DiracBilinears.jl: A package for computing Dirac bilinears in solids

TL;DR

The work presents DiracBilinears.jl, a Julia package to compute Dirac bilinears in solids in the non-relativistic regime, interfacing with first-principles tools such as Quantum ESPRESSO, Wannier90, and wan2respack. By leveraging the $1/m$ expansion and Bloch/Wannier formalisms, the package provides quantitative spatial distributions and Wannier matrix elements of the bilinears, enabling studies of charge, spin, chirality, and polarization in materials. The authors demonstrate the approach with Te and BaTiO$_3$, illustrating spatial maps of electron chirality and spin-derived polarization, respectively, and show how to obtain total quantities via Wannier interpolation. This work bridges relativistic Dirac bilinear concepts with practical first-principles calculations, offering a concrete toolkit for exploring electronic asymmetry in chiral and polar materials.

Abstract

DiracBilinears.jl is a Julia package for computing Dirac bilinears, which are fundamental physical quantities of electrons in relativistic quantum theory, using first-principles calculations for solids. In relativistic quantum theory, 16 independent bilinears can be defined using the four-component Dirac field. We take the non-relativistic limit for the bilinears, which corresponds to the $1/m$ expansion, and focus on the low-energy physics typically considered in condensed matter physics. This package can evaluate the spatial distributions and Wannier matrix elements of the Dirac bilinears in solids quantitatively by connecting to the external first-principles calculation packages, including Quantum ESPRESSO, Wannier90, and wan2respack.

Figures (4)

• Figure 1: Calculation flow of Diracbilinears.jl. The blue solid arrows represent the calculation procedure for spatial distributions (Sec. \ref{['sec:usage_spatial']}), while the red dashed arrows represent the procedure for computing Wannier matrix elements (Sec. \ref{['sec:usage_wannier']}). These two calculations can be performed independently.
• Figure 2: Crystal structure of right-handed Te and $\mathrm{BaTiO_3}$. The crystal Te has the helical structure, and $\mathrm{BaTiO_3}$ has a Ti atom displaced from the center of the surrounding O atoms, as indicated by the red arrows.
• Figure 3: Spatial distribution of electron chirality for (a) left-handed Te and (b) right-handed Te. (c) Chemical potential dependence of total electron chirality for R-crystal Te. The lines correspond to calculations with different interpolated $\bm k$-mesh sizes. We use the same lattice constant as that employed in Ref. Miki24. The black shades in (a) and (b) are the cross-section of the cell.
• Figure 4: (a) Spatial distribution and (b) total value of spin-derived electric polarization. We use the same lattice constant as that employed in Ref. Miki24. The black shades in (a) are the cross-section of the cell.