Clifford Modules and Symmetries of Topological Insulators

Abstract

We complete the classification of symmetry constraints on gapped quadratic fermion hamiltonians proposed by Kitaev. The symmetry group is supposed compact and can include arbitrary unitary or antiunitary operators in the Fock space that conserve the algebra of quadratic observables. We analyze the multiplicity spaces of real irreducible representations of unitary symmetries in the Nambu space. The joint action of intertwining operators and antiunitary symmetries provides these spaces with the structure of Clifford module: we prove a one-to-one correspondence between the ten Altland-Zirnbauer symmetry classes of fermion systems and the ten Morita equivalence classes of real and complex Clifford algebras. The antiunitary operators, which occur in seven classes, are projectively represented in the Nambu space by unitary "chiral symmetries". The space of gapped symmetric hamiltonians is homotopically equivalent to the product of classifying spaces indexed by the dual object of the group of unitary symmetries.

On the K-theoretic classification of topological phases of matter

“…Two gapped Hamiltonians which are nonhomotopic may posses homotopic valence bands; indeed, remembering only the data of the valence band means that one loses information about the presence or absence of a charge-conjugation symmetry. (9) The precise treatment of charge-conjugation symmetry differs among authors, and the related notion of particle-hole symmetry in a Dirac-Nambu formalism is often thrown into the mix as well [19,24,1,46]. The Dirac-Nambu space is a vector space of second-quantized creation operators and annihilation operators, and is a useful auxiliary space often used for studying Bogoliubov de Gennes Hamiltonians.…”

“…In many treatments of the tenfold way [1,2,19,24,34,46,47], the single-particle "Hamiltonian" in certain symmetry classes is taken to act on a Nambu space W = V ⊕ V rather than a single-particle Hilbert space V . A Nambu space has a canonical real structure Σ :…”

“…In many of our applications, A is trivially graded, i.e., purely even, and the only complication comes from the data of c : G → {±1}. Let φ : G → {±1} be a continuous homomorphism, and σ be a U(1)-valued 2-cocycle as in (1). A projective unitary-antiunitary representation…”

“…A Periodic Table was partially drawn up, with each symmetry class in each spatial dimension having one of the K-theory groups of a point as its classification group. Little detail was provided in the original paper, which led to a series of authors providing their own accounts [46,50,1,19]. Upon careful investigation, these contain inequivalent treatments of distinct families of free-fermion systems.…”

“…For any two x, y ∈ G, their representatives θ x , θ y satisfy θ x θ y = σ(x, y)θ xy , with σ : G × G → U(1) a generalised 2-cocycle, (1) σ(x, y)σ(xy, z) = σ(y, z) x σ(x, yz),…”