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An operator theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity

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 نشر من قبل Shuji Watanabe
 تاريخ النشر 2016
  مجال البحث فيزياء
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 تأليف Shuji Watanabe




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We first show some properties such as smoothness and monotone decreasingness of the solution to the BCS-Bogoliubov gap equation for superconductivity. Moreover we give the behavior of the solution with respect to the temperature near the transition temperature. On the basis of these results, dealing with the thermodynamic potential, we then show that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator theory. Here we have no magnetic field and we need to introduce a cutoff $varepsilon>0$, which is sufficiently small and fixed (see Remark ref{rmk:varepsilon}). Moreover we obtain the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature.



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We show that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator theory. Here we have no magnetic field. Moreover we obtain the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature. To this end, we have to differentiate the thermodynamic potential with respect to the temperature two times. Since there is the solution to the BCS-Bogoliubov gap equation in the form of the thermodynamic potential, we have to differentiate the solution with respect to the temperature two times. Therefore, we need to show that the solution to the BCS-Bogoliubov gap equation is differentiable with respect to the temperature two times as well as its existence and uniqueness. We carry out its proof on the basis of fixed point theorems.
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