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BCS Critical Temperature on Half-Spaces

Barbara Roos, Robert Seiringer

TL;DR

The paper analytically demonstrates that a boundary can raise the BCS critical temperature relative to the bulk across dimensions $d=1,2,3$ under Dirichlet or Neumann conditions, with strict increase proven in the weak-coupling regime (and in small $\mu$ for $d=3$). The authors leverage Birman-Schwinger theory, a reduction to the zero-total-momentum sector, and carefully constructed trial states to compare half-space and bulk spectra, establishing both a strict inequality and the vanishing of the relative shift as the coupling tends to zero. They provide a detailed treatment for the challenging 3D case, including explicit half-space boundary kernels, rigorous conditions ensuring boundary superconductivity, and a thorough analysis of the relative-temperature shift. The work extends boundary-superconductivity results beyond 1D, clarifying the mechanisms by which boundaries can enhance pairing and outlining precise spectral criteria and asymptotics that govern the effect. The results hold significant implications for understanding surface superconductivity and for the design of low-dimensional superconducting systems in continuum models.

Abstract

We study the BCS critical temperature on half-spaces in dimensions $d=1,2,3$ with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on $\mathbb{R}^d$, at least at weak coupling in $d=1,2$ and weak coupling and small chemical potential in $d=3$. Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.

BCS Critical Temperature on Half-Spaces

TL;DR

The paper analytically demonstrates that a boundary can raise the BCS critical temperature relative to the bulk across dimensions under Dirichlet or Neumann conditions, with strict increase proven in the weak-coupling regime (and in small for ). The authors leverage Birman-Schwinger theory, a reduction to the zero-total-momentum sector, and carefully constructed trial states to compare half-space and bulk spectra, establishing both a strict inequality and the vanishing of the relative shift as the coupling tends to zero. They provide a detailed treatment for the challenging 3D case, including explicit half-space boundary kernels, rigorous conditions ensuring boundary superconductivity, and a thorough analysis of the relative-temperature shift. The work extends boundary-superconductivity results beyond 1D, clarifying the mechanisms by which boundaries can enhance pairing and outlining precise spectral criteria and asymptotics that govern the effect. The results hold significant implications for understanding surface superconductivity and for the design of low-dimensional superconducting systems in continuum models.

Abstract

We study the BCS critical temperature on half-spaces in dimensions with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on , at least at weak coupling in and weak coupling and small chemical potential in . Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.
Paper Structure (23 sections, 33 theorems, 345 equations, 5 figures, 2 tables)

This paper contains 23 sections, 33 theorems, 345 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction and Results
  2. Preliminaries
  3. Proof of Lemma \ref{['H1_esspec']}
  4. Proof of Lemma \ref{['ti_reduction']}
  5. Ground State of $H_{T_c^0(\lambda)}^0$
  6. Proof of Lemma \ref{['psiT_asymptotic_d']}
  7. Regularity and convergence of $\Phi_\lambda$
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Lemma \ref{['H1-Tcont']}
  10. Proof of Lemma \ref{['plugin_triaL_state']}
  11. Proof of Lemma \ref{['claim_weak_lambda_asy']}
  12. Boundary Superconductivity in 3d
  13. Proof of Lemma \ref{['lea:t1_expr']}
  14. Proof of Lemma \ref{['lea:tj_reg']}
  15. Relative Temperature Shift
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $d\in\{1,2,3\}$, $\mu>0$ and let $V$ satisfy Assumption aspt_V_halfspace. Assume either Dirichlet or Neumann boundary conditions. For $d=3$ additionally assume that Then there is a $\lambda_1>0$, such that for all $0<\lambda<\lambda_1$, $T_c^{\Omega_1}(\lambda)>T_c^{\Omega_0}(\lambda)$.

Figures (5)

  • Figure 1: Plot of $m_3^D$ for $\mu=1$, created using inc_mathematica_2022.
  • Figure 2: Plot of $m_3^N$ for $\mu=1$, created using inc_mathematica_2022.
  • Figure 3: Two circles of radius $\sqrt{\mu}$, centered at $(-\vert q \vert,0)$ and $(\vert q \vert,0)$. In $d=2$ the function $B_T(p,(\vert q \vert,0))$ diverges on the two circles as $T\to0$ and approaches zero in the shaded area. Given an angle $\varphi$, the numbers $r_\pm(e_\varphi)$ are the distances between zero and the intersections of the circles with the ray tilted by an angle $\varphi$ with respect to the $p_1$-axis.
  • Figure 4: In the proof of Lemma \ref{['Dmu1mu2_bound']}, in the case $0<\mu_1\leq\mu_2$ we split the domain of $p_1,q_1$ into ten different regions. The solid lines indicate the boundaries between these regions.
  • Figure 5: Domains occurring in the proof of Lemma \ref{['lea:IGIto0']}.

Theorems & Definitions (74)

  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • ...and 64 more