Fermionic Topological Order on Generic Triangulations

Abstract

Consider a finite triangulation of a surface M of genus g and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if ${{\mathcal {P}}}$ is any of its spectral projections, the Booleanization of the fundamental group $\pi _1(M)$ can be embedded inside the group of invertible elements of the corner algebra ${{\mathcal {P}}}\, \mathrm{CAR} \, {{\mathcal {P}}}$. As a consequence, ${{\mathcal {P}}}$ decomposes in $4^g$ lower projections. Furthermore, a projective representation of ${{\mathbb {Z}}}_2^{4g}$ is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group $(2^X,\Delta )$, where X is the set of fermion sites and $\Delta$ is the symmetric difference of its sub-sets.