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arXiv:2606.07041

Nonlinear sigma models, antiperiodic boundary conditions, spin chains, and 't Hooft anomalies

N. Read, Hubert Saleur

Published Jun 5, 2026• 0 citations
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Abstract

We consider two sets of related models: initially, these are $SU(2)$ antiferromagnetic spin chains with $N$ sites of spin $S$, and the $O(3)$ nonlinear sigma model in two dimensions with topological coefficient $Θ$ a multiple of $π$ (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with $Θ=2πS$. We study these models with $N$ odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the $\mathbb{Z}_2$ inversion symmetry $\vec{n}\to-\vec{n}$, and amounts to a flux of a $\mathbb{Z}_2$ gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the $\mathbb{Z}_2$ inversion symmetry. We show directly that, if and only if $(-1)^{Θ/π}=-1$, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space $\mathbb{R}\mathbb{P}^2\cong \mathbb{S}^2/\mathbb{Z}_2$, and hence this model does not exist for $Θ=π$ (mod $2π$). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on $N$ modulo $4$ and $2S$ modulo $2$. For a large class of spin chains and associated sigma models we find similar results, but now $(-1)^{Θ/π}$ is replaced by the value $\pm 1$ of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.

DOI
10.48550/arxiv.2606.07041

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We consider two sets of related models: initially, these are SU(2)SU(2)SU(2) antiferromagnetic spin chains with NNN sites of spin SSS, and the O(3)O(3)O(3) nonlinear sigma model in two dimensions with topological coefficient ΘΘΘ a multiple of πππ (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with Θ=2πSΘ=2πSΘ=2πS. We study these models with NNN odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the Z2\mathbb{Z}_2Z2​ inversion symmetry n⃗→−n⃗\vec{n}\to-\vec{n}n→−n, and amounts to a flux of a Z2\mathbb{Z}_2Z2​ gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the Z2\mathbb{Z}_2Z2​ inversion symmetry. We show directly that, if and only if (−1)Θ/π=−1(-1)^{Θ/π}=-1(−1)Θ/π=−1, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space RP2≅S2/Z2\mathbb{R}\mathbb{P}^2\cong \mathbb{S}^2/\mathbb{Z}_2RP2≅S2/Z2​, and hence this model does not exist for Θ=πΘ=πΘ=π (mod 2π2π2π). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on NNN modulo 444 and 2S2S2S modulo 222. For a large class of spin chains and associated sigma models we find similar results, but now (−1)Θ/π(-1)^{Θ/π}(−1)Θ/π is replaced by the value ±1\pm 1±1 of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.
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CCC-PPP-TTT anomaly in bosonic systems
We consider the O(3)O(3)O(3) nonlinear sigma model with the θ\thetaθ-term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any θ\thetaθ, and enjoys the periodicity in θ→θ+2π\theta\rightarrow \theta+2\piθ→θ+2π. At θ=0,π\theta=0,\piθ=0,π it also has a charge-conjugation CCC-symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable RP2\mathbb RP^2RP2 manifold, after which the 2π2\pi2π periodicity of θ\thetaθ and the CCC symmetry at θ=π\theta=\piθ=π are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation CCC, parity PPP, and time-reversal TTT symmetries when θ=π\theta=\piθ=π. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as CCC, PPP and TTT are all good symmetries of the theory. We make several consistency checks with various semi-classical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity and time reversal (but not involving the SO(3)SO(3)SO(3)-spin symmetry), requiring that the ground state is never trivially gapped, even if SO(3)SO(3)SO(3) spin symmetry is explicitly and completely broken. We also consider generalizations to CPN−1\mathbb{C}P^{N-1}CPN−1 models and show that the CCC-PPP-TTT anomaly exists for even NNN.
It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear $\ensuremath{\sigma}$ models $(\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms})$ of the symmetry group GGG with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology Hd(G,R/Z){\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})Hd(G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}) \mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^{d}[G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in ddd-dimensional space-time are described by ${E}^{d}(G)\ensuremath{\rtimes}{\ensuremath{\bigoplus}}_{k=1}^{d\ensuremath{-}1}{\mathcal{H}}^{k}(G,{\text{iTO}}_{L}^{d\ensuremath{-}k})\ensuremath{\bigoplus}{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ where GGG may contain time reversal. Here iTOLd{\text{iTO}}_{L}^{d}iTOLd​ is the set of the topologically ordered phases in ddd-dimensional space-time that have no topological excitations, and one has iTOL1=iTOL2=iTOL4=iTOL6=0,iTOL3=Z,iTOL5=Z2,iTOL7=2Z{\text{iTO}}_{L}^{1}={\text{iTO}}_{L}^{2}={\text{iTO}}_{L}^{4}={\text{iTO}}_{L}^{6}=0,\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{3}=\mathbb{Z},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{5}={\mathbb{Z}}_{2},\phantom{\rule{0.16em}{0ex}}{\text{iTO}}_{L}^{7}=2\mathbb{Z}iTOL1​=iTOL2​=iTOL4​=iTOL6​=0,iTOL3​=Z,iTOL5​=Z2​,iTOL7​=2Z. For $G=U(1)\ensuremath{\rtimes}{Z}_{2}^{T}$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^{d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},\mathbb{R}/\mathbb{Z}]$ are described by Z2{\mathbb{Z}}_{2}Z2​ in 3 + 1D, Z\mathbb{Z}Z in 4 + 1D, 3Z23{\mathbb{Z}}_{2}3Z2​ in 5 + 1D, and 4Z24{\mathbb{Z}}_{2}4Z2​ in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.
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We analyze lattice Hamiltonian systems whose global symmetries have ’t Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of “on-site action.”) In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the c=1 c = 1 compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an ’t Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.