Higher-dimensional Wannier functions of multiparameter Hamiltonians

Abstract

When using Wannier functions to study the electronic structure of multi-parameter Hamiltonians H (k,λ) carrying a dependence on crystal momentum k and an additional periodic parameter λ, one usually constructs several sets of Wannier functions for a set of values of λ. We present the concept of higher dimensional Wannier functions (HDWFs), which provide a minimal and accurate description of the electronic structure of multi-parameter Hamiltonians based on a single set of HDWFs. The obstacle of non-orthogonality of Bloch functions at different λ is overcome by introducing an auxiliary real space, which is reciprocal to the parameter λ. We derive a generalized interpolation scheme and emphasize the essential conceptual and computational simplifications in using the formalism, for instance, in the evaluation of linear response coefficients. We further implement the necessary machinery to construct HDWFs from ab initio within the full-potential linearized augmented plane-wave method (FLAPW). We apply our implementation to accurately interpolate the Hamiltonian of a one-dimensional magnetic chain of Mn atoms in two important cases of λ: (i) the spin-spiral vector q, and (ii) the direction of the ferromagnetic magnetizationm. Using the generalized interpolation of the energy, we extract the corresponding values of magneto-crystalline anisotropy energy, Heisenberg exchange constants, and spin stiffness, which compare very well with the values obtained from direct first principles calculations. For toy models we demonstrate that the method of HDWFs can also be used in applications such as the virtual crystal approximation, ferroelectric polarization and spin torques.

Mixed Weyl semimetals and low-dissipation magnetization control in insulators by spin–orbit torques

“…Starting from the converged charge density, the Kohn–Sham equations were solved on an equidistant mesh of 8 × 8 k -points (6 × 6 in case (1)) for 8 different magnetization directions = (sin θ , 0, cos θ ), where the angle θ covers the unit circle once. Based on the resulting wave-function information in the composite phase space, we constructed a single set of higher-dimensional Wannier functions 54 (HDWFs) for each of the systems by employing our extension of the wannier90 code 55 . In case (1), we generated 274 HDWFs from 360 bands with the frozen window up to 4 eV above the Fermi level, and in the case (2), we extracted from 28 bands 14 HDWFs for a frozen window that extends to 2 eV above the Fermi energy.…”

Higher-dimensional Wannier Interpolation for the Modern Theory of the Dzyaloshinskii–Moriya Interaction: Application to Co-based Trilayers

“…This has motivated us to develop an efficient generalization of Wannier functions 28) for higher phase-space dimensions by performing Fourier transformations not only with respect to k but also with respect to additional parameters λ (for example, the angles ϕ and θ) entering the Hamiltonian. In the following, we briefly review the main aspects of these higher-dimensional Wannier functions (HDWFs).…”

“…As a consequence, the full product state (9) is an eigenstate of the Hamiltonian H with the eigenvalue E kλn , and assumes the desired Bloch-like form |Φ kλn = e ik·r e iλ·ξ |φ kλn with the latticeperiodic function |φ kλn = |u kλn ⊗ |ρ λ . Localization in real space Discrete Fourier transformations of the orthogonal product states with respect to k and λ can then be employed to define HDWFs as a meaningful generalization of Wannier functions: 28) …”

“…Using our adapted version of wannier 90, we are able to construct a single set of maximally-localized HDWFs from the electronic structure given on a coarse uniform grid of (k, λ)-points. 28) …”

On The Bloch Theorem and Orthogonality Relations

“…Since there are infinitely many choices for ( ), Wannier functions would not be unique, and normally need to be constructed in such a way to be maximally-localized. This is normally done through global minimization of a spread functional; the reader is referred to literature for in-depth discussion of this issue [15], [17][18][19][20], [21], [9,22] for plasmonic, photonic, phononic, and electronic crystals, respectively.…”