Bose-Einstein quantum phase transition in an optical lattice model

Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half of the maximum density and the optical lattice is modeled by a periodic potential of strength $\lambda$. For small $\lambda$ and temperature, BEC is proved to occur, while at large $\lambda$ or temperature there is no BEC. At large $\lambda$ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the $\mathbf{p}=\mathbf{0}$ mode in this model, although the density itself has the periodicity of the imposed potential.

Figure 1

Schematic phase diagram at half filling.

Figure 2

Loop gas describing paths of quasiparticles for particle number $N=\mid \Lambda \mid ∕2–1$. A line on an $A$ site means presence of a particle, while on a $B$ site it means absence. The horizontal rungs correspond to hopping of a particle.