Source-linked AI summary
WannierTools: An open-source software package for novel topological materials
QuanSheng Wu, ShengNan Zhang, Hai-Feng Song, Matthias Troyer, Alexey A. Soluyanov
TL;DR
The paper addresses the need for software to investigate and classify novel topological materials. It presents WannierTools, a tight-binding open-source package that computes bulk topological and Berry properties, searches for nodes, and obtains surface spectra. The package is demonstrated on HfPtGe, identified as a Dirac nodal line semimetal.
Problem
Existing software support for investigating novel topological materials is limited, motivating tools that classify bulk topology and study corresponding surface states and nodal structures.
Method
WannierTools uses tight-binding models with Wilson-loop and Wannier-center calculations, Berry-property computations, nodal searches, and iterative Green’s functions for surface spectra.
Results
HfPtGe was identified as a Dirac nodal line semimetal, with a smallest spin–orbit-coupling gap of about 0.1meV.
Takeaways & Limitations
WannierTools provides a user-friendly open-source workflow for identifying topological properties and obtaining surface spectra that can be compared with experimental data.
Takeaways & Limitations
WannierTools is limited to tight-binding models with orthogonal basis functions and requires a Wannier90-format Hamiltonian plus an input file.
Abstract
from arXiv · showhide
We present an open-source software package WannierTools, a tool for investigation of novel topological materials. This code works in the tight-binding framework, which can be generated by another software package Wannier90. It can help to classify the topological phase of given materials by calculating the Wilson loop and can get the surface state spectrum which is detected by angle-resolved photoemission (ARPES) and in scanning tunneling microscopy (STM) experiments. It also identifies positions of Weyl/Dirac points and nodal line structures, calculates the Berry phase around a closed momentum loop and Berry curvature in a part of the Brillouin zone.
PROGRAM SUMMARY
WannierTools is an open-source Fortran 90 package for classifying crystalline topological phases and studying related slab and ribbon electronic properties. It uses tight-binding models, Wilson loops, and iterative Green’s functions.
- The distributed program contains 17118 lines and 7421589 bytes of code and test data.
- WannierTools is written in Fortran 90 and runs on Unix, Linux, and Mac OS X systems with a Fortran 90 compiler.
- WannierTools identifies topological classifications of crystalline insulators, semimetals, and metals.
- The package uses the tight-binding method as an approximation for solid-state systems.
- Wilson loops classify the topological phase, while iterative Green’s functions obtain surface-state spectra.
1. Introduction
WannierTools addresses the growing investigation of novel topological materials amid limited software support. It combines topological classification with surface-state, nodal-structure, and Berry-property calculations in a user-friendly tight-binding package.
- Novel topological phases include three-dimensional topological insulators, Dirac and Weyl semimetals, nodal-line semimetals, and nodal-chain metals.
- Topologically protected surface states provide experimentally detectable evidence for non-trivial bulk topology through bulk-edge correspondence.
- Existing packages such as Z2pack and PythTB provide Wannier-center, tight-binding, Berry-phase, and related calculations, but software options remain limited.
- WannierTools calculates bulk Z2 numbers and Wilson loops, searches for Weyl/Dirac points and nodal loops, and computes Berry phases and Berry curvature.
- The package also studies slab and ribbon electronic structures and surface states for comparison with experiments.
2. Methods
The methods formulate orthogonal tight-binding Hamiltonians in momentum space and compute bulk topology, Berry properties, and surface spectra. WannierTools uses Wannier-center techniques, discrete overlap matrices, and iterative Green’s functions for these calculations.
- 2.1. TB method: Tight-binding models project solid-state Hamiltonians onto local orbitals and can be constructed using Slater–Koster methods, MLWFs, or discretized k·p models.
- 2.1. TB method: WannierTools handles orthogonal tight-binding bases, including MLWF models generated with Wannier90.
- 2.1. TB method: The real-space tight-binding Hamiltonian is Fourier transformed into momentum space, using the convention whose eigenvectors correspond to periodic Bloch functions.
- 2.1. TB method: Slab Hamiltonians use two periodic surface directions and a confined third direction represented by layer-indexed intra-plane and inter-plane blocks.
- 2.1. TB method: Ribbon systems confine two directions, enlarging the Hamiltonian relative to slab systems.
- 2.2. Wannier charge center calculation: Wannier charge centers and Wilson loops classify topology through their evolution across momentum-space strings, using overlap matrices between neighboring Bloch states.
- 2.3. Berry phase and Berry curvature: Berry phase and Berry curvature are formulated for isolated bands and generalized to occupied multi-band manifolds using non-Abelian connections.
- 2.4. Calculation of surface states: Surface spectra can be calculated for slabs or semi-infinite systems using iterative Green’s functions and surface Green’s functions.
3. Capabilities of WannierTools
WannierTools supports two complementary tasks: classifying bulk band topology and exploring surface-state properties associated with that topology.
- Bulk topology studies: WannierTools classifies bulk topology using band-structure, Fermi-surface, DOS, WCC, node-searching, energy-gap, Berry-phase, and Berry-curvature calculations.These calculations distinguish insulating or metallic behavior and identify topological indices, Weyl/Dirac points, and nodal-line structures.
- Bulk topology studies: The package computes Z2 topological indices or Chern numbers for band insulators through Wannier charge-center calculations.
- Related responses from the bulk topology: It explores bulk-topology-related responses through JDOS, slab and wire electronic structures, surface-state spin textures, and QPI patterns.The listed responses connect bulk topology to calculated surface and related electronic properties.
- Capabilities overview: The two capability groups cover both determining a material’s topological classification and studying corresponding surface-state properties.
4. Installation and usage
WannierTools is an open-source Fortran 90 package installed with numerical linear-algebra libraries and run from a tight-binding input file plus a master control file.
- Installation: WannierTools is open-source software released on GitHub under the GNU General Public Licence 3.0.
- Installation: Building WannierTools requires a Fortran 90 compiler, BLAS, and LAPACK; MPI-enabled compilation supports parallel execution.The package is compatible with several MPI implementations, including MPICH, MVAPICH, and Intel MPI.
- Input files: Users provide a Wannier90 tight-binding parameter file and wt.in, the master input file, before running the program.The tight-binding file can come from Wannier90, a toy model, or a discretized k · p model, and its filename is specified in wt.in.
- Running: The executable can be launched with MPI using a command such as mpirun -np 4 wt.x.
- Outputs: WannierTools records status information in WT.out and produces data files plus gnuplot scripts for visualizing results.A bulk band-structure example generates bulkek.dat and bulkek.gnu after successful completion.
5. Examples
The HfPtGe example uses WannierTools to connect first-principles and tight-binding calculations with nodal-line, Z2, surface-state, and spin-texture analyses. HfPtGe is identified as a strong topological semimetal with a weak-SOC gap and linked electron–hole pockets.
- 5.1. Crystal structure and Band structure: First-principles calculations show that Hf and Pt 5d orbitals dominate near the Fermi level, with hybridization from Ge 4p and Pt s orbitals.The resulting tight-binding model is used for the WannierTools analysis.
- 5.2. Energy gap shape: Two Dirac-like cones occur along the Y-Γ-Z direction, originating from a mirror-symmetry-protected nodal line without SOC.With SOC, the nodal line becomes gapped because the material’s SOC strength is weak.
- 5.2. Energy gap shape: 0.1 meV is the smallest SOC-opened gap, located near k points (±0.4, 0.0, ±0.229).The opened gap is small because the SOC strength is very weak.
- 5.2. Energy gap shape: Six nodes cross the Fermi level, producing linked compensated electron and hole pockets on the Fermi surface.The paper associates these pockets with extremely large positive magnetoresistance.
- 5.3. Wannier charge center: Wannier charge-center calculations give Z2 = 1 on kx = 0, ky = 0, and kz = 0 planes and zero on the other time-reversal-invariant planes, yielding (1, 000).The result classifies HfPtGe as a strong topological material in all three reciprocal-lattice directions.
- 5.4. Surface state spectrums: On the (010) surface, the SOC surface spectrum contains a highly anisotropic, tilted Dirac-like cone at Γ.At E = EF, the surface states originate from momenta with the smallest bulk gap.
- 5.5. Surface state spin texture: The calculated surface-state spin texture reflects SOC-induced spin–momentum locking in momentum space.The paper presents the spin texture for the fixed-energy surface-state spectrum.
6. Conclusions
WannierTools is an open-source, user-friendly Fortran90 package that uses MPI for performance and investigates topological properties and surface spectra. Its HfPtGe case study identifies the material as a Dirac nodal line semimetal.
- WannierTools is an open-source, user-friendly Fortran90 package using MPI techniques for strong performance on computer clusters.
- The package identifies topological properties and calculates surface state spectra that can be compared with experimental data.
- HfPtGe is identified as a Dirac nodal line semimetal through the package’s exploration of a new topological material.
Appendix A. wt.in for HfPtGe
The HfPtGe input file enables bulk, surface-state, arc, spin-texture, gap, and three-dimensional topological calculations using specified energy, sampling, and threshold parameters.
- The control block enables bulk bands, bulk Fermi surfaces, bulk gap planes, three-dimensional Z2 calculations, slab surface states, slab arcs, and slab spin textures.
- The parameter block sets E_arc to 0.0, OmegaMin to -0.6, OmegaMax to 0.6, and Gap_threshold to 0.05.
- The calculation uses OmegaNum = 100 and momentum-grid sizes Nk1 = 101, Nk2 = 201, and Nk3 = 101.
- The input identifies Hf, Ge, and Pt orbital components for the tight-binding calculation.
- KCUBE_BULK is set to -0.50 -0.50 -0.50 in the supplied configuration.