### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 075, 19 pages      arXiv:1402.4606      https://doi.org/10.3842/SIGMA.2014.075
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

### Energy Spectrum and Phase Transition of Superfluid Fermi Gas of Atoms on Noncommutative Space

Yan-Gang Miao a, b and Hui Wang a
a) School of Physics, Nankai University, Tianjin 300071, China
b) Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China

Received February 25, 2014, in final form July 07, 2014; Published online July 10, 2014

Abstract
Based on the Bogoliubov non-ideal gas model, we discuss the energy spectrum and phase transition of the superfluid Fermi gas of atoms with a weak attractive interaction on the canonical noncommutative space. Because the interaction of a BCS-type superfluid Fermi gas originates from a pair of Fermionic quasi-particles with opposite momenta and spins, the Hamiltonian of the Fermi gas on the noncommutative space can be described in terms of the ordinary creation and annihilation operators related to the commutative space, while the noncommutative effect appears only in the coefficients of the interacting Hamiltonian. As a result, we can rigorously solve the energy spectrum of the Fermi gas on the noncommutative space exactly following the way adopted on the commutative space without the use of perturbation theory. In particular, different from the previous results on the noncommutative degenerate electron gas and superconductor where only the first order corrections of the ground state energy level and energy gap were derived, we obtain the nonperturbative energy spectrum for the noncommutative superfluid Fermi gas, and find that each energy level contains a corrected factor of cosine function of noncommutative parameters. In addition, our result shows that the energy gap becomes narrow and the critical temperature of phase transition from a superfluid state to an ordinary fluid state decreases when compared with that in the commutative case.

Key words: noncommutative space; Fermi gas of atoms; superfluidity; energy spectrum; critical temperature of phase transition.

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