Fractionalization of Itinerant Anyons in One-Dimensional Chains

We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains, focusing, in particular, on itinerant Ising anyon chains, and derive effective anyonic $t\mathrm{\text{−}}J$ models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (non-Abelian) anyonic degrees of freedom—a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum by combining spectra for charge and anyonic sectors requires a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization.

Figure 1

A chain with $N$ itinerant anyonic quasiparticles. The (red) dark dots represent the sigma quasiparticles, open circles represent empty sites, and the (light) shaded dots describe the fusion channels of the nonlocal states and can take the value $\mathbf{1}$, $\sigma$, or $\psi$, as dictated by the fusion rules.

Figure 2

Low-energy spectrum of a 24-site Ising anyon chain at $J=0$ and density $\rho =2/3$. Shaded (yellow) circles correspond to the spectrum at ${\varphi }_{\mathrm{ext}}=0$ as a function of momentum $K$. The (red) crosses correspond to the spectrum at ${\varphi }_{\mathrm{ext}}=\pi /4$ restricting $K\in [-\pi /3,\pi /6]$ (so that $\tilde{K}\in [-\pi /6,\pi /3]$). The parent charge excitations are shown by solid (black) circles.

Figure 3

Low-energy spectra of a 24-site Ising anyon $t\mathrm{\text{−}}J$ chain at density $\rho =2/3$ (a) for small $J/t$ and (b) for $J=t$. The ${\varphi }_{\mathrm{ext}}=0$ spectra are shown by shaded (yellow) circles as a function of momentum $K$. The parent charge excitations ($K=K_p$ and ${\varphi }_{\mathrm{ext}}=0$) are shown by black dots. The corresponding charge excitation branches (obtained by varying ${\varphi }_{\mathrm{ext}}$) are shown by continuous lines of different colors. The $+$ and $\times$ symbols correspond to the sum of the charge and (expected) anyonic excitation spectra, respectively (see the text). The colors of these symbols are the same as their corresponding charge excitation parabola.