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REVIEW 3 major objections 5 minor 113 references

Infrared properties of two-dimensional $\mathrm{SU}(N)/H$ nonlinear $\sigma$ models at nonzero $\theta$ angles

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A single OPE sign decides whether theta=pi SU(N)/H sigma models are gapless or gapped

desk verdict A careful Affleck-Haldane extension that makes the sign of the marginal current-current coupling the pivot; the sign is load-bearing and the point-splitting prescription is unproven, but the results align with independent checks and the strategy is worth engaging. read the letter →

arxiv 2412.17493 v3 pith:AZ735OB7 submitted 2024-12-23 cond-mat.str-el hep-th

Philippe Lecheminant , Yuya Tanizaki , Keisuke Totsuka This is my paper · ORCID
classification cond-mat.str-elhep-th
keywords nonlinearsigmamodelthetatermWZNWconformalfieldtheorycurrent-currentperturbationflagmanifoldGrassmannianpoint-splittingregularizationanomalymatching
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 proposes a general route to decide the fate of two-dimensional nonlinear $\sigma$ models with topological $\theta$ terms. The route starts from the $\mathrm{SU}(N)_1$ Wess–Zumino–Novikov–Witten (WZNW) conformal field theory, a massless theory of a unitary matrix field, and connects it to the $\sigma$ model by adding a potential whose classical minima are the desired target space. The paper then reads off the low-energy behaviour from the short-distance operator product of that potential: for the flag $\mathrm{SU}(N)/\mathrm{U}(1)^{N-1}$ and $\mathrm{SU}(N)/\mathrm{SO}(N)$ cases the leading term is a marginally irrelevant current-current coupling, so the theory flows back to the $\mathrm{SU}(N)_1$ CFT at long distances. For complex Grassmannian and $\mathbb{C}P^{N-1}$ models at $\theta=\pi$, the same computation gives a relevant perturbation, producing a mass gap with a two-fold degenerate ground state. The payoff is that case-by-case conjectures are replaced by one prescription whose sign is fixed by the OPE.

What carries the argument

The load-bearing object is the $\mathrm{SU}(N)_1$ WZNW conformal field theory, realized exactly by $N-1$ compactified free bosons on the root lattice. One adds a double-trace potential $\sum_n \lambda_n|\operatorname{Tr}G^n|^2$ (or $\operatorname{Tr}|G-G^T|^2$ for the $\mathrm{SO}(N)$ coset) whose classical minima reproduce the target space; sending $\lambda_n\to\infty$ recovers the nonlinear $\sigma$ model with its $\theta$ terms. The decisive step is a point-splitting regularization of the potential combined with short-distance OPEs in the free-boson representation, which isolates the leading current-current operator and fixes its sign, e.g. $\gamma_{\rm eff}>0$ in the flag case. That sign determines whether the deformation is a marginally irrelevant perturbation flowing back to $\mathrm{SU}(N)_1$, or a relevant perturbation that pins the bosons and produces gapped vacua.

What would settle it

Measure the low-energy central charge of the flag $\mathrm{SU}(4)/\mathrm{U}(1)^3$ $\sigma$ model at $\theta=(\pi/2,\pi,3\pi/2)$ on a lattice realization: finding $c=3$ with gapless excitations would confirm the predicted massless flow to $\mathrm{SU}(4)_1$, while finding a gap or a different central charge would falsify it.

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Extended reading notes

Core claim

The paper's central claim is that the infrared phase of each model is controlled by the sign of one operator product expansion: the coefficient of the $\mathrm{SU}(N)_1$ current-current interaction generated when the ultraviolet-regulated potential is expanded at short distances. For the flag-manifold $\mathrm{SU}(N)/\mathrm{U}(1)^{N-1}$ model at $\theta_a=2\pi a/N$ and for $\mathrm{SU}(N)/\mathrm{SO}(N)$ at $\theta=\pi$, this coefficient is positive, the perturbation is marginally irrelevant, and the long-distance limit is the $\mathrm{SU}(N)_1$ fixed point with central charge $c=N-1$. For $\mathbb{C}P^{N-1}$ with $N>2$ and for the Grassmannian $\mathrm{Gr}(2k,k)$ at $\theta=\pi$, the leading perturbation is relevant, the spectrum is fully gapped, and the two degenerate vacua come from spontaneous breaking of the charge-conjugation symmetry. The strategy is presented as a systematic deformation route from a level-1 WZNW theory to any $\mathrm{SU}(N)/H$ $\sigma$ model, and it does not require integrability.

Load-bearing premise

The fragile premise is that the sign of the leading interaction is independent of the detailed shape of the short-distance regulator used to define the potential near coincident points; the paper itself says it has no rigorous proof that this prescription always yields the correct answer.

Editorial extensions

If this is right

  • The flag-manifold $\mathrm{SU}(N)/\mathrm{U}(1)^{N-1}$ sigma model at $\theta_a=2\pi a/N$ is massless and flows logarithmically to the $\mathrm{SU}(N)_1$ fixed point.
  • The $\mathrm{SU}(N)/\mathrm{SO}(N)$ model at $\theta=\pi$ reaches the same $\mathrm{SU}(N)_1$ criticality, independently of whether the flow is integrable.
  • For $N>2$, $\mathbb{C}P^{N-1}$ at $\theta=\pi$ is gapped with a two-fold degenerate ground state from spontaneous charge-conjugation breaking.
  • The Grassmannian $\mathrm{Gr}(2k,k)$ at $\theta=\pi$ is likewise gapped and two-fold degenerate.
  • Because the argument uses only OPEs and symmetry, it decides the infrared phase without needing the model to be integrable.

Reading between the lines

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

  • The same point-splitting and OPE recipe should decide the infrared phases of real Grassmannian and $\mathrm{Sp}(N)$ sigma models at $\theta=\pi$, which the paper lists as future work.
  • If the sign rule survives a rigorous lattice definition, it offers a shortcut for classifying the infrared fate of any symmetric-space sigma model with a level-1 WZNW ancestor.
  • The predicted two-fold degeneracy for $\mathbb{C}P^{N-1}$ at $\theta=\pi$ could be probed in cold-atom ladder realizations by measuring dimerization order.
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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 / 5 minor

Summary. The manuscript proposes a general strategy for determining the infrared properties of two-dimensional SU(N)/H nonlinear sigma models with topological theta terms. The authors start from the SU(N)_1 WZNW CFT and add double-trace deformations designed so that in the strong-coupling limit the classical minima realize the desired target manifolds. The key technical step is a point-splitting regularization of the double-trace operators combined with OPE computations in a free-field representation to extract the leading marginal current-current operator and its sign. For the flag manifold SU(N)/U(1)^{N-1} and the SU(N)/SO(N) model at theta=pi, the extracted sign is such that the current-current perturbation is marginally irrelevant, leading to a massless RG flow back to the SU(N)_1 fixed point. For Gr(2k,k) and CP^{N-1} at theta=pi, the perturbations are relevant and produce a massive two-fold degenerate ground state. The paper also discusses 't Hooft anomaly matching as a consistency constraint.

Significance. If the central sign determination is correct, the paper provides a unified, parameter-free framework that confirms and extends important conjectures: the massless SU(N)_1 flows for flag and SU(N)/SO(N) models, and the gapped two-fold degenerate phases for Grassmannian and CP^{N-1} models. The strength of the paper lies in its explicit OPE computations (Appendices A and B), its consistency with known exact/numerical results (the CP^1 benchmark, the integrable SU(N)/SO(N) results, and lattice spin-chain data), and its broader applicability to non-integrable cases. No free parameters are fitted; the sign of the current-current coupling is derived from CFT data rather than imposed.

major comments (3)
  1. [Secs. 2.3 and 6] The central conclusion that the flag and SU(N)/SO(N) models flow to the SU(N)_1 fixed point rests on the sign of the marginal current-current coefficient extracted via the point-splitting prescription (19)/(21) and computed in Eqs. (40) and (63). The authors explicitly concede in Sec. 6 that they 'do not have a rigorous proof that this prescription always yields the correct answer.' Positivity and locality of F_epsilon alone do not guarantee that the OPE coefficient c1 equals the physical coupling generated by the double-trace deformation; scheme-dependent normal-ordering or improvement terms could, in principle, alter the sign. Since flipping the sign would change the predicted IR phase from a massless flow to a gapped regime (Secs. 3.2 and 4.3), this assumption is load-bearing. The authors should either provide a sharper justification (e.g., a non-renormalization argument or a comparison with an exact calculation) or clearly state the result as a well-supported conjecture.
  2. [Sec. 3.2 and Eq. (40)] The OPE computation leading to Eq. (40) is performed in a free-field representation and neglects non-rotation-symmetric terms in the short-distance expansion (see Eq. (B.4) and the sentence preceding Eq. (B.13)). While the point-splitting weight is rotationally symmetric, the regularization of the composite operator |Tr G^n|^2 may introduce scheme-dependent finite parts not fixed by this procedure. The consistency with the classical analysis on the opposite side (lambda_n<0) is reassuring, but it does not establish the scheme independence of the sign on the lambda_n>0 side.
  3. [Sec. 2.3 and Fig. 1] The equivalence between the deformed WZNW model and the nonlinear sigma model is established only semiclassically, in the lambda -> infinity limit. The inference of the sigma model's IR behavior from the sign of a weak perturbation of the WZNW fixed point assumes that no strong-coupling effects change the direction of the flow. This is a standard limitation of the Affleck-Haldane strategy, but the paper should be explicit that the argument is a controlled conjecture rather than a derivation from the sigma-model action.
minor comments (5)
  1. [Abstract] The abstract contains a typo: '\text{SU($N$)})/\text{SO($N$)}' has an extra closing parenthesis; it should read '\text{SU($N$)}/\text{SO($N$)}'.
  2. [Sec. 2.3] The phrase 'the exact details of the weight function would not be important' is an unproven assertion; a brief discussion of the expected scheme independence of the leading OPE coefficient would make the argument more complete.
  3. [Eq. (24) and footnote 3] The potential is written as a sum over |Tr[G^n]|^2, while the proposed strategy of Sec. 2.3 is formulated for sums over |O|^2 with O a primary operator. The footnote 3 explains the connection, but the main text would benefit from a more direct statement that the two potentials share the same minima.
  4. [Sec. 4.2] The symmetry group for even N is written as \widetilde{SU(N)}_V \times (Z_N)_L over (Z_2)_{\widetilde V} \times Z_{N/2}; the notation is dense and a brief explanation of the quotient structure would improve readability.
  5. [Sec. 4.3, Eq. (65)] The beta function is given as \dot g = -N/(4\pi) g^2; it would be helpful to define the RG time l explicitly (e.g., l = \ln(\mu_0/\mu)) to make the direction of the flow unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained: the massless/massive distinction follows from explicit OPE signs, not from fitted inputs or load-bearing self-citation.

full rationale

The central claims are not circular. For the flag model, the sign of the current-current coupling is computed by point-splitting (Eq. (37)) and the free-boson OPE (Eq. (40)), giving gamma_eff > 0 in Eq. (42); this is then fed into the known integrable SU(N) Gross-Neveu analysis. For SU(N)/SO(N), the corresponding sign is obtained from Eq. (63) and the one-loop beta function (65), with the marginal direction checked independently by the duality to Eq. (70). No parameter is fitted to the target phase; the OPE computation is parameter-free and does not assume the SU(N)_1 fixed point. The semiclassical identification of the sigma-model target with the strong-coupling limit of the deformed WZNW action is the standard Affleck-Haldane strategy, and the paper explicitly notes this is not a rigorous proof: "While we do not have a rigorous proof that this prescription always yields the correct answer..." (Sec. 6). That is a limitation and a correctness risk, not circularity, because the sign that determines the phase is not imported from the conclusion. The paper also concedes in footnote 5 that the prescription in (21) is UV-cutoff dependent when comparing magnitudes of different deformations, again a limitation rather than a circular step. Self-citations (Refs. [60], [89], [91]) are used for anomaly matching, the one-loop beta function, and a duality transformation; these are parameter-free auxiliary results that do not encode the target massless/massive dichotomy and do not raise the circularity score. The Grassmannian and CP^{N-1} massive results follow from strongly relevant perturbations (Eqs. (83), (91)), again by direct scaling dimension rather than by construction. Accordingly, there are no circular steps.

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

No parameters are fitted to data. The coupling constants lambda_n and lambda in the deformations are arbitrary positive constants; the conclusions depend only on their signs. The UV cutoff epsilon enters the effective coupling gamma_eff but is not fitted. The axioms are standard CFT results plus the semiclassical Affleck-Haldane connection and the unproven point-splitting prescription.

assumptions (5)
  • standard math SU(N)_1 WZNW CFT is equivalent to N-1 free compact bosons with the stated periodicity.
    Used throughout Secs. 3.2, 4.3, 5, and Appendix A.2; a standard result in CFT (Ref. [67]).
  • domain assumption The strong-coupling limit lambda to infinity of the deformed WZNW model restricts the field to the classical moduli space of the potential, and the WZNW term becomes the theta term.
    Core of the Affleck-Haldane argument, stated in Secs. 2.1 and 2.3; standard semiclassical reasoning, not rigorous for finite lambda.
  • domain assumption Only operators compatible with the discrete chiral symmetry (Z_N)_L can be generated; this forbids relevant scalar primaries, so the current-current term is the leading perturbation.
    Used in Secs. 3.2 and 4.3; the argument depends on (Z_N)_L being an exact symmetry.
  • standard math The SU(N) Gross-Neveu model is integrable and is gapless for lambda' > 0, gapped for lambda' < 0.
    Cited from Refs. [63-66]; used to translate the sign of the current-current coupling into an IR phase.
  • ad hoc to paper Point-splitting regularization with a positive, local weight function preserves the sign of the potential and yields the leading OPE term.
    The method's key assumption; not proven and explicitly flagged by the authors in Sec. 6.

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Pith. "Pith review of Infrared properties of two-dimensional $\mathrm{SU}(N)/H$ nonlinear $\sigma$ models at nonzero $\theta$ angles." pith.science (2026). https://pith.science/paper/AZ735OB7

@misc{pith2026241217493,
  author       = {Pith},
  title        = {Pith review of: Infrared properties of two-dimensional $\mathrmSU(N)/H$ nonlinear $\sigma$ models at nonzero $\theta$ angles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AZ735OB7}},
  note         = {Machine review of arXiv:2412.17493}
}
abstract

A general strategy is proposed to explore the low-energy properties of two-dimensional nonlinear $\sigma$ models with $\theta$ terms. We demonstrate its application to nonlinear $\sigma$ models with the target space $\text{SU($N$)}$/H, which include $\mathbb{C}P^{N-1}$, complex Grassmannian manifolds as well as the flag $\text{SU($N$)}/\text{U(1)}^{N-1}$ and $\text{SU($N$)})/\text{SO($N$)}$ manifolds. By analyzing the symmetry and its anomaly content, we realize these nonlinear $\sigma$ models through perturbations added to the SU(N)$_1$ conformal field theory. For the flag-manifold $\text{SU($N$)}/\text{U(1)}^{N-1}$ and $\text{SU($N$)})/\text{SO($N$)}$ models, those perturbations are shown to correspond to the marginal current-current operator with the specific sign which leads to a massless renormalization group flow to the SU(N)$_1$ fixed point. In contrast, a massive regime with a two-fold ground-state degeneracy is found for the $\mathbb{C}P^{N-1}$ ($N >2$) and Grassmannian nonlinear $\sigma$ models at $\theta=\pi$.

Figures

Figures reproduced from arXiv: 2412.17493 by the authors.

Figure 1
Figure 1. Illustration of the strategy. We start from the WZNW model (the prin [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Two directions of marginal deformations J A L J A L and T A αβ T B αβ J A L J B R around the SU(N)1 fixed point and the corresponding G/H nonlinear σ models [G = SU(N)]. When the IR phases are gapped, the ground-state degeneracy (GSD) is also shown. The second one is nothing but (66). However, the periodicity for the Φ⃗ field, (38), should also transform the Θ⃗ field, and this plays the pivotal role to correctly fin… view at source ↗

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Works this paper leans on

113 extracted references · 55 canonical work pages

  1. [1]

    A. M. Polyakov, Interaction of goldstone particles in two dimensions. applica- tions to ferromagnets and massive yang-mills fields , Phys. Lett. B 59, 79 (1975), doi:https://doi.org/10.1016/0370-2693(75)90161-6

    work page doi:10.1016/0370-2693(75)90161-6 1975

  2. [2]

    Sachdev, Quantum Phase Transitions, Cambridge University Press (1999)

    S. Sachdev, Quantum Phase Transitions, Cambridge University Press (1999)

    1999

  3. [3]

    Tsvelik, Quantum Field Theory in Condensed Matter Physics , Cambridge University Press (1995)

    A. Tsvelik, Quantum Field Theory in Condensed Matter Physics , Cambridge University Press (1995)

    1995

  4. [4]

    Fradkin, Field Theories of Condensed Matter Physics , Cambridge University Press, 2 edn., doi:10.1017 /CBO9781139015509 (2013)

    E. Fradkin, Field Theories of Condensed Matter Physics , Cambridge University Press, 2 edn., doi:10.1017 /CBO9781139015509 (2013)

    2013

  5. [5]

    Altland and B

    A. Altland and B. D. Simons, Condensed Matter Field Theory , Cambridge University Press, Cambridge, 2 edn., doi:10.1017 /cbo9780511789984 (2010)

    2010

  6. [6]

    Evers and A

    F . Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008), doi:10.1103/RevModPhys.80.1355

    doi:10.1103/revmodphys.80.1355 2008

  7. [7]

    M. Fava, L. Piroli, T . Swann, D. Bernard and A. Nahum, Nonlinear sigma mod- els for monitored dynamics of free fermions , Phys. Rev. X 13, 041045 (2023), doi:10.1103/PhysRevX.13.041045

    doi:10.1103/physrevx.13.041045 2023

  8. [8]

    Poboiko, P

    I. Poboiko, P . Pöpperl, I. V . Gornyi and A. D. Mirlin, Theory of free fermions under random projective measurements , Phys. Rev. X 13, 041046 (2023), doi:10.1103/PhysRevX.13.041046

    doi:10.1103/physrevx.13.041046 2023

Show all 113 references
  1. [9]

    C.-M. Jian, H. Shapourian, B. Bauer and A. W . W . Ludwig, Measurement-induced en- tanglement transitions in quantum circuits of non-interacting fermions: Born-rule versus forced measurements (2023), 2302.09094

    2023 arXiv

  2. [10]

    M. Fava, L. Piroli, D. Bernard and A. Nahum, Monitored fermions with conserved U (1) charge, Phys. Rev. Res. 6, 043246 (2024), doi:10.1103 /PhysRevResearch.6.043246

    2024

  3. [11]

    Tiutiakina, H

    A. Tiutiakina, H. Lóio, G. Giachetti, J. D. Nardis and A. D. Luca, Field theory for moni- tored Brownian SYK clusters (2024), 2410.08079

    2024 arXiv

  4. [12]

    Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model, Phys

    F . Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model, Phys. Lett. A 93, 464 (1983), doi:10.1016/0375- 9601(83)90631-X

    1983 doi

  5. [13]

    F . D. M. Haldane,Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semi- classically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50, 1153 (1983), doi:10.1103 /PhysRevLett.50.1153

    1983

  6. [14]

    S. Ryu, A. P . Schnyder, A. Furusaki and A. W . W . Ludwig,Topological insulators and su- perconductors: tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010), doi:10.1088/1367-2630/12/6/065010

    2010 doi

  7. [15]

    C.-K. Chiu, J. C. Y. Teo, A. P . Schnyder and S. Ryu, Classification of topo- logical quantum matter with symmetries , Rev. Mod. Phys. 88, 035005 (2016), doi:10.1103/RevModPhys.88.035005. 38 SciPost Physics Submission

    2016 doi

  8. [16]

    Z. Bi, A. Rasmussen, K. Slagle and C. Xu, Classification and description of bosonic sym- metry protected topological phases with semiclassical nonlinear sigma models, Phys. Rev. B 91, 134404 (2015), doi:10.1103 /PhysRevB.91.134404

    2015

  9. [17]

    Levine, S

    H. Levine, S. B. Libby and A. M. M. Pruisken, Electron delocalization by a magnetic field in two dimensions, Phys. Rev. Lett.51, 1915 (1983), doi:10.1103/PhysRevLett.51.1915

    1983 doi

  10. [18]

    Pruisken, On localization in the theory of the quantized Hall effect: A two-dimensional realization of the θ-vacuum, Nucl

    A. Pruisken, On localization in the theory of the quantized Hall effect: A two-dimensional realization of the θ-vacuum, Nucl. Phys. B 235, 277 (1984), doi:10.1016 /0550- 3213(84)90101-9

    1984

  11. [19]

    Tanaka and X

    A. Tanaka and X. Hu, Effective field theory with aθ-vacua structure for two-dimensional spin systems, Phys. Rev. B 74, 140407 (2006), doi:10.1103 /PhysRevB.74.140407

    2006

  12. [20]

    Senthil and M

    T . Senthil and M. P . A. Fisher, Competing orders, nonlinear sigma models, and topological terms in quantum magnets , Phys. Rev. B 74, 064405 (2006), doi:10.1103/PhysRevB.74.064405

    2006 doi

  13. [21]

    Senthil, A

    T . Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P . A. Fisher,Deconfined quantum critical points, Science 303, 1490 (2004), doi:10.1126/science.1091806, https://www. science.org/doi/pdf/10.1126/science.1091806

    2004 doi

  14. [22]

    Senthil,Deconfined quantum critical points: a review (2023), 2306.12638

    T . Senthil,Deconfined quantum critical points: a review (2023), 2306.12638

    2023 arXiv

  15. [23]

    Wiegmann, Exact solution of the O(3) nonlinear σ-model, Phys

    P . Wiegmann, Exact solution of the O(3) nonlinear σ-model, Phys. Lett. B 152, 209 (1985), doi:10.1016 /0370-2693(85)91171-2

    1985

  16. [24]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, time reversal and tem- perature, JHEP 2017, 91 (2017), doi:10.1007 /JHEP05(2017)091

    2017

  17. [25]

    M. A. Metlitski and R. Thorngren,Intrinsic and emergent anomalies at deconfined critical points, Phys. Rev. B 98, 085140 (2018), doi:10.1103 /PhysRevB.98.085140

    2018

  18. [26]

    ’t Hooft,Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, vol

    G. ’t Hooft,Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, vol. Recent Developments in Gauge Theories. Proceedings of Nato Advanced Study Institute in Cargese of NATO Advanced Study Institutes Series, p. 135, Plenum (1980)

    1980

  19. [27]

    E. Lieb, T . Schultz and D. Mattis,Two soluble models of an antiferromagnetic chain, Ann. Phys. 16, 407 (1961), doi:DOI: 10.1016 /0003-4916(61)90115-4

    1961

  20. [28]

    Affleck and F

    I. Affleck and F . D. M. Haldane,Critical theory of quantum spin chains, Phys. Rev. B 36, 5291 (1987), doi:10.1103 /PhysRevB.36.5291

    1987

  21. [29]

    Witten, Non-abelian bosonization in two dimensions , Comm

    E. Witten, Non-abelian bosonization in two dimensions , Comm. Math. Phys. 92, 455 (1984), doi:10.1007 /BF01215276

    1984

  22. [31]

    Shankar and N

    R. Shankar and N. Read, Theθ =π nonlinear sigma model is massless , Nucl. Phys. B 336, 457 (1990), doi:https: //doi.org/10.1016/0550-3213(90)90437-I

    1990 doi

  23. [32]

    Zamolodchikov and A

    A. Zamolodchikov and A. Zamolodchikov, Massless factorized scattering and sigma models with topological terms , Nucl. Phys. B 379, 602 (1992), doi:https://doi.org/10.1016/0550-3213(92)90136-Y. 39 SciPost Physics Submission

    1992 doi

  24. [33]

    Bietenholz, A

    W . Bietenholz, A. Pochinsky and U. J. Wiese, Meron-cluster simulation of the θ vacuum in the 2D O(3) model , Phys. Rev. Lett. 75, 4524 (1995), doi:10.1103/PhysRevLett.75.4524

    1995 doi

  25. [34]

    Azcoiti, G

    V . Azcoiti, G. Di Carlo, E. Follana and M. Giordano,Critical behavior of the O(3) nonlinear sigma model with topological term atθ =π from numerical simulations, Phys. Rev. D 86, 096009 (2012), doi:10.1103 /PhysRevD.86.096009

    2012

  26. [35]

    Allés, M

    B. Allés, M. Giordano and A. Papa, Behavior near θ = π of the mass gap in the two-dimensional O(3) nonlinear sigma model , Phys. Rev. B 90, 184421 (2014), doi:10.1103/PhysRevB.90.184421

    2014 doi

  27. [36]

    Caspar and H

    S. Caspar and H. Singh, From asymptotic freedom to θ vacua: Qubit embed- dings of the O(3) nonlinear σ model, Phys. Rev. Lett. 129, 022003 (2022), doi:10.1103/PhysRevLett.129.022003

    2022 doi

  28. [37]

    W . Tang, X. C. Xie, L. Wang and H.-H. Tu, Tensor network simulation of the ( 1 + 1)- dimensional O(3) nonlinearσ-model withθ =π term, Phys. Rev. D104, 114513 (2021), doi:10.1103/PhysRevD.104.114513

    2021 doi

  29. [38]

    J. Y. Araz, S. Schenk and M. Spannowsky , Toward a quantum simulation of non- linear sigma models with a topological term , Phys. Rev. A 107, 032619 (2023), doi:10.1103/PhysRevA.107.032619

    2023 doi

  30. [39]

    Fukushima, T

    K. Fukushima, T . Shimazaki and Y. Tanizaki, Exploring the θ-vacuum struc- ture in the functional renormalization group approach , JHEP 2022, 40 (2022), doi:10.1007/JHEP04(2022)040

    2022 doi

  31. [40]

    Haruna, K

    J. Haruna, K. Shimizu and M. Yamada, Functional renormalization group analysis of O(3) nonlinear sigma model and non-abelian bosonization duality (2024), 2405.14627

    2024 arXiv

  32. [41]

    M. R. Zirnbauer, On the infrared limit of the O(3) nonlinear σ-model atθ =π (2024), 2408.12215

    2024 arXiv

  33. [42]

    A. J. McRoberts, C. Hooley and A. G. Green,The J1-J2 spin chain and the O(3) non-linear sigma model atθ =π (2024), 2411.08095

    2024

  34. [44]

    Affleck, D

    I. Affleck, D. Bykov and K. Wamer, Flag manifold sigma models: Spin chains and inte- grable theories, Phys. Rep. 953, 1 (2022), doi:10.1016 /j.physrep.2021.09.004

    2022

  35. [45]

    Affleck, The quantum hall effects,σ-models atθ =π and quantum spin chains, Nucl

    I. Affleck, The quantum hall effects,σ-models atθ =π and quantum spin chains, Nucl. Phys. B 257, 397 (1985), doi:https: //doi.org/10.1016/0550-3213(85)90353-0

    1985 doi

  36. [46]

    Read and S

    N. Read and S. Sachdev, Some features of the phase diagram of the square lattice SU(N) antiferromagnets, Nucl. Phys. B 316, 609 (1989), doi:10.1016 /0550-3213(89)90061- 8

    1989

  37. [47]

    S.-Y. Wang, D. Wang and Q.-H. Wang, Critical exponents of the nonlinear sigma model on a grassmann manifold U(n)/U(m)U(n− m) by the 1/n-expansion, Phys. Rev. B 99, 165142 (2019), doi:10.1103 /PhysRevB.99.165142. 40 SciPost Physics Submission

    2019

  38. [48]

    B. B. Beard, M. Pepe, S. Riederer and U.-J. Wiese, Study of CP(n− 1) θ-vacua by cluster simulation of SU(n) quantum spin ladders, Phys. Rev. Lett. 94, 010603 (2005), doi:10.1103/PhysRevLett.94.010603

    2005 doi

  39. [49]

    Laflamme, W

    C. Laflamme, W . Evans, M. Dalmonte, U. Gerber, H. Mejía-Díaz, W . Bietenholz, U.-J. Wiese and P . Zoller,CP(N−1) quantum field theories with alkaline-earth atoms in optical lattices, Ann. Phys. 370, 117 (2016), doi:https://doi.org/10.1016/j.aop.2016.03.012

    2016 doi

  40. [50]

    Evans, U

    W . Evans, U. Gerber, M. Hornung and U.-J. Wiese,SU(3) quantum spin ladders as a regu- larization of the CP(2) model at non-zero density: From classical to quantum simulation, Ann. Phys. 398, 94 (2018), doi:https: //doi.org/10.1016/j.aop.2018.09.002

    2018 doi

  41. [51]

    Nguyen, Y

    M. Nguyen, Y. Tanizaki and M. Ünsal, Windingθ and destructive interference of instan- tons, JHEP 2023, 33 (2023), doi:10.1007 /JHEP09(2023)033

    2023

  42. [52]

    Bykov, Haldane limits via Lagrangian embeddings , Nucl

    D. Bykov, Haldane limits via Lagrangian embeddings , Nucl. Phys. B 855, 100 (2012), doi:https://doi.org/10.1016/j.nuclphysb.2011.10.005

    2012 doi

  43. [53]

    Bykov, The geometry of antiferromagnetic spin chains, Comm

    D. Bykov, The geometry of antiferromagnetic spin chains, Comm. Math. Phys. 322, 807 (2013), doi:10.1007 /s00220-013-1702-5

    2013

  44. [54]

    Lajkó, K

    M. Lajkó, K. Wamer, F . Mila and I. Affleck, Generalization of the Hal- dane conjecture to SU(3) chains , Nucl. Phys. B 924, 508 (2017), doi:https://doi.org/10.1016/j.nuclphysb.2017.09.015

    2017 doi

  45. [55]

    Wamer, M

    K. Wamer, M. Lajkó, F . Mila and I. Affleck, Generalization of the Hal- dane conjecture to su(n) chains , Nucl. Phys. B 952, 114932 (2020), doi:https://doi.org/10.1016/j.nuclphysb.2020.114932

    2020

  46. [56]

    Wamer and I

    K. Wamer and I. Affleck, Flag manifold sigma models from SU(n) chains, Nucl. Phys. B 959, 115156 (2020), doi:https: //doi.org/10.1016/j.nuclphysb.2020.115156

    2020

  47. [57]

    Fendley ,Integrable sigma models with θ = π, Phys

    P . Fendley ,Integrable sigma models with θ = π, Phys. Rev. B 63, 104429 (2001), doi:10.1103/PhysRevB.63.104429

    2001 doi

  48. [58]

    Fendley ,Integrable sigma models and perturbed coset models, JHEP 2001, 050 (2001), doi:10.1088/1126-6708/2001/05/050

    P . Fendley ,Integrable sigma models and perturbed coset models, JHEP 2001, 050 (2001), doi:10.1088/1126-6708/2001/05/050

    2001 doi

  49. [59]

    Mariño, R

    M. Mariño, R. Miravitllas and T . Reis, Instantons, renormalons and the theta angle in integrable sigma models , SciPost Phys. 15, 184 (2023), doi:10.21468/SciPostPhys.15.5.184

    2023 doi

  50. [60]

    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 generalizations, Phys. Rev. B 98, 115126 (2018), doi:10.1103 /PhysRevB.98.115126

    2018

  51. [61]

    Ohmori, N

    K. Ohmori, N. Seiberg and S.-H. Shao, Sigma models on flags , SciPost Phys. 6, 17 (2019), doi:10.21468 /SciPostPhys.6.2.017

    2019

  52. [62]

    Seiberg, Topology in strong coupling , Phys

    N. Seiberg, Topology in strong coupling , Phys. Rev. Lett. 53, 637 (1984), doi:10.1103/PhysRevLett.53.637

    1984 doi

  53. [63]

    Andrei and J

    N. Andrei and J. H. Lowenstein, Diagonalization of the chiral-invariant gross-neveu hamiltonian, Phys. Rev. Lett. 43, 1698 (1979), doi:10.1103 /PhysRevLett.43.1698. 41 SciPost Physics Submission

    1979

  54. [65]

    Berg and P

    B. Berg and P . Weisz,Exact s-matrix of the chiral invariant su(n) thirring model , Nucl. Phys. B 146, 205 (1978), doi:https: //doi.org/10.1016/0550-3213(78)90438-8

    1978 doi

  55. [66]

    Köberle, V

    R. Köberle, V . Kurak and J. A. Swieca, Scattering theory and 1 N expansion in the chiral gross-neveu model, Phys. Rev. D 20, 897 (1979), doi:10.1103 /PhysRevD.20.897

    1979

  56. [67]

    Di Francesco, P

    P . Di Francesco, P . Mathieu and D. Sénéchal,Conformal Field Theory, Springer Verlag (1996)

    1996

  57. [68]

    S. C. Furuya and M. Oshikawa, Symmetry protection of critical phases and a global anomaly in 1 + 1 dimensions, Phys. Rev. Lett. 118, 021601 (2017), doi:10.1103/PhysRevLett.118.021601

    2017 doi

  58. [69]

    Kutasov, String theory and the non-abelian thirring model , Phys

    D. Kutasov, String theory and the non-abelian thirring model , Phys. Lett. B 227, 68 (1989), doi:https: //doi.org/10.1016/0370-2693(89)91285-9

    1989 doi

  59. [70]

    Gerganov, A

    B. Gerganov, A. LeClair and M. Moriconi, Beta function for anisotropic current interac- tions in 2d, Phys. Rev. Lett. 86, 4753 (2001), doi:10.1103 /PhysRevLett.86.4753

    2001

  60. [72]

    Johannesson, The structure of low-lying excitations in a new integrable quantum chain model, Nucl

    H. Johannesson, The structure of low-lying excitations in a new integrable quantum chain model, Nucl. Phys. B 270, 235 (1986)

    1986

  61. [73]

    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, 180201 (2019), doi:10.1103/PhysRevLett.123.180201

    2019 doi

  62. [74]

    Nataf, S

    P . Nataf, S. Gozel and F . Mila, Edge states and universality class of the crit- ical two-box symmetric SU(3) chain , Phys. Rev. B 104, L180411 (2021), doi:10.1103/PhysRevB.104.L180411

    2021 doi

  63. [75]

    Affleck, Exact critical exponents for quantum spin chains, non-linear σ-models at θ = π and the quantum Hall effect , Nucl

    I. Affleck, Exact critical exponents for quantum spin chains, non-linear σ-models at θ = π and the quantum Hall effect , Nucl. Phys. B 265, 409 (1986), doi:http://dx.doi.org/10.1016/0550-3213(86)90167-7

    1986 doi

  64. [76]

    Gogolin, A

    A. Gogolin, A. Nersesyan and A. Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press (1998)

    1998

  65. [77]

    A. J. A. James, R. M. Konik, P . Lecheminant, N. J. Robinson and A. M. Tsvelik, Non- perturbative methodologies for low-dimensional strongly-correlated systems: From non- Abelian bosonization to truncated spectrum methods , Rep. Prog. Phys. 81, 046002 (2018), doi:10.1088 /1361...

    2018

  66. [78]

    Coleman, J

    S. Coleman, J. Wess and B. Zumino,Structure of phenomenological lagrangians. i, Phys. Rev.177, 2239 (1969), doi:10.1103 /PhysRev.177.2239

    1969

  67. [79]

    C. G. Callan, S. Coleman, J. Wess and B. Zumino, Structure of phenomenological la- grangians. ii, Phys. Rev. 177, 2247 (1969), doi:10.1103 /PhysRev.177.2247. 42 SciPost Physics Submission

    1969

  68. [80]

    D. S. Freed and G. W . Moore, Setting the quantum integrand of M-theory , Commun. Math. Phys. 263, 89 (2006), doi:10.1007 /s00220-005-1482-7, hep-th /0409135

    2006

  69. [81]

    D. S. Freed and M. J. Hopkins, Reflection positivity and invertible topological phases , Geom. Topol. 25, 1165 (2021), doi:10.2140 /gt.2021.25.1165, 1604.06527

    2021 arXiv

  70. [82]

    D. S. Freed, Z. Komargodski and N. Seiberg, The Sum Over Topological Sectors and θ in the 2 +1-Dimensional CP1 σ-Model, Comm. Math. Phys. 362, 167 (2018), doi:10.1007/s00220-018-3093-0, 1707.05448

    2018 arXiv

  71. [83]

    Y. Lee, K. Ohmori and Y. Tachikawa,Revisiting Wess-Zumino-Witten terms, SciPost Phys. 10, 061 (2021), doi:10.21468 /SciPostPhys.10.3.061, 2009.00033

    2021 arXiv

  72. [84]

    Kobayashi, Y

    R. Kobayashi, Y. Lee, K. Shiozaki and Y. Tanizaki, Topological terms of (2 +1)d flag- manifold sigma models, JHEP 08, 075 (2021), doi:10.1007 /JHEP08(2021)075, 2103. 05035

    2021

  73. [85]

    Wen, Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders, Phys

    X.-G. Wen, Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders, Phys. Rev. D 88, 045013 (2013), doi:10.1103 /PhysRevD.88.045013, 1303.1803

    2013 arXiv

  74. [86]

    Kapustin, Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology (2014), 1403.1467

    A. Kapustin, Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology (2014), 1403.1467

    2014 arXiv

  75. [87]

    Yonekura, On the cobordism classification of symmetry protected topological phases , Comm

    K. Yonekura, On the cobordism classification of symmetry protected topological phases , Comm. Math. Phys. 368, 1121 (2019), doi:10.1007 /s00220-019-03439-y, 1803. 10796

    2019

  76. [88]

    Polyakov and P

    A. Polyakov and P . Wiegmann, Goldstone fields in two dimensions with multival- ued actions , Phys. Lett. B 141, 223 (1984), doi:https: //doi.org/10.1016/0370- 2693(84)90206-5

    1984 doi

  77. [89]

    Lecheminant and K

    P . Lecheminant and K. Totsuka, The SU(N) self-dual sine-gordon model and competing orders, JSTAT 2006, L12001 (2006), doi:10.1088 /1742-5468/2006/12/L12001

    2006

  78. [90]

    Itoi and M.-H

    C. Itoi and M.-H. Kato, Extended massless phase and the Haldane phase in a spin-1 isotropic antiferromagnetic chain , Phys. Rev. B 55, 8295 (1997), doi:10.1103/PhysRevB.55.8295

    1997 doi

  79. [91]

    Boulat, P

    E. Boulat, P . Azaria and P . Lecheminant, Duality approach to one- dimensional degenerate electronic systems , Nucl. Phys. B 822, 367 (2009), doi:http://dx.doi.org/10.1016/j.nuclphysb.2009.06.020

    2009 doi

  80. [92]

    Komatsu, R

    S. Komatsu, R. Mahajan and S.-H. Shao, An index for quantum integrability , SciPost Phys. 7, 065 (2019), doi:10.21468 /SciPostPhys.7.5.065

    2019

  81. [93]

    Corboz, A

    P . Corboz, A. M. Läuchli, K. Totsuka and H. Tsunetsugu, Spontaneous trimeriza- tion in a bilinear-biquadratic s = 1 zig-zag chain, Phys. Rev. B 76, 220404 (2007), doi:10.1103/PhysRevB.76.220404

    2007 doi

  82. [94]

    Babichenko, Quantum integrability ofσ models on aii and cii symmetric spaces, Phys

    A. Babichenko, Quantum integrability ofσ models on aii and cii symmetric spaces, Phys. Lett. B 554, 96 (2003), doi:https: //doi.org/10.1016/S0370-2693(02)03268-9

    2003 doi

  83. [95]

    Babichenko and R

    A. Babichenko and R. Tateo,Thermodynamic bethe ansatz for the aii sigma-models, Phys. Lett. B 573, 239 (2003), doi:https: //doi.org/10.1016/j.physletb.2003.08.044. 43 SciPost Physics Submission

    2003 doi

  84. [96]

    D’Adda, M

    A. D’Adda, M. Lüscher and P . Di Vecchia,A 1/n expandable series of non-linearσ models with instantons, Nucl. Phys. B 146, 63 (1978), doi:10.1016 /0550-3213(78)90432-7

    1978

  85. [97]

    MacFarlane, Generalizations ofσ-models and cpN models, and instantons, Phys

    A. MacFarlane, Generalizations ofσ-models and cpN models, and instantons, Phys. Lett. B 82, 239 (1979), doi:https: //doi.org/10.1016/0370-2693(79)90745-7

    1979 doi

  86. [98]

    R. D. Pisarski, Nonlinearσ models of symmetric spaces, Phys. Rev. D 20, 3358 (1979), doi:10.1103/PhysRevD.20.3358

    1979 doi

  87. [99]

    Dubois-Violette and Y

    M. Dubois-Violette and Y. Georgelin, Gauge theory in terms of projector valued fields , Phys. Lett. B 82, 251 (1979), doi:https: //doi.org/10.1016/0370-2693(79)90748-2

    1979 doi

  88. [100]

    Brézin, C

    E. Brézin, C. Itzykson, J. Zinn-Justin and J.-B. Zuber, Remarks about the exis- tence of non-local charges in two-dimensional models , Phys. Lett. B 82, 442 (1979), doi:https://doi.org/10.1016/0370-2693(79)90263-6

    1979 doi

  89. [102]

    A. M. M. Pruisken, B. Danu and R. Shankar, Superuniversality of dimerized SU(n + m) spin chains, Phys. Rev. B 105, 155111 (2022), doi:10.1103 /PhysRevB.105.155111

    2022

  90. [103]

    Nguyen and H

    M. Nguyen and H. Singh,Lattice regularizations ofθ vacua: Anomalies and qubit models, Phys. Rev. D107, 014507 (2023), doi:10.1103 /PhysRevD.107.014507

    2023

  91. [104]

    Sulejmanpasic and Y

    T . Sulejmanpasic and Y. Tanizaki, C-P-T anomaly matching in bosonic quantum field theory and spin chains , Phys. Rev. B 97, 144201 (2018), doi:10.1103/PhysRevB.97.144201

    2018 doi

  92. [105]

    Kikuchi and Y

    Y. Kikuchi and Y. Tanizaki, Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics, PTEP 2017, 113B05 (2017), doi:10.1093 /ptep/ptx148, https: //academic.oup.com/ptep/article-pdf/2017/11/113B05/22090624/ptx148.pdf

    2017

  93. [106]

    Komargodski, A

    Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou,Comments on abelian Higgs mod- els and persistent order, SciPost Phys.6, 003 (2019), doi:10.21468/SciPostPhys.6.1.003

    2019 doi

  94. [107]

    Affleck, Exact results on the dimerisation transition in SU(n) antiferromagnetic chains, J

    I. Affleck, Exact results on the dimerisation transition in SU(n) antiferromagnetic chains, J. Phys. Condens. Matter 2, 405 (1990), doi:10.1088 /0953-8984/2/2/016

    1990

  95. [108]

    Klümper, The spectra of q-state vertex models and related antiferromagnetic quantum spin chains, J

    A. Klümper, The spectra of q-state vertex models and related antiferromagnetic quantum spin chains, J. Phys. A: Math. Gen.23, 809 (1990), doi:10.1088/0305-4470/23/5/023

    1990 doi

  96. [109]

    G. V . Dunne, Y. Tanizaki and M. Ünsal, Quantum distillation of Hilbert spaces, semi-classics and anomaly matching , JHEP 2018, 68 (2018), doi:10.1007/JHEP08(2018)068

    2018 doi

  97. [110]

    Nonne, P

    H. Nonne, P . Lecheminant, S. Capponi, G. Roux and E. Boulat, Competing orders in one-dimensional half-filled multicomponent fermionic cold atoms: The Haldane-charge conjecture, Phys. Rev. B 84, 125123 (2011), doi:10.1103 /PhysRevB.84.125123

    2011

  98. [111]

    Dufour, P

    J. Dufour, P . Nataf and F . Mila,Variational monte carlo investigation ofSU(n) Heisenberg chains, Phys. Rev. B 91, 174427 (2015), doi:10.1103 /PhysRevB.91.174427

    2015

  99. [112]

    A. V . Onufriev and J. B. Marston,Enlarged symmetry and coherence in arrays of quantum dots, Phys. Rev. B 59, 12573 (1999), doi:10.1103 /PhysRevB.59.12573. 44 SciPost Physics Submission

    1999

  100. [113]

    Assaraf, P

    R. Assaraf, P . Azaria, E. Boulat, M. Caffarel and P . Lecheminant,Dynamical symmetry enlargement versus spin-charge decoupling in the one-dimensional SU(4) Hubbard model, Phys. Rev. Lett.93, 016407 (2004), doi:10.1103 /PhysRevLett.93.016407

    2004

  101. [114]

    Paramekanti and J

    A. Paramekanti and J. B. Marston, SU(N) quantum spin models: a variational wave- function study , J. Phys. Condens. Matter 19, 125215 (2007), doi:10.1088 /0953- 8984/19/12/125215

    2007

  102. [115]

    Boyanovsky and R

    D. Boyanovsky and R. Holman, Critical behavior and duality in extended sine- Gordon theories, Nucl. Phys. B 358, 619 (1991), doi:https: //doi.org/10.1016/0550- 3213(91)90427-Y

    1991 doi

  103. [116]

    Assaraf, P

    R. Assaraf, P . Azaria, M. Caffarel and P . Lecheminant, Metal-insulator transition in the one-dimensional SU(N ) Hubbard model , Phys. Rev. B 60, 2299 (1999), doi:10.1103/PhysRevB.60.2299

    1999 doi

  104. [117]

    S. R. Manmana, K. R. A. Hazzard, G. Chen, A. E. Feiguin and A. M. Rey , SU(N ) mag- netism in chains of ultracold alkaline-earth-metal atoms: Mott transitions and quantum correlations, Phys. Rev. A 84, 043601 (2011), doi:10.1103 /PhysRevA.84.043601

    2011

  105. [118]

    Fuji and P

    Y. Fuji and P . Lecheminant,Non-Abelian SU (N−1)-singlet fractional quantum Hall states from coupled wires, Phys. Rev. B95, 125130 (2017), doi:10.1103/PhysRevB.95.125130. 45

    2017 doi

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