Symmetry-conserving maximally projected Wannier functions

To obtain a local description with a small basis size from density functional theory codes often a transformation to a local orthonormal Wannier function basis is desirable. In order to do so while enforcing the constraints of the space group symmetry the symmetry conserving maximally projected Wannier functions (SCMPWF) approach has been implemented in the full potential-local orbital code, FPLO. SCMPWFs are obtained from the initial step of the maximally localized Wannier function algorithm, projecting a subset of wave functions onto a set of suitably chosen local trial functions with prescribed symmetry properties with subsequent orthonormalization. The particular nature of the local orbitals in FPLO makes them an ideal set of projectors since they are constructed to be a small basis adapted to the occupied sector and the lowest-lying unoccupied states. While in many cases projection onto the FPLO basis orbitals is sufficient, the option is there to choose particular local linear combinations as projectors, in order to treat cases of bond centered Wannier functions. This choice turns out to lead to very localized Wannier functions, which obey the space group symmetry of the crystal by construction. Furthermore we discuss the interplay of the Berry connection and position operator and especially its possible approximation, symmetries, and the optimal choice of Bloch sum phase gauge in cases where the basis is not explicitly known. We also introduce various features, which are accessible via the FPLO implementation of SCMPWFs, discuss and compare performance and provide example applications.

Figure 1

The work flow to construct and use SCMPWFs in the FPLO implementation.

Figure 2

(a) Band structure (black) of ${\mathrm{CaCuO}}_2$, the FPLO orbital $3d_{x^2-y^2}$ band character (red) and the single band Wannier fit (cyan). (b) Unit cell containing the Wannier function at isovalue 0.04. (c) Maximum of orbital contributions to the WF in logarithmic scale as a function of WF-orbital distance. (d) WF printed along the (100) axis through the WF center (black), the Cu $3d_{x^2-y^2}$ (orange), and the O $2p_x$ (red) orbitals scaled according to their amplitude. The inset shows $\left|w\right|$ along the same path in logarithmic scale.

Figure 3

(a) Band weights of the FPLO calculation in pastel colors and Fe $3d$ WF fit in bright colors (see text). (b) $3d_{x^2-y^2}$ WF printed along the (100) axis through the WF center (black), the Fe $3d_{x^2-y^2}$ local orbital (orange). Left inset shows $\left|w\right|$ along the same path in logarithmic scale. Right inset shows the maximal orbital contributions to the WFs in logarithmic scale as a function of WF-LO distance.

Figure 4

(a) Band weights of the FPLO calculation in pastel colors and the WF fit in bright colors. (b) (Top down) $3d_{x^2-y^2}, 4s$ and $4p$ WFs printed along the (100) axis through the WF center (black), the FPLO local orbitals (orange). (c) Corresponding WF isosurfaces for isovalue 0.09. (d) Maximal orbital contributions to the WFs in logarithmic scale as a function of WF-LO distance. (e) $\left|w\right|$ along the same path in logarithmic scale.

Figure 5

Berry curvature of spin polarized full relativistic bcc Fe. (a) Thin lines: Fermi surface contour in the $k_y=0$ plane. Color plot: $-{\mathit{F}}_z$. (b) The Berry curvature components along the path $\mathrm{\Gamma }\mathrm{HP}$ (see text). The left panel shows term-grouping of Ref. [21]: $\nabla \nabla =i{\langle Si\nabla \rangle }_C\times {\langle Si\nabla \rangle }_C, \nabla a=i{\langle Si\nabla \rangle }_C\times {\langle \mathit{A}\rangle }_C+\text{H.c.,}$ and $f={\langle \mathit{f}\rangle }_C$ The right panel shows our preferred grouping $AA=i{\mathit{A}}_{\mathrm{\Psi }}\times {\mathit{A}}_{\mathrm{\Psi }}$ and $f-aa={\langle \mathit{f}\rangle }_C-i{\langle \mathit{A}\rangle }_C\times {\langle \mathit{A}\rangle }_C$.

Figure 6

(Left) Berry curvature of spin polarized full relativistic B2 FeAl. (Right) Berry curvature of non-spin-polarized full relativistic HgS. (Top) Term grouping of Ref. [21] for periodic gauge. (Middle) Term grouping of Ref. [21] for relative gauge. (Bottom) Preferred term grouping. The terms are explained in Fig. 5 except for $A{A}_{\text{approx}}$, which is the $AA$ term with approximated basis connection.

Figure 7

(a) FPLO band weights in pastel colors compared to the $s{p}^2$-hybrid Wannier fit in black and the energy window plotted at the right side. (b) WF profile along a path through the Wannier center as indicated by the thin line in (c). The left inset shows $\left|w\right|$ along the same path in a logarithmic plot, while the right inset shows $max\left(\right|a_{\mathit{R}\mathit{s},\mathit{0}\mathit{c}}\left|\right)$, i.e., the orbital contributions to the WF, also in a logarithmic plot. (c) Isosurface of the WF for isovalue 0.4. (d) Schematic of the MO projectors used to obtain the WFs.

Figure 8

Isosurfaces of the four WFs of the minimum model for ${\mathrm{H}}_2\mathrm{O}$ for an isovalue of 0.09. The character is denoted at the side. The atoms are shown as white (H) and red (O) spheres.

Figure 9

Surface spectral densities (SD) for ${\mathrm{TaIrTe}}_4$ of a semi infinite slab with (001) (left column) and $\left(00\overline{1}\right)$ (right column) termination. The crosses mark the position of the Weyl point. (Top) Fermi surface SD, the arrows mark the Fermi arcs. (Bottom) Corresponding energy resolve SD along a line $k_y=0.125$. The arrows mark the surface bands forming the arcs.

Figure 10

(Left) LDA band structure of ${\mathrm{CaCuO}}_2$ computed with quantum espresso and Wannier fit for a one-orbital model performed using symmetry-adapted scheme implemented in wannier90. Right: maximal contributions to the WF in a logarithmic scale as a function of WF-orbital distance.

Figure 11

(Top) Wannier fit of the LDA band structure of bcc Fe computed with quantum espresso. The Wannier model is constructed by projections onto $3d$ states. Right panel shows maximal contributions to the WF in a logarithmic scale as a function of WF-orbital distance. Only the orbital-diagonal contributions are shown. (Bottom) Same for the model comprising $3d, 4s$, and $4p$ states.

Figure 12

Berry curvature of spin polarized full relativistic bcc Fe computed using wannier90.

Figure 13

Berry curvature of spin polarized full relativistic FeAl (left) and nonmagentic HgS (right) computed using wannier90.

Figure 14

(a) Berry curvature dipole of TaAs. The solid and dashed lines represent the $xy$ and $yx$ components, respectively. The Berry curvature dipole of (b) the $xy$ component and (c) the $yx$ component of ${\mathrm{TaIrTe}}_4$.