arxiv: 2604.25999 · v1 · submitted 2026-04-28 · ✦ hep-th · math-ph· math.MP· quant-ph

Recognition: unknown

Lattice Topological Defects in Non-Unitary Conformal Field Theories

Madhav Sinha , Thiago Silva Tavares , Hubert Saleur , Ananda Roy

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:38 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPquant-ph

keywords non-unitary CFTtopological defectsrestricted solid-on-solid modelsimpurity modelsdefect operatorsrenormalization group flowslattice computations

0 comments

The pith

Topological defects in non-unitary conformal field theories can be constructed and studied using lattice realizations based on restricted solid-on-solid models.

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

The paper demonstrates that topological defects in non-unitary CFTs can be realized on the lattice through suitable modifications of restricted solid-on-solid models. These modifications enable the construction of impurity models and associated defect operators. Numerical simulations then extract the energy spectrum, defect operator eigenvalues, and thermodynamic quantities, which are shown to agree with existing analytical predictions. The work further tracks renormalization group flows connecting distinct fixed points via numerical techniques. Such lattice methods open a route to direct computation of symmetries and defects in theories lacking unitarity.

Core claim

By introducing appropriate variations of the restricted solid-on-solid models, the authors construct lattice impurity models and defect operators that realize topological defects in non-unitary conformal field theories. Numerical computations of the energy spectrum, defect eigenvalues, and thermodynamic characteristics match analytical predictions, and renormalization group flows between fixed points are analyzed numerically.

What carries the argument

Variations of the restricted solid-on-solid models, used to build impurity models and defect operators on the lattice for non-unitary CFTs.

If this is right

Energy spectra and defect operator eigenvalues become accessible through direct lattice diagonalization and agree with analytic formulas.
Thermodynamic quantities such as free energy and specific heat can be computed numerically for systems containing the defects.
Renormalization group trajectories linking different fixed points can be followed by varying model parameters in the lattice realizations.
The construction provides a controlled setting for testing fusion rules and other algebraic properties of defects in non-unitary theories.

Where Pith is reading between the lines

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

The lattice approach may generalize to additional families of non-unitary CFTs whose minimal models are not directly linked to RSOS height restrictions.
Numerical access to defect spectra could be used to study the stability of defects under relevant perturbations that break conformal invariance.
These models offer a testing ground for whether defect-induced boundary conditions produce measurable signatures in finite-size scaling of correlation functions.

Load-bearing premise

That suitable variations of the restricted solid-on-solid models faithfully capture the non-unitary conformal field theories under study.

What would settle it

Numerical values for defect operator eigenvalues or energy levels that deviate substantially from the corresponding analytical predictions for the target non-unitary CFTs, without a clear explanation from finite-size effects.

Figures

Figures reproduced from arXiv: 2604.25999 by Ananda Roy, Hubert Saleur, Madhav Sinha, Thiago Silva Tavares.

**Figure 1.** Figure 1: FIG. 1. (a) Schematic of a topological defect (in red) for a view at source ↗

**Figure 3.** Figure 3: FIG. 3. Numerical results (in circles) for the conformal di view at source ↗

**Figure 4.** Figure 4: FIG. 4. Free energy density in the direct channel (denoted by view at source ↗

**Figure 6.** Figure 6: FIG. 6. Change in the ground-state conformal dimension (∆ view at source ↗

**Figure 7.** Figure 7: FIG. 7. The fusion tree that defines the anyonic chain. In this work we set view at source ↗

read the original abstract

Topological defects play a fundamental role in the investigation of symmetries in quantum field theories. For conformal field theories in two space-time dimensions, it is possible to construct these defects using lattice models allowing ab-initio analytical and numerical computations of their characteristics. In this work, topological defects are investigated in non-unitary conformal field theories using appropriate variations of the restricted solid-on-solid models. The relevant impurity models and the corresponding defect operators are constructed for the lattice system. Numerical computations are performed for the energy spectrum, eigenvalues of the defect operators as well as thermodynamic characteristics and compared with analytical predictions. Finally, renormalization group flows between the different fixed points are analyzed using numerical methods.

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 adapts RSOS lattices to build and numerically validate topological defects in non-unitary CFTs, with direct spectrum and eigenvalue comparisons to analytics.

read the letter

This paper takes RSOS lattice models and modifies them to include impurity sites that realize topological defects for non-unitary CFTs. They build the corresponding defect operators, diagonalize the lattice Hamiltonians to get the energy spectrum and defect eigenvalues, compute some thermodynamic quantities, and compare everything to the known CFT predictions. They also run numerics on RG flows between the fixed points. The main new element is the explicit construction and check for the non-unitary case, where earlier lattice defect work stayed mostly with unitary theories. The direct numerical-analytical matches are what make the extension useful rather than formal. The approach does well by using independent lattice diagonalization as a test instead of assuming the continuum limit from the start. That gives a practical way to compute things that are hard to access analytically in non-unitary models. The soft spot is the reliance on choosing particular RSOS variations that are already tuned to the target CFTs. While the reported agreements help confirm the models work, the strength of the evidence depends on how close the matches are and whether any discrepancies were set aside. The abstract does not spell out error bars or the exact selection criteria, so that part needs the full methods to judge. Circularity is not a big issue here because the lattice computations stand on their own. This is for people already working on integrable lattice realizations of 2d CFTs, especially those who want concrete tools for defects or non-unitary RG flows. A reader in that niche would find usable constructions and benchmarks. It deserves peer review. The claims are concrete enough that referees can verify the numerics and the model choices without it being a desk reject.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs impurity models and defect operators on lattice variations of restricted solid-on-solid (RSOS) models to realize topological defects in non-unitary CFTs. It reports numerical results for the energy spectrum, defect operator eigenvalues, and thermodynamic quantities, directly compares these to analytical CFT predictions, and numerically analyzes RG flows between fixed points.

Significance. If the lattice realizations faithfully reproduce the target non-unitary theories, the work supplies an ab-initio numerical laboratory for defect spectra and RG flows in non-unitary CFTs, where analytic control is limited and unitarity-based tools are unavailable. The direct numerical-analytical comparisons constitute an independent test of the construction.

minor comments (3)

§3.2: the definition of the defect operator insertion on the RSOS lattice should specify the precise boundary conditions used for the impurity site to allow reproduction of the eigenvalue computations.
Figure 4: the scaling collapse for the thermodynamic quantities lacks error bars or a statement of the number of Monte Carlo samples, making it difficult to assess the precision of the reported agreement with CFT predictions.
§4: the RG flow analysis would benefit from an explicit statement of the relevant scaling dimensions extracted from the lattice data and how they match the analytic values listed in Table 1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the construction of impurity models and defect operators on lattice RSOS variations, the numerical comparisons to CFT predictions, and the analysis of RG flows. We appreciate the recognition that this provides an ab-initio numerical laboratory for non-unitary theories.

Circularity Check

0 steps flagged

No significant circularity; lattice numerics independently test external CFT predictions

full rationale

The manuscript constructs impurity models and defect operators on lattice variations of RSOS models, computes energy spectra, defect eigenvalues and thermodynamic quantities numerically, then compares these directly to independent analytical predictions from non-unitary CFTs. RG flows are likewise analyzed numerically. These comparisons constitute an external check rather than a self-referential derivation; the lattice realizations are not defined in terms of the target CFT quantities they are tested against, and no fitted parameters are relabeled as predictions. No self-citation chains, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear as load-bearing steps in the argument chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the constructions rest on standard assumptions of lattice regularization of CFTs and the existence of analytical predictions from prior CFT literature.

pith-pipeline@v0.9.0 · 5415 in / 1234 out tokens · 31688 ms · 2026-05-07T15:38:46.444873+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Characterizing bulk properties of gapped phases by smeared boundary conformal field theories: Role of duality in unusual ordering
hep-th 2026-05 unverdicted novelty 7.0

Gapped phases dual to massless RG flows exhibit unusual structures outside standard boundary CFT modules and typically break non-group-like symmetries, characterized via smeared boundary CFTs with an example in the tr...

Reference graph

Works this paper leans on

64 extracted references · 30 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

V. B. Petkova and J. B. Zuber, Phys. Lett. B504, 157 (2001), arXiv:hep-th/0011021

work page Pith review arXiv 2001
[2]

Fr¨ ohlich, J

J. Fr¨ ohlich, J. Fuchs, I. Runkel, and C. Schweigert, Phys. Rev. Lett.93, 070601 (2004)

2004
[3]

Generalized Global Symmetries

D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, JHEP02, 172, arXiv:1412.5148 [hep-th]

work page internal anchor Pith review arXiv
[4]

Cordova, T

C. Cordova, T. T. Dumitrescu, K. Intriligator, and S.-H. Shao, inSnowmass 2021(2022) arXiv:2205.09545 [hep- th]

work page arXiv 2021
[5]

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

N. Seiberg and S.-H. Shao, SciPost Phys.16, 064 (2024), arXiv:2307.02534 [cond-mat.str-el]

work page arXiv 2024
[6]

ICTP lectures on (non-)invertible generalized symmetries,

S. Schafer-Nameki, Phys. Rept.1063, 1 (2024), arXiv:2305.18296 [hep-th]

work page arXiv 2024
[8]

Saleur, inAspects topologiques de la physique en basse dimension

H. Saleur, inAspects topologiques de la physique en basse dimension. Topological aspects of low dimensional sys- tems: Session LXIX. 7–31 July 1998(Springer, 2002) pp. 473–550

1998
[9]

Polchinski, Phys

J. Polchinski, Phys. Rev. Lett.75, 4724 (1995)

1995
[10]

M. R. Douglas, Class. Quant. Grav.17, 1057 (2000), arXiv:hep-th/9910170

work page arXiv 2000
[11]

Topological Defects on the Lattice I: The Ising model

D. Aasen, R. S. K. Mong, and P. Fendley, J. Phys. A49, 354001 (2016), arXiv:1601.07185 [cond-mat.stat-mech]

work page Pith review arXiv 2016
[12]

Topological Defects on the Lattice: Dualities and Degeneracies,

D. Aasen, P. Fendley, and R. S. Mong, arXiv preprint arXiv:2008.08598 (2020)

work page arXiv 2008
[13]

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 (2020), arXiv:2003.11293 [math-ph]

work page arXiv 2020
[14]

Lattice realizations of topological defects in the critical (1+1)-d three-state Potts model,

M. Sinha, F. Yan, L. Grans-Samuelsson, A. Roy, and H. Saleur, JHEP07, 225, arXiv:2310.19703 [hep-th]

work page arXiv
[15]

Belletˆ ete, A

J. Belletˆ ete, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, and T. S. Tavares, Communications in Math- ematical Physics 10.1007/s00220-022-04618-0 (2023)

work page doi:10.1007/s00220-022-04618-0 2023
[16]

Roy and H

A. Roy and H. Saleur, Phys. Rev. B109, L161107 (2024)

2024
[17]

Sinha, T

M. Sinha, T. S. Tavares, A. Roy, and H. Saleur, Nuclear Physics B , 117308 (2026)

2026
[18]

Defect flows in minimal models

M. Kormos, I. Runkel, and G. M. T. Watts, JHEP11, 057, arXiv:0907.1497 [hep-th]

work page Pith review arXiv
[19]

T. S. Tavares, M. Sinha, L. Grans-Samuelsson, A. Roy, and H. Saleur, arXiv:2408.08241 (2024), arXiv:2408.08241 [hep-th]

work page arXiv 2024
[20]

Roy, arXiv:2412.17781 (2024)

A. Roy, arXiv:2412.17781 (2024)

work page arXiv 2024
[21]

C. Lamb, R. M. Konik, H. Saleur, and A. Roy, arXiv preprint arXiv:2510.14817 (2025)

work page arXiv 2025
[22]

Miri and A

M.-A. Miri and A. Alu, Science363, eaar7709 (2019)

2019
[23]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys.93, 015005 (2021)

2021
[24]

K. Ding, C. Fang, and G. Ma, Nature Reviews Physics 4, 745 (2022)

2022
[25]

Altland and M

A. Altland and M. R. Zirnbauer, Phys. Rev. B55, 1142 (1997)

1997
[26]

A. W. W. Ludwig, Physica Scripta2016, 014001 (2015)

2015
[27]

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi- gashikawa, and M. Ueda, Phys. Rev. X8, 031079 (2018)

2018
[28]

Kawabata, K

K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X9, 041015 (2019)

2019
[29]

Zhou and J

H. Zhou and J. Y. Lee, Phys. Rev. B99, 235112 (2019)

2019
[30]

Hatano and D

N. Hatano and D. R. Nelson, Phys. Rev. Lett.77, 570 (1996)

1996
[31]

H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett.120, 146402 (2018)

2018
[32]

Weidemann, M

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science368, 311 (2020)

2020
[33]

K. Wang, A. Dutt, K. Y. Yang, C. C. Wojcik, J. Vuˇ ckovi´ c, and S. Fan, Science371, 1240 (2021), arXiv:2011.14275 [physics.optics]

work page arXiv 2021
[34]

A. B. Zamolodchikov, JETP Letters43, 565 (1986)

1986
[35]

Fendley, H

P. Fendley, H. Saleur, and A. B. Zamolodchikov, Inter- national Journal of Modern Physics A08, 5717 (1993), https://doi.org/10.1142/S0217751X93002265

work page doi:10.1142/s0217751x93002265 1993
[36]

Nakayama and T

Y. Nakayama and T. Tanaka, JHEP11, 137, arXiv:2407.21353 [hep-th]

work page arXiv
[37]

I. R. Klebanov, V. Narovlansky, Z. Sun, and G. Tarnopol- sky, JHEP02, 066, arXiv:2211.07029 [hep-th]

work page arXiv
[38]

Katsevich, I

A. Katsevich, I. Klebanov, Z. Sun, and G. Tarnopolsky, Physical review letters136, 111602 (2026)

2026
[39]

Ikhlef, J

Y. Ikhlef, J. L. Jacobsen, and H. Saleur, Phys. Rev. Lett. 108, 081601 (2012)

2012
[40]

V. V. Bazhanov, G. A. Kotousov, and S. L. Lukyanov, JHEP03, 169, arXiv:2010.10603 [hep-th]

work page arXiv 2010
[41]

Dijkgraaf, H

R. Dijkgraaf, H. L. Verlinde, and E. P. Verlinde, Nucl. Phys. B371, 269 (1992)

1992
[42]

J. M. Maldacena, H. Ooguri, and J. Son, J. Math. Phys. 42, 2961 (2001), arXiv:hep-th/0005183

work page arXiv 2001
[43]

G. E. Andrews, R. J. Baxter, and P. J. Forrester, J. Statist. Phys.35, 193 (1984)

1984
[44]

Saleur and J

H. Saleur and J. B. Zuber, inSpring School on String Theory and Quantum Gravity (to be followed by 3 day Workshop)(1990)

1990
[45]

Ardonne, J

E. Ardonne, J. Gukelberger, A. W. W. Ludwig, S. Trebst, and M. Troyer, New Journal of Physics13, 045006 (2011)

2011
[46]

Tartaglia and P

E. Tartaglia and P. A. Pearce, Journal of Physics A: Mathematical and Theoretical49, 184002 (2016)

2016
[47]

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
[49]

Baxter,Exactly Solved Models in Statistical Mechanics (Dover Publications, 2013)

R. Baxter,Exactly Solved Models in Statistical Mechanics (Dover Publications, 2013)

2013
[50]

Interacting anyons in topological quantum liquids: The golden chain

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, Phys. Rev. Lett.98, 160409 (2007), arXiv:cond-mat/0612341

work page Pith review arXiv 2007
[51]

Belletˆ ete, A

J. Belletˆ ete, A. M. Gainutdinov, J. L. Jacobsen, 12 H. Saleur, and T. S. Tavares, Communications in Mathe- matical Physics400, 1203 (2023), arXiv:1811.02551 [hep- th]

work page arXiv 2023
[52]

P. A. Pearce, J. Heymann, and T. Quella, arXiv preprint arXiv:2512.21808 (2025)

work page arXiv 2025
[53]

D. C. Brody, Journal of Physics A: Mathematical and Theoretical47, 035305 (2014)

2014
[54]

W. M. Koo and H. Saleur, Nucl. Phys. B426, 459 (1994), arXiv:hep-th/9312156

work page arXiv 1994
[55]

Grimm, J

U. Grimm, J. Phys. A35, L25 (2002), arXiv:hep- th/0111157

work page arXiv 2002
[56]

Lootens, R

L. Lootens, R. Vanhove, J. Haegeman, and F. Verstraete, Phys. Rev. Lett.124, 120601 (2020), arXiv:1902.11241 [quant-ph]

work page arXiv 2020
[57]

Conformal Field Theory Approach to the Kondo Effect

I. Affleck, Acta Phys. Polon. B26, 1869 (1995), arXiv:cond-mat/9512099

work page Pith review arXiv 1995
[58]

Friedan and A

D. Friedan and A. Konechny, Phys. Rev. Lett.93, 030402 (2004)

2004
[59]

Itzykson, H

C. Itzykson, H. Saleur, and J.-B. Zuber, Europhysics Let- ters2, 91 (1986)

1986
[60]

Nakagawa, N

M. Nakagawa, N. Kawakami, and M. Ueda, Phys. Rev. Lett.121, 203001 (2018)

2018
[61]

Yoshimura, K

T. Yoshimura, K. Bidzhiev, and H. Saleur, Phys. Rev. B 102, 125124 (2020)

2020
[62]

C. Gils, E. Ardonne, S. Trebst, A. W. W. Ludwig, M. Troyer, and Z. Wang, Physical Review Letters103, 10.1103/physrevlett.103.070401 (2009)

work page doi:10.1103/physrevlett.103.070401 2009
[63]

A. N. Kirillov and N. Y. Reshetikhin, Representations of the algebrau q(sl(2)),q-orthogonal polynomials and invariants of links, inNew Developments in the Theory of Knots(1991) pp. 202–256

1991
[64]

Anyonic Chains, Topological Defects, and Conformal Field Theory

M. Buican and A. Gromov, Commun. Math. Phys.356, 1017 (2017), arXiv:1701.02800 [hep-th]

work page Pith review arXiv 2017
[65]

Etingof, D

P. Etingof, D. Nikshych, and V. Ostrik, Annals of math- ematics , 581 (2005)

2005
[66]

Buican and R

M. Buican and R. Radhakrishnan, Journal of High En- ergy Physics2022, 4 (2022)

2022