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

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 $(-1)<sup>{Θ/π}=-1$, 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 $$\mathbb{R}\mathbb{P}^2\cong \mathbb{S}<sup>2/\mathbb{Z}_2$$, 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 $(-1)<sup>{Θ/π}$ 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.

• The paper establishes that antiperiodic boundary conditions in spin chains induce a pure 't Hooft anomaly obstructing the gauging of Z2 inversion symmetry.
• It presents a detailed construction of path integral invariants in the O(3) NLSM, showing sign ambiguities of topological terms at Θ = π.
• Semiclassical analysis reveals that ground state properties, such as crystal momentum and degeneracies, depend sensitively on system size and spin value.

Nonlinear Sigma Models, Antiperiodic Boundary Conditions, Spin Chains, and 't Hooft Anomalies

Overview

This paper provides a rigorous and comprehensive analysis of the relationship between quantum antiferromagnetic $SU(2)$ spin chains and the $O(3)$ nonlinear sigma model (NLSM) in 1+1 dimensions, particularly under the influence of antiperiodic boundary conditions (b.c.s) and their interplay with 't Hooft anomalies. The study systematically generalizes the paradigmatic mapping between spin chain systems and effective field theories with topological terms. It introduces and explores the effects of antiperiodic (A) b.c.s (in the spatial—corresponding to odd-length spin chains—or temporal directions), addressing their consequences for gauging discrete symmetries, modular invariance, and the existence of well-defined orbifold (gauged) theories.

Main Contributions and Results

The analysis begins with a review of the well-established correspondence: at large spin $S$ and large system size $N$, the low-energy limit of quantum antiferromagnetic $SU(2)$ spin chains maps to the $O(3)$ NLSM with topological coefficient $\Theta = 2\pi S$. For even $N$ and periodic b.c.s, the model is well-understood; the spectrum and criticality depend on whether $\Theta=0$ (integer $S$) or $O(3)$0 (half-integer $O(3)$1), with the latter yielding gapless behavior in accordance with the Lieb-Schultz-Mattis theorem.

The paper's major advance is a comprehensive study of the implications and obstructions that arise for odd $O(3)$2 and antiperiodic boundary conditions (in either space or time), specifically when considering the discrete $O(3)$3 (inversion) symmetry $O(3)$4 of the NLSM. The authors establish:

• Existence of a Pure 't Hooft Anomaly: When $O(3)$5 (mod $O(3)$6), antiperiodic b.c.s introduce a global anomaly that obstructs the gauging of the $O(3)$7 inversion symmetry; the partition function in the presence of an antiperiodic flux fails to be gauge invariant. Consequently, the orbifold theory by this symmetry (the real projective plane $O(3)$8 sigma model with topological term) cannot be consistently defined for $O(3)$9. This obstruction is also manifest as a modular anomaly of the partition functions: modular invariance for the full set of boundary conditions is impossible except for the trivial ($S$0) case.
• Explicit Construction of Path Integral Invariants: The authors provide a detailed prescription for extending the definition of the topological term (the theta term, instanton number) in the presence of branch cuts (antiperiodic b.c.s), leading to a topological invariant $S$1 which is only gauge invariant for $S$2 (mod $S$3). For $S$4 (mod $S$5), they show that this invariant acquires a sign ambiguity under certain gauge transformations.
• Semiclassical and Exact Results for Spin Chains: Employing semiclassical analysis, the work derives the crystal momentum and discrete symmetry properties of the ground state of spin chains with odd and even $S$6, matching and explaining rigorous and numerical results. The ground-state crystal momentum and degeneracies are shown to depend sensitively on $S$7 (modulo $S$8) and $S$9 (modulo $N$0), arising from Berry phase effects associated with the non-trivial topology of the order parameter trajectory.
• Lifts of Discrete Symmetries and Double Cover Groups: The inversion symmetry, while absent as a microscopic symmetry in the spin chain, emerges as an effective symmetry in the low-energy field theory and is promoted to an element of the double cover of $N$1, specifically of $N$2 for the NLSM with $N$3 and A b.c.s. The modular properties of partition functions and the resulting symmetry algebra require this double cover, leading to half-integer spin multiplets and non-trivial projective actions.
• Consistency with Conformal Field Theory in the IR: For half-integer spin chains ($N$4), the IR fixed point maps to $N$5 WZW conformal field theory, and the anomaly analysis at the field theory level matches Gepner and Witten's and subsequent studies on orbifolds, modular invariance, and symmetry-gauged partition functions.
• Generalization to Other Lie Groups and Anisotropies: The framework is broadened to chains with other continuous symmetry groups (general semisimple Lie groups), identifying criteria (in terms of the square of the time-reversal operator on a single site and the behavior under the center of the group) that determine when a pure $N$6 anomaly—and thus the obstruction to a well-defined gauged sigma model—arises.

Detailed Discussion and Technical Claims

The paper's construction is anchored by the following crucial technical points:

• Path Integral Formulation With Branch Cuts: The impact of A b.c.s is precisely formulated through the introduction of $N$7 gauge field background fluxes (encoded as branch cuts in the path integral), and a careful analysis demonstrates that for $N$8 (mod $N$9), the obstruction to full gauge invariance is encoded in the non-invariance of the “Berry phase” (topological term) factor.
• Spectral Properties and Projective Representations: Detailed semiclassical quantization of the NLSM with A b.c.s shows that all states carry half-integer spin and appear in representations of $SU(2)$0, with translation by $SU(2)$1 acting projectively. The spectrum splits into sectors with quantized momenta $SU(2)$2, consistent with the anomaly in modular invariance.
• Operator Content in Conformal and Massive Phases: The mapping from spin chain operators to continuum primaries and the identification of twists and inversion symmetries are described in the context of both conformal limits ($SU(2)$3; CFT) and massive phases ($SU(2)$4; massive $SU(2)$5 NLSM and its $SU(2)$6 orbifold), with explicit expressions for partition function decompositions in terms of conformal characters. Only for $SU(2)$7 is the full set of boundary conditions modular and gauge invariant, allowing consistent orbifold/gauged theory construction.
• Implications for General Symmetry Groups: In chains with other continuous symmetry groups, the presence or absence of the pure inversion ($SU(2)$8) anomaly is shown to be governed by the square of the time reversal transformation on the representation at a single site, and its non-triviality is dictated by the values of the associated topological angles.

Implications and Future Directions

The results have significant consequences for the classification and realization of topologically ordered phases and symmetry-protected topological (SPT) states in one dimension. Most notably:

• The explicit demonstration of the pure $SU(2)$9 anomaly in the $O(3)$0 NLSM with $O(3)$1 provides a field-theoretic underpinning for the impossibility of consistently orbifolding/gauging certain symmetries, a fact that is crucial in understanding the protection mechanisms for gapless edge states and criticality in SPT chains.
• The mapping between lattice translation and emergent inversion symmetry clarifies the nature of effective low-energy symmetries and their projective character, informing the construction of effective field theories for higher-symmetry chains and their low-energy limits.
• The modular properties and consistency conditions on partition functions highlight constraints on possible boundary conditions and the allowable operator content in CFTs arising from SPT systems.
• The extension to general symmetry groups and the explicit criterion for the anomaly in terms of the square of time-reversal symmetries open avenues for classifying anomalies in more complex sigma models and potentially in higher-dimensional analogs.

Conclusion

The paper accomplishes a rigorous and explicit characterization of the pure $O(3)$2 (inversion) anomaly in the $O(3)$3 NLSM with topological term $O(3)$4 and the corresponding lattice spin chains with odd $O(3)$5. It identifies the precise field-theoretic origin of the anomaly, demonstrates its manifestation in modular and gauge noninvariance, and connects these topological features to semiclassical and quantum properties of spin chains. The work provides a unified framework for anomalies in sigma models associated with antiferromagnetic spin chains and their generalizations, with direct implications for the structure of SPT phases, orbifold conformal field theories, and the interplay of lattice and continuum symmetries.

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References:

• For foundational background: Haldane, F.D.M., "Nonlinear field theory of large-spin Heisenberg antiferromagnets: semi-classically quantized solitons of the one-dimensional easy-axis Néel state", Phys. Lett. A 93, 464 (1983) [HALDANE1983464].
• Discussion of modular invariance and orbifold construction: Gepner, D. and Witten, E., "String Theory on Group Manifolds", Nucl. Phys. B 278 (1986) 493 [gw86].
• Context for anomaly inflow: Seiberg, N. et al., “Anomalies, Group Cohomology, and Topological Phases”, (Govindarajulu et al., 2017) [Seiberg1703].
• For related discussion in the context of 1D SPT phases and anomalies: Cheng, M. and Seiberg, N., “Fermionic Quotients of Quantum Field Theories and Spin Structures,” (Benestad et al., 2023) [ChengSeiberg].

See also (2606.07041) for the full paper text and technical appendices.