Order by disorder in spin-orbit-coupled Bose-Einstein condensates

Motivated by recent experiments, we investigate the system of isotropically interacting bosons with Rashba spin-orbit coupling. At the noninteracting level, there is a macroscopic ground-state degeneracy due to the many ways bosons can occupy the Rashba spectrum. Interactions treated at the mean-field level restrict the possible ground-state configurations, but there remains an accidental degeneracy not corresponding to any symmetry of the Hamiltonian, indicating the importance of fluctuations. By finding analytical expressions for the collective excitations in the long-wavelength limit and through numerical solution of the full Bogoliubov-de Gennes equations, we show that the system condenses into a single-momentum state of the Rashba spectrum via the mechanism of order by disorder. We show that in three dimensions the quantum depletion for this system is small, while the thermal depletion has an infrared logarithmic divergence, which is removed for finite-size systems. In two dimensions, on the other hand, thermal fluctuations destabilize the system.