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The BCS-Bogoliubov gap equation with external magnetic field and the first-order phase transition

Shuji Watanabe

TL;DR

This paper analyzes a type I superconductor in a constant external magnetic field within the BCS-Bogoliubov mean-field framework. It derives the field- and temperature-dependent gap equation and proves, via the implicit function theorem, the existence of a unique, smooth critical field $H_c(T)$ and a gap function $ riangle(T,H)$, both defined implicitly by a single function $F$ and its derivatives. Using the grand potential, it shows that the normal-to-superconducting transition in the $T$-$H$ plane is of first order, and provides an explicit expression for the entropy gap $oldsymbol{\nabla} oldsymbol{S}$ across the transition; it also discusses how FFLO-like effects can arise from the field dependence. The results are obtained under simplifying assumptions (e.g., constant DOS and constant interaction) and point toward extensions to more general DOS and potentials via fixed-point methods, with potential implications for understanding magnetic-field–driven superconducting transitions.

Abstract

We deal with a type I superconductor in a constant external magnetic field. We obtain the BCS-Bogoliubov gap equation with external magnetic field and apply the implicit function theorem to it. We show that there is a unique magnetic field (the critical magnetic field) given by a smooth function of the temperature and that there is also a unique nonnegative solution (the gap function) given by a smooth function of both the temperature and the external magnetic field. Using the grand potential, we show that the transition from the normal state to the superconducting state in a type I superconductor is of the first order. Moreover we obtain the explicit expression for the entropy gap.

The BCS-Bogoliubov gap equation with external magnetic field and the first-order phase transition

TL;DR

This paper analyzes a type I superconductor in a constant external magnetic field within the BCS-Bogoliubov mean-field framework. It derives the field- and temperature-dependent gap equation and proves, via the implicit function theorem, the existence of a unique, smooth critical field and a gap function , both defined implicitly by a single function and its derivatives. Using the grand potential, it shows that the normal-to-superconducting transition in the - plane is of first order, and provides an explicit expression for the entropy gap across the transition; it also discusses how FFLO-like effects can arise from the field dependence. The results are obtained under simplifying assumptions (e.g., constant DOS and constant interaction) and point toward extensions to more general DOS and potentials via fixed-point methods, with potential implications for understanding magnetic-field–driven superconducting transitions.

Abstract

We deal with a type I superconductor in a constant external magnetic field. We obtain the BCS-Bogoliubov gap equation with external magnetic field and apply the implicit function theorem to it. We show that there is a unique magnetic field (the critical magnetic field) given by a smooth function of the temperature and that there is also a unique nonnegative solution (the gap function) given by a smooth function of both the temperature and the external magnetic field. Using the grand potential, we show that the transition from the normal state to the superconducting state in a type I superconductor is of the first order. Moreover we obtain the explicit expression for the entropy gap.
Paper Structure (6 sections, 5 theorems, 72 equations, 3 figures)

This paper contains 6 sections, 5 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The mean field BCS Hamiltonian with external magnetic field
  3. The grand potential with external magnetic field
  4. The BCS-Bogoliubov gap equation with external magnetic field
  5. A proof of the first-order phase transition
  6. Discussion and conclusions

Key Result

Lemma 4.3

Let $D$ be as in eqn:domain-d. (1) The function F is uniformly continuous on $D$. (2) $F \in C^1(D)$. Moreover, ${ \frac{\partial F}{\, \partial H \,}(T, \, H,\, Y)<0 }$, and hence $F$ is strictly decreasing with respect to $H$. (3) Let $T_0 \le T <\tau_1$. Then ${ F(T, \, 0,\, 0)>0 }$.

Figures (3)

  • Figure 1: Density of states (DOS).
  • Figure 2: The critical magnetic field $H_c(T)$.
  • Figure 3: The graph of the squared gap function $Y=f(T,\, H)$.

Theorems & Definitions (14)

  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Proposition 4.5
  • proof
  • ...and 4 more