BEC with two ground-state filling rates for BCS Cooper-pairs

TL;DR

This work analyzes a 3D Bose gas with a temperature-dependent gap $\Delta(T)$ between the ground state and first excited state, altering standard Bose-Einstein condensation. The authors derive closed-form expressions for the BEC critical temperature $T_c$, the condensate fraction $N_0/N$, the chemical potential $\mu$, the internal energy $U$, and the iso­choric specific heat $C_V$ for six gap scenarios (three undamped gaps and their damped counterparts), focusing on a BCS-like gap. A key finding is that for $T_B \le T_0$, $T_c = T_0$, while for $T_B > T_0$ the gap raises $T_c$ and induces a two-step condensate filling in the ungapped BCS case, with a finite $C_V$ jump at $T_c$ and a $\alpha=1/2$ divergence at $T_B$; damping smooths these singular features. The results connect Cooper-pair physics to Bose condensation, revealing how a temperature-dependent gap shapes phase transitions and thermodynamic responses in systems with quadratic dispersion.

Abstract

We report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap between the ground state and the first excited state of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is absent but whose properties are very different when it is present. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps which abruptly go to zero at a temperature $T_B$, as well as for the damped counterparts with a smoothed drop to zero. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature $T_c$ is equal to that of the ideal Bose gas $T_0$, for every $T_B \leq T_0$; surprisingly, the condensate fraction presents two different filling rates of the ground state, one at $T_c = T_0$ and another with a higher rate at $T_B < T_0$, suggesting something like a {\it two-step} BEC; also, its specific heat shows a finite jump at $T_c$ and a divergence at $T_B$ with a critical exponent $α=1/2$, which is inherited from the divergence of the temperature derivative of the BCS gap at $T_B$.

Figures (17)

• Figure 1: Examples of three temperature-dependent gaps studied here with $\Delta_0=k_BT_0$ and $T_B=1.5T_0$.
• Figure 2: The same gaps of Fig. \ref{['fig:TresGapsSinFermi']} but now multiplied by the damping function $f(T)=1/(\exp[b(T/T_D - 1)] + 1)$. For these cases $\Delta_0 = k_BT_0$, $T_B = 10 \, T_0$, $b = 10$, $T_D = 0.75 \, T_0$. In particular for $\Delta_{DB}$ the resulting $T_c = 1.09 \, T_0$. Subscripts $DL$, $DS$ and $DB$ mean Damped Linear, Damped Step and Damped BCS, respectively.
• Figure 3: Bose-Einstein critical temperature $T_c/T_0$ as function of $T_B/T_0$ for the three undamped gaps. Dashed line is $T_c=T_B$.
• Figure 4: Bose-Einstein critical temperature $T_c/T_0$ as a function of the damping temperature $T_D/T_0$ using the tree damped gaps of Eq. (\ref{['eq:Deltan']}), where $\Delta_0 = k_BT_0$, $T_B = 10 \, T_0$, and $b = 10$. Dashed line is $T_c=T_D$. In the inset we show the $T_D$ temperatures for which $T_D = T_C$.
• Figure 5: Condensate fractions: the dashed lines correspond to the zero gap (IBG) case; for BCS, step and linear gaps $T_B =T_0$ was taken; for $\Delta_{DB}(T)$, $\Delta_{DL}(T)$, and $\Delta_{DS}(T)$ gaps the values $T_B=10\, T_0$, $b=10$, and $T_D=0.75\, T_0$ were used. For all cases $\Delta_0=k_BT_0$.