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.

Appendix A: Alphabetic Presentation of Superdivision Algebras

We extend here the alphabetic presentation (described in Sect. 2 for Clifford algebras) to the cases of ${\mathbb Z}_2$ and ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superdivision algebras.

Any homogeneous element g of a superdivision algebra is represented by an invertible real matrix which takes the form $g=M\otimes N$, where the matrix M encodes the information of the grading, either ${\mathbb Z}_2$ or ${\mathbb Z}_2\times {\mathbb Z}_2$, while the matrix N encodes the information of the real, complex or quaternionic structure. The matrix size for M is

$$\begin{aligned}&{\mathbb Z}_2{\text {-grading}}: ~(2\times 2);\qquad \qquad {\mathbb Z}_2\times {\mathbb Z}_2{\text {-grading}}: ~(4\times 4).&\end{aligned}$$

(A.1)

The matrix size for N is

$$\begin{aligned}&{\mathbb R}{\text {-series}}: ~(1\times 1);\qquad \quad {\mathbb C}{\text {-series}}: ~(2\times 2);\qquad \quad {\mathbb H}{\text {-series}}: ~(4\times 4).&\nonumber \\ \end{aligned}$$

(A.2)

Concerning the ${\mathbb Z}_2$ grading, the even (odd) sector is denoted as $M_0$ ($M_1$); the nonvanishing elements are accommodated according to

$$\begin{aligned} \left( \begin{array}{cc}*&{}0\\ 0&{}*\end{array}\right) \in M_0,&\qquad&\left( \begin{array}{cc}0&{}*\\ *&{}0\end{array}\right) \in M_1. \end{aligned}$$

(A.3)

The tensor products of the ${\mathbb Z}_2$-graded matrices (A.3) produce the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded matrices $M_{ij}$ (ij denotes the grading) according to

$$\begin{aligned} \left( \begin{array}{cc}*&{}0\\ 0&{}*\end{array}\right) \otimes \left( \begin{array}{cc}*&{}0\\ 0&{}*\end{array}\right) \mapsto \left( \begin{array}{cccc}*&{}0&{}0&{}0\\ 0&{}*&{}0&{}0\\ 0&{}0&{}*&{}0\\ 0&{}0&{}0&{}*\end{array}\right) \in M_{00},{} & {} \left( \begin{array}{cc}*&{}0\\ 0&{}*\end{array}\right) \otimes \left( \begin{array}{cc}0&{}*\\ *&{}0\end{array}\right) \mapsto \left( \begin{array}{cccc}0&{}*&{}0&{}0\\ *&{}0&{}0&{}0\\ 0&{}0&{}0&{}*\\ 0&{}0&{}*&{}0 \end{array}\right) \in M_{01},\nonumber \\ \left( \begin{array}{cc}0&{}*\\ *&{}0\end{array}\right) \otimes \left( \begin{array}{cc}*&{}0\\ 0&{}*\end{array}\right) \mapsto \left( \begin{array}{cccc}0&{}0&{}*&{}0\\ 0&{}0&{}0&{}*\\ *&{}0&{}0&{}0\\ 0&{}*&{}0&{}0 \end{array}\right) \in M_{10},{} & {} \left( \begin{array}{cc}0&{}*\\ *&{}0\end{array}\right) \otimes \left( \begin{array}{cc}0&{}*\\ *&{}0\end{array}\right) \mapsto \left( \begin{array}{cccc}0&{}0&{}0&{}*\\ 0&{}0&{}*&{}0\\ 0&{}*&{}0&{}0\\ *&{}0&{}0&{}0 \end{array}\right) \in M_{11}.\nonumber \\{} & {} \end{aligned}$$

(A.4)

In the ${\mathbb Z}_2$-grading the $M_0$, $M_1$ sectors can be spanned by the matrices denoted by the letters, with the (1) identification, given by:

$$\begin{aligned} M_{0}: \quad I,~X;{} & {} M_{1}: \quad Y, ~A. \end{aligned}$$

(A.5)

In the ${\mathbb Z}_2\times {\mathbb Z}_2$-grading the $M_{00}, M_{01}, M_{10}, M_{11}$ sectors can be spanned by the matrices denoted by the 2-character words

$$\begin{aligned} M_{00}: \quad II,~IX,~XI, ~XX;{} & {} M_{01}: \quad IA, ~IY, ~XA, ~XY;\nonumber \\ M_{10}: \quad AI,~AX,~YI, ~YX;{} & {} M_{11}: \quad AA, ~AY, ~YA, ~YY. \end{aligned}$$

(A.6)

In Sect. 4 we showed that each one of the 7 inequivalent ${\mathbb Z}_2$-graded superdivision algebras admits an alphabetic presentation in terms of equal-length words. Without loss of generality (up to similarity transformations) the even sector ${\mathbb D}_0^{[1]}$ can be expressed as

$$\begin{aligned}&{\mathbb R}{\text {-series}}: ~ I;\qquad ~{\mathbb C}{\text {-series}}: ~II,~IA;\qquad ~{\mathbb H}{\text {-series}}: ~III, ~IIA,~IAX,~ IAY.&\nonumber \\ \end{aligned}$$

(A.7)

The odd sectors ${\mathbb D}_1^{[1]}$ are presented in table (17) (for the ${\mathbb R}$-series), table (18) (for the ${\mathbb C}$-series), table (19) (for the ${\mathbb H}$-series).

The alphabetic presentation is extended to the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superdivision algebras by taking into account that:

(i) without loss of generality (up to similarity transformations) the even sector ${\mathbb D}_{00}^{[2]}$ can be expressed as

$$\begin{aligned}&{\mathbb R}{\text {-series}}: ~ II;\quad ~{\mathbb C}{\text {-series}}: ~III,~IIA;\quad ~{\mathbb H}{\text {-series}}: ~IIII, ~IIIA,~IIAX,~ IIAY;&\nonumber \\ \end{aligned}$$

(A.8)

(ii) each one of the three subalgebras ${\mathbb S}_{10}, {\mathbb S}_{01}, {\mathbb S}_{11}\subset {\mathbb D}^{[2]}$, given by the direct sums

$$\begin{aligned}&{\mathbb S}_{01}:= {\mathbb D}_{00}^{[2]}\oplus {\mathbb D}_{01}^{[2]},\qquad {\mathbb S}_{10}:= {\mathbb D}_{00}^{[2]}\oplus {\mathbb D}_{10}^{[2]},\qquad {\mathbb S}_{11}:= {\mathbb D}_{00}^{[2]}\oplus {\mathbb D}_{11}^{[2]},&\nonumber \\ \end{aligned}$$

(A.9)

is isomorphic to one (of the seven) ${\mathbb Z}_2$-graded superdivision algebra;

(iii) the alphabetic presentation can be assumed for ${\mathbb S}_{01}$ and, since the second ${\mathbb Z}_2$ grading is independent from the first one, ${\mathbb S}_{10}$. The closure under multiplication for any $g\in {\mathbb D}_{01}^{[2]},~ g'\in {\mathbb D}_{10}^{[2]}$ implies that $gg'\in {\mathbb D}_{11}^{[2]}$ is alphabetically presented.

As discussed in Sect. 5, a ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superdivision algebra can be associated with its ${\mathbb Z}_2$-graded superdivision algebra projections ${\mathbb S}_{01},~{\mathbb S}_{10},~{\mathbb S}_{11}$.