Bose-Einstein condensation and superfluidity on a fuzzy sphere

According to Hohenberg's theorem, Bose-Einstein condensation (BEC) in two dimensions is impossible for any temperature $T>0$. By contrast, superfluidity does occur in two dimensions at finite temperatures; it emerges due to the breaking of Galilei invariance. Here we consider BEC and superfluidity on a compact two-dimensional space taking the form of a non-commutative ("fuzzy") sphere, where the scalar bosonic fields are promoted to $N\times N$ matrices. The dimension $N$ is related to the non-commutativity parameter of space and introduces an additional scale into the system. We find that non-commutativity favors ordered phases and so enhances BEC and superfluidity. We analyze BEC in ideal and weakly interacting Bose gases on a fuzzy sphere, finding in each case that the critical temperature of BEC is greater compared to that found in the case of a commutative sphere $S^2$. Then we investigate the superfluid response of weakly interacting Bose systems. To account for vortices in a superfluid, we show that, even on an ordinary sphere, the collective coordinates of vortices induce non-commutativity. With this in mind, we extend the definition of vortex defects to an inherently non-commutative sphere studied here, where the notion of a point is untenable. The non-commutativity is expected to be experimentally relevant to BEC and superfluidity since the fuzzy sphere has a thermodynamic limit distinct from the one defined over a plane, unlike the $S^2$ case. The significance of this difference is illustrated by the superfluid density calculation indicating that, in the large sphere limit, the normal fluid fraction on the fuzzy sphere yields a linear in $T$ dependence, while on a commutative $S^2$ it exhibits the usual two-dimensional $\sim T^3$ behavior. This linear dependence, arising directly from non-commutativity, is reminiscent of Uemura's law in cuprate high-$T_c$ superconductors.