High-order and adaptive optical conductivity calculations using Wannier interpolation

The optical conductivity provides a comprehensive view of the electronic response of materials to electromagnetic fields, offering insights into transport phenomena, optoelectronic properties, and other fundamental aspects of condensed matter physics. We present an automatic, high-order accurate, and adaptive Brillouin zone integration algorithm for the calculation of the optical conductivity using the Kubo formula, with a nonzero but small broadening factor $\ensuremath{\eta}$, focusing on the case in which a Hamiltonian in a downfolded model can be evaluated efficiently using Wannier interpolation. The algorithm uses iterated adaptive integration to exploit the localization of the transport distribution near energy and energy-difference isosurfaces, yielding polylogarithmic computational complexity with respect to $\ensuremath{\eta}$, rather than the algebraic complexity of uniform integration rules. To demonstrate the method, we compute the AC optical conductivity of a three-band tight-binding model, and are able to resolve the Drude and interband peaks with broadening in the sub-meV regime to several digits of accuracy. Our algorithm automates convergence testing to a user-specified error tolerance, providing an important tool in black-box first-principles calculations of electrical transport phenomena and other response functions.