Nontopological parafermions in a one-dimensional fermionic model with even multiplet pairing

We discuss a one-dimensional fermionic model with a generalized ${\mathbb{Z}}_N$ even multiplet pairing extending Kitaev ${\mathbb{Z}}_2$ chain. The system shares many features with models believed to host localized edge parafermions, the most prominent being a similar bosonized Hamiltonian and a ${\mathbb{Z}}_N$ symmetry enforcing an $N$-fold degenerate ground state robust to certain disorders. Interestingly, we show that the system supports a pair of parafermions but they are nonlocal instead of being boundary operators. As a result, the degeneracy of the ground state is only partly topological and coexists with spontaneous symmetry breaking by a (two-particle) pairing field. Each symmetry-breaking sector is shown to possess a pair of Majorana edge modes encoding the topological twofold degeneracy. Surrounded by two band insulators, the model exhibits for $N=4$ the dual of an $8\pi$ fractional Josephson effect highlighting the presence of parafermions.

Figure 1

Top: Sketch of model (1). A one-dimensional tight-binding chain of spinless fermions is proximity coupled to clusters of $N=4$ fermions. Bottom: Sketch of the spectrum of model (1). The fourfold degeneracy for open boundary conditions (left) related to the presence of a ${\mathbb{Z}}_4$ symmetry, $\widehat{Q}$ (see text), is lifted to a twofold degeneracy for periodic boundary conditions (right).

Figure 2

DMRG data of model (1) for $\mathrm{\Delta }=t$ with bond link $m=150$. Top left: Standard deviation $s$ of the GS energies for fixed $L$ and different values of $\widehat{Q}$. Top right: Energy gap between the ground state and the first excited state for $\widehat{Q}=1$. Bottom left: Entanglement spectrum for a bipartition of the system of $L=72$ with $\widehat{Q}=1$. Bottom right: Standard deviation $s$ of the GS energies for fixed $L$ and different values of $\widehat{Q}$ in the presence of on-site ${\sum }_j{\delta }_j{\widehat{n}}_j$ and interacting disorder ${\sum }_j{\delta }_j^{\prime }{\widehat{n}}_j{\widehat{n}}_{j+1}$. ${\delta }_j,{\delta }_j^{\prime }$ are taken with uniform probability in the range $[-0.1t,0.1t]$.

Figure 3

DMRG data of the model ${\widehat{H}}_P+\lambda {\widehat{H}}_A$ ($m=150$). Left: Gap of the model as a function of $\lambda$ extracted with a finite-size scaling for $L$ between 24 and 72. Right: Standard deviation of the GS energies at $\lambda =0.5$ for fixed $L$ and different values of $Q$. A small on-site disorder ${\sum }_j{\delta }_j{\widehat{n}}_j$ is introduced with ${\delta }_j\in [-0.1t,0.1t]$.