Gradient expansion technique for inhomogeneous, magnetized quark matter

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = 0$$\end{document}) in benchmark studies. It is found that the region of the μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-T plane (where μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is the chemical potential) occupied by the inhomogeneous phase expands, as H increases and T decreases.