Bulk-Defect Correspondence in Topological Materials

• The paper establishes a framework using real-space invariants to correlate bulk topology with quantized defect charges in crystalline systems.
• Bulk-defect correspondence is defined as the rigorous link between bulk RSIs and localized observables such as fractional corner charges in topological insulators.
• The methodology employs group-theoretic induction and Smith normal form to diagnose fragile topology, guiding predictions of surface and defect states.

Bulk-defect correspondence generalizes traditional bulk-boundary correspondence by precisely connecting bulk real-space invariants (RSIs) of crystalline or engineered systems to quantized observables at lower-dimensional defects—such as corners, hinges, or terminations. In the paradigmatic case of time-reversal-symmetric (TRS), spin-orbit coupled insulators, certain topological obstructions hidden in the bulk wavefunction do not yield gapless edge or surface states, but instead manifest as quantized, often fractional, defect-localized (e.g., corner) charges or bound states. The bulk-defect correspondence is governed by topological invariants that are not accessible to conventional Berry-phase, symmetry-indicator, or momentum-space approaches, necessitating the construction of refined RSIs anchored in the real-space crystalline symmetry and local site content of the insulator. These invariants quantitatively diagnose the presence of fractional corner charges, strong or fragile bulk topology, and classify the equivalence of atomic and obstructed atomic insulators across all crystalline space groups.

1. Real-Space Invariants and Site-Symmetry Formalism

The foundational objects in bulk-defect correspondence are partial real-space invariants (RSIs), which encode the excess number of Wannier (Kramers') pairs centered at high-symmetry Wyckoff positions in a (possibly gauge-fixed) sector of the filled bands. For an insulator with $N_F$ occupied bands, $n$-fold rotational symmetry $C_n$, and time-reversal $\Theta$, the occupied space can be partitioned into two $\Theta$-related sectors, each resembling a (possibly fragile) Chern insulator. The partition is constructed via a U($N_F$)-gauge preserving both $C_n$ and $\Theta$, leading to RSIs $\nu^I_{mx}$ for each Wyckoff position $mx$ in sector $n$0.

For example, in $n$1-symmetric TRS insulators, the RSIs are computed using multiplicities of $n$2-eigenvalues at time-reversal-invariant momenta: $n$3 with similar formulas for $n$4 and $n$5 symmetry (Tab. I, (Kooi et al., 2020)).

Notably, in spinful TRS systems, conventional symmetry indicators and Berry phase formulations fail: all single-point eigenvaule-based indicators vanish, and partial Berry phases (Fu-Kane-Mele line invariants) are insufficient to capture the full $n$6 classification of corner phenomena. The construction of RSIs relies on the structure of the sewing matrix for $n$7-combined symmetry, whose U($n$8)/O($n$9) gauge class yields invariants not accessible by Berry-Zak or Wilson loop integrals.

2. Bulk-Corner Correspondence: Quantized Corner Charge

The principal physical outcome of these RSIs is the quantization of corner charge localized at symmetry-related corners of finite systems: $C_n$0 where $C_n$1 denotes the $C_n$2-th Wyckoff position (Kooi et al., 2020). Because local physical observables can only shift $C_n$3 by even integers (e.g., by populating or depopulating in-gap corner states), the parity of $C_n$4 is a robust bulk invariant, insensitive to boundary condition details or filling of corner modes. This correspondence holds for

• atomic insulators (exponentially localized bulk Wannier functions)
• fragile topological insulators (gapped edge, non-Wannierizable bulk, yet well-defined quantized corner charges)
• heterostructures of symmetry-compatible insulators without protected in-gap boundary modes

The existence of fractional quantized corner charge—signaled by nonzero $C_n$5—is thus a direct, physically measurable manifestation of bulk-related, higher-order topology.

3. Detecting Fragile Topology and the Fragility Discriminant

Fragile crystalline phases exhibit gapped edges but cannot be adiabatically deformed to a pure atomic insulator without adding extraneous trivial bands. The bulk-defect correspondence provides a symmetry-indicator–free criterion for fragility based on the "fragility discriminant": $C_n$6 where $C_n$7 counts the essential defect charge at each corner. A negative discriminant ($C_n$8) certifies fragile topology, since the band filling is insufficient to realize all measured corner charges within the atomic limit (Kooi et al., 2020).

As an explicit example, in a $C_n$9-symmetric s–d model with properly tuned hopping and spin-orbit terms, calculated RSIs (e.g., $\Theta$0, $\Theta$1, etc.) and observed corner charges ($\Theta$2, $\Theta$3, $\Theta$4) yield a fragility discriminant $\Theta$5, confirming the fragile character.

4. Mathematical Structure: Induction Matrices and Smith Normal Form

The mathematical underpinning of bulk-defect correspondence within the RSI framework derives from group-theoretic induction:

• For each relevant Wyckoff position $\Theta$6 with site-symmetry group $\Theta$7, the induction of irreps from maximal subgroups to $\Theta$8 yields an integer “induction matrix” $\Theta$9.
• The Smith decomposition $\Theta$0 provides a canonical form where the diagonal entries of $\Theta$1 yield the torsion indices $\Theta$2 associated with individual RSIs.
• The vector of actual orbital multiplicities $\Theta$3 at each site defines the local RSI values

$\Theta$4

signaling obstructions to moving all Wannier centers off $\Theta$5 (Xu et al., 2021, Hwang et al., 14 May 2025).

Composite (multi-site) invariants emerge by taking appropriate mod-2, mod-4, or mod-$\Theta$6 linear combinations, especially when the symmetry allows for moving pairs or subsets of orbitals among different positions. This group-theoretic approach permits the construction of global topological invariants that fully diagnose both stable atomic equivalence and fragile/obstructed topology.

5. Physical Applications: Obstructed Atomic Insulators and Surface/Defect States

Bulk-defect correspondence applies beyond higher-order crystalline topological insulators. In the classification of obstructed atomic insulators (OAIs), RSIs detect situations where atomic-like bands are forced to place Wannier centers at empty Wyckoff positions. Such OAIs invariably exhibit "obstructed surface states" (OSSs) at terminations that cut through these positions, resulting in surface-localized (possibly metallic) modes even when the edge is otherwise trivial (Xu et al., 2021).

Concrete diagnostic procedure:

• Extract symmetry data from first-principles or tight-binding calculations.
• Compute induction matrices and identify nontrivial local RSIs at relevant Wyckoff positions.
• A nonzero RSI at an empty Wyckoff position signals an OAI, and predicts OSSs terminated through the obstructed site.

This approach has been validated through high-throughput computational screening, identifying thousands of OAIs and correlating calculated OSSs with experimental surface activity (e.g., edge-catalyzed hydrogen evolution in 2H-MoS$\Theta$7) (Xu et al., 2021).

6. Algorithmic and Computational Frameworks

The unified computational flow for diagnosing bulk-defect correspondence is as follows:

1. Obtain Bloch states (tight-binding or ab initio) and symmetry eigenvalues.
2. Construct rotation- and $\Theta$8-preserving gauges along relevant Brillouin zone boundaries as necessary.
3. Compute sector Berry connections, rotation-eigenvalue multiplicities, and sector sewing-matrix windings if required (Kooi et al., 2020).
4. Apply Smith decomposition to induction matrices for each Wyckoff position to extract all local and composite RSIs (Hwang et al., 14 May 2025, Xu et al., 2021).
5. Insert RSI values into the analytic formulas for defect (e.g., corner) charge, fragility discriminant, or surface state presence.

In complex space groups and with spin-orbit coupling, this is automated using tabulated induction matrices (e.g., Bilbao Crystallographic Server) and TQC tools (Hwang et al., 14 May 2025, Xu et al., 2021).

7. Significance, Extensions, and Future Directions

Bulk-defect correspondence rooted in real-space invariants fundamentally extends the scope of bulk-boundary correspondence:

• Provides sharp, symmetry-based criteria for anomalous defect observables (fractional charge, higher-order modes) beyond Berry-phase or symmetry-indicator diagnosis.
• Unifies the diagnosis of both strong and fragile topology in the real-space (Wannier) formalism.
• Enables fully group-theoretic, combinatorial computation of all topological classes (atomic, obstructed, fragile, stable) across the 1651 Shubnikov space groups with or without spin-orbit coupling (Xu et al., 2021, Hwang et al., 14 May 2025).
• Offers predictive power for physical properties of heterostructures, surface functionality, and response in the presence of geometric terminations.
• Forms the basis for machine learning mapping of local topology and identification of defect states in large, disordered, or engineered metamaterials.

Research continues towards systematic inclusion of interactions, disorder, projective symmetries (as in Hofstadter-like flux phases), and the design of new materials or devices exploiting robust bulk-defect quantization.

References:

(Kooi et al., 2020): Kooi–van Miert–Ortix, "Bulk–corner correspondence of time-reversal symmetric insulators: deduplicating real-space invariants" (Xu et al., 2021): "Three-Dimensional Real Space Invariants, Obstructed Atomic Insulators and A New Principle for Active Catalytic Sites" (Hwang et al., 14 May 2025): "Stable Real-Space Invariants and Topology Beyond Symmetry Indicators"