Wire deconstructionism of two-dimensional topological phases

A scheme is proposed to construct integer and fractional topological quantum states of fermions in two spatial dimensions. We devise models for such states by coupling wires of nonchiral Luttinger liquids of electrons that are arranged in a periodic array. Which interwire couplings are allowed is dictated by symmetry and the compatibility criterion that they can simultaneously acquire a finite expectation value, opening a spectral gap between the ground state(s) and all excited states in the bulk. First, with these criteria at hand, we reproduce the tenfold classification table of integer topological insulators, where their stability against interactions becomes immediately transparent in the Luttinger liquid description. Second, we construct an example of a strongly interacting fermionic topological phase of matter with short-range entanglement that lies outside of the tenfold classification. Third, we expand the table to long-range entangled topological phases with intrinsic topological order and fractional excitations.

Figure 1

The boundary conditions determine whether a topological phase has protected gapless modes or not. (a) With open boundary conditions, gapless modes exist near the wires $j=1$ and $N$, the scattering between them is forbidden by imposing locality in the limit $N\to \infty$. (b) Periodic boundary conditions allow the scattering vector ${\mathcal{T}}^{\left(0\right)}$ that gaps modes, which were protected by locality before.