Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 2

Let ${\eta }_i=+1$ and $-1$ on the two sublattices of the square lattice. A unit-vector decomposition that separates staggered and uniform magnetization iswhereThe alternating part is the Néel order parameter, while $\mathbf{m}$ is the slowly varying uniform magnetization generated by canting the two sublattices.

For a smooth configuration, neighboring sites lie on opposite sublattices and contribute Spin coherent-state Berry phases with opposite orientations. The terms $S\Gamma [\tilde{\mathbf{n}}]$ and $S\Gamma [-\tilde{\mathbf{n}}]$ therefore cancel pairwise modulo the quantized solid-angle ambiguity. Their smooth bulk contribution vanishes in the continuum limit.

Singular spacetime configurations can leave lattice Berry phases attached to hedgehog events, but those are outside the smooth sector used in this continuum derivation. Thus the order-parameter-only Wess–Zumino term may be dropped here.

At a site with background ${\eta }_i\tilde{\mathbf{n}}$, the linear variation is $\delta {\mathbf{n}}_i=\mathbf{m}$. The supplied variation formula givesReplacing the lattice sum by $a^{-2}\int d^{2}x$ yieldsThis Berry-phase term makes the uniform canting field the momentum conjugate to rotations of the Néel order parameter.

For a nearest-neighbor bond in the positive coordinate direction $\mu$, smoothness and orthogonality giveThere are two such bonds per site on the square lattice. Omitting the constant ground-state energy,The staggered part of the field coupling cancels between the two sublattices, whereasIncluding the overall minus sign of the Hamiltonian contribution to the real-time action, the continuum Lagrangian density is therefore

SetAfter adding $\lambda \mathbf{m}\cdot \tilde{\mathbf{n}}$, the terms involving the massive canting field areCompleting the square givesThe Gaussian functional integral over $\mathbf{m}$ contributes only a field-independent determinant. Hence

Because $\tilde{\mathbf{n}}\cdot (\tilde{\mathbf{n}}\times {\partial }_t\tilde{\mathbf{n}})=0$, the component of $\mathbf{q}$ parallel to $\tilde{\mathbf{n}}$ is $-\mathbf{B}\cdot \tilde{\mathbf{n}}$. The Gaussian integral over $\lambda$ removes precisely this longitudinal component. ThusUsing the vector triple-product identity and $|\tilde{\mathbf{n}}|=1$, this becomes the $O(3)$ nonlinear sigma model in a background field:The magnetic field acts as the temporal component of an $O(3)$ background gauge connection.

Choose the orientation of spherical coordinates aswhich differs from the opposite azimuth convention only by $\varphi \mapsto -\varphi$. With $\mathbf{B}=B\hat{\mathbf{x}}$, $\theta =\pi /2-\delta \theta$, and $\varphi =\delta \varphi$,Substitution into the effective action giveswhere

Let $\psi =\delta \theta +i\delta \varphi$ and define the zero-field spin-wave velocityThe linearized equation isFor a plane wave, the two circular polarizations therefore obeyor, with signed-frequency branches, $\omega =ck\pm B$ and their negative-frequency partners. At $B=0$ these are the two degenerate, linearly dispersing antiferromagnetic spin waves. The field Zeeman-splits the two opposite circular polarizations by shifting their frequencies in opposite directions.