AutoBZ.jl

This Julia package defines integrands for multi-dimensional Brillouin zone (BZ) using Wannier interpolation for the calculation of density of states, chemical potential, and optical conductivity accounting for electronic interactions through frequency-dependent self energies. It uses algorithms which automatically compute BZ integrals to a specified error tolerance by resolving smooth yet highly localized integrands.

In many-body Green's function methods, BZ integrands are localized at a scale determined by a non-zero, but possibly small, system- and temperature-dependent scattering rate. For example, the single-particle retarded Green's function of an electronic system for frequency $\omega$ and reciprocal space vector $\bm{k}$ with chemical potential $\mu$, Hermitian Hamiltonian matrix $H(\bm{k})$, and self-energy matrix $\Sigma(\omega)$, which is given by

\[G(\omega) = \int_{\text{BZ}} d\bm{k}\ \operatorname{Tr} \left[ (\hbar\omega - H(\bm{k}) - \Sigma(\omega))^{-1} \right]\]

is localized about the manifold defined by $\det(\hbar\omega - H(\bm{k}))=0$ (i.e. the Fermi surface when $\hbar\omega=\mu$) by a scattering rate depending on $\operatorname{Im}\ \Sigma(\omega)$.

AutoBZ.jl is built using a lower-level interface defined in AutoBZCore.jl intended for general-purpose integration and Wannier interpolation. If you are interested in defining your own integrals, please visit the AutoBZCore.jl documentation.

Package features

• Automatic and adaptive algorithms
  • Equispace integration (PTR) as described by Kaye et al. ^[1]
    • Automatic p-adaptive algorithm that refines $k$-grid to meet requested error
  • Iterated adaptive integration (IAI) with nested calls to QuadGK.jl
    • H-adaptive algorithm with logarithmic complexity for increasingly localized integrands
    • Irreducible Brillouin zone (IBZ) integration for the cubic lattice
    • Auxiliary integration for nearly-singular integrands as described in Ref.^[2]
• Support for Wannier-interpolated integrands
  • User-defined integrands based on Bloch Hamiltonians
  • Density of states (DOS) calculations
  • Transport calculations based on TRIQS DFTTools
    • Calculation of transport function and kinetic coefficients
    • Option to separate intra-band and inter-band contributions
  • Wannier90-based parsers Hamiltonians (*_hr.dat files) and position operators (*_r.dat files)
  • PythTB interface for using tight-binding models
  • Automated interpolation for frequency-dependent self-energy data in text files, using EquiBaryInterp.jl and HChebInterp.jl.
• IBZ integration for arbitrary symmetry groups (via an interface to SymmetryReduceBZ.jl)