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Persistence of translational symmetry in the BCS model with radial pair interaction

Andreas Deuchert, Alissa Geisinger, Christian Hainzl, Michael Loss

TL;DR

The paper addresses whether translational symmetry is broken in the two-dimensional BCS model with a radial pair potential. It polishes a strategy based on angular-momentum sector decomposition and a relative-entropy method to compare the full BCS functional with its translation-invariant restriction, establishing a temperature interval $[\tilde{T}, T_c)$ where minimizers remain translation-invariant and lie in a specific angular-momentum sector $\ell_0$. If $\ell_0 = 0$, the minimizer is unique up to a phase and radial; if $\ell_0 \neq 0$, there are two symmetric minimizers $\alpha_{\pm\ell_0}$ corresponding to $\pm\ell_0$. The results extend to the 3D case for $\ell_0 = 0$ under appropriate regularity on $V$, and rely on spectral analysis of $K_T + V$ together with norm-resolvent convergence as $T\uparrow T_c$. Collectively, the work clarifies conditions under which the BCS minimizers preserve translational symmetry and connects full periodic minimizers to those in the translation-invariant sectors.

Abstract

We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. In the case of vanishing angular momentum our results carry over to the three-dimensional case.

Persistence of translational symmetry in the BCS model with radial pair interaction

TL;DR

The paper addresses whether translational symmetry is broken in the two-dimensional BCS model with a radial pair potential. It polishes a strategy based on angular-momentum sector decomposition and a relative-entropy method to compare the full BCS functional with its translation-invariant restriction, establishing a temperature interval where minimizers remain translation-invariant and lie in a specific angular-momentum sector . If , the minimizer is unique up to a phase and radial; if , there are two symmetric minimizers corresponding to . The results extend to the 3D case for under appropriate regularity on , and rely on spectral analysis of together with norm-resolvent convergence as . Collectively, the work clarifies conditions under which the BCS minimizers preserve translational symmetry and connects full periodic minimizers to those in the translation-invariant sectors.

Abstract

We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. In the case of vanishing angular momentum our results carry over to the three-dimensional case.

Paper Structure

This paper contains 4 sections, 10 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preparations
  4. Proof of Theorem \ref{['thm:vollesFunktional']} and Theorem \ref{['thm:3dim']}

Key Result

Theorem 1

Let $V \in L^2(\mathbb{R}^2)$ with $\hat{V} \in L^{r}(\mathbb{R}^2)$, where $r \in [1,2)$, be radial and such that $T_c > 0$. Suppose that $T_c = T_c(\ell_0)$ and that the lowest eigenvalue of $K_{T_c} + V$ is at most twice degenerate. If minimizes $\mathcal{F}_{\ell_0}^{\mathrm{ti}}$, then there exists $\tilde{T} < T_c$ such that where $\hat{\alpha}_{\pm \ell_0}(p) = e^{\pm i \ell_0 \varphi} \s

Figures (1)

  • Figure 1: Schematic picture of the lowest eigenvalues of $K_{T} + V$ as a function of the temperature $T$. The lowest two lines represent eigenvalues in the sector of angular momentum $\ell_0$. The third line corresponds to the lowest eigenvalue in the angular momentum $\ell_1$ sector. The red dots highlight the temperatures at which one of the eigenvalues crosses the $T$-axis.

Theorems & Definitions (27)

  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2
  • Remark 2.7
  • Remark 2.8
  • ...and 17 more