Multi-component Ginzburg-Landau theory: microscopic derivation and examples

TL;DR

This paper derives a multi-component Ginzburg-Landau theory from microscopic BCS theory for unconventional superconductors, providing explicit examples and interactions that produce degenerate ground states with arbitrary angular momentum.

Contribution

It offers a microscopic derivation of multi-component GL theory allowing for degenerate ground states, with explicit interaction models and mathematical insights on Bessel functions.

Findings

Derived multi-component GL theory from BCS for systems with degenerate ground states.

Constructed specific interactions producing arbitrary angular momentum degeneracies.

Proved a new property of half-integer Bessel functions at their maxima.

Abstract

This paper consists of three parts. In part I, we microscopically derive Ginzburg--Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator $K_{T_c}+V$ to be $n$-fold degenerate and the resulting GL theory then couples $n$ order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure $d$-wave order parameters and (b) mixed $(s+d)$-wave order parameters, in two and three dimensions. In part III, we present explicit choices of spherically symmetric interactions $V$ which produce the examples in part II. In fact, we find interactions $V$ which produce ground state sectors of $K_{T_c}+V$ of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schr\"odinger operators $-\nabla^2+V$, for which the ground state is always non-degenerate. Along the way, we prove the following fact about Bessel functions: At its first maximum, a half-integer Bessel function is strictly larger than all other half-integer Bessel functions.