Determine finite-temperature compensation for unequal next-nearest-neighbor couplings

Determine whether the auxiliary-magnetization sum \(\Phi_{\mathrm{a}}+\Phi_{\mathrm{b}}\) remains equal to \(-1\) at finite temperatures in the square-lattice Heisenberg \(J_{1}-J_{2}-J_{2}^{\prime}\) model with easy-axis single-ion anisotropy when \(J_{2}\neq J_{2}^{\prime}\).

Background

The paper derives a relation between the auxiliary quantities $\Phi_{\mathrm{a}}$ and $\Phi_{\mathrm{b}}$ that enter the two-sublattice magnetization equations. For the symmetric case $J_{2}=J_{2}^{\prime}$, the two spin-wave branches satisfy $\omega_{1}=-\omega_{2}$, implying $\Phi_{\mathrm{a}}+\Phi_{\mathrm{b}}=-1$, as in a conventional antiferromagnet and yielding exact compensation of the two sublattice magnetizations.

When $J_{2}\neq J_{2}^{\prime}$, the spin-wave branches are nondegenerate. The authors establish that the auxiliary-magnetization sum remains $-1$ at zero and very low temperatures, but explicitly leave unresolved whether it retains this value at finite temperatures. Resolving this behavior would determine the precise onset of uncompensated net magnetization and temperature-dependent weak ferromagnetism in the asymmetric model.