Projected Holstein-Primakoff boson representation of quantum spins for spin wave theory

TL;DR

The work identifies a critical limitation of the standard Holstein-Primakoff spin-wave approach: neglecting projection onto the physical spin space can distort finite-$S$ magnon interactions. It develops a fully projected Holstein-Primakoff framework with exact projection operators $\hat{P}_S$ and projected spin operators $\hat{S}_a^{(P)}$, yielding normal-ordered infinite-series representations and a locally projected Hamiltonian $\hat{H}^{(P)}$ that preserves exact physics for physical states. In ferromagnetic settings, the second-order projected HP theory reproduces the exact bimagnon spectrum (with a single unphysical zero-energy state removed), outperforming the conventional HP approach; in antiferromagnetic contexts, however, non-conservation of HP boson number and symmetry issues hinder improvements, though preliminary Gutzwiller-type analyses offer insight. Overall, the projected HP formalism provides a rigorous route to incorporate projection effects in spin-wave theory and may enhance modeling for systems with strong quantum fluctuations, with future work needed to reconcile AFM cases and symmetry requirements.

Abstract

The Holstein-Primakoff boson representation of quantum spins and associated large-$S$ expansion have been the standard framework for describing the spin wave excitations in magnetically order phases of quantum spin systems. However, we will show that the omission of projection operators and normal-ordering in this representation can produce incorrect magnon hamiltonians for finite $S$. We will present the exact normal-ordered forms of the finite-$S$ projection operators and projected Holstein-Primakoff boson representations of spin and quadrupole operators, which can produce exact two-magnon interaction terms under ferromagnetic or fully polarized states. We will also discuss the difficulties of applying this projected representation to antiferromagnetic spin wave theory.

Figures (2)

• Figure 1: Calculated bimagnon energies at momentum $\boldsymbol{q}=(q,q)$ for spin-$1/2$ FM XXZ model Eq. (\ref{['equ:H-XXZ-FM']}) on $12\times 12$ square lattice with parameters $h=1$, $J_z=6$, $J_\perp=1$, and $D_z=4$. Lines in figures are upper and lower bounds of the bimagnon continuum in LSWT. Insets are detailed dispersions of the two bimagnon bound states below the continuum. See Appendix \ref{['appsec:BimagnonSpinHamiltonian']} and \ref{['appsec:BimagnonBosonHamiltonian']} for calculation details. (a) Results using projected HP representation Eq. (\ref{['equ:FM-Interactions-P']}). Note the zero-energy unphysical states. (b) Results using original HP representation Eq. (\ref{['equ:FM-Interactions-HP']}). The lowest energy bimagnon state is spurious (with more than $70$% spectral weight in unphysical double-occupancy state). (c) Results using original HP representation with $D_z$ term removed. The unphysical state merges into the continuum and slightly changes the spectrum. (d) Exact bimagnon energies from the spin model Eq. (\ref{['equ:H-XXZ-FM']}).
• Figure 2: Some LSWT and 2nd-order NLSWT results for honeycomb AFM easy-axis($J_z\geq J_\perp=J_{xy}=1$) XXZ model Eq. (\ref{['equ:H-AFM']}) versus the anisotropy $J_z/J_{xy}-1$. (a) LSWT expectation values of boson density $n\equiv \langle \hat{a}^\dagger_i\hat{a}^{\newline}_i\rangle$ and NN pairing amplitude $\Delta\equiv \langle \hat{a}^{\newline}_i\hat{a}^{\newline}_j\rangle$ and their sum. (b) The LSWT and 2nd-order NLSWT magnon gap. There is a spurious magnon gap closing in NLSWT for $1< J_z/J_{xy} < 1.0133$ (see inset).