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Irvsp: to obtain irreducible representations of electronic states in the VASP

Jiacheng Gao, Quansheng Wu, Clas Persson, Zhijun Wang

arXiv:2002.04032v3cond-mat.mtrl-scicond-mat.str-el

TL;DR

The paper addresses the need to determine electronic-band irreducible representations throughout Brillouin zones, including nonsymmorphic crystals and all 230 space groups. It presents irvsp, which processes VASP wavefunctions and space-group operators to compute matrix traces and IRs, with SOC, library, and tight-binding extensions. Applied to PdSb2, the analysis identifies a strain-sensitive topology and a four-fold Dirac nodal-line metal under tiny strains.

  • Problem

    Determining irreducible representations for every k-point in all 230 space groups, especially nonsymmorphic Brillouin-zone surface points, remained unsupported by existing tools.

  • Method

    irvsp reads VASP plane-wave wavefunctions and space-group operators, computes matrix-presentation traces, and assigns irreducible representations, including SOC double groups and orthogonal tight-binding Hamiltonians.

  • Results

    For PdSb2, irreducible-representation analysis finds a topological insulating phase without strain and a symmetry-enforced semimetallic phase with tiny tensile strains.

  • Takeaways & Limitations

    The BCS-labeled irreducible representations can be compared directly with elementary band representations to investigate band topology and band connectivities.

Abstract

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We present an open-source program irvsp, to compute irreducible representations of electronic states for all 230 space groups with an interface to the Vienna ab-initio Simulation Package. This code is fed with plane-wave-based wavefunctions (e.g. WAVECAR) and space group operators (listed in OUTCAR), which are generated by the VASP package. This program computes the traces of matrix presentations and determines the corresponding irreducible representations for all energy bands and all the k-points in the three-dimensional Brillouin zone. It also works with spin-orbit coupling (SOC), i.e., for double groups. It is in particular useful to analyze energy bands, their connectivities, and band topology, after the establishment of the theory of topological quantum chemistry. Accordingly, the associated library - irrep_bcs.a - is developed, which can be easily linked to by other ab-initio packages. In addition, the program has been extended to orthogonal tight-binding (TB) Hamiltonians, e.g. electronic or phononic TB Hamiltonians. A sister program ir2tb is presented as well.

I. INTRODUCTION

The paper develops irvsp to determine irreducible representations (IRs) of crystal electronic bands across all 230 space groups, addressing limitations of existing tools for nonsymmorphic and non-maximal k-points. It computes symmetry characters from wavefunctions and space-group operators, supporting SOC and extensions to tight-binding workflows.

  • Motivation: Topological quantum chemistry classifies isolated-band topology using IRs at maximal high-symmetry k-points and compatibility relations.The existing vasp2trace program computes IRs at maximal points but is unsuitable for non-maximal high-symmetry points.
  • Motivation: Existing PNG-based IR labels cannot describe many Brillouin-zone surface k-points in nonsymmorphic crystals.This leaves IR determination for every k-point across all 230 space groups unresolved.
  • Approach: irvsp uses wavefunctions at k-points and character tables for k-little groups to determine IRs in the BCS convention.The code computes matrix-presentation traces, whose characters determine the corresponding little-group IRs.
  • Capabilities: The implementation supports plane-wave and tight-binding bases, including single and double groups and spin-orbit coupling.SOC is handled by doubling the basis with spin degrees of freedom.
  • Implementation: The program interfaces with VASP and can link the irrep bcs.a library to other ab-initio packages.VASP supplies plane-wave coefficients and space-group operators used in the calculations.

B. Orthogonal tight-binding basis

The TB extension represents Wannier functions in an orthogonal, exponentially localized orbital basis and uses symmetry operations on local orbitals to formulate the Hamiltonian representation. This implementation is provided through the sister program ir2tb for orthogonal electronic or phononic TB Hamiltonians.

  • Wannier functions are represented using exponentially localized orthogonal orbitals indexed by atoms, orbitals, lattice vectors, and home-cell positions.
  • At each k-point, the Wannier functions are obtained as eigenvectors of the tight-binding Hamiltonian.
  • Local-orbital rotational symmetries are encoded by D-matrices, which are real in bases of real spherical harmonics with fixed integer angular momentum.
  • The TB extension is implemented in ir2tb, which accepts case hr.dat and tbbox.in for orthogonal electronic or phononic Hamiltonians.case hr.dat may come from symmetrized Wannier90 parameters, toy models, Slater-Koster methods, or discretized k · p models.

III. GENERAL PROCESS OF THE CODE

The main code, irvsp, operates in the plane-wave basis through an interface to VASP, using reconstructed all-electron wavefunctions and coefficients supplied in WAVECAR. The workflow accounts for plane-wave cutoffs, trace calculations, wavefunction normalization, and separate spin components in SOC calculations.

  • irvsp is primarily based on the plane-wave basis and interfaces with VASP through summarized key subroutines.
  • VASP reconstructs each all-electron wavefunction by applying a linear operator T to the corresponding pseudo-wavefunction.
  • The pseudo-wavefunction is expanded in plane waves whose G_j vectors satisfy a kinetic-energy cutoff condition set by E_cutoff.
  • Wavefunctions are renormalized before traces are evaluated, and the resulting quantities are sufficient for calculating traces of matrix presentations of space-group operators.
  • The plane-wave coefficients output in WAVECAR are read by irvsp, with separate coefficient arrays for spin-up and spin-down components when SOC is enabled.

B. Space group operators of a 3D crystal

irvsp reads VASP-generated space group operators and lattice data from OUTCAR, representing them in lattice and spin spaces for subsequent symmetry analysis.

  • irvsp reads space group operators directly from VASP's OUTCAR rather than generating them from POSCAR.The operators are listed under “Space group operators:” and encode determinant, rotation angle, axis, and translation information.
  • The program obtains primitive real-space and reciprocal-space lattice vectors from OUTCAR for the crystal representation.Bi2Se3 is used as the example in the supplied figures.
  • The code generates O(3) and SU(2) matrix presentations in spin-1 and spin-1/2 bases, respectively.The stated bases are {x, y, z} for spin-1 and {↑, ↓} for spin-1/2.
  • Rotational operators are transformed into 3 × 3 integer matrices in the primitive lattice-vector basis.These matrices, Z and Z^-1, are stored and used throughout the code.
  • Users can modify the OUTCAR operator list and specify a corresponding space group number for sub-space-group symmetry calculations.The paper gives parity eigenvalues as an example using only identity and inversion with space group #2.

C. Little group of a certain k-point

For each k-point, irvsp first identifies the little group and then determines irreducible representations by comparing computed matrix traces with character tables. It supports both point-group and BCS conventions, including time-reversal and SOC-dependent degeneracy rules.

  • Little group determination: irvsp determines the little group LG(k) before analyzing the irreducible representations supported by eigen-wavefunctions at a given k-point.The implementation uses integer matrices Z^-1 together with the momentum-space symmetry relation.
  • Character-table conventions: The PNG convention is suitable for symmorphic space groups and inner k-points, whereas the BCS convention covers k-points of all 230 space groups.PNG labels derive from point-group irreducible representations with phase factors, while BCS character tables are listed for space-group little groups.
  • Character-table conventions: At Γ in Bi2Se3, both the D3d PNG table and the SG 166 BCS table can determine the corresponding irreducible representations.The BCS table includes irreducible-representation labels and characters for different space group operators.
  • Time reversal and SOC: Time-reversal symmetry doubles irreducible-representation degeneracies in cases (b) and (c) without SOC, and in cases (a) and (b) with SOC.The cases correspond respectively to essentially complex, pseudo-real, and potentially real representations as defined in the paper.
  • Irreducible-representation determination: The code compares traces of matrix presentations computed from normalized VASP pseudo-wavefunctions with character-table entries to identify irreducible representations.The traces are treated as characters in the determination procedure.

VI. EXAMPLES

The examples show how irvsp identifies irreducible representations and uses them with topological quantum chemistry to analyze strain-dependent topology in PdSb2 and band representations in Bismuth.

  • PdSb2: irvsp distinguishes the irreducible representations at PdSb2’s R point for unstrained and tensile-strained crystal structures.Figure 6 reports band indices, degeneracies, energies, operator characters, and assigned IR labels for the three structures.
  • PdSb2: A tiny gap of approximately 10 meV separates two sixfold degeneracies at R in unstrained PdSb2, while tensile strain primarily changes the bands near R.The calculations use Δa/a = 0.31% and 0.62% tensile strains.
  • PdSb2: Comparing IRs at maximal high-symmetry points with elementary band representations identifies unstrained PdSb2 as topological insulating and strained PdSb2 as symmetry-enforced semimetallic.The strained phase has tiny tensile strains.
  • PdSb2: Along R–X, irvsp assigns different S3 and S4 representations at P and Q, establishing a robust fourfold crossing on the Brillouin-zone boundary.The double degeneracy follows from time-reversal symmetry, while the crossing is protected by the gy symmetry.
  • PdSb2: Considering the full SG 205 symmetry, the strained PdSb2 bands form three Dirac nodal rings.The paper notes that these nodal rings can be further checked experimentally.
  • Bismuth: For Bismuth in SG 166, irvsp obtains IRs at maximal high-symmetry points for comparison with the complete SOC elementary-band-representation list.The EBR list is organized by maximal Wyckoff site, site-symmetry IR, and IRs at maximal HSK points.

APPENDIX

The appendix provides multiple code versions tailored to different wavefunction types and character-table conventions.

  • Appendix: Different irvsp-related codes are available because wavefunction types and character-table conventions differ.The appendix lists these variants in Table S1 and provides their repository location.

2. The brief description of ir2tb

ir2tb extends irreducible-representation analysis to orthogonal tight-binding Hamiltonians using Wannier-based or user-generated tight-binding inputs.

  • The brief description of ir2tb: ir2tb is based on tight-binding wavefunctions and requires BLAS and LAPACK to diagonalize the tight-binding Hamiltonian.It uses the library irrep bcs.a during compilation.
  • The brief description of ir2tb: The program requires two input files: case hr.dat containing tight-binding parameters and tbbox.in describing the Hamiltonian.The parameters may come from Wannier90, a toy model, Slater–Koster construction, or discretized k · p models.
  • The brief description of ir2tb: tbbox.in specifies whether spin-orbit coupling is absent or present and controls local-orbital ordering conventions.The corresponding lda or soc hr.dat file must be supplied consistently.
  • The brief description of ir2tb: The appendix documents tbbox.in and the local orbitals used by ir2tb, including orbital conventions and vector definitions.These materials are summarized in Tables S2 and S3 and related equations.

3. The standard settings for POSCAR and maximal HSK points

The appendix specifies standard POSCAR conventions, recommends symmetry standardization before VASP calculations, and explains conversion between conventional and primitive reciprocal lattices.

  • Standard POSCAR settings: The standard POSCAR settings use unique axis b for monoclinic space groups, the obverse triple hexagonal cell for rhombohedral groups, and origin choice two for centrosymmetric groups.Origin choice two places the inversion center at (0,0,0).
  • POSCAR standardization: The authors recommend running phonopy to obtain the space-group number and standardize POSCAR before VASP calculations.The stated command uses --tolerance 0.01 and copies PPOSCAR to POSCAR.
  • Reciprocal-lattice transformations: Maximal HSK points from the BCS use conventional reciprocal lattice vectors, whereas VASP commonly uses primitive-cell lattice vectors.The required transformation depends on the lattice type among P, C, B, A, R, F, and I.

4. The character tables for R-little group and S-little group

Figs. S1 and S2 provide character tables for the R-little and S-little groups in the BCS convention. The supplementary expressions encode complex phase factors at k-points.

  • Figs. S1 and S2 show the character tables for the R-little group and S-little group, respectively, using the BCS convention.
  • At k-point coordinates (u, v, w) in the conventional reciprocal basis, a block x + iy acquires a symmetry-dependent complex phase.
  • The phase factor is expressed as (x + iy) · exp[iπ(au + bv + cw)].

5. Other versions of irvsp

irvsp provides four implemented versions to represent irreducible representations under different character-table conventions. The versions address nonsymmorphic boundary k-points where the basic PNG-based approach can fail.

  • Version I presents irreducible representations with point-group symmetries and works similarly to the irrep program in WIEN2k.
  • Version I cannot classify special Brillouin-zone boundary k-points in nonsymmorphic crystals when exp[−ik(Rsvt + vs)] ≠ 1 for some little-group operations.
  • Version II handles nonsymmorphic-space-group k-points where Version I does not, while Version III combines Versions I and II.
  • The command irvsp -sg $sgn [-v $nv] [-nb $m $n] > outir & runs irvsp with a selected space group, version, and band range.
  • Four versions of irvsp are implemented, with the reference character-table convention and descriptions summarized in Table S4.

6. The library – irrep bcs.a

The irrep bcs.a library packages the irreducible-representation calculations for reuse by DFT and tight-binding codes. Its routines accept crystal, symmetry, wavefunction, band, and basis data for plane-wave or orthogonal tight-binding calculations.

  • The irrep bcs.a library can be linked to other DFT packages, and ir2tb demonstrates calling its main irrep_bcs subroutine.
  • The library is compiled after configuring the environment and running make lib, which creates irrep bcs.a in the current folder.
  • The main routines are irrep_bcs, pw_setup, and tb_setup, covering irreducible-representation evaluation and plane-wave or tight-binding setup.
  • The interface takes space-group number and operations, k-point coordinates, phase factors, band energies, band ranges, and a spinor flag.
  • Wavefunction inputs include plane-wave coefficients and optional plane-wave phase and transformation data, or tight-binding transformation matrices in an orthogonal basis.
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