# Bosons with incommensurate potential and spin-orbit coupling

## TL;DR

This paper explores the phase diagram of spin-half bosons in a 1D optical lattice with incommensurate potential and spin-orbit coupling, revealing novel localized phases, spin-split momentum distributions, and spectral statistics transitions.

## Contribution

It provides new insights into the effects of spin-orbit coupling and Aubry-André potential on bosonic phases, including the identification of a spin-split momentum distribution as a transition signature.

## Key findings

- Spin-split momentum distribution in localized phase
- Transition of level statistics from GUE to GOE
- Identification of density wave phase with NNI

## Abstract

We chart out the phase diagram of ultracold `spin-half' bosons in a one-dimensional optical lattice in the presence of Aubry-Andr\'e (AA) potential and with spin-orbit (SO) and Raman couplings investigating the transition from superfluid (SF) to localized phases and the existence of density wave phase for nearest-neighbor interaction (NNI). We show that the presence of SO coupling and AA potential leads to a novel spin-split momentum distribution of the bosons in the localized phase near the boundary with the SF phase, which can act as a signature of such a transition. We also obtain the level statistics of the bosons in the superfluid phase with finite NNI and demonstrate its change from Gaussian Unitary Ensemble (GUE) to Gaussian Orthogonal Ensemble (GOE) as a function of the Raman coupling. We discuss experiments which can test our theory.