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.

磁场H中两种风味非均质夸克物质自由能的夸克-磁性 Ginzburg-Landau (qHGL) 梯度扩展是解析导出的。它可以远离 Lifshitz 点应用，推广标准的 Ginzburg-Landau 技术。热力学势被写为热贡献、非热最低朗道能级贡献和非热 qHGL 泛函的总和，该泛函将手性凝聚的任意位置相关周期性调制作为输入处理。qHGL 近似有两个主要的实用特性：(1) 计算速度快；(2) 它适用于非平面波调制，例如孤子，即使当凝聚物的幅度及其梯度很大时（与标准的 Ginzburg-Landau 技术不同）。它与基于标准正则化方案的数值技术的输出一致，并减少到零温度下的已知结果 ( \(T = 0\)) 在基准研究中。发现由不均匀相占据的\(\mu\) - T平面的区域(其中\(\mu\)是化学势)随着H的增加和T的减小而扩大。