Gradient expansion technique for inhomogeneous, magnetized quark matter

TL;DR

This paper introduces a gradient expansion method for analyzing inhomogeneous, magnetized quark matter, enabling efficient and accurate modeling of complex condensate modulations in magnetic fields.

Contribution

A novel analytical gradient expansion technique (qHGL) for inhomogeneous quark matter in magnetic fields, applicable beyond the Lifshitz point and accommodating large amplitude modulations.

Findings

The inhomogeneous phase region expands with increasing magnetic field.

The qHGL method agrees with numerical results and simplifies calculations.

It extends Ginzburg-Landau techniques to non-plane-wave modulations.

Abstract

A quark-magnetic Ginzburg-Landau (qHGL) gradient expansion of the free energy of two-flavor inhomogeneous quark matter in a magnetic field $H$ is derived analytically. It can be applied away from the Lifshitz point, generalizing standard Ginzburg-Landau techniques. The thermodynamic potential is written as a sum of the thermal contribution, the non-thermal lowest Landau level contribution, and the non-thermal qHGL functional, which handles any arbitrary position-dependent periodic modulation of the chiral condensate as an input. The qHGL approximation has two main practical features: (1) it is fast to compute; (2) it applies to non-plane-wave modulations such as solitons even when the amplitude of the condensate and its gradients are large (unlike standard Ginzburg-Landau techniques). It agrees with the output of numerical techniques based on standard regularization schemes and reduces to known results at zero temperature ($T = 0$) in benchmark studies. It is found that the region of the $\mu$-$T$ plane (where $\mu$ is the chemical potential) occupied by the inhomogeneous phase expands, as $H$ increases and $T$ decreases.