Energy Spectrum of the Anisotropic Magnetic Chain

The energy spectrum of the lowest (one-cluster) multiplet of the linear anisotropic spin-½ ferromagnetic chain is obtained analytically in the absence of transverse mean exchange ($j^x+j^y=0\mathrm{}$) and neglecting coupling to other states. In the limit where the external field tends to zero, the semi-infinite discrete spectrum exhibits an algebraic singularity of order $\frac{2}{3}$ as a function of the field, and an algebraic singularity as a function of the transverse anisotropy parameter [$j^a=\frac{1}{2}(j^x-j^y)$] of order $\frac{1}{3}$. At zero field the spectrum is bounded and continuous. In the high-field limit the system approaches the Ising model. A magnetic excitation spectrum of that character has recently been observed by Torrance and Tinkham in the magnetic salt Co${\mathrm{Cl}}_2$. 2${\mathrm{H}}_2$O.