${\mathbb{Z}}_4$ parafermions in one-dimensional fermionic lattices

Parafermions are emergent excitations which generalize Majorana fermions and are potentially relevant to topological quantum computation. Using the concept of Fock parafermions, we present a mapping between lattice ${\mathbb{Z}}_4$ parafermions and lattice spin-$1/2$ fermions which preserves the locality of operators with ${\mathbb{Z}}_4$ symmetry. Based on this mapping, we construct an exactly solvable, local, and interacting one-dimensional fermionic Hamiltonian which hosts zero-energy modes obeying parafermionic algebra. We numerically show that this parafermionic phase remains stable in a wide range of parameters, and discuss its signatures in the fermionic spectral function.

Figure 1

Energy gap (units $J$) of $\overline{H}(U,V)$ as a function of $U$ and $V$. The triangle, square, and star correspond to $H_J, H_A$, and $H^{\left(2\right)}$, respectively. DMRG simulations on 16 sites.

Figure 2

(a) ${\mathcal{A}}_j$ for different points in the $(U,V)$ parameter space. Green plots feature a dependence on the parity of the site $j$. (b) ${\mathcal{A}}_1$ (blue dots) along the straight paths in parameter space connecting $H^{\left(2\right)}$ (star), $H_A$ (square), and $H_J$ (triangle). The energy gap (units $J$) is shown in red to help in identifying the phase transition (here around $U\sim 0.7$) between the WI phase (red fade) and the SI one (green fade). DMRG simulations on 16 sites.