Source-linked AI summary
Catalogue of Topological Electronic Materials
Tiantian Zhang, Yi Jiang, Zhida Song, He Huang, Yuqing He, Zhong Fang, Hongming Weng, Chen Fang
TL;DR
Identifying topological materials is hindered by the complexity of calculating invariants and nodes. The paper combines first-principles symmetry data with complete representation-to-topology mappings in an automated diagnosis algorithm. It scans materials databases and reports thousands of topological materials across semimetal, insulator, and crystalline-insulator classes.
Problem
Calculating all topological invariants or identifying all topological nodes is lengthy and specialized, hindering thorough screening of materials.
Method
The algorithm combines first-principles band and symmetry data in spin-orbital-coupling and non-spin-orbital-coupling settings with mappings from occupied-band representations to topology.
Results
The scan classifies materials into high-symmetry and generic-momentum semimetals, topological insulators, topological crystalline insulators, magnetic materials, conventional metals, and trivial insulators.
Takeaways & Limitations
The resulting catalogue includes materials with unusual topological features, including candidate Z2-nontrivial nodal rings and noncentrosymmetric topological-insulator behavior.
Abstract
from arXiv · showhide
Topological electronic materials are new quantum states of matter hosting novel linear responses in the bulk and anomalous gapless states at the boundary, and are for scientific and applied reasons under intensive research in physics and in materials sciences. The detection for such materials has so far been hindered by the level of complication involved in the calculation of the so-called topological invariants, and is hence considered a specialized task that requires both experience with materials and expertise with advanced theoretical tools. Here we introduce an effective, efficient and fully automated algorithm in obtaining the topological invariants for all non-magnetic materials that are known to human, based on recently developed principles that allow for exhaustive mappings between the symmetry representation of occupied bands and the topological invariants. Our algorithm requires as input only the occupied-band information (energy and wavefunction) at a handful (up to eight) of high-symmetry points in the Brillouin zone, which is readily calculable with any first-principles software. In return, it is capable of providing a detailed topological classification of all non-magnetic materials. Equipped with this method we have scanned through a total of 39519 materials available in structural databases, and found that as many as 8056 of them are actually topological (8889 if spin-orbital coupling is neglected). These are further catalogued into classes of 5005 topological semimetals,1814 topological insulators and 1237 topological crystalline insulators, most of which are new to human knowledge. All the results are available and searchable at http://materiae.iphy.ac.cn/ .
I. CONTEXT
Topological materials are characterized by symmetry-protected invariants or robust band crossings, but identifying them requires lengthy calculations that have limited systematic searches. Recent complete mappings from valence-band representations to topology enable automated database scanning.
- I. CONTEXT: Topological invariants characterize materials with direct gaps throughout momentum space and depend on dimensionality and crystal symmetries.Time-reversal and crystalline symmetries protect different invariants, whose complete set characterizes a crystal’s topology.
- I. CONTEXT: Topological semimetals contain symmetry-robust crossings between conduction and valence bands, classified as nodal-point or nodal-line semimetals.Their classification depends on the degeneracy and dimensionality of the band crossings.
- I. CONTEXT: Evaluating all invariants or identifying all topological nodes requires lengthy, involved calculations, hindering thorough materials screening.Previously successful discoveries were often attributed to experienced researchers’ intuition.
- I. CONTEXT: Complete mappings from valence-band irreducible representations to topological invariants and nodes provide the basis for automated searches of large materials databases.The paper combines these theoretical mappings with first-principles numerical inputs.
II. THE ALGORITHM
The algorithm filters materials by magnetism and electron filling, computes band and symmetry data in two coupling settings, and diagnoses topology from high-symmetry representations and compatibility relations. Symmetry-based indicators then determine the final material classification.
- II. THE ALGORITHM: Materials are first filtered using magnetization and electron-count criteria because the symmetry-to-topology mapping applies to non-magnetic materials.The implementation rejects materials whose total magnetization in one unit cell is not below 10^-1 µB, then checks electron filling.
- II. THE ALGORITHM: First-principles calculations produce energy levels and Bloch wavefunctions at high-symmetry points in both soc-setting and nsoc-setting.These settings represent spin-orbital coupling on and off, respectively, and correspond to different symmetry classes.
- II. THE ALGORITHM: The analysis identifies irreducible representations of valence bands or degenerate multiplets and checks for partially filled representations at each high-symmetry point.Valence bands are defined as the lowest N bands, where N is the number of electrons per unit cell.
- II. THE ALGORITHM: Symmetry indicators quantify departures from atomic-insulator symmetry data and classify nontrivial materials when their values are nonzero.In the soc-setting, nonzero indicator combinations correspond to topological crystalline insulators; in the nsoc-setting, they correspond to generic-momentum semimetals.
- II. THE ALGORITHM: Materials without partially filled high-symmetry representations are tested against compatibility relations before indicator calculation.The flowchart labels failures as high-symmetry point or line semimetals and routes compatible data to the indicator calculator.
III. RESULTS
The scan classifies materials into five topological categories and documents representative candidates, including semimetals, a topological insulator, and a topological crystalline insulator.
- The algorithm labels materials as high-symmetry point semimetals, high-symmetry line semimetals, generic-momenta semimetals, topological insulators, or topological crystalline insulators.
- BaPPt is a high-symmetry point semimetal with conduction and valence bands meeting at Γ and R, including a sixfold degeneracy at R stabilized by nonsymmorphic symmetries.
- The representative materials include YCoC2, Sr2NiOsO6, NaCuO, and ZrTi2H4 for high-symmetry line, generic-momenta, insulating, and crystalline-insulating classes, respectively.Figure 2 presents band structures, Brillouin zones, and zoomed regions where needed.
- Sr2NiOsO6 has 2 mod 4 nodal rings at generic momenta, each carrying Z2 topological charge.
- NaCuO is a noncentrosymmetric topological insulator with three d–s band inversions at Γ and a band gap of approximately 0.1 eV.
- ZrTi2H4 becomes a topological crystalline insulator when spin-orbital coupling opens an approximately 10 meV full gap.
A. Setting up the first principles numerics
The numerical workflow uses VASP-based first-principles calculations with specified convergence settings, while warning that symmetry-based diagnosis and GGA have material-dependent limitations.
- Calculations use VASP with the PBE generalized-gradient approximation and pseudopotentials listed on the project website.
- Materials whose calculations fail to converge within 300 self-consistency steps are labeled and discarded.
- The scheme cannot identify topological materials sharing symmetry data with atomic insulators, which may require Wilson-loop diagnosis.
- GGA results may be inaccurate when strong correlation affects the Fermi energy, so rare-earth compounds with partially filled f orbitals are deferred.
B. Proof that band crossings in YCoC2 are nodal lines
YCoC2 exhibits band crossings on mirror-invariant planes whose opposite mirror eigenvalues establish two intersecting nodal rings.
- Compatibility relations fail along four high-symmetry lines where YCoC2 conduction and valence bands touch.
- Opposite mirror eigenvalues show that the crossings form nodal lines on two mirror-invariant planes.
- The two nodal rings are centered at Z and intersect at two points along Z-T and −Z-T.
C. Determine all topological invariants of ZrTi2H4
For ZrTi2H4, the authors use its symmetry-indicator set and an ab-initio Wilson-loop calculation to resolve the allowed topological invariants. The resulting symmetry-protected invariants imply one- and two-dimensional anomalous boundary states on suitably oriented surfaces.
- Invariant determination: ZrTi2H4 has symmetry indicators (z2w,1, z2w,2, z2w,3, z4) = (0002), yielding two possible invariant assignments.The two cases differ by whether the mirror Chern number on the 1¯10 plane is 2 (mod 4) or 0 (mod 4).
- Boundary states: Screw and inversion invariants protect one-dimensional helical modes along screw axes in an inversion-preserving cubic sample.The cubic sample’s top, bottom, and side surfaces are fully gapped in the stated orientation, while four helical modes run along each screw axis.
- Boundary states: Glide and two-fold rotation invariants protect two-dimensional surface states, including hourglass modes and 2 (mod 4) Dirac nodes on the 1¯10 surface.The 001 surface is fully gapped, whereas the 1¯10 and equivalent 110 surfaces are gapless.
- Invariant determination: The mirror Chern number is determined from the winding number of a mirror-eigenvalue-projected Wilson loop.Because the mirror representation commutes with the Wilson loop on the mirror-invariant plane, the loop can be projected into the +i mirror subspace.
- Invariant determination: The projected Wilson loop has winding number 0, placing ZrTi2H4 in the case with mirror Chern number 0 (mod 4).This ab-initio result selects the second possibility associated with the (0002) symmetry-indicator set.
SUPPLEMENTARY MATERIALS: TABLES OF TOPOLOGICAL MATERIALS
The supplementary tables catalogue topological materials by symmetry-based classification, band-crossing location, and indicators, while documenting notation and ordering conventions. They also distinguish SOC and NSOC results and use Fermi-level density of states to organize candidates by ideality.
- Classification: The tables catalogue high-symmetry point, high-symmetry line, generic-momentum semimetals, topological insulators, and topological crystalline insulators.These are the listed topological classes associated with band crossings or symmetry indicators.
- Notation: Table II uses irreducible-representation symbols, including dimensions and time-reversal-enforced degenerate pairs, to describe high-symmetry point semimetals.The notation follows the Bilbao Crystallographic Server conventions for double-group representations.
- Semimetals: High-symmetry line entries identify lines violating compatibility relationships, revealing isolated crossings, nodal lines, or nodal surfaces.These correspond respectively to Weyl/Dirac, nodal-line, and nodal-surface semimetals.
- Indicators: Tables IV–VI list symmetry-based indicator groups, but their generators can be constrained or annihilated by additional space-group symmetries.The catalogue therefore lists all three indices for centrosymmetric space groups even when they are not independent.
- Ordering: Materials sharing classification data are ordered by density of states at the Fermi energy, with 0.5 eV−1 per unit cell separating lower- and higher-density entries in Tables II–IV.Tables V and VI likewise order materials by Fermi-level density of states and separate zero-density entries from the rest.
- Candidate quality: Many insulating candidates have vanishing indirect gaps despite requiring only a full direct gap for well-defined band topology.Fermi-level density of states is used to indicate how ideal each candidate is.