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Published September 2016 | Submitted
Journal Article Open

Multi-Component Ginzburg-Landau Theory: Microscopic Derivation and Examples

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_(Tc) +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_(Tc) +V of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schrödinger operators −∇^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.

Additional Information

© 2016 Springer International Publishing. Communicated by Vieri Mastropietro. Received: April 26, 2015. Accepted: November 10, 2015. First Online: 15 March 2016. The authors would like to thank Egor Babaev, Christian Hainzl, Edwin Langmann and Robert Seiringer for helpful discussions. R.L.F. was supported by the U.S. National Science Foundation through grants PHY-1347399 and DMS-1363432.

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