Crystalline phases by an improved gradient expansion technique

TL;DR

This paper introduces an improved gradient expansion technique to analyze inhomogeneous phases with broken symmetry, demonstrating its effectiveness in studying crystalline structures in quark matter and confirming the 1D real kink crystal as the favored modulation.

Contribution

The paper presents a novel, efficient gradient expansion method that enhances the analysis of inhomogeneous phases, especially near phase transition points, applied to quark matter.

Findings

Accurately reproduces known 1D and 2D modulation results.

Identifies the 1D real kink crystal as the most energetically favored structure.

Provides a qualitative pairing mechanism explanation.

Abstract

We develop an innovative technique for studying inhomogeneous phases with a spontaneous broken symmetry. The method relies on the knowledge of the exact form of the free energy in the homogeneous phase and on a specific gradient expansion of the order parameter. We apply this method to quark matter at vanishing temperature and large chemical potential, which is expected to be relevant for astrophysical considerations. The method is remarkably reliable and fast as compared to performing the full numerical diagonalization of the quark Hamiltonian in momentum space, and is designed to improve the standard Ginzburg-Landau expansion close to the phase transition points. For definiteness we focus on inhomogeneous chiral symmetry breaking, accurately reproducing known results for 1D and 2D modulations and examining novel crystalline structures, as well. Consistently with previous results, we find that the energetically favored modulation is the so-called 1D real kink crystal. We propose a qualitative description of the pairing mechanism to motivate this result.