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Ono and Shiozaki Map Superconductor Topology in 159 Space Groups
Seishiro Ono and Ken Shiozaki present a method for constructing topological invariants using the Atiyah-Hirzebruch spectral sequence. Applied to time-reversal-symmetric spinful superconductors with conventional pairing, their invariants com
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“Remarkably, the constructed topological invariants completely characterize K groups in 159 space groups.”
Seishiro Ono and Ken Shiozaki, authors of the paper, write
A method for topology beyond high-symmetry points
Seishiro Ono and Ken Shiozaki published a systematic method for constructing topological invariants in Physical Review X on October 23, 2026. Their paper applies the method to time-reversal-symmetric spinful superconductors with conventional pairing symmetries, a setting in which symmetry indicators are silent. The authors report that their invariants completely characterize K groups in 159 of the 230 space groups.
Symmetry-indicator methods use information such as wave-function representations at high-symmetry points to diagnose topology. They provide an efficient partial diagnosis, but some phases cannot be detected using information from those points alone. Ono and Shiozaki’s approach uses the Atiyah-Hirzebruch spectral sequence (AHSS) in momentum space to develop a systematic way to construct invariants.
From momentum-space regions to invariants
The framework organizes momentum space into subregions. In a three-dimensional system, a fundamental domain of the Brillouin zone can be divided into points, line segments and faces, with symmetry operations generating the rest of the zone. The researchers use the AHSS to identify which symmetry representations on these regions enter the construction and how quantities associated with them can be combined.
The paper describes two broad steps: identifying the relevant representation sectors on subregions, then finding quantities that characterize their topological properties. The authors formulate the first step for subregions of arbitrary dimension. For the second, they focus on constructing invariants associated with zero- and one-dimensional subregions. These include information from lines as well as points, enabling the method to address phases that point-based diagnosis alone can miss.
Ono and Shiozaki present matrices that encode information used in the construction, including combinations of representations on line segments and points. Their paper also describes how quantities defined on these subregions combine into topological invariants. The method is presented as a systematic alternative to constructing invariants one case at a time.
Testing the construction and defining its scope
As a demonstration, the authors construct invariants for time-reversal-symmetric spinful superconductors with conventional pairing symmetries across 230 space groups. They say symmetry indicators are always trivial in this setting. To test whether their constructed quantities work as topological invariants, they compute them for randomly generated symmetric Hamiltonians.
The reported 159 space groups are those in which the constructed invariants completely characterize the corresponding K groups. The paper defines its scope as stable K-theoretic classification: the invariants characterize the K group, rather than the finer classifications of fixed-size Hamiltonians or fixed-rank vector bundles. The authors note that the invariants can also serve as diagnostic tools for phases in those finer classifications.
The paper also addresses normal conducting phases. It says the invariants constructed for these phases are defined under certain gauge conditions, and discusses deriving gauge-independent invariants from gauge-fixed ones through examples. The authors describe the method as a framework for constructing topological invariants and demonstrate it through the superconducting classification and numerical checks.