue expressions. The numerical checks compensate, but the abstract claims to 'calculate analytically all the associated eigenvalues (and degeneracy factors)'. The analytic claim should be qualified accordingly, or a proof/completeness argument should be supplied for the low-lying spectrum relevant to the CFT identification.
minor comments (6)
  1. [Introduction, p. 5] The statement that only the (1,s) family is lattice-topological is qualified later by the remark on the dilute-Temperley-Lieb regularization, where the roles of (1,s) and (r,1) are exchanged. Please make this qualification in the first occurrence to avoid a flat contradiction.
  2. [Notation, Table 1] The conventions for τ, ̃τ, bD, bD^(latt), and bD^(latt),0 are introduced across several sections. A short summary of the parity and momentum-shift conventions used in the crossed channel would improve readability.
  3. [Tables 8–24] Finite-size estimates of h+̄h and h−̄h are presented without error bars or extrapolation details. Please state the extrapolation procedure and uncertainties, especially for the DMRG data in Tables 12, 19, 21, and 22.
  4. [Figure 30] The label “Λ/(Cosh(v)^(2L-1)Cosh(v+v_I))” should use consistent typesetting; the two asymptotics discussed in the text are easier to follow if the intermediate plateau for 0≪−v≪−v_I is marked explicitly.
  5. [Appendix D, §6.1] The verification that the fused weights (1J)W and (J1)W satisfy the Yang-Baxter and unitarity relations is only indicated ('It can be checked'). Since this underpins the higher-spin construction, at least one explicit check or a precise pointer to the verification would improve reproducibility.
  6. [Eq. (8.33)] The five-parameter fit c0,...,c4 is described with errors for c0 and c1, but no covariance or stability under changing the fit range is reported. Given that the central conclusion depends on c0 being consistent with 0, please add this information.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central eigenvalue results are parameter-free Bethe-ansatz/NLIE computations benchmarked against the external Verlinde S-matrix; self-citations are algebraic or independent.

full rationale

The central claim (Eq. 7.59) identifies lattice defect operators by computing their eigenvalues via Bethe ansatz and NLIE, then comparing with the CFT Verlinde S-matrix ratios (Eq. 7.58). The matching is a genuine prediction, not a fit: the defect spectral parameter is fixed (e.g., u~=±π/2) and the O(1) eigenvalues are derived from the T-system/Y-system, not imposed from the S-matrix. The (1,s) lTDL identification uses the algebraic fact that the center of the affine Temperley-Lieb algebra is generated by the Y operators, cited to [11]; that theorem is parameter-free and independent of the present paper's target result, so it is real support rather than circular self-citation. The paper explicitly discloses its reliance on precursors [9,17,19,20] and the Introduction states that the fixed-spectral-parameter RG flow to the IR defect is not addressed there ('We are not concerned by this aspect here...'). This is a rigor/correctness limitation, not a circular reduction: the paper does provide independent direct-channel NLIE and numerical evidence for the (r,1) identifications. The generalized T-system is inferred from eigenvalue expressions in Sec. 7.3.2, with a stated completeness caveat, but the same bilinear relations are known in the cited literature and are used only as consistency constraints. The nonzero extrapolated commutator intercepts in Secs. 8.3.1/8.4.1 (0.0209/0.0378 for A4 and 0.0124/0.0118 for A5) are a potential correctness concern about exact topological behavior in the continuum limit, but they are not a circularity: the paper does not define the defects in terms of those commutators. Overall, no step in the derivation reduces by construction to a fitted input or to an unverified self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard integrability results (Yang-Baxter, Bethe ansatz) and on the literature result that the aTL center generates the continuum Virasoro commutant. Two assumptions are specific to this paper: the fused weights form valid YBE solutions, and the finite-spectral-parameter impurities flow to a single defect fixed point. No new physical entities are introduced and no target constants are fitted; the only fitted numbers are finite-size extrapolation coefficients used for numerical presentation.

free parameters (1)
  • Finite-size extrapolation coefficients c_0..c_4 (Eq. 8.33) = c0 ~ -3.1e-5, c1 ~ 0.32448 for the |3/80,3/80> state of A4
    Polynomial fit in (2R)^(-1/2) used to extract continuum expectation values of T(pi/2); not part of the analytic construction. The commutator fits in Figs. 36/39 are similar post-processing.
assumptions (5)
  • domain assumption The A_p RSOS model at criticality flows to the diagonal minimal model M(p+1,p) with c = 1 - 6/[p(p+1)]
    Invoked throughout as the lattice-to-CFT correspondence (Sections 2 and 2.3); standard results [31,37,38].
  • domain assumption Central elements of the affine Temperley-Lieb algebra approximate the commutant of Vir x Vir on the lattice
    Basis of the (1,s) lTDL construction via Y operators (Sections 3.2, 5.2); taken from [11], with sign corrections noted in Section 7.4.1 and Appendix B.
  • standard math The Bethe ansatz / NLIE eigenvalue expressions and their scaling limits are valid, including the assumed zero and twist-sector structure
    Used to compute all defect eigenvalues (Section 7); the zero structure is established partly by numerics ('Thorough numerical study leads us to conclude...', Section 7.1.1).
  • ad hoc to paper The fused weights (1J)W and (J1)W satisfy Yang-Baxter and unitarity
    Needed for all higher (r,s) defects; derived by the paper's own projection construction in Appendix D and checked indirectly through the T-system relations.
  • ad hoc to paper The fixed-spectral-parameter impurity models flow to a single defect fixed point in the thermodynamic limit
    The Introduction brackets off the defect RG flow of [20]; the claim that the IR fixed point is the pure (r,1) defect is what the numerics and NLIE are testing.

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Pith. "Pith review of Integrability and lattice discretizations of all Topological Defect Lines in minimal CFTs." pith.science (2026). https://pith.science/paper/E3BKAK23

@misc{pith2026250904257,
  author       = {Pith},
  title        = {Pith review of: Integrability and lattice discretizations of all Topological Defect Lines in minimal CFTs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3BKAK23}},
  note         = {Machine review of arXiv:2509.04257}
}
abstract

We discuss in this paper the lattice discretizations of all topological defect lines (TDLs) for diagonal, minimal CFTs, using integrable restricted solid-on-solid (RSOS) models. For these CFTs, the TDLs can be labeled by the Kac labels. In the case of $(1,s)$ TDLs, lines that are exactly topological on the lattice can be obtained using the centralizer of the underlying Temperley-Lieb algebra, all the other lines become topological in the continuum limit only. Our general construction relies on insertions of rows/columns of faces with modified spectral parameters, and can therefore be studied using integrability techniques. We determine the regions of spectral parameters realizing the different $(r,s)$ TDLs, and in particular calculate analytically all the associated eigenvalues (and degeneracy factors). We also show how fusion of TDLs can be obtained from fusion hierarchies in the algebraic approach to the Bethe-ansatz. All our results are checked numerically in detail for several minimal CFTs.

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

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Reference graph

Works this paper leans on

83 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [20]

    T. S. Tavares, M. Sinha, L. Grans-Samuelsson, A. Roy, and H. Saleur, Integrable RG flows on topological defect lines in 2D conformal field theories , JHEP 04 (2025) 148, [arXiv:2408.08241]

    arXiv 2025

  2. [9]

    Sinha, F

    M. Sinha, F. Yan, L. Grans-Samuelsson, A. Roy, and H. Saleur, Lattice realizations of topological defects in the critical (1+1)-d three-state Potts model , JHEP 07 (2024) 225, [arXiv:2310.19703]

    arXiv 2024

  3. [17]

    C. H. O. Chui, C. Mercat, and P. A. Pearce, Integrable and conformal twisted boundary conditions for sl(2) a-d-e lattice models , Journal of Physics A: Mathematical and General 36 (mar, 2003) 2623

    work page 2003

  4. [19]

    Belletˆ ete, A

    J. Belletˆ ete, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, and T. S. Tavares, Topological defects in periodic RSOS models and anyonic chains , arXiv:2003.11293

    arXiv 2003

  5. [1]

    Oshikawa and I

    M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line , Nucl. Phys. B 495 (1997) 533–582, [cond-mat/9612187]

    arXiv 1997

  6. [2]

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

    U. Grimm, Spectrum of a duality twisted Ising quantum chain , J. Phys. A 35 (2002) L25–L30, [hep-th/0111157]

    arXiv 2002

  7. [3]

    Aasen, P

    D. Aasen, P. Fendley, and R. S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies, arXiv:2008.08598

    arXiv 2008

  8. [4]

    Roy and H

    A. Roy and H. Saleur, Entanglement Entropy in the Ising Model with Topological Defects , Phys. Rev. Lett. 128 (2022), no. 9 090603, [ arXiv:2111.04534]

    arXiv 2022

Show all 83 references
  1. [5]

    Roy and H

    A. Roy and H. Saleur, Topological interfaces of luttinger liquids , Phys. Rev. B 109 (Apr,

  2. [6]

    V. B. Petkova and J. B. Zuber, Generalized twisted partition functions , Phys. Lett. B 504 (2001) 157–164, [ hep-th/0011021]

    2001 arXiv

  3. [7]

    Bachas and M

    C. Bachas and M. Gaberdiel, Loop operators and the Kondo problem , JHEP 11 (2004) 065, [hep-th/0411067]

    2004 arXiv

  4. [8]

    Kormos, I

    M. Kormos, I. Runkel, and G. M. T. Watts, Defect flows in minimal models , JHEP 11 (2009) 057, [ arXiv:0907.1497]

    2009 arXiv

  5. [12]

    Bhardwaj, L

    L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Lattice Models for Phases and Transitions with Non-Invertible Symmetries , arXiv:2405.05964

    arXiv

  6. [13]

    Vanhove, L

    R. Vanhove, L. Lootens, H.-H. Tu, and F. Verstraete, Topological aspects of the critical three-state Potts model, J. Phys. A 55 (2022), no. 23 235002, [ arXiv:2107.11177]

    2022 arXiv

  7. [14]

    Aasen, R

    D. Aasen, R. S. K. Mong, and P. Fendley, Topological Defects on the Lattice I: The Ising model, J. Phys. A 49 (2016), no. 35 354001, [ arXiv:1601.07185]

    2016 arXiv

  8. [15]

    Aasen, P

    D. Aasen, P. Fendley, and R. S. K. Mong, Topological defects on the lattice: Dualities and degeneracies, 2020

    2020

  9. [16]

    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

    1984

  10. [18]

    Belletˆ ete, A

    J. Belletˆ ete, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, and T. S. Tavares, Topological – 148 – Defects in Lattice Models and Affine Temperley–Lieb Algebra , Commun. Math. Phys. 400 (2023), no. 2 1203–1254, [ arXiv:1811.02551]

    2023 arXiv

  11. [21]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory . Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997

    1997

  12. [22]

    Chang, Y.-H

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, Topological Defect Lines and Renormalization Group Flows in Two Dimensions , JHEP 01 (2019) 026, [arXiv:1802.04445]

    2019 arXiv

  13. [23]

    Seiberg and S.-H

    N. Seiberg and S.-H. Shao, Majorana chain and Ising model – (non-invertible) translations, anomalies, and emanant symmetries , SciPost Phys. 16 (2024) 064, [ arXiv:2307.02534]

    2024 arXiv

  14. [24]

    Seiberg, S

    N. Seiberg, S. Seifnashri, and S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space , arXiv:2401.12281

    arXiv

  15. [25]

    Zhang and G

    H.-C. Zhang and G. Sierra, Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective , JHEP 25 (2020) 157, [arXiv:2410.06727]

    2020 arXiv

  16. [26]

    P. E. Finch, M. Flohr, and H. Frahm, Integrable anyon chains: From fusion rules to face models to effective field theories , Nuclear Physics B 889 (2014) 299–332

    2014

  17. [27]

    Braylovskaya, P

    N. Braylovskaya, P. E. Finch, and H. Frahm, Exact solution of the D3 non-abelian anyon chain, Phys. Rev. B 94 (Aug, 2016) 085138

    2016

  18. [28]

    Cappelli, C

    A. Cappelli, C. Itzykson, and J. B. Zuber, Modular invariant partition functions in two dimensions, Nucl. Phys. B 280 (1987) 445–465

    1987

  19. [29]

    B. M. McCoy, Advanced statistical mechanics. Oxford University Press, 2010

    2010

  20. [30]

    S. O. Warnaar and B. Nienhuis, Solvable lattice models labelled by Dynkin diagrams , J. Phys. A 26 (1993) 2301–2316, [ hep-th/9301026]

    1993 arXiv

  21. [31]

    Feiguin, S

    A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, Interacting anyons in topological quantum liquids: The golden chain , Phys. Rev. Lett. 98 (2007), no. 16 160409, [ cond-mat/0612341]

    2007 arXiv

  22. [32]

    Saleur and J

    H. Saleur and J. B. Zuber, Integrable lattice models and quantum groups , in Spring School on String Theory and Quantum Gravity (to be followed by 3 day Workshop) , 4, 1990

    1990

  23. [33]

    Baxter, Exactly Solved Models in Statistical Mechanics

    R. Baxter, Exactly Solved Models in Statistical Mechanics . Dover Books on Physics. Dover Publications, 2013

    2013

  24. [34]

    J. B. Kogut, An introduction to lattice gauge theory and spin systems , Rev. Mod. Phys. 51 (Oct, 1979) 659–713

    1979

  25. [35]

    Frahm and D

    H. Frahm and D. Westerfeld, Density matrices in integrable face models , SciPost Phys. 11 (2021), no. 3 057, [ arXiv:2104.06269]

    2021 arXiv

  26. [36]

    Ardonne, J

    E. Ardonne, J. Gukelberger, A. W. W. Ludwig, S. Trebst, and M. Troyer, Microscopic models of interacting yang–lee anyons , New Journal of Physics 13 (Apr., 2011) 045006. – 149 –

    2011

  27. [37]

    C. Gils, E. Ardonne, S. Trebst, A. W. W. Ludwig, M. Troyer, and Z. Wang, Collective states of interacting anyons, edge states, and the nucleation of topological liquids , Physical Review Letters 103 (aug, 2009)

    2009

  28. [38]

    M. S. Zini and Z. Wang, Conformal Field Theories as Scaling Limit of Anyonic Chains , Commun. Math. Phys. 363 (2018), no. 3 877–953, [ arXiv:1706.08497]

    2018 arXiv

  29. [39]

    Koo and H

    W. Koo and H. Saleur, Representations of the virasoro algebra from lattice models , Nuclear Physics B 426 (sep, 1994) 459–504

    1994

  30. [40]

    Cheng and N

    M. Cheng and N. Seiberg, Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems , SciPost Phys. 15 (2023), no. 2 051, [ arXiv:2211.12543]

    2023 arXiv

  31. [41]

    Grans-Samuelsson, L

    L. Grans-Samuelsson, L. Liu, Y. He, J. L. Jacobsen, and H. Saleur, The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic Q, arXiv:2007.11539

    2007 arXiv

  32. [42]

    J. L. Jacobsen and H. Saleur, Non-invertible symmetries and RG flows in the two-dimensional O(n) loop model , JHEP 12 (2023) 090, [ arXiv:2305.05746]

    2023 arXiv

  33. [43]

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

    S. Seifnashri, Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries, SciPost Phys. 16 (2024) 098, [ arXiv:2308.05151]

    2024 arXiv

  34. [44]

    Buican and A

    M. Buican and A. Gromov, Anyonic Chains, Topological Defects, and Conformal Field Theory, Commun. Math. Phys. 356 (2017), no. 3 1017–1056, [ arXiv:1701.02800]

    2017 arXiv

  35. [45]

    E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Exactly solvable sos models: Local height probabilities and theta function identities , Nuclear Physics B 290 (1987) 231–273

    1987

  36. [46]

    Klumper and P

    A. Klumper and P. A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchies, Physica A 183 (1992) 304

    1992

  37. [47]

    G¨ ohmann, A

    F. G¨ ohmann, A. Seel, and J. Suzuki, Correlation functions of the integrable isotropic spin-1 chain at finite temperature , Journal of Statistical Mechanics: Theory and Experiment 2010 (nov, 2010) P11011

    2010

  38. [48]

    P. P. Kulish, N. Y. Reshetikhin, and E. K. Sklyanin, Yang-baxter equation and representation theory: I , Letters in Mathematical Physics 5 (1981), no. 5 393–403

    1981

  39. [49]

    L. H. Kauffman and S. Lins, Chapter 3. Jones-Wenzl Projectors , pp. 13–21. Princeton University Press, Princeton, 1994

    1994

  40. [50]

    Martins, Integrable three-state vertex models with weights lying on genus five curves , Nuclear Physics B 874 (2013), no

    M. Martins, Integrable three-state vertex models with weights lying on genus five curves , Nuclear Physics B 874 (2013), no. 1 243–278

    2013

  41. [51]

    Bazhanov and N

    V. Bazhanov and N. Reshetikhin, Restricted Solid on Solid Models Connected With Simply Based Algebras and Conformal Field Theory , J. Phys. A 23 (1990) 1477

    1990

  42. [52]

    P. P. Kulish and E. K. Sklyanin, Quantum spectral transform method. Recent developments , Lect. Notes Phys. 151 (1982) 61–119

    1982

  43. [53]

    Belletˆ ete, A

    J. Belletˆ ete, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, and R. Vasseur, On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories, J. Phys. A 50 (2017), no. 48 484002, [ arXiv:1705.07769]

    2017 arXiv

  44. [54]

    F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, Conformal invariance and the spectrum of the xxz chain , Phys. Rev. Lett. 58 (Feb, 1987) 771–774. – 150 –

    1987

  45. [55]

    Gehrmann, G

    S. Gehrmann, G. A. Kotousov, and S. L. Lukyanov, Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin - 12 chain , Nucl. Phys. B 1006 (2024) 116624, [arXiv:2406.12102]

    2024 arXiv

  46. [56]

    Destri and H

    C. Destri and H. J. de Vega, New thermodynamic bethe ansatz equations without strings , Phys. Rev. Lett. 69 (Oct, 1992) 2313–2317

    1992

  47. [57]

    T. S. Tavares, I. R. Passos, and A. Kl¨ umper, Canonical formulation for the thermodynamics of sln-invariant integrable spin chains , Nucl. Phys. B 998 (2024) 116417, [arXiv:2311.02238]

    2024 arXiv

  48. [58]

    A. N. Kirillov, Dilogarithm identities, Progress of Theoretical Physics Supplement 118 (01,

  49. [59]

    Takahashi, One-dimensional heisenberg model at finite temperature , Prog

    M. Takahashi, One-dimensional heisenberg model at finite temperature , Prog. Theor. Phys. 46 (1971) 401

    1971

  50. [60]

    Trotter, On the product of semi-groups of operators , Proc

    H. Trotter, On the product of semi-groups of operators , Proc. Am. Math. Soc. 10 (1959) 545

    1959

  51. [61]

    M. Suzuki, Relationship between d-dimensional quantal spin systems and (d+1)-dimensional ising systems: Equivalence, critical exponents and systematic approximants of the partition function and spin correlations , Prog. Theor. Phys. 56 (1976) 1454

    1976

  52. [62]

    Kl¨ umper,Thermodynamics of the anisotropic spin-1/2 heisenberg chain and related quantum chains , Zeitschrift f¨ ur Physik B Condensed Matter91 (1993), no

    A. Kl¨ umper,Thermodynamics of the anisotropic spin-1/2 heisenberg chain and related quantum chains , Zeitschrift f¨ ur Physik B Condensed Matter91 (1993), no. 4 507

    1993

  53. [63]

    Takahashi, M

    M. Takahashi, M. Shiroishi, and A. Kl¨ umper,Equivalence of tba and qtm , Journal of Physics A: Mathematical and General 34 (apr, 2001) L187

    2001

  54. [64]

    Kuniba, T

    A. Kuniba, T. Nakanishi, and J. Suzuki, T-systems and y-systems in integrable systems* , Journal of Physics A: Mathematical and Theoretical 44 (feb, 2011) 103001

    2011

  55. [65]

    Krichever, O

    I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, Quantum integrable models and discrete classical hirota equations , Communications in Mathematical Physics 188 (Sep, 1997) 267–304

    1997

  56. [66]

    Tsuboi, Analytic Bethe Ansatz and functional relations related to tensor like representations of type-II Lie superalgebras B(r—s) and D(r—s) , J

    Z. Tsuboi, Analytic Bethe Ansatz and functional relations related to tensor like representations of type-II Lie superalgebras B(r—s) and D(r—s) , J. Phys. A 32 (1999) 7175–7206, [arXiv:0911.5393]

    1999 arXiv

  57. [67]

    Morin-Duchesne, A

    A. Morin-Duchesne, A. Kl¨ umper, and P. A. Pearce, Groundstate finite-size corrections and dilogarithm identities for the twisted, and models , Journal of Statistical Mechanics: Theory and Experiment 2021 (2021), no. 3 033105

    2021

  58. [68]

    Kuniba, K

    A. Kuniba, K. Sakai, and J. Suzuki, Continued fraction TBA and functional relations in XXZ model at root of unity , Nucl. Phys. B 525 (1998) 597–626, [ math/9803056]

    1998 arXiv

  59. [69]

    Kuniba and J

    A. Kuniba and J. Suzuki, Analytic bethe ansatz for fundamental representations of yangians , Communications in Mathematical Physics 173 (Oct, 1995) 225–264

    1995

  60. [70]

    L. Li, M. Oshikawa, and Y. Zheng, Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases , Phys. Rev. B 108 (2023), no. 21 214429, [ arXiv:2301.07899]

    2023 arXiv

  61. [71]

    Grimm and G

    U. Grimm and G. M. Schutz, The Spin 1/2 XXZ Heisenberg chain, the quantum algebra U(q)[sl(2)], and duality transformations for minimal models , J. Statist. Phys. 71 (1993) 921–964, [hep-th/0111083]. – 151 –

    1993 arXiv

  62. [72]

    Makabe and G

    I. Makabe and G. M. T. Watts, Defects in the Tri-critical Ising model , JHEP 09 (2017) 013, [arXiv:1703.09148]

    2017 arXiv

  63. [73]

    A. M. Gainutdinov, J. L. Jacobsen, N. Read, H. Saleur, and R. Vasseur, Logarithmic Conformal Field Theory: a Lattice Approach , J. Phys. A 46 (2013) 494012, [arXiv:1303.2082]

    2013 arXiv

  64. [74]

    Rogerson, F

    D. Rogerson, F. Pollmann, and A. Roy, Entanglement entropy and negativity in the Ising model with defects , JHEP 06 (2022) 165, [ arXiv:2204.03601]

    2022 arXiv

  65. [75]

    Roy, Variational Quantum Simulation of Anyonic Chains , arXiv:2412.17781

    A. Roy, Variational Quantum Simulation of Anyonic Chains , arXiv:2412.17781

    arXiv

  66. [76]

    Pasquier and H

    V. Pasquier and H. Saleur, Common Structures Between Finite Systems and Conformal Field Theories Through Quantum Groups , Nucl. Phys. B 330 (1990) 523–556

    1990

  67. [77]

    A. M. Gainutdinov, N. Read, and H. Saleur, Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra, Commun. Math. Phys. 341 (2016), no. 1 35–103, [ arXiv:1207.6334]

    2016 arXiv

  68. [78]

    J. J. Graham and G. I. Lehrer, Diagram algebras, hecke algebras and decomposition numbers at roots of unity , Annales scientifiques de l’ ´Ecole Normale Sup´ erieure36 (2003), no. 4 479–524

    2003

  69. [79]

    C. Gils, E. Ardonne, S. Trebst, D. A. Huse, A. W. W. Ludwig, M. Troyer, and Z. Wang, Anyonic quantum spin chains: Spin-1 generalizations and topological stability , Phys. Rev. B 87 (2013), no. 23 235120, [ arXiv:1303.4290]

    2013 arXiv

  70. [80]

    Lootens, C

    L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners, PRX Quantum 4 (2023), no. 2 020357, [ arXiv:2112.09091]

    2023 arXiv

  71. [81]

    Inamura and K

    K. Inamura and K. Ohmori, Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories, arXiv:2305.05774

    arXiv

  72. [82]

    A. N. Kirillov and N. Y. Reshetikhin, Representations of the algebra U(q)(sl(2)), q orthogonal polynomials and invariants of links ,

  73. [83]

    Bernard and A

    D. Bernard and A. Leclair, Residual quantum symmetries of the restricted sine-gordon theories, Nuclear Physics B 340 (1990), no. 2 721–751

    1990

  74. [84]

    Bernard and A

    D. Bernard and A. LeClair, Quantum group symmetries and nonlocal currents in 2d qft , Comm. Math. Phys. 142 (1991), no. 1 99–138. – 152 –

    1991

  75. [1995]

    61–142, [https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.118.61/5200829/118-61.pdf]

    doi

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