Ten Classes Were Never Enough

Plain-language explainer for doi:10.5281/zenodo.21480491 (#460)

The Altland-Zirnbauer tenfold way

Altland and Zirnbauer (1997) classified all Hermitian topological phases by asking: which combinations of time-reversal T, particle-hole conjugation C, and chiral symmetry S = TC can a Hamiltonian H = H† have? The answer gives exactly 10 symmetry classes, organised by whether each symmetry is present and whether it squares to +1 or −1. This is the “tenfold way” — the complete periodic table of topological insulators and superconductors.

Kawabata, Shiozaki, Ueda, and Sato (2019) extended this to non-Hermitian Hamiltonians and found 38 symmetry classes. The extra 28 arise because for H ≠ H†, Hermitian conjugation (†) and transposition (T) are independent operations — the standard identification H* = H^T fails. This doubles the set of anti-unitary symmetries available.

How PT symmetry fits in

The AZ classification uses T (anti-unitary time reversal), C (anti-unitary particle-hole), and S (unitary chiral). PT symmetry introduces a fourth symmetry: 𝒫 (unitary spatial parity). The combined operation 𝒫𝒯 is anti-unitary and squares to +1, placing it in a class with no AZ counterpart.

The AZ operator T and the PT operator 𝒯 are the same anti-unitary operator — they agree on their action on H. The difference is the context: AZ imposes T on a Hermitian Hamiltonian; PT imposes 𝒫𝒯 jointly on a non-Hermitian one. The parity 𝒫 is the genuinely new ingredient.

Two new ISA phenomena

The main contribution of this paper is translating the 38-fold way into the Origami ISA opcode language. This reveals two phenomena that have no Hermitian counterpart:

LABEL failure (PT phase transition). The LABEL opcode projects a state onto a definite symmetry sector — a definite parity, charge, or spin quantum number. In Hermitian systems, symmetry sectors are always orthogonal and LABEL always works. In PT-symmetric systems, as the system approaches an exceptional point, the 𝒫𝒯 eigenstates rotate from real (orthogonal sectors) to complex-conjugate pairs (non-orthogonal sectors). At the EP itself, the two sectors coalesce: LABEL becomes undefined. This is the ISA description of the PT phase transition.

Exceptional point at the H¹/H² tier boundary. In the PT-unbroken phase, the system lives in the H¹ tier: real eigenvalues, well-defined eigenstates, Berry phase accumulation, TWIST operates correctly. At the exceptional point, the eigenstates coalesce into a Jordan block — the eigenbundle is ill-defined, TWIST cannot be performed, and SNAP↑ fires. The system crosses into the H² tier (PT-broken phase): complex eigenvalues, gain-loss dynamics, dissipation dominant. The EP is the β*₁₂ snap threshold of the ISA.

Reading the 38 classes in ISA language

Each of the 38 non-Hermitian symmetry classes corresponds to a different combination of which ISA opcodes remain well-defined:

  • Classes where LABEL works but TWIST approaches failure: near-EP classes in the H¹ tier, approaching the snap threshold
  • Classes where LABEL fails (PT-broken): in the H² tier, past the snap threshold
  • Classes where both LABEL and TWIST are well-defined throughout parameter space: topologically protected, no EP accessible via smooth deformation
  • Classes with additional structure (C symmetry, chiral symmetry): the Hermitian AZ classes embedded in the non-Hermitian landscape

The ISA provides a dictionary between symmetry class (abstract algebra — which anti-unitary symmetries are present and how they compose) and computational capability (which opcodes fire and which fail).

See also