Researchers Achieve Efficient Local Classification of Parity-Based Material Topology

Classifying the topological properties of aperiodic materials presents a significant hurdle for physicists, as traditional methods rely on the symmetry found in crystals. Researchers Stephan Wong, Ichitaro Yamazaki, and Chris Siefert, alongside colleagues including Iain Duff, Terry A. Loring, and Alexander Cerjan from institutions such as Sandia National Laboratories, Rutherford Appleton Laboratory, and the University of New Mexico, have now developed a new, numerically efficient framework to address this challenge. Their work bypasses the need for translational symmetry , a key requirement of existing techniques , by employing a real-space approach based on spectral localizers and direct computation of a Pfaffian, offering a local and energy-resolved topological invariant. This advancement is particularly crucial for understanding complex systems like quasicrystals and heterostructures, and promises a robust tool for diagnosing parity-based topological phases where conventional band theory falls short.

Pfaffian sign reveals aperiodic topological phases

Researchers Stephan Wong, Ichitaro Yamazaki, and Chris Siefert, alongside colleagues including Iain Duff, Terry A. Loring, and Alexander Cerjan from institutions such as Sandia National Laboratories, Rutherford Appleton Laboratory, and the University of New Mexico, have now developed a new, numerically efficient framework to address the challenge of classifying the topological properties of aperiodic materials. Scientists face an outstanding challenge in topologically classifying aperiodic materials using space-based approaches, as the absence of translational symmetry renders conventional methods inapplicable. This framework is based on the spectral localizer framework and the direct computation of the sign of a Pfaffian associated with a large sparse skew-symmetric matrix.
Unlike projector-based or momentum-space-based approaches, this method does not rely on translational symmetry, spectral gaps in the Hamiltonian’s bulk, or gapped auxiliary operators. Instead, it provides a local, energy-resolved topological invariant accompanied by an intrinsic measure of topology. Scientists have developed a scalable sparse factorization algorithm for reliably determining the Pfaffian’s sign for large sparse matrices, making it practical for realistic physical materials. Topological insulators are characterized by non-trivial topological invariants and exhibit robust boundary states protected against perturbations.

This robustness is key to enabling new technological domains, such as spintronics and fault-tolerant quantum computing . In crystalline systems, topological classification is traditionally performed using band theory, but this approach is inapplicable to non-periodic systems due to the lack of translational symmetry! Real-space approaches have been developed, based on projected position operators or the spectral localizer, and successfully applied to quasicrystalline, amorphous, and disordered systems! However, these existing local markers assume a spectral gap and sometimes a gap in a spin projection, failing in gapless systems!

Moreover, they often lack an independent measure of topological protection and are limited by the need for low-energy approximations. Here, researchers develop a numerically efficient approach rooted in the spectral localizer framework to classify Z2 parity-based topology across different symmetry classes and demonstrate its application to realistic aperiodic systems without needing a low-energy approximation. The algorithm can handle materials and heterostructures in arbitrary dimensions described by wave equations. The utility of the approach is illustrated by classifying the quantum spin Hall effect in a quasicrystalline heterostructure and identifying the fragile topology of a 2D photonic quasicrystal.

More broadly, the numerical approach allows for the efficient computation of Z2 parity-based invariants in many other AZ classes, enabling the prediction of boundary-localized protected states in natural, photonic, and acoustic materials, as well as identifying candidate materials that may exhibit novel correlated phases. The spectral localizer is Hermitian, and its symmetry manifests as (T ⊗I)L(x,E)(T ⊗I)−1 = L(x,E). The incorporation of the Pauli matrices introduces an additional symmetry to L(x,E). Perturbations to L(x,E) must possess the same invariant as that of the unperturbed system so long as the spectral gap of L(x,E) at 0 remains open, as a matrix’s Pfaffian cannot change sign without two of its eigenvalues reaching.

As L(x,E) is Hermitian, the movement of its eigenvalues under such perturbations is limited by Weyl’s inequality. Thus, the local index ξL (x,E) is accompanied by a measure of robustness given by the local gap μ(x,E)(X, H) = min |Spec L(x,E)(X, H) |. Locations where μ(x,E) ≈0 indicate the existence of an approximately localized state. Overall, this analysis shows that the topology of a 2D class AII system at a given location in position-energy space (x, E) can be locally classified by ξL (x,E), which carries with it a measure of protection μ(x,E), and at locations where the topology changes and μ(x,E) →0, the system exhibits an approximately localized state, yielding a bulk-boundary correspondence.

Pfaffian sign determination via sparse factorization

Scientists have developed a new real-space framework for classifying parity-based topology in aperiodic systems, overcoming limitations of conventional momentum-space approaches. This method, based on the spectral localizer framework and direct computation of a Pfaffian, doesn’t require translational symmetry, spectral gaps, or auxiliary operators, features often necessary for existing techniques. Instead, it offers a local, energy-resolved topological invariant alongside a measure of topological protection, proving particularly useful where band theory fails. A central achievement of this research is a scalable sparse factorization algorithm capable of reliably determining the sign of the Pfaffian for very large sparse matrices, a significant hurdle in applying these methods to realistic physical system.

Researchers successfully applied this framework to identify the spin Hall effect in quasicrystalline systems and to diagnose fragile topology in a photonic quasicrystal, demonstrating its versatility across electronic, photonic, and acoustic materials. The authors acknowledge a limitation in the size of systems currently addressable, dependent on computational resources and discretization density. Future work could focus on extending the algorithm to even larger systems and exploring its application to more complex aperiodic materials. This research establishes a robust, energy- and position-resolved understanding of topology, remaining meaningful even in gapless or inhomogeneous systems, and connects real-space classification to homotopy classes of sparse operators. The accompanying local gap quantifies topological protection and reinforces a bulk, boundary correspondence, guaranteeing localized states at topological transitions, a valuable tool for materials science and beyond.