Fractional Fermions with Non-Abelian Statistics

We introduce a novel class of low-dimensional topological tight-binding models that allow for bound states that are fractionally charged fermions and exhibit non-Abelian braiding statistics. The proposed model consists of a double (single) ladder of spinless (spinful) fermions in the presence of magnetic fields. We study the system analytically in the continuum limit as well as numerically in the tight-binding representation. We find a topological phase transition with a topological gap that closes and reopens as a function of system parameters and chemical potential. The topological phase is of the type BDI and carries two degenerate midgap bound states that are localized at opposite ends of the ladders. We show numerically that these bound states are robust against a wide class of perturbations.

Figure 1

Double-ladder tight-binding model, consisting of a lower ($\sigma =-1$) and an upper ($\sigma =1$) ladder, each lying in the $xy$ plane and held at chemical potentials ${\mu }_{\sigma }$. Here, $t_x$ (red links) is the intrachain, $t_y$ (green links) the intraladder, and $t_z$ (blue links) the interladder hopping amplitude, and $a_{x,y,z}$ are the corresponding lattice constants. A uniform magnetic field $\mathbf{B}={\mathbf{B}}_1+{\mathbf{B}}_2$ is applied in the $yz$ plane. The associated magnetic flux results in phases ${\varphi }_1$ and ${\varphi }_2$ in the hopping amplitudes between different chains; see Eqs. (2, 3).

Figure 2

The spectrum of the upper (orange) and lower (red) chains in the first Brillouin zone. The chemical potentials are chosen such that the system is at the charge-neutrality point. The filled (empty) states are indicated by dark-colored (light-colored) lines. The two Fermi wave vectors are given by $k_{F,1}=\pi /4a_x$ and $k_{F,\overline{1}}=3\pi /4a_x$. The intraladder hopping $t_y$ (green dashed line) and the interladder hopping $t_z$ (blue dotted lines) lead to opening of gaps at the Fermi level ($ϵ=0$).

Figure 3

(a) The relevant part of the spectrum for a double-ladder system of length $L/a_x=401$ found by numerical diagonalization of the tight-binding Hamiltonian $H$ [see Eq. (4)]. The energy ${ϵ}_N$ corresponds to the $N$th energy level. The parameters $t_z/t_x=0.05$, $t_y/t_x=0.1$, and $\delta \mu =0$ are chosen to satisfy the topological criterion Eq. (12). In the middle of the gap, $ϵ=0$, there are two degenerate bound states (red dots). (b) The probability density $|{\psi }_{\tau ,\sigma }{|}^2$ of the left state in the ($\tau$, $\sigma$) chain. The density patterns are the same for all chains, in full agreement with the continuum solution Eq. (13).

Figure 4

Single-ladder model. A uniform magnetic field ${\overline{\mathbf{B}}}_1$ is applied perpendicular to the ladder. A spatially periodic magnetic field ${\overline{\mathbf{B}}}_2$ with period $2a_x$ is applied in the $xy$ plane.