Antiferromagnets, Sigma Models, and Theta Terms

The nonlinear sigma model describes slow rotations of an antiferromagnetic order parameter after the uniform magnetization has been integrated out. Its gradient terms encode stiffness and susceptibility; Berry phases leave a theta term whose value depends on the microscopic spin. The reduction is powerful only when its field normalization, topological charge, and gradient control are stated.

Required background. Spin Coherent States and Berry Phases supplies the lattice Berry phase; Sigma-Model Dynamics, Geometry, and Control supplies constrained fields and continuum dynamics. Helpful background. Theta and Topological Effects in Sigma Models supplies theta periodicity and nonperturbative sectors.

The one-dimensional antiferromagnetic map

For the nearest-neighbor chain

$$H=J\sum_j\mathbf S_j\cdot\mathbf S_{j+1}, \qquad J>0,$$

separate the slowly varying Néel unit vector $\mathbf n$ and transverse uniform magnetization $\mathbf L$,

$$\frac{\mathbf S_j}{S}\simeq(-1)^j\mathbf n(x_j) \sqrt{1-\frac{a^2\mathbf L^2}{S^2}} +\frac aS\mathbf L(x_j), \quad \mathbf n^2=1,\quad\mathbf n\cdot\mathbf L=0.$$

Expanding exchange to two gradients and combining neighboring Berry phases gives a term $i\mathbf L\cdot(\mathbf n\times\partial_\tau\mathbf n)$ plus a residual topological phase. The Gaussian integral over $\mathbf L$ yields

$$S_E=\frac1{2g}\int d\tau\,dx \left[c^{-1}(\partial_\tau\mathbf n)^2 +c(\partial_x\mathbf n)^2\right]+i\theta Q[\mathbf n],$$

with the leading large-$S$ matching

$$g=\frac2S, \qquad c=2JSa, \qquad \theta=2\pi S.$$

For spacetime compactified to a sphere,

$$Q[\mathbf n]=\frac1{4\pi}\int d\tau\,dx\, \mathbf n\cdot(\partial_\tau\mathbf n\times\partial_x\mathbf n)\in\mathbb Z.$$

Reversing the orientation of $(\tau,x)$ reverses $Q$ and hence the displayed sign of $\theta$; physics is unchanged after the convention is translated. Haldane obtained this integer/half-integer distinction directly from the coherent-state chain Haldane 1983, pp. 464–468.

What the theta angle changes

Because $Q$ is integer, the bulk partition function is periodic under $\theta\mapsto\theta+2\pi$. Integer $S$ maps to $\theta=0$ modulo $2\pi$ and the O(3) model is massive. Half-integer $S$ maps to $\theta=\pi$; destructive interference between topological sectors prevents the same featureless massive outcome. For the translation-invariant half-integer Heisenberg chain, the infrared theory is the critical SU(2)$_1$ WZW model with a marginally irrelevant interaction Affleck 1986, pp. 409–447.

This argument distinguishes the two classes but does not prove that every half-integer spin chain is gapless. Translation symmetry can break spontaneously, producing degenerate gapped states, and additional interactions can change the route. Nor does $\theta=0$ imply that all integer-spin phases are topologically equivalent when boundaries and protecting symmetries are retained.

Higher-dimensional antiferromagnets

For a bipartite antiferromagnet in $d$ spatial dimensions the smooth action has the form

$$S_E=\frac{\rho_s}{2}\int d\tau\,d^dx \left[c^{-2}(\partial_\tau\mathbf n)^2+(\nabla\mathbf n)^2\right],$$

with stiffness $\rho_s$ and velocity $c$. In $2+1$ dimensions, Berry phases are concentrated on lattice-scale hedgehog events rather than captured by simply appending the $1+1$-dimensional $Q$. Their sublattice phases can suppress some monopoles and influence valence-bond order. A smooth continuum treatment that discards such events is therefore conditional, not a proof that Berry phases are absent.

At nonzero temperature a two-dimensional isotropic magnet lacks true continuous-symmetry order in the thermodynamic limit, yet the renormalized-classical regime can have an exponentially large correlation length. The sigma-model matching of stiffness, susceptibility, and correlation length for quantum antiferromagnets is developed by Chakravarty, Halperin, and Nelson 1989, §§ II–IV.

Control and matching

The bare $g=2/S$ suggests semiclassical control at large $S$. For small $S$, the sigma model remains a symmetry-based long-distance theory, but numerical or exact information is needed to match its parameters and establish the phase. Gradient expansion requires correlation lengths large compared with $a$; frustration, strong anisotropy, or several soft ordering wave vectors may require additional fields. Topological terms must be derived before coarse graining singular configurations.

Exercises

1. Evaluate the theta phase for integer and half-integer $S$ in a sector of charge $Q=1$.

Solution

The factor is $e^{-i\theta Q}=e^{-i2\pi S}$. For integer $S$ it equals $1$; for half-integer $S$ it equals $-1$. The relative minus sign between neighboring topological sectors is the essential distinction.

2. Check the dimensions of the two gradient terms when $\hbar=1$.

Solution

$c$ has dimension length/time. Thus $d\tau\,dx\,c^{-1}(\partial_\tau\mathbf n)^2$ and $d\tau\,dx\,c(\partial_x\mathbf n)^2$ are both dimensionless in one spatial dimension. Hence $g$ is dimensionless.

References

• Affleck, I. “Exact Critical Exponents for Quantum Spin Chains, Nonlinear Sigma Models at $\theta=\pi$ and the Quantum Hall Effect.” Nuclear Physics B 265 (1986): 409–447. DOI.
• Chakravarty, S., B. I. Halperin, and D. R. Nelson. “Two-Dimensional Quantum Heisenberg Antiferromagnet at Low Temperatures.” Physical Review B 39 (1989): 2344–2371. DOI.
• Haldane, F. D. M. “Continuum Dynamics of the 1-D Heisenberg Antiferromagnet: Identification with the O(3) Nonlinear Sigma Model.” Physics Letters A 93 (1983): 464–468. DOI.