Two-dimensional Topological Quantum Chemistry and Catalog of Topological Materials

Technical Summary: Two-dimensional Topological Quantum Chemistry and Catalog of Topological Materials

Problem Statement
While the field of topological materials has expanded significantly in three dimensions (3D), with thousands of topological insulators (TIs) and semimetals identified via symmetry-based methods like Topological Quantum Chemistry (TQC) and Symmetry Indicators, the landscape of two-dimensional (2D) topological materials remains comparatively underexplored. Prior to this work, fewer than 300 topological or obstructed atomic insulator monolayers had been identified, a stark contrast to the over 9,000 known 3D counterparts. Existing high-throughput studies on 2D materials have utilized diverse techniques but have not provided a comprehensive, systematic catalog based on the full TQC formalism adapted specifically for 2D symmetries. Furthermore, the library of 2D materials is rapidly growing through computational databases like C2DB and MC2D, creating an urgent need for a unified framework to diagnose their band topology.

Methodology
The authors adapted the TQC formalism, originally developed for 3D space groups, to layer groups (LGs), which describe the symmetry of 2D monolayers and multilayers. This adaptation involved several key technical steps:

1. Development of New Software Tools: A suite of new programs was implemented in the Bilbao Crystallographic Server (BCS) to handle layer groups. These include:

   • LBANDREP: Calculates band representations (BRs) induced from atomic orbitals and identifies Elementary Band Representations (EBRs).
   • DLCOMPREL: Provides compatibility relations between irreducible representations (irreps) of little groups for pairs of k-vectors.
   • DLSITESYM: Calculates site-symmetry group multiplicities.
   • Check Topological Layer Mat.: Determines the topological classification of a material by analyzing the symmetry data vector (SDV) against EBRs.
   • RSI-layer: Calculates Real Space Invariants (RSIs) to identify Obstructed Atomic Insulators (OAIs).
   • Representations DLSG: Tabulates irreducible representations for layer groups with and without time-reversal symmetry.

2. Data Processing and Workflow: The authors processed 8,872 entries from the computational 2D materials databases C2DB and MC2D.

   • Structures were symmetrized and assigned to specific layer groups.
   • Density Functional Theory (DFT) calculations (using VASP) were performed to obtain wavefunctions and band structures, including spin-orbit coupling (SOC).
   • The symmetry data vector (traces of symmetry operations at high-symmetry k-points) was extracted and fed into the Check Topological Layer Mat. program.

3. Classification Scheme: Materials were classified based on the decomposition of their valence bands into EBRs:

   • Trivial Insulators (LCEBR): Bands are a linear combination of EBRs with positive integer coefficients.
   • Topological Insulators (TI): Includes SEBR (Split EBR, where bands are a linear combination of EBRs and parts of decomposable EBRs) and NLC (Not a Linear Combination, strong topology).
   • Topological Semimetals (TSM): Includes ES (Enforced Semimetal) and ESFD (Enforced Semimetal with Fermi level Degeneracy).
   • Obstructed Atomic Insulators (OAI/OOAI): Identified via non-vanishing RSIs at empty Wyckoff positions, indicating Wannier centers pinned at non-atomic sites.

Key Results
The high-throughput analysis yielded a substantial expansion of the known library of 2D topological materials:

• Total Non-Trivial Entries: 4,073 entries were identified as topologically non-trivial or possessing an obstructed atomic limit, representing 46.27% of the 8,872 materials studied.
• Specific Classifications:
  • 905 Topological Insulators: Comprising 647 SEBR and 258 NLC types (139 with a non-zero direct gap).
  • 2,129 Topological Semimetals: Including 19 ES and 2,110 ESFD types.
  • 1,003 Obstructed Atomic Insulators (OAI).
  • 36 Orbital-selected Obstructed Atomic Insulators (OOAI).
• Experimental Relevance: Through a literature review, the authors identified approximately 170 entries corresponding to experimentally fabricated non-magnetic materials (monolayer or few-layer). Among these, 86 were identified as topological or possessing an OAI.
• Selected Case Studies:
  • Bi₂Br₂ (LG 15): Predicted to be a thermodynamically stable NLC Quantum Spin Hall (QSH) insulator with a large gap (0.64 eV) and protected helical edge states.
  • Re₄Se₈ (LG 2): An experimentally existing monolayer identified as an OAI. The study demonstrated that cutting the material into ribbons along specific planes (passing through empty Wyckoff positions with non-zero RSIs) results in mid-gap obstructed edge states and potential fractional corner charges.
  • C₃Ta₄ (LG 72): An experimentally realized MXene identified as a SEBR QSH insulator.
  • Br₂Hg₃ (LG 80): A computationally exfoliated TI with a "dual" topological character (both QSHI and Topological Crystalline Insulator).

Significance and Claims
The paper claims to have largely expanded the library of known topological and obstructed materials in two dimensions, increasing the number of identified topological monolayers by an order of magnitude compared to previous literature. The authors emphasize that their work provides:

1. A Comprehensive Catalog: The creation of the Topological 2D Materials Database (2D-TQCDB), which contains band structures, topological indices, RSIs, and detailed properties for all studied materials. This serves as an "atlas" for experimentalists to guide synthesis efforts.
2. Methodological Advancement: The successful adaptation of TQC to layer groups enables the systematic study of not only monolayers but also multilayer 2D materials, which are described by the same symmetry groups.
3. Discovery of New Phenomena: The identification of materials with fragile topology, hourglass dispersions, and specific obstructed edge states (e.g., in Re₄Se₈) highlights the richness of 2D topological phases beyond standard TIs.

The authors conclude that their work facilitates the study of topology in new 2D systems and provides a necessary resource for the continuous progress in building and analyzing computationally predicted monolayers, while noting that further work is required to predict synthesis feasibility and analyze edge states in specific experimental configurations.