The Principle Of Local Rotational Invariance And The Coexistence Of Magnetism, Charge And Superconductivity

TL;DR

This paper introduces a macroscopic, gauge-invariant framework for superconductivity that incorporates local rotational invariance, spin-charge intertwining, and external magnetic fields, extending Cartan's geometric formalism to condensed matter physics.

Contribution

It develops a novel first-order differential equation approach based on Cartan's formalism, revealing local rotational invariance as a gauge symmetry in superconductors with intertwined spin and charge.

Findings

Describes superconducting states with external magnetic fields using first-order equations.

Unveils local rotational invariance as a gauge symmetry in the order parameter.

Connects geometric Cartan formalism to superconductivity theory.

Abstract

We propose a macroscopic description of the superconducting state in presence of an applied external magnetic field in terms of first order differential equations. They describe a corrugated two-component order parameter intertwined with a spin-charged background, caused by spin correlations and charged dislocations. The first order differential equations are a consequence of a Weitzenb\"ock-Liechnorowitz identity which renders a SUL(2) \otimes UL(1) invariant ground state, based on (L) local rotational and electromagnetic gauge symmetry. The proposal is based on a long ago developed formalism by \'Elie Cartan to investigate curved spaces, viewed as a collection of small Euclidean granules that are translated and rotated with respect to each other. \'Elie Cartan's formalism unveils the principle of local rotational invariance as a gauge symmetry because the global SU(2) invariance of the order parameter is turned local by the interlacement of spin and charge to pairing.