Topological phases in gapped edges of fractionalized systems

Recently, it has been proposed that exotic one-dimensional phases can be realized by gapping out the edge states of a fractional topological insulator. The low-energy edge degrees of freedom are described by a chain of coupled parafermions. We introduce a classification scheme for the phases that can occur in parafermionic chains. We find that the parafermions support both topological symmetry fractionalized phases as well as phases in which the parafermions condense. In the presence of additional symmetries, the phases form a non-Abelian group. As a concrete example of the classification, we consider the effective edge model for a $\nu =1/3$ fractional topological insulator for which we calculate the entanglement spectra numerically and show that all possible predicted phases can be realized.

Figure 1

Illustration of a bipartition of a chain into a segment ($S$) of length $L_S$ and an environment ($E$). A symmetry operation $\Omega$ acting on the important eigenstates of the reduced density matrix fractionalizes into two operators ${\Omega }^A$ and ${\Omega }^B$.

Figure 2

Two different dimerization patterns corresponding to the two phases of the parafermionic chain. The trivial case ($u=1$, $t=0$) is shown in (a) where pairs of parafermions ${\chi }_j$ are localized on each site, yielding a unique and gapped ground state. Panel (b) shows a nontrivial state ($u=0$, $t=1$) with uncoupled parafermions at the two ends of the chain. This state is gapped in the bulk but has gapless excitations resulting from the unpaired parafermions and the edges.

Figure 3

(a) Overlap of a ground state of the ${\mathbb{Z}}_4$ parafermionic chain [Hamiltonian (7) with $N=4$ for different $L$] transformed by ${\chi }_1^{†}$ and the ground state of the next $q$ sector. The state $|{\psi }_0\rangle$ is the ground state for $q=1$ and $|{\phi }_0\rangle$ for $q=-i$. (b) Matrix elements of ${\chi }_1^{†}$ between two degenerate eigenstates $|{\varphi }_{\alpha }\rangle$ and $|{\varphi }_{{\alpha }^{\prime }}\rangle$ of the reduced density matrix ${\rho }^S$, which lie in $q$ sectors that differ exactly by one charge. Calculations are performed for different chains of length $2L$ with periodic boundary conditions. The segment $S$ always includes the sites 1 to $L$.

Figure 4

Combination of two chains. The hopping in the two independent chains is shown in (a). After combining the chains (b) not all hopping processes occur below the chain. The anyons of former chain $\varphi$ pass above the sites of former chain $\psi$ (c).

Figure 5

Physical setup: The edge modes of the two-dimensional fractional topological insulator are gapped out by coupling to superconducting and ferromagnetic domains. At the domain walls, localized parafermionic modes emerge.

Figure 6

Physical realization of the Hamiltonian $H_0$ (45) on the edge of a fractional topological insulator (FTI). Changing the geometry can be used to tune the effective tunneling amplitudes for different particles.

Figure 7

Entanglement spectra for different values of the parameters $(t_1,u_1,t_3,u_3)$ in $H_0$.