Two-step Bose-Einstein condensation coming from a temperature dependent energy gap.*

Regardless of the microscopic mechanism that generates it, electron pairing is fundamental to observe superconductivity where Cooper pairs show a temperature dependent energy gap (TDEG) which abruptly disappears at the superconducting critical temperature.

Here we explore the effects of a TDEG on the properties of a Bose gas whose energy spectrum is that of an ideal Bose gas (IBG) with a BCS TDEG at its lower edge.  

The first thing we observe is that the Bose-Einstein condensation (BEC) critical temperature T_c of our gas is equal to the BEC critical temperature T₀ of the IBG if the temperature T_B at which the gap dissapears is lower than T₀; If T_B ≥ T₀, T_c ≥T₀. Secondly, for T_B  < T₀, the condensed fraction as a function of temperature presents two different rates to populate the ground state: one at T₀ and a higher one at T_B, generating a kind of two-step BEC. Third, the specific heat as a function of temperature exhibits a finite jump as when the gap is temperature independent [1] but also an infinite jump at the temperature T_B where the BCS gap abruptly disappears with an infinite slope [2]. We discuss possible connections of our model results with what has been observed in superconductivity.

[1]   J.G. Martinez-Herrera et al., Phys. Scri. 94 (2019) 075002.

[2] J.J. Valencia and M.A. Solís, Two-step Bose-Einstein condensation coming from a temperature dependent energy gap, to be published.