Crystalline and Higher-Order Topological Matter

Crystalline topology uses spatial symmetry to obstruct a symmetric atomic description. A higher-order phase is additionally characterized by gapped boundaries of codimension one and protected states or responses on hinges, corners, or crystalline defects. Neither a symmetry indicator nor a corner-localized eigenstate alone establishes that complete statement.

Required background. Time-reversal topological insulators supplies stable band topology and boundary parity; Bloch and Wannier theory supplies orbital positions, sewing representations, and localization obstructions.

Atomic limits and symmetry indicators

An atomic insulator has exponentially localized, symmetry-compatible Wannier orbitals transforming in site-symmetry representations at allowed Wyckoff positions. Band representations induced from these orbitals determine symmetry eigenvalues at high-symmetry momenta. A symmetry indicator is a quotient: occupied-band representation data modulo those generated by chosen atomic limits Po, Vishwanath, and Watanabe 2017.

A nonzero indicator proves an obstruction within its assumptions, but a zero indicator does not prove triviality: some phases require Wilson loops or real-space invariants. Conversely, an obstructed atomic limit can carry quantized boundary charge for a fixed ionic reference without being stably topological after arbitrary atomic bands are added. Fragile topology is removed by adding selected trivial bands, whereas stable topology is not. The allowed band additions must therefore be stated.

Higher-order boundary structure

In a two-dimensional second-order insulator, one-dimensional edges are gapped while symmetry-related mass domains bind zero-dimensional corner charge or states. In three dimensions, gapped surfaces can meet at one-dimensional hinge modes. A continuum surface Dirac theory makes the mechanism transparent: if adjacent surfaces carry masses $m_1$ and $m_2$ whose signs are forced to differ by a protecting symmetry, their hinge is a mass domain wall and supports a lower-dimensional mode.

The boundary signature requires:

• a bulk gap and gapped lower-codimension boundaries;
• the spatial symmetry globally or in the explicitly stated statistical sense;
• a termination compatible with the symmetry comparison;
• a filling and ionic background used consistently when quoting corner charge;
• separation of in-gap topology from accidental boundary orbitals.

For the reflection-symmetric quadrupole model, nested Wilson loops can diagnose quantized edge polarizations and corner charge, but the construction requires gapped Wannier bands in addition to the energy gap Benalcazar, Bernevig, and Hughes 2017. If the Wannier spectrum closes, the nested invariant is not defined even when a corner state remains visible.

Disorder, interactions, and defects

A single crystal defect can probe rotation or translation response more robustly than a particular surface termination. Disclination or dislocation quantum numbers must be compared with the same crystalline symmetry and ionic reference. Generic disorder breaks spatial symmetry locally; an average-symmetry claim needs an ensemble definition and a mobility-gap test. Interactions can preserve a quantized many-body multipole or defect response, reduce a free classification, or enable intrinsically ordered surfaces. Symmetry eigenvalues of single-particle bands are then not a complete diagnostic.

Exercise

Four identical corners of a charge-conserving, fourfold-symmetric square sample each carry excess charge $q_c$ modulo $e$. If the total excess charge is one electron modulo $4e$, what values of $q_c$ are symmetry compatible?

Solution

Fourfold symmetry gives $4q_c=e$ modulo $4e$, hence $q_c=e/4$ modulo $e$. Adding an integer electron locally at every corner changes $q_c$ by $e$ and does not change the fractional corner class.

References

• Wladimir A. Benalcazar, B. Andrei Bernevig, and Taylor L. Hughes, “Quantized Electric Multipole Insulators,” Science 357 (2017) 61–66, doi:10.1126/science.aah6442.
• Hoi Chun Po, Ashvin Vishwanath, and Haruki Watanabe, “Symmetry-Based Indicators of Band Topology in the 230 Space Groups,” Nature Communications 8 (2017) 50, doi:10.1038/s41467-017-00133-2.