Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2016 / Paper 1 / 1 / h / Solution

In this part $0<\Delta =1+\alpha <1$, so the ordered reference lies in the $xy$ plane. Choosing one direction breaks the continuous rotation symmetry within that plane. A gapless, linearly dispersing phase mode is therefore expected in the spin-wave approximation, while tilting the Néel axis out of the plane costs anisotropy energy. It is incorrect to continue the easy-axis expansion about an unstable $z$-ordered reference and interpret its imaginary frequencies as physical modes.

For the easy-plane XXZ antiferromagnetic spin-wave spectrum, take the Néel direction along old $x$, then rotate alternating local spins so that all longitudinal components point up. One such local frame gives the full-zone quadratic coefficientsand thereforeNear $k=0$, ${ϵ}_k\simeq JS\sqrt{2(1+\Delta )}|k|$. The zone-edge mode has gap ${ϵ}_{\pi }=2JS\sqrt{2(1-\Delta )}$. A different sublattice phase convention shifts the zero by $\pi$; folding into the magnetic zone displays an acoustic and a gapped polarization. The figure uses this explicitly chosen frame. One-dimensional long-wavelength fluctuations again limit the ordered-reference interpretation.