We review the formalisms of the self-consistent GW approximation to many-body perturbation theory and of the generation of optimally-localized Wannier functions from groups of energy bands. We show that the quasiparticle Bloch wave functions from such GW calculations can be used within this Wannier framework. These Wannier functions can be used to interpolate the many-body band structure from the coarse mesh of Brillouin zone points on which it is known from the initial calculation to the usual symmetry lines, and we demonstrate that this procedure is accurate and efficient for the self-consistent GW band structure. The resemblance of these Wannier functions to the bond orbitals discussed in the chemical community led us to expect differences between density-functional and many-body functions that could be qualitatively interpreted. However, the differences proved to be minimal in the cases studied. Detailed results are presented for SrTiO_3 and solid argon.
Proof that the maximally localized Wannier functions are real
The maximally localized Wannier functions play a very important role in the study of chemical bonding, ballistic transport and strongly-correlated system, etc. A significant development in this branch was made in 1997 and conjectured that the maximally localized Wannier functions are real. In this paper, we prove the conjecture. A key to this proof is that the real parts of complex Wannier functions equal to the algebraic average of two real Wannier functions and the spread functional of one of the two real Wannier functions is less than one of complex Wannier functions. If one starts with ini…
We show that an optimized projection functions method can automatically construct maximally localized Wannier functions even for bands with nontrivial topology. We demonstrate this method on a tight-binding model of a two-dimensional $\mathbb{Z}_2$ topological insulator, on a three-dimensional strong $\mathbb{Z}_2$ topological insulator, as well as on first-principles density functional theory calculated valence states of Bi$_2$Se$_3$. In all cases, the resulting Wannier functions contain large imaginary components and are more extended than those in the topologically trivial phase.
Maximally localized Wannier functions are localized orthogonal functions that can accurately represent given Bloch eigenstates of a periodic system at a low computational cost, thanks to the small size of each orbital. Tight-binding models based on the maximally localized Wannier functions obtained from different systems are often combined to construct tight-binding models for large systems such as a semi-infinite surface. However, the corresponding maximally localized Wannier functions in the overlapping region of different systems are not identical, and this discrepancy can introduce serious…
We propose an algorithm to determine Maximally Localized Wannier Functions (MLWFs). This algorithm, based on recent theoretical developments, does not require any physical input such as initial guesses for the Wannier functions, unlike popular schemes based on the projection method. We discuss how the projection method can fail on fine grids when the initial guesses are too far from MLWFs. We demonstrate that our algorithm is able to find localized Wannier functions through tests on two-dimensional systems, simplified models of semiconductors, and realistic DFT systems by interfacing with the …
We present a robust algorithm that computes (maximally localized) Wannier functions (WFs) without the need of providing an initial guess. Instead, a suitable starting point is constructed automatically from so-called local orbitals which are fundamental building blocks of the basis set within (linearized) augmented planewave methods. Our approach is applied to a vast variety of materials such as metals, bulk and low-dimensional semiconductors, and complex inorganic-organic hybrid interfaces. For the interpolation of electronic single-particle energies, an accuracy in the meV range can be easil…
Using a realistic tight-binding Hamiltonian based on maximally localized Wannier functions, we investigate the two-ion magnetocrystalline anisotropy energy (MAE) in $L1_0$ transition metal compounds. MAE contributions from throughout the Brillouin zone are obtained using magnetic force theorem calculations with and without perturbation theory. The results from either method agree with one another, and both reflect features of the Fermi surface. The intra-sublattice and inter-sublattice contributions to MAE are evaluated using a Green's function method. We find that the sign of the inter-sublat…
The EPW (Electron-Phonon coupling using Wannier functions) software is a Fortran90 code that uses density-functional perturbation theory and maximally localized Wannier functions for computing electron-phonon couplings and related properties in solids accurately and efficiently. The EPW v4 program can be used to compute electron and phonon self-energies, linewidths, electron-phonon scattering rates, electron-phonon coupling strengths, transport spectral functions, electronic velocities, resistivity, anisotropic superconducting gaps and spectral functions within the Migdal-Eliashberg theory. Th…
We present a method to compute the intrinsic spin Hall conductivity from first principles using an interpolation scheme based on maximally-localized Wannier functions. After obtaining the relevant matrix elements among the ab initio Bloch states calculated on a coarse k-point mesh, we Fourier transform them to find the corresponding matrix elements between Wannier states. We then perform an inverse Fourier transform to interpolate the velocity and spin-current matrix elements onto a dense k-point mesh, and use them to evaluate the spin Hall conductivity as a Brillouin-zone integral. This strat…
Meta-optimization of maximally-localized Wannier functions
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Department of Physics, The University of Texas at Austin（德克萨斯大学奥斯汀分校物理系）
Maximally localized Wannier function within linear combination of pseudo-atomic orbital method: Implementation and applications to transition-metal-benzene complex
Spin-orbit interaction is an important vehicle for spin relaxation. At finite temperature lattice vibrations modulate the spin-orbit interaction and thus generate a mechanism for spin-phonon coupling, which needs to be incorporated in any quantitative analysis of spin transport. Starting from a density functional theory \textit{ab initio} electronic structure, we calculate spin-phonon matrix elements over the basis of maximally localized Wannier functions. Such coupling terms form an effective Hamiltonian to be used to extract thermodynamic quantities, within a multiscale approach particularly…
Real-time, time-dependent density functional theory (RT-TDDFT) has gained popularity as a first-principles approach to study a variety of excited-state phenomena such as optical excitations and electronic stopping. Within RT-TDDFT simulations, the gauge freedom of the time-dependent electronic orbitals can be exploited for numerical and scientific convenience while the unitary transformation does not alter physical properties calculated from the quantum dynamics of electrons. Exploiting this gauge freedom, we demonstrate propagation of maximally localized Wannier functions within RT-TDDFT. We …
Within this paper we outline a method able to generate truly minimal basis sets which describe either a group of bands, a band, or even just the occupied part of a band accurately. These basis sets are the so-called NMTOs, Muffin Tin Orbitals of order N. For an isolated set of bands, symmetrical orthonormalization of the NMTOs yields a set of Wannier functions which are atom-centered and localized by construction. They are not necessarily maximally localized, but may be transformed into those Wannier functions. For bands which overlap others, Wannier-like functions can be generated. It is show…
Spin-orbit coupling in organic crystals is responsible for many spin-relaxation phenomena, going from spin diffusion to intersystem crossing. With the goal of constructing effective spin-orbit Hamiltonians to be used in multiscale approaches to the thermodynamical properties of organic crystals, we present a method that combines density functional theory with the construction of Wannier functions. In particular we show that the spin-orbit Hamiltonian constructed over maximally localised Wannier functions can be computed by direct evaluation of the spin-orbit matrix elements over the Wannier fu…
Computing electron-defect (e-d) interactions from first principles has remained impractical due to computational cost. Here we develop an interpolation scheme based on maximally localized Wannier functions (WFs) to efficiently compute e-d interaction matrix elements. The interpolated matrix elements can accurately reproduce those computed directly without interpolation, and the approach can significantly speed up calculations of e-d relaxation times and defect-limited charge transport. We show example calculations of vacancy defects in silicon and copper, for which we compute the e-d relaxatio…
We describe and implement a first-principles algorithm based on maximally-localized Wannier functions for calculating the shift-current response of piezoelectric crystals in the independent-particle approximation. The proposed algorithm presents several advantages over existing ones, including full gauge invariance, low computational cost, and a correct treatment of the optical matrix elements with nonlocal pseudopotentials. Band-truncation errors are avoided by a careful formulation of $k\cdot p$ perturbation theory within the subspace of wannierized bands. The needed ingredients are the matr…
