General two-order-parameter Ginzburg-Landau model with quadratic and quartic interactions

TL;DR

This paper introduces a geometric method to analyze a complex two-order-parameter Ginzburg-Landau model, enabling the study of its minima, symmetries, and phase diagram without explicit minimization.

Contribution

A novel geometric approach simplifies the analysis of the most general U(1)-symmetric Ginzburg-Landau potential with quadratic and quartic terms.

Findings

Number of minima determined

Symmetry classification achieved

Conditions for spontaneous symmetry breaking established

Abstract

Ginzburg-Landau model with two order parameters appears in many condensed-matter problems. However, even for scalar order parameters, the most general U(1)-symmetric Landau potential with all quadratic and quartic terms contains 13 independent coefficients and cannot be minimized with straightforward algebra. Here, we develop a geometric approach that circumvents this computational difficulty and allows one to study properties of the model without knowing the exact position of the minimum. In particular, we find the number of minima of the potential, classify explicit symmetries possible in this model, establish conditions when and how these symmetries are spontaneously broken, and explicitly describe the phase diagram.