Interaction driven exotic quantum phases in spin-orbit coupled spin$-1$ bosons

TL;DR

This paper explores the complex phase diagram of spin-1 bosons with spin-orbit coupling in a 2D optical lattice, revealing new symmetry-breaking phases and the effects of interactions on spin density wave order.

Contribution

It introduces a detailed analysis of the interplay between spin-orbit coupling, interactions, and lattice effects in spin-1 bosons, identifying novel phases not previously characterized.

Findings

Discovery of a phase with simultaneous condensation at multiple Brillouin zone points.

Interactions suppress spin density wave order, favoring edge superfluidity.

The spin density wave superfluid phase persists in harmonic traps, relevant for experiments.

Abstract

We study the interplay between large-spin, spin-orbit coupling, and superfluidity for bosons in a two dimensional optical lattice, focusing on the spin-1 spin-orbit coupled system recently realized at the Joint Quantum Institute [Campbell et. al., arXiv:1501.05984]. We find a rich quantum phase diagram, where, in addition to the conventional phases ---superfluid and insulator--- contained in the spin-$1$ Bose-Hubbard model, there are new lattice symmetry breaking phases. For weak interactions, the interplay between two length scales, the lattice momentum and the spin-orbit wave-vector induce a phase transition from a uniform superfluid to a phase where bosons simultaneously condense at the center and edge of the Brillouin zone at a non-zero spin-orbit strength. This state is characterized by spin density wave order, which arises from the spin-$1$ nature of the system. Interactions suppress spin density wave order, and favor a superfluid \textit{only} at the Brillouin zone edge. This state has spatially oscillating mean field order parameters, but a homogeneous density. We show that the spin density wave superfluid phase survives in a two dimensional harmonic trap, and thus establish that our results are directly applicable to experiments on $^{87}$Rb, $^7$Li, and $^{41}$K.