[Paper Review] Vortices in a Ginzburg Landau Theory of Superconductors with Nematic Order

R. S. Severino, Pablo D. Mininni+4 more|arXiv (Cornell University)|Apr 14, 2022

Physics of Superconductivity and Magnetism67 references5 citations

TL;DR

This study investigates vortex-vortex interactions in superconductors with nematic order using a Ginzburg-Landau theory coupled to a real nematic order parameter. Employing a pseudo-spectral method to solve time-dependent Ginzburg-Landau equations, the authors find that a biquadratic coupling induces attractive vortex interactions, while a coupling to the superconductor's covariant derivative always generates repulsion—revealing a competing interplay between nematicity and vortex dynamics in Fe-based superconductors.

ABSTRACT

In this work we explore the interplay between superconductivity and nematicity in the framework of a Ginzburg Landau theory with a nematic order parameter coupled to the superconductor order parameter, often used in the description of superconductivity of Fe based materials. In particular, we focus on the study of the vortex-vortex interaction in order to determine the way nematicity affects its attractive or repulsive character. To do so, we use a dynamical method based on the solutions of the Time Dependent Ginzburg Landau equations in a bulk superconductor. An important contribution of our work is the implementation of a pseudo-spectral method to solve the dynamics, known to be highly efficient and of very high order in comparison to the usual finite differences/elements methods. The coupling between the superconductor and the (real) nematic order parameters is represented by two terms in the free energy: a biquadratic term and a coupling of the nematic order parameter to the covariant derivatives of the superconductor order parameter. Our results show that there is a competing effect: while the former independently of its competitive or cooperative character generates an attractive vortex-vortex interaction, the latter always generates a repulsive interaction.

Motivation & Objective

• To understand how nematic order influences vortex-vortex interactions in superconductors.
• To determine whether nematicity alters the critical Ginzburg-Landau parameter (κc = 1/√2) that separates Type I and Type II superconductivity.
• To analyze the role of two distinct coupling terms between nematic and superconducting order parameters in shaping vortex dynamics.
• To implement and validate a high-order pseudo-spectral method for solving time-dependent Ginzburg-Landau equations in bulk superconductors.

Proposed method

• Uses a Ginzburg-Landau free energy functional with two coupling terms: a biquadratic interaction and a coupling to the covariant derivative of the superconducting order parameter.
• Applies a pseudo-spectral method for solving the time-dependent Ginzburg-Landau equations, enabling high-order accuracy and efficiency compared to finite difference/element methods.
• Performs numerical simulations in a bulk 3D superconductor geometry, assuming spatially uniform and time-independent nematic order in the external field limit.
• Solves the full dynamical system to compute vortex-vortex interaction forces, tracking the evolution of vortex configurations over time.
• Analyzes the resulting vortex dynamics to classify interactions as attractive or repulsive based on relative motion and energy changes.
• Validates results by recovering the known self-dual point (κ = 1/√2) when the nematic order is fixed and the system reduces to the standard GL model.

Experimental results

Research questions

• RQ1 How does the presence of a nematic order parameter modify the vortex-vortex interaction in superconductors?
• RQ2 Does the biquadratic coupling between nematic and superconducting order parameters induce attractive or repulsive vortex interactions?
• RQ3 What is the role of the nematic coupling to the covariant derivative of the superconducting order parameter in determining vortex interaction character?
• RQ4 Can the self-dual point (κ = 1/√2) be preserved or modified in the presence of dynamic nematic order?
• RQ5 How does the choice of numerical method (pseudo-spectral vs. finite difference) affect the accuracy and efficiency of vortex dynamics simulations?

Key findings

• The biquadratic coupling term between the nematic and superconducting order parameters generates an attractive vortex-vortex interaction, regardless of whether the coupling is cooperative or competitive.
• The coupling of the nematic order parameter to the covariant derivative of the superconducting order parameter always produces a repulsive vortex-vortex interaction.
• The two coupling terms act in competition: one promotes attraction, the other repulsion, leading to a complex interplay that can alter vortex lattice stability.
• When the nematic order is held constant in space and time, the system retains a self-dual point at κ = 1/√2, confirming consistency with the standard GL model.
• The pseudo-spectral method enables high-accuracy, high-order numerical solutions of the time-dependent Ginzburg-Landau equations, significantly improving computational efficiency and precision over traditional finite difference methods.
• The results suggest that nematic fluctuations can drive a transition between Type I and Type II behavior by shifting the effective κ parameter, even in bulk superconductors.