Edge ${\mathbb{Z}}_3$ parafermions in fermionic lattices

Parafermion modes are non-Abelian anyons which were introduced as ${\mathbb{Z}}_N$ generalizations of ${\mathbb{Z}}_2$ Majorana states. In particular, ${\mathbb{Z}}_3$ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on ${\mathbb{Z}}_3$ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of ${\mathbb{Z}}_3$ parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t-J$ model with a Kitaev-like chain supporting free ${\mathbb{Z}}_3$ parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing ${\mathbb{Z}}_3$ parafermions in strongly interacting electronic systems.

Figure 1

Gap (black) and entanglement entropy (red) as a function of interaction strength (a) $W_6$ and (b) $W_3$ for the models described by $H_{\mathrm{I}}$ and $H_{\mathrm{II}}$, respectively. While the phase transition occurs at $W_6>2t$ in $H_{\mathrm{I}}$, any $W_3>0$ induces the ${\mathbb{Z}}_3$ phase in $H_{\mathrm{II}}$. The inset in panel (a) shows the difference in energy level between the five states with lower energy ($E$) and the ground-state energy $E_{\mathrm{GS}}$. Note that all the states converge around the phase transition.

Figure 2

Characterization of $H_{\mathrm{I}}$ with $W_6=3.2t$. (a) Entanglement spectrum for the ground state in the sector $\langle {\widehat{P}}_{{\mathbb{Z}}_3}\rangle =1$ for a chain with $L=100$ sites; the spectrum has a threefold degeneracy and is the same for other sectors. (b) Gap between the ground states due to the finite size of a chain with $L$ sites; note that there are two different behaviors for $L\lesssim 60$ and $L\gtrsim 70$ indicating two mechanisms of intraedge interaction.

Figure 3

Gap (black) and entanglement entropy (red) as a function of deformation parameter $x$, Eq. (9). We consider the case $W_6=3.2t$ and $W_3=t$.

Figure 4

Dependence of the gap with respect to local operators. (a) Doping ($\mu$), (b) $z$-direction ($V_z$), (c) $x$-direction ($V_x$), and (d) $y$-direction ($V_y$) Zeeman terms. Panels (a) and (b) show a topological phase transition at $\mu =3t$ and $V_z=2.3t$, respectively. Panel (c) has a phase transition between a twofold degenerate state and a normal state for $V_x\approx 2.5J$. Finite-size effects are responsible for the discontinuity in the gap. Panel (d) shows no phase transition as a function of $V_y$, and the gap always increases. The inset in panel (a) shows the exponential dependence of the gap between the parafermionic modes with the chain length for $\mu =2.5t$ (black circles) and $\mu =2.9t$ (red squares).

Figure 5

Dependence of (a) the gap to the intensity and angle of the magnetic field and (b) its ground-state degeneracy, $n_{g}s$. For fixed $V_{xy}<2.5t$ the gap is minimum at angles $0,\pm 2\pi /3$ and the ground state is twofold degenerate. The dotted circles in panel (a) correspond to the transversal cut shown in Fig. 6.

Figure 6

Transversal cut of Fig. 5 with $V_{xy}=1.5t$, solid black, and $V_{xy}=2.4t$, dashed red.

Figure 7

Mean energy splitting between the ground states due to random potential with maximum intensity for the chemical potential $\mu$ (solid black) and Zeeman field at $z$ direction $V_z$ (dashed red). The mean was calculated based on 20 different distributions of impurities. Note the splitting happens for values of $\langle \mu \rangle /t$ or $\langle V_z\rangle /t$ of the order of the critical value seen in Fig. 4.

Figure 8

FPF spectral function ${\mathcal{A}}_j\left(0\right)$ for different interactions. (a) Spatial spread of the ground state over the chain for $H_{\mathrm{I}}$ with $W_6=2.2t$ (solid black) and $W_6=3.2t$ (dashed red). (b, c) Spatial spread of the ground state of $H_{\mathrm{II}}$ for $W_3=t$ and (b) $\mu =0.1t$ (solid black), $\mu =2.5t$ (dashed red), and $\mu =2.8t$ (dot-dashed blue) and (c) for $V_z=0.1t$ (solid black), $V_z=2.1t$ (dashed red), and $V_z=2.2t$ (dot-dashed blue). The green dots mark the analytical value (2/9) for perfectly localized ${\mathbb{Z}}_3$ edge parafermions of $H_{\mathrm{PF}}$.

Figure 9

Entanglement entropy (E.E.) dependence with doping for different chain lengths of 16 (solid black), 48 (dashed red), and 100 (dotted blue). Note that far away from the phase transition all of them have the same value.

Figure 10

Gap dependency with Hubbard interaction in a double occupancy basis. The gaps of the 16-site chain (red dashed) and 100-site chain (solid black) have minor differences only in the low interaction regime, $U_H\approx t$.