Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 337 / 2

In the long-wavelength $O(N)$ description of an antiferromagnet, $v$ is the antiferromagnetic spin wave velocity and $g$ controls the stiffness or strength of quantum fluctuations. The field $\lambda$ is an auxiliary Lagrange multiplier enforcing the fixed-length constraint on the order parameter. At the translation-invariant saddle, $i\lambda =m^2$ shifts every propagator denominator and $m$ is the excitation gap, which explains the name gap equation.

The Large-N expansion makes fluctuations of the auxiliary field relatively small. Its leading saddle-point condition is then self-consistent and becomes the displayed gap equation.

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As $T\to 0$, the Matsubara sum becomes $\int d\omega /(2\pi )$. Put $p_0=\omega /v$ and $M=m/v$. The factors of $v$ cancel from both sides, leavingUsing four-dimensional spherical coordinates,At the onset of spontaneous symmetry breaking, the symmetric-phase gap closes. ThusFor $g<g_c$ the constraint is instead satisfied by an ordered condensate; a positive symmetric gap exists for $g>g_c$.

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Subtracting the critical equation from the massive equation givesAs $g\downarrow g_c$ from above,Therefore, up to constants inside the slowly varying logarithm,The square-root mean-field power is modified by a logarithm. It is therefore not a pure quantum-critical power law; this is an upper-critical-dimension logarithmic correction.

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The first term is a gauge choice for the Spin coherent-state Berry phase. For a closed spin trajectory, changing the surface used to evaluate the solid angle changes the action byThe path-integral phase $e^{i{S}_B}$ must be independent of that choice, soHenceand the spin is integer or half-integer.

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Write $\theta =\pi /2-\vartheta$ and $\varphi =\phi$. To quadratic order,After dropping the total derivative $S{\phi }_t$, the quadratic Lagrangian density isIts Euler--Lagrange equations areFor modes proportional to $e^{i(kx-\omega t)}$, nontrivial amplitudes requireThusAt short wavelength the anisotropy is negligible and the quadratic dispersion is that of a conventional ferromagnetic magnon. At long wavelength, easy-plane anisotropy makes the out-of-plane fluctuation the conjugate density of the in-plane phase; eliminating it produces the linear Goldstone sound mode characteristic of a superfluid.

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