One-dimensional itinerant interacting non-Abelian anyons

We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains. We focus on itinerant Ising (Majorana) and Fibonacci anyons, which are, respectively, related to SU${\left(2\right)}_2$ and SU${\left(2\right)}_3$ anyons and also, respectively, describe quasiparticles of the Moore-Read and ${\mathbb{Z}}_3$-Read-Rezayi fractional quantum Hall states. Following the derivation of the electronic large-U effective Hubbard model, we derive effective anyonic $t$-$J$ models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (neutral) anyonic degrees of freedom—a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum can be performed by combining spectra for charge and anyonic sectors. The anyonic sector is that of a squeezed chain of localized interacting anyons and, hence, is described by the same conformal field theory (CFT), with central charge $c=1/2$ for Ising anyons and $c=7/10$ or $c=4/5$ for Fibonacci anyons with antiferromagnetic or ferromagnetic coupling, respectively. The charge sector is the spectrum of a chain of hard-core bosons subject to phase shifts which coincide with the momenta of the combined anyonic eigenstates, revealing 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. The combined central charge extracted from the entanglement entropy between segments of the chain is shown to be $1+c$, where $c$ is the central charge of the underlying CFT of the localized anyon (squeezed) chain.

Figure 1

Anyonic fusion trees for (a) electrons, (b) Ising $\sigma$ anyons, and (c) Fibonacci anyons. Sites are denoted by the (filled or open) circles at the top of the diagrams. Open circles denote vacant sites, which carry the vacuum or trivial topological charge 0 or $I$. The bond labels $\left\{x_i\right\}$ encode nonlocal information about the state and their possible values are specified for each model. We note that, for Ising anyons, our model excludes $\psi$ anyons on the sites but not on the links (see the text).

Figure 2

Charts of the quasiparticle contents of (a) the electronic Hubbard model, (b) the MR state, and (c) the $k=3$ RR state. The elementary electric charges of the MR quasiparticles are multiples of $e/4$, while those of the RR quasiparticles are multiples of $e/5$. Dark (green) circles correspond to the different particle types. Filled black circles represent electrons/holes, which are identified with the vacuum $(I,0)$ in (b) and the vacuum $({\psi }_0,0)$ in (c). Gray circles correspond to particles which are identified with one of the particles corresponding to a green symbol, as explained in the text.

Figure 3

(a) All two-anyon processes can be represented as a general “tunneling” term, where topological (and/or electrical) charge is transferred from one localized quasiparticle to the other. Special cases of this include (b) an “interaction” between the two anyons, for which the localized charges are unchanged, and (c) a “hopping” term, where a localized anyon moves to a vacant site.

Figure 4

Hopping and interaction terms for (a) electrons, (b) Ising anyons, and (c) Fibonacci anyons, expressed in the notations used in Fig. 3.

Figure 5

Schematic energy spectra (arbitrary scale) in the presence of a parabolic Coulomb charging energy for the case of (a) electrons, (b) Ising anyons, and (c) Fibonacci anyons. The Hubbard $U$ energy is shown in (a). Upward arrows show the shifts corresponding to the topological contribution to the bare energies of the quasiparticles, e.g., ${\Delta }_{\sigma }=1/8$ and ${\Delta }_{\psi }=1$ in (b) and ${\Delta }_{{\tau }_0}=4/5$, ${\Delta }_{{\tau }_1}={\Delta }_{{\tau }_2}=2/15$, and ${\Delta }_{{\psi }_1}={\Delta }_{{\psi }_2}=4/3$ in (c).

Figure 6

(a, b) Second-order exchange processes for two electrons at nearest-neighbor sites via the virtual creation of a “vacuum” quasiparticle $I$ (i.e., a vacancy) and a “doublon” $d$. These two exchange diagrams can be replaced by the first-order magnon exchange diagram shown in (c), leading to a renormalization of the magnon exchange interaction.

Figure 7

Change of basis involving the fusion channel of two neighboring electrons or anyons and the $F$ symbol.

Figure 8

Matrix elements describing (exchange) interactions between nearest neighbors of (a) electrons, (b) Ising anyons, and (c) Fibonacci anyons. In (b), when $x=z=\sigma$, $y$ and $y^{\prime }$ can take the value $I$ or $\psi$, making $h_{\sigma }^{\sigma ,\sigma ,\sigma }$ a $2\times 2$ matrix, while when neither $x$ nor $z$ equals $\sigma$, then $y=y^{\prime }=\sigma$, making $h_z^{x,\sigma ,\sigma }$ a $1\times 1$ matrix.

Figure 9

Tunneling process (or “hopping”) of (a) an electron, (b) an Ising anyon, and (c) a Fibonacci anyon.

Figure 10

Spectra of 16 (hardcore) bosons moving on a 24-site chain with periodic boundary conditions. (a) Spectrum for zero magnetic flux through the ring. The linear dispersions vs momentum $K$ are shown at $K=0$ and $K=\pm K_c$. (b) Spectrum vs external (continuous) magnetic flux ${\varphi }_{\mathrm{ext}}$ (multiplied by the density $\rho$). Each discrete level [filled (black) circles] leads to a (parabolic) branch of excitations.

Figure 11

Low-energy spectrum of a 24-site $t$-$J$ chain at $J=0$ and density $\rho =2/3$, obtained numerically using Ising anyons. Yellow circles correspond to the spectrum at ${\varphi }_{\mathrm{ext}}=0$ as a function of momentum $K$. The “parent” charge excitations (HCB at ${\varphi }_{\mathrm{ext}}=0$) are shown by filled (black) circles. Adding an external flux ${\varphi }_{\mathrm{ext}}$ through the ring is equivalent to shifting $K$ by $\rho {\varphi }_{\mathrm{ext}}$: the (red) crosses correspond to the spectrum at ${\varphi }_{\mathrm{ext}}=\pi /4$ restricting $K\in [-\pi /3,\pi /6]$ (so that $\tilde{K}\in [-\pi /6,\pi /3]$).

Figure 12

Low-energy spectra vs momentum $K$ of an 18-site $t$-$J$ chain with $J=0$ at anyon densities of (a) $\rho =1/2$ and (b) $\rho =2/3$. The notations here are the same as in Fig. 11, but these results were obtained numerically using Fibonacci anyons. The low-energy spectra of the 18-site HCB chain at the same densities (parent excitations) are shown by filled (black) circles. Data for an external flux ${\varphi }_{\mathrm{ext}}=\pi /5$ are shown in (b). The minimum of the spectrum occurs at momentum $K=0$ or $K=\pi$, depending on the parity of the number $N=\rho L$ of quasiparticles.

Figure 13

Energy spectrum of a dense Ising-anyon chain ($\rho =1$) of length $L_a=16$. The ground-state energy has been subtracted.

Figure 14

A zoom-in on the low-energy spectra vs $K$ of a 24-site Ising anyon $t$-$J$ chains at density $\rho =2/3$ for (a) $J=0$ and (b) the small value of $J/t=\tan (\pi /50)\simeq 0.06$. The Lanczos algorithm with 800 iterations is used so that, in the energy window shown, most of the eigenenergies have converged to within a relative error of ${10}^{-16}$ (a few not fully converged levels are not shown). The (red) crosses correspond to the sum of the (computed) lowest charge branch [solid (red) line] with all the expected anyonic excitations. See text for more details.

Figure 15

Low-energy spectra vs $K$ of a 24-site Ising $t$-$J$ chain at density $\rho =2/3$ for (a) $J=0$ and (b) $J=t=1/\sqrt{2}$. Data at ${\varphi }_{\mathrm{ext}}=0$ are shown by shaded (yellow) circles. Parent charge excitations are shown by filled (black) circles (${\varphi }_{\mathrm{ext}}=0$) and, as a function of pseudomomentum $\tilde{K}=K+{\varphi }_{\mathrm{ext}}\rho$ (varying ${\varphi }_{\mathrm{ext}}$), by solid lines of different colors. The cross and X symbols added for comparison correspond to the sum of the charge and expected anyonic excitation spectra. (See text for more details.) The colors of these symbols are the same as their parent charge excitation parabolas.

Figure 16

Low-energy spectra vs $K$ of an 18-site Fibonacci $t$-$J$ chain at density $\rho =2/3$ and $\left|J\right|=t=1/\sqrt{2}$ for both (a) $J<0$ and (b) $J>0$. Data at ${\varphi }_{\mathrm{ext}}=0$ are shown by shaded (yellow) circles. Parent charge excitations are shown by filled (black) circles (${\varphi }_{\mathrm{ext}}=0$) and, as a function of pseudomomentum $\tilde{K}=K+\rho {\varphi }_{\mathrm{ext}}$ (varying ${\varphi }_{\mathrm{ext}}$), by solid lines of different colors. The cross and X symbols added for comparison correspond to the sum of the charge and (expected) anyonic excitation spectra (see text). The color of these symbols is the same as their parent charge excitation parabolas.

Figure 17

Renormalization parameter $\gamma$ of the energy scale of the anyonic degrees of freedom for Ising and Fibonacci anyons at $\rho =2/3$, computed on $L=24$ and $L=18$ chains, respectively.

Figure 18

Entanglement entropy obtained using DMRG for an open chain of length $L=72$, with $N=48$ Fibonacci anyons and $J/t=0.3$. The entanglement entropy predicted for a CFT with central charge $c=1.7$ is plotted and the agreement is seen to be excellent.