Effects of spatial noncommutativity on energy spectrum of a trapped Bose-Einstein condensate

In noncommutative space, we examine the problem of a noninteracting and harmonically trapped Bose-Einstein condensate, and derive a simple analytic expression for the effect of spatial noncommutativity on energy spectrum of the condensate. It indicates that the ground-state energy incorporating the spatial noncommutativity is reduced to a lower level, which depends upon the noncommutativity parameter $\theta$. The appeared gap between the noncommutative space and commutative one for the ground-state level of the condensate should be a signal of spatial noncommutativity.Comment: 7 pages; revtex

Effective Field Theories and Finite-temperature Properties of Zero-dimensional Superradiant Quantum Phase Transitions

The existence of zero-dimensional superradiant quantum phase transitions seems inconsistent with conventional statistical physics, which has not been explained so far. Here we demonstrate the corresponding effective field theories and finite-temperature properties of light-matter interacting systems, and show how this zero-dimensional quantum phase transition occurs. We first focus on the Rabi model, which is a minimum model that hosts a superradiant quantum phase transition. With the path integral method, we derive the imaginary-time action of the photon degrees of freedom. We also define a dynamical exponent as the rescaling between the temperature and the photon frequency, and perform dimensional analysis to the effective action. Our results show that the effective theory becomes a free scalar field or $\phi^4$-theory for a proper dynamical exponent, where a true second-order quantum phase transition emerges. These results are also verified by numerical simulations of imaginary-time correlation functions of the order parameter. Furthermore, we also generalize this method to the Dicke model. Our results make the zero-dimensional superradiant quantum phase transition compatible with conventional statistical physics, and pave the way to understand it in the perspective of effective field theories.Comment: 6+4 pages, 2 figure