Condensation energy for a Bose gas with gaped quadratic dispersion relation in 1, 2 and 3 dimensions*

In superconductivity it is common to calculate the condensation energy (CE) which is the difference between the superconducting and normal Helmholtz free energies.

In this work we calculate the condensation energy for a Bose gas with quadratic dispersion relation plus a constant energy gap between the ground state and the first excited state (gaped IBG) for which Bose-Einstein condensation (BEC) exist in $d > 0$ dimensions [1], defined as the difference between the Helmholtz free energy of the gaped IBG minus that of the gapless IBG.  After  plotting the condensation energy from $T = 0$ to the BEC critical temperature of the gaped IBG, we observe the same functional form as that reported for the CE of conventional superconductor [2, 3] although with scaled magnitudes as we are dealing with Bose gases.  

[1] J.G. Martínez et al., Physica Scripta 94 (2019) 75002; [2] I. Chávez et al., Physica C 600, 1354090 (2022);  

[3] C. Kittel, Introduction to Solid State Physics (Wiley, NY, 2005) p. 267.