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

TL;DR

This paper introduces a generalized spin-wave theory (GSWT) extending traditional methods from SU(2) to SU(N) to analyze complex spin systems with multipolar orderings, demonstrated on a bilinear-biquadratic model.

Contribution

The paper develops a new GSWT framework for arbitrary spin S, enabling analysis of multipolar orders and instabilities in quantum spin systems.

Findings

GSWT extends spin-wave theory to SU(N) for arbitrary spins.

GSWT can detect multipolar orderings and instabilities.

Application to bilinear-biquadratic model demonstrates effectiveness.

Abstract

We present a generalized spin-wave theory (GSWT) for treating spin Hamiltonians of arbitrary spin $S$. The generalization consists of an extension of the traditional spin-wave theory from SU(2) to SU($N$). 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 GSWT are described with $N-1$ different bosons. The generalization allows treating quantum spin systems whose ground state exhibit multipolar ordering as well as detecting instabilities of magnetically ordered states (dipolar ordering) towards higher multipolar orderings. We illustrate these advantages by applying the GSWT to a bilinear-biquadratic model of arbitrary spin $S$ on hypercubic lattices.