Generalized spin-wave theory: Application to the bilinear-biquadratic model

Abstract

We present a mathematical framework for the multi-boson approach that has been used several times for treating spin systems. We demonstrate that the multi-boson approach corresponds to a generalization of the traditional spin-wave theory from SU(⁠|$2$|⁠) to SU(⁠|$N$|⁠), where |$N$| is the number of states of the local degree of freedom. Low-energy excitations are waves of the local order parameter that fluctuates in the SU(⁠|$N$|⁠) space of unitary transformations of the local spin states, instead of the SU(⁠|$2$|⁠) space of local spin rotations. Since the generators of the SU(⁠|$N$|⁠) group can be represented as bilinear forms in |$N$|-flavored bosons, the low-energy modes of the generalized spin-wave theory (GSWT) are described with |$N-1$| different bosons, which provide a more accurate description of low-energy excitations even for the usual ferromagnetic and antiferromagnetic phases. The generalization enables the treatment of quantum spin systems whose ground states exhibit multipolar ordering as well as the detection of instabilities of magnetically ordered states (dipolar ordering) towards higher multipolar orderings. We illustrate the advantages of the GSWT by applying it to a bilinear–biquadratic model of arbitrary spin |$S$| on hypercubic lattices, and then analyzing the spectrum of dipolar phases in order to find their instabilities. In contrast to the known results for |$S=1$| when the biquadratic term in the Hamiltonian is negative, we find that there is no nematic phase between the ferromagnetic or antiferromagnetic orderings for |$S>1$|⁠.