Topological Phases of Parafermions: A Model with Exactly Solvable Ground States

Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.

Figure 1

Phase space of the model (1). The ground state is exactly solvable along the red line, which is parametrized $\varphi$ according to Eq. (2). The points $\varphi =\pm \infty$ and $\varphi =0$ are highlighted. For better reference, the well-studied critical point $f=J$, $b=0$ with central charge $c=4/5$ is also highlighted. Inset: Perturbative analysis of the size scaling of the degeneracies ${\mathrm{\Delta }}_m$ of the first three excited states ($m=1$, black circles) and of a higher excited triplet ($m=4$, red squares), exhibiting, respectively, exponential and polynomial energy splitting. The polynomial scaling demonstrates the absence of strong edge modes. Three values of $\varphi$ are considered: $\varphi ={10}^{-4}$ (solid line), $\varphi ={10}^{-3}$ (dashed line), and $\varphi ={10}^{-2}$ (dashed-dotted line).

Figure 2

(Top left) Correlation function $G_0(x)$ and (top right) $|⟨n{⟩}_0-⟨n{⟩}_2|$ for several values of $\varphi$. Violet lines represent the exponential scalings extracted from the analytical formulas. (Bottom left) Correlation length $\xi$. (Bottom right) We show one typical example of the ${\mathbb{Z}}_3$-breaking observable $F_i(x)$ for $i=0$.

Figure 3

Top left: Entanglement spectrum as a function of $\ell$ for $\varphi =2$. Top right: von Neumann entropy $S({\widehat{\rho }}_{\ell })$ as a function of $\ell$ for three values of $\varphi$. At large $\ell$, it saturates to a finite value corresponding to an area law. Bottom: DMRG calculation of the gap of the model obtained with length $L=168$; the maximal number of retained states is $m=250$.