Static and Dynamic Properties of Interacting Spin-1 Bosons in an Optical Lattice

TL;DR

This paper investigates the static and dynamic properties of interacting spin-1 bosons in an optical lattice using a variational Gutzwiller approach, analyzing phase transitions and excitation spectra across different magnetic interactions.

Contribution

It extends the Gutzwiller method to derive excitation spectra and incorporates Schwinger boson mean-field theory to capture spin-wave Goldstone modes in ferromagnetic Mott insulators.

Findings

Gapped, quadratically dispersing modes in nematic Mott phase

Non-dispersive gapped mode in singlet Mott phase

Gapless ferromagnetic spin-wave mode in ferromagnetic Mott insulator

Abstract

We study the physics of interacting spin-$1$ bosons in an optical lattice using a variational Gutzwiller technique. We compute the mean-field ground state wave-function and discuss the evolution of the condensate, spin, nematic, and singlet order parameters across the superfluid-Mott transition. We then extend the Gutzwiller method to derive the equations governing the dynamics of low energy excitations in the lattice. Linearizing these equations, we compute the excitation spectra in the superfluid and Mott phases for both ferromagnetic and antiferromagnetic spin-spin interactions. In the superfluid phase, we recover the known excitation spectrum obtained from Bogoliubov theory. In the nematic Mott phase, we obtain gapped, quadratically dispersing particle and hole-like collective modes, whereas in the singlet Mott phase, we obtain a non-dispersive gapped mode, corresponding to the breaking of a singlet pair. For the ferromagnetic Mott insulator, the Gutzwiller mean-field theory only yields particle-hole like modes but no Goldstone mode associated with long range spin order. To overcome this limitation, we supplement the Gutzwiller theory with a Schwinger boson mean-field theory which captures super-exchange driven fluctuations. In addition to the gapped particle-hole-like modes, we obtain a gapless quadratically dispersing ferromagnetic spin-wave Goldstone mode. We discuss the evolution of the singlet gap, particle-hole gap, and the effective mass of the ferromagnetic Goldstone mode as the superfluid-Mott phase boundary is approached from the insulating side. We discuss the relevance and validity of Gutzwiller mean-field theories to spinful systems, and potential extensions of this framework to include more exotic physics which appears in the presence of spin-orbit coupling or artificial gauge fields.