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

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.