nLab ten-fold way

Idea

Various structures in algebra and algebraic topology have a classification into 10 classes, more or less explicitly related to the 2 + 8 topological K-theory-groups of the point up to Bott periodicity: 2 for KU and 8 for KO, which are unified in KR.

The specific term “10-fold way” is a variation of the term “threefold way” used in Dyson 1962, which is referred to as inspiration by Heinzner, Huckleberry & Zirnbauer 2004 and Zirnbauer 2010 in reference to work going back to Altland & Zirnbauer 1997. While the term “10-fold way” is now often attributed to these authors, they may not actually have used it as a term (Zirnbauer 2010 finally speaks at least of the “10-way”).

The term became rather popular in the wake of the seminal suggestion by Kitaev 2009, following Schnyder, Ryu, Furusaki & Ludwig 2008 table 1, that “free” topological phases of matter, specifically free crystalline topological insulator-phases, are classified by some form of topological K-theory: see at K-theory classification of topological phases of matter for more on this.

(Though Kitaev 2009 does not use the term “10-fold way”, either, nor the now standard labels for the ten classes — all this appears later, cf. Chiu, Teo, Schnyder & Ryu 2016 table 1.)

Kitaev’s suggestion was made more precise by Freed & Moore 2013 (who, however and ironically, doubted, on p. 57, its application to topological phases – but see pp. 2 of SS23 for resolution) and it is these authors who very much amplify (and may have actually coined) the term “10-fold way”. Moreover, they point out (pp. 75) that a 10-fold classification is already contained by Dyson 1962(!), which the authors re-interpret as the classification of super division algebras.

This 10-fold way of super division algebras is further amplified in [Moore 2013, p. 129] and Geiko & Moore 2021.

The 10-fold way was also approached from a purely representation theoretic point of view by Rumynin and Taylor 2021, who also considered the same theory but with an action of odd elements by bilinear forms instead of anti-linear maps (cf. Rumynin and Taylor 2022).

Realizations

PCT Quantum Symmetries

We discuss how PCT quantum symmetries (Def. below) are classified (Prop. below) by a “10-fold way” (Cor. below).

The definition and the argument is quite straightforward. Just for completeness we offer some Preliminaries, but the reader may want to skip right ahead to the Definition and Classification.

The 10-fold classification of these PCT quantum symmetries immediately induces other incarnations of the 10-fold way, notably:

• the classification of flavors of topological K-theory (KU-theory, KO-theory, hence KR-theory in any degree)

  (by the Atiyah-Jänich theorem in terms of graded Fredholm operators, see there)

and with that

• the 10-fold way in the K-theory classification of topological phases of matter.

Preliminaries

Let:

• $\mathscr{H}$ be a countably infinite-dimensional complex Hilbert space,

• $\mathrm{U}(\mathscr{H})$ denote its unitary group,

• $T$ be a real structure on $\mathscr{H}$, hence a complex antilinear involution,

• $C_2^{(t)} \coloneqq \{id, T\}$ denote the group generated by $T$.

Observe that the semidirect product of the unitary group with that generated by $T$ is isomorphic the group consisting of unitary operators and anti-unitary operators:

(1)$$\begin{array}{ccc} \mathrm{U}(\mathscr{H}) \rtimes C_2^{(t)} &\overset{\sim}{\longrightarrow}& \mathrm{U}(\mathscr{H}) \sqcup \mathrm{U}_{anti}(\mathscr{H}) \\ (U, id) &\mapsto& U \\ (U, T) &\mapsto& U \circ T \mathrlap{\,.} \end{array}$$

This relation passes to projective unitary groups:

$$\begin{aligned} P \mathrm{U}(\mathscr{H}) &\coloneqq\; \mathrm{U}(\mathscr{H})/\mathrm{U}(1) \\ P \mathrm{U}_{anti}(\mathscr{H}) &\coloneqq\; \mathrm{U}_{anti}(\mathscr{H})/\mathrm{U}(1) \end{aligned}$$

in that we have an analogous isomorphism

$$\begin{array}{ccc} P \mathrm{U}(\mathscr{H}) \rtimes C_2^{(t)} &\overset{\sim}{\longrightarrow}& P\mathrm{U}(\mathscr{H}) \sqcup P\mathrm{U}_{anti}(\mathscr{H}) \\ ([U], id) &\mapsto& [U] \\ ([U], T) &\mapsto& [U \circ T] \mathrlap{\,.} \end{array}$$

This group of projective unitary/antiunitary operators is the group of quantum symmetries as usually considered (Wigner's theorem).

We next enlarge this a little more to include what may be thought of as particle/antiparticle symmetry, often referred to as charge conjugation symmetry.

To that end, let

• $\mathscr{H}_{gr} \coloneqq \mathscr{H} \ominus \mathscr{H}$ denote the $\mathbb{Z}_2$-graded Hilbert space,

• $P$ the grading involution on $\mathscr{H}$:

  $$\begin{array}{ccc} \mathscr{H} &\overset{P}{\longrightarrow}& \mathscr{H} \\ \left( \begin{matrix} \psi_+ \\ \psi_- \end{matrix} \right) &\mapsto& \left( \begin{matrix} \psi_- \\ \psi_+ \end{matrix} \right) \end{array}$$

• $C_2^{(p)} \coloneqq \{id, P\}$ denote the group generated by $P$,

• $C_2^{(c)} \coloneqq \{id, C \coloneqq P T\}$ denote the group generated by $P \circ T$,

• $\mathrm{U}_{gr}(\mathscr{H}_{gr}) \coloneqq \mathrm{U}(\mathscr{H})^2 \rtimes C_2^{(p)}$ denote the semidirect product which is isomorphic to the subgroup of $\mathrm{U}(\mathscr{H}_{gr})$ on the operators which are of homogeneous degree (either even or odd),

• $P\mathrm{U}_{gr}(\mathscr{H}_{gr}) \coloneqq \frac{\mathrm{U}(\mathscr{H})^2}{\mathrm{U}(1)} \rtimes C_2^{(p)}$ denote the corresponding projective group

Finally, combine all this to consider the following:

Definition and Classification

Definition

The group of graded quantum symmetries is the semidirect product

(2)$$QS \;\coloneqq\; \frac{ \mathrm{U}(\mathscr{H})^2 }{ \mathrm{U}(1) } \rtimes \big( C_2^{(t)} \times C_2^{(c)} \big) \mathrlap{\,,}$$

of the even graded projective unitary group with the operations of degree involution $P$ and complex involution $T$.

This is a group extension

$$\tfrac{ \mathrm{U}(\mathscr{H})^2 }{ \mathrm{U}(1) } \hookrightarrow QS \twoheadrightarrow C_2^{(t)} \times C_2^{(c)}$$

of

(3)$$C_2^{(t)} \times C_2^{(c)} \;=\; \big\{ id, T, C, P \coloneqq C T \big\} \,,$$

which we may call the group of PCT symmetries.

Definition

A PCT quantum symmetry is a lift of a subgroup of PCT symmetries (3) to a quantum symmetry (2), hence a dashed group homomorphism making the following diagram commute:

where for $g \in G$ we denote by

(4)$$\widehat{g} \;\in\; \mathrm{U}(\mathscr{H})^2 \rtimes \big( C_2^{(t)} \times C_2^{(c)} \big)$$

a representative of the $\mathrm{U}(1)$-coset equivalence class $\big[\widehat{g}\big]$.

Proposition

Given a PCT quantum symmetry $\widehat{G}$ (Def. ), we have

1. if $G = C_2^{(p)} = \big\{id, P = C T\big\}$, then $\big[\widehat{P}\big]$ has a representative $\widehat{P}$ (4) such that

   $$\widehat{P}{}^2 = id \mathrlap{\,,}$$

2. if $T \in G$ then $\big[\widehat{T}\big]$ has a representative $\widehat{T}$ (4) such that

   $$\widehat{T}{}^2 \in \{\pm id\} \mathrlap{\,,}$$

3. if $C \in G$ then $\big[\widehat{C}\big]$ has a representative $\widehat{C}$ (4) such that

   $$\widehat{C}{}^2 \in \{\pm id\} \mathrlap{\,,}$$

and all these cases occur.

Proof

1. That $\big[\widehat{(-)}\big]$ is a group homomorphism means equivalently that

   $$\begin{aligned} & \big[\widehat{P}\big]^2 \;=\; id \\ \Leftrightarrow \;\; & \widehat{P}^2 \;=\; \omega\, id \;\;\; \text{for some} \; \omega \in \mathrm{U}(1) \,. \end{aligned}$$

   But since for $G = C_2^{(p)}$ the operator $\widehat{P}$ must be unitary according to (1) and hence in particular complex-linear, its rescaling by any square root $\sqrt{\omega} \in \mathrm{U}(1)$ yields an alternative representative

   $$\big[ \tfrac{1}{\sqrt{\omega}} \widehat{P} \big] \;=\; \big[ \widehat{P} \big]$$

   with the desired property:

   $$\begin{aligned} \big( \tfrac{1}{\sqrt{\omega}} \widehat{P} \big)^2 & = \tfrac{1}{\sqrt{\omega}} \widehat{P} \tfrac{1}{\sqrt{\omega}} \widehat{P} \\ & = \tfrac{1}{\sqrt{\omega}} \tfrac{1}{\sqrt{\omega}} \widehat{P} \widehat{P} \\ & = id \mathrlap{\,.} \end{aligned}$$

2. Group homomorphy again requires that

   $$\begin{aligned} & \big[\widehat{T}\big]^2 \;=\; id \\ \Leftrightarrow \;\; & \widehat{T}^2 \;=\; \omega\, id \;\;\; \text{for some} \; \omega \in \mathrm{U}(1) \,, \end{aligned}$$

   but now that $\widehat{T}$ is anti-unitary and hence in particular complex-antilinear, there is first all a further constraint, namely

   $$\begin{aligned} & \widehat{T} \widehat{T}^2 = \widehat{T}^2 \widehatT \\ \Rightarrow\;\; & \omega^\ast \widehat{T} = \omega \widehat{T} \\ \Leftrightarrow\;\; & \omega^\ast = \omega \\ \Leftrightarrow\;\; & \omega \,\in\, \mathrm{U}(1) \cap \mathbb{R} \\ \Leftrightarrow\;\; & \omega \,\in\, \{\pm 1\} \mathrlap{\,.} \end{aligned}$$

   On the other hand, for the same reason the single non-trivial value for $\omega$ may no longer be scaled away as before, since now

   $$\begin{aligned} \big( \pm \mathrm{i} \widehat{T} \big)^2 & = (\pm\mathrm{i})\widehat{T} (\pm\mathrm{i})\widehat{T} \\ & = (\pm\mathrm{i})(\mp\mathrm{i})\widehat{T}\widehat{T} \\ & = \widehat{T}^2 \mathrlap{\,.} \end{aligned}$$

3. Same argument as in (2.).

Corollary

(10-fold way of PCT quantum symmetries)
The set of PCT quantum symmetries (Def. ) falls, by Prop. , into ten classes, according to the following table:

Remark

The labels in the last row of the table in Cor. are traditional — early appearance in Schnyder, Ryu, Furusaki & Ludwig 2008 table 1 (where, beware, the labels “DII” and “CII” are swapped with regards to the modern convention), reviewed in Chiu, Teo, Schnyder & Ryu 2016), with historical origin in Cartan’s 1926 classification of symmetric spaces in terms of simple Lie algebras (see there).

Further

• Atiyah-Jänich theorem– In terms of graded Fredholm operators

• …

Related concepts

• Bott periodicity

• K-theory classification of topological phases of matter

• fermionic group

References

• Freeman Dyson, The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics, J. Math. Phys. 3 1199 (1962) [doi:10.1063/1.1703863]

• Alexander Altland and Martin R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142 [doi:10.1103/PhysRevB.55.1142]

• P. Heinzner, A. Huckleberry, M.R. Zirnbauer, Symmetry classes of disordered fermions [arXiv:math-ph/0411040]

• Martin R. Zirnbauer, Symmetry Classes [arXiv:1001.0722]

• Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, Andreas W. W. Ludwig: Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 195125 (2008) $[$doi:10.1103/PhysRevB.78.195125, arXiv:0803.2786]

• Alexei Kitaev: Periodic table for topological insulators and superconductors, talk at: L.D.Landau Memorial Conference “Advances in Theoretical Physics”, June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) (arXiv:0901.2686, doi:10.1063/1.3149495)

• Daniel Freed, Gregory Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013) 1927-2023 [arXiv:1208.5055, doi:10.1007/s00023-013-0236-x]

• Gregory Moore: Quantum symmetries and compatible Hamiltonians (2013) [pdf]

• Ching-Kai Chiu, Jeffrey C.Y. Teo, Andreas P. Schnyder, Shinsei Ryu: Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88 (2016) 035005 [arXiv:1505.03535, doi:10.1103/RevModPhys.88.035005]

• Roman Geiko, Gregory W. Moore, Dyson’s classification and real division superalgebras, Journal of High Energy Physics 2021 4 (2021) 299 [doi: 10.1007/jhep04(2021)299]

• Dmitriy Rumynin, James Taylor: Real Representations of $C_2$-Graded Groups: The Antilinear Theory, Linear Algebra and its Applications 610 (2021) 135-168. [doi:10.1016/j.laa.2020.09.040, arXiv:2006.09765]

• Dmitriy Rumynin, James Taylor: Real Representations of $C_2$-Graded Groups: The Linear and Hermitian Theories, Higher Structures 6 1 (2022) 359-374. [doi:10.21136/HS.2022.07, arXiv:2008.07846]

• Luuk Stehouwer: Free phases of Majorana fermions: Tenfold ways compared [arXiv:2507.08694]

• Lucas C.P.A.M. Müssnich, Renato Vasconcellos Vieira: The weakly interacting tenfold way [arXiv:2603.16799]

Review:

• John Baez, The Tenfold Way (2020) [webpage]