Phase transitions on a ladder of braided non-Abelian anyons

Non-Abelian anyons can exist as pointlike particles in two-dimensional systems and have particle exchange statistics which are neither bosonic nor fermionic. Like in spin systems, the role of fusion (Heisenberg-like) interactions between anyons has been well studied. However, unlike our understanding of the role of bosonic and fermionic statistics in the formation of different quantum phases of matter, little is known concerning the effect of non-Abelian braid statistics. We explore this physics using an anyonic Hubbard model on a two-legged ladder which includes braiding and nearest-neighbor Heisenberg interactions among anyons. We study two of the most prominent non-Abelian anyon models: the Fibonacci and Ising types. We discover rich phase diagrams for both anyon models and show the different roles of their fusion and braid statistics.

Figure 1

A two-leg ladder of interacting and itinerant non-Abelian anyons. An empty site is equivalent to a vacuum charge. When a site is occupied with a charge $a$, the charge can be either a Fibonacci anyon $\tau$ or an Ising anyon $\sigma$, depending on the model studied. (The fermion $\psi$ in the particle spectrum ${\mathcal{A}}^{\mathrm{Ising}}$ will not be physically supported on-site.)

Figure 2

Diagrammatic representations of anyonic charges. (a) Two types of lines representing the two types of charges permitted on each site: a vacuum charge $(\mathbb{I},0)$ and a nontrivial charge $(a,1)$, where $a=\tau \text{or}\sigma$. (b) The charge fusion outcomes on each rung, where the lines carry the same meaning as in (a). The charge label $b$ in the last vertex diagram can be $b=\mathbb{I}$ if $a$ is either a $\tau$ or $\sigma$, as both can annihilate to vacuum, or $b=\tau$ if $a=\tau$, or $b=\psi$ if $a=\sigma$, all based on the fusion rules of the individual anyon model. Note that as Fibonacci and Ising anyons are self-dual charges, lines do not need arrowheads, but in the presence of a U(1) number charge, which is not self-dual, the lines need arrowheads.

Figure 3

(a) Orientation of ladder: the ladder is viewed at an angled position from the bottom to top, where particles on the top leg are regarded to be “behind” those on the bottom leg. (b) Linear ordering: the ladder is projected to one dimension; the ladder is “stretched out” into a chain. (c) Fusion ordering: the particles on the sites of the ladder are fused in the usual way, from left to right. The charge label of site $i$ is denoted as $a_i$ (which can be a vacuum charge or a nontrivial charge). The intermediate fusion outcomes on the links of the fusion tree have been suppressed for simplicity.

Figure 4

Phase diagrams of both anyon models: (a) Fibonacci and (b) Ising anyons. The horizontal axis is the leg hopping ratio $t_{\parallel }/t_{\perp }$, the vertical axis is the chemical potential $\mu /t_{\perp }$, and the plotted quantity is the filling fraction $\nu =\langle \widehat{n}\rangle$. Six distinct phases are apparent for Fibonacci anyons, and four for Ising anyons. We describe these phases in terms of the properties of their ground states, including the nature of filling and average particle density in parentheses, the central charge $c$ for gapless phases, and the types of spin and charge excitations in the critical phases. For convenience of reference, we assign some names to distinct sectors of the phase diagrams based on the following mnemonics: “ZF” for zero filling $\left(\nu =0\right)$, “$\frac{1}{2}\mathrm{F}$” for half-filling $\left(\nu =1/2\right)$ at low leg hopping, “FF” for fully filled $\left(\nu =1\right)$, “LF” for low filling $\left(0<\nu <\frac{1}{4}\right)$, “IF” for intermediate filling $\left(\frac{1}{4}<\nu <\frac{3}{4}\right)$, “HF” for high filling $\left(\frac{3}{4}<\nu <1\right)$, “$\mathrm{\Delta }$”for a gapped phase, “GC” for effective golden chain [22], “IC” for effective Ising chain, “FC” for effective “Fibonacci crystal” structure, “${\mathrm{LL}}_{{\tau }_1}$” for a Luttinger liquid of hopping ${\tau }_1$ charges, “${\mathrm{LL}}_{{\sigma }_1}$” for a Luttinger liquid of hopping ${\sigma }_1$ charges, and “${\mathrm{LL}}_{U\left(1\right)}$” for a Luttinger liquid of hopping bosonic U(1) charges. Some of the terms which are unfamiliar are explained in the text.

Figure 5

Three-dimensional versions of Fig. 4. The vertical axis is the filling fraction, which was collapsed in Fig. 4.

Figure 6

Plots of block scaling of the entanglement entropy for Fibonacci anyons. The individual subplot belongs to a distinct phase in the Fibonacci anyons phase diagram Fig. 4. The same set of labels as used in Fig. 4 are also used here for easy identification. These plots are for some representative states within the phases at some value of $(\mu ,t_{\parallel })$. For $\frac{1}{2}\mathrm{F},(-1,0.25)$; for LF, $(-1.5,0.75)$; for IF, $(-1,1.5)$; and for HF, $(-0.45,0.75)$. The solid lines are the data, and the dashed lines are the linear fit to the plots. The saturation seen in the plots is the effect of the limited bond dimension of $\chi =200$ used to capture a critical state in the MPS and is not an indication of a gapped phase in this case.

Figure 7

Plots of block scaling of the entanglement entropy for Ising anyons. The individual subplot belongs to a distinct phase in the Ising anyon phase diagram Fig. 4. The same set of labels as used in Fig. 4 are also used here for easy identification. These plots are for some representative states within the phases at some value of $(\mu ,t_{\parallel })$. For $\frac{1}{2}\mathrm{F},(-1,0.25)$; and for IF, $(-1,1)$. The solid lines are the data, and the dashed lines are the linear fit to the plots. The saturation is the effect of finite bond dimension of $\chi =200$ used in the MPS.

Figure 8

Schematic representation of a potential experimental setup to realize our braided anyonic ladder model using a chained version of the double point contact interferometer of the type of fractional quantum Hall systems proposed in Ref. [37] and realized in Ref. [47]. The tunneling matrix $T$ and the Aharonov-Böhm phase can be controlled by adjusting the gate voltages of the “thin” and “wide” electrodes.

Figure 9

The block scaling of the entanglement entropy $S\left(r\right)$ for the Fibonacci comblike structure in Eq. (A5). No effect of finite bond dimension is visible in this plot up to 800 sites, the maximum block we considered in computing $S\left(r\right)$.