Beyond the 10-fold Way: 13 Associative $$ {\mathbb Z}_2\times {\mathbb Z}_2$$-Graded Superdivision Algebras

Abstract

The “10-fold way” refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, $${\mathbb Z}_2$$ -graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in $${\mathbb Z}_2\times {\mathbb Z}_2$$ -graded physics (classical and quantum invariant models, parastatistics) we classify the associative $${\mathbb Z}_2\times {\mathbb Z}_2$$ -graded superdivision algebras and show that 13 inequivalent cases have to be added to the 10-fold way. Our scheme is based on the “alphabetic presentation of Clifford algebras”, here extended to graded superdivision algebras. The generators are expressed as equal-length words in a 4-letter alphabet (the letters encode a basis of invertible $$2\times 2$$ real matrices and in each word the symbol of tensor product is skipped). The 13 inequivalent $${\mathbb Z}_2\times {\mathbb Z}_2$$ -graded superdivision algebras are split into real series (4 subcases with 4 generators each), complex series (5 subcases with 8 generators) and quaternionic series (4 subcases with 16 generators). As an application, the connection of $${\mathbb Z}_2\times {\mathbb Z}_2$$ -graded superdivision algebras with a parafermionic Hamiltonian possessing time-reversal and particle-hole symmetries is presented.