Assembling Fibonacci anyons from a ${\mathbb{Z}}_3$ parafermion lattice model

Recent concrete proposals suggest it is possible to engineer a two-dimensional bulk phase supporting non-Abelian Fibonacci anyons out of Abelian fractional quantum Hall systems. The low-energy degrees of freedom of such setups can be modeled as ${\mathbb{Z}}_3$ parafermions “hopping” on a two-dimensional lattice. We use the density matrix renormalization group to study a model of this type interpolating between the decoupled-chain, triangular-lattice, and square-lattice limits. The results show clear evidence of the Fibonacci phase over a wide region of the phase diagram, most notably including the isotropic triangular-lattice point. We also study the broader phase diagram of this model and show that elsewhere it supports an Abelian state with semionic excitations.

Figure 1

Model of ${\mathbb{Z}}_3$ parafermions with interactions forming an anisotropic triangular lattice with intrachain couplings $t_3$ (horizontal) and interchain couplings $t_1$ (vertical) and $t_2$ (slanted).

Figure 2

(a) A trench dividing two regions of an Abelian quantum Hall phase supports gapless modes. (b) Gapping these modes via electron backscattering recovers the original bulk phase. (c) Introducing either Cooper pairing, or “crossed” tunneling in a bilayer system, provides an inequivalent way of gapping the trench. (d) Domain walls between incompatibly gapped regions support protected zero modes.

Figure 3

One-dimensional chain of parafermionic zero modes arising from inequivalently gapped regions of a trench. Hybridization among nearest neighbors in the chain is described by the Hamiltonian in Eq. (3).

Figure 4

A series of one-dimensional parafermion chains interact by tunneling fractional charge through the quantum Hall fluid [Eq. (4)]. For this work, we consider anisotropic next-neighbor tunneling processes for parafermions arranged in a triangular lattice, allowing us to study (among other limits) the isotropic triangular lattice ($t_1=t_2=t_3$), the isotropic square lattice ($t_1=t_3, t_2=0$), and weakly coupled chains ($t_3\gg t_1,t_2$).

Figure 5

Two-chain limit of the Hamiltonian in Eq. (5), with open boundary conditions along the $y$ direction.

Figure 6

Pairing of sites in the ground state for the two-chain ladder system in the limits of (a) dominant $t_1>0$, which produces a trivial gapped phase, and (b) dominant $t_2>0$, which produces a topological gapped phase with a threefold ground-state degeneracy.

Figure 7

Bulk gap of the two-chain system (Fig. 5) as a function of $\theta$ with $t_1=sin\theta , t_2=cos\theta$, and $t_3=1$. The vertical dashed line marks the $t_1=t_2(=1/\sqrt{2})$ critical point separating topological and trivial gapped phases. For the topological phase, the first-excited state is in the quantum number sector with triality 1; for the trivial phase, the first-excited states are twofold degenerate and lie in the $\omega$ or ${\omega }^2$ triality sectors.

Figure 8

Entanglement entropy $S_L\left(x\right)$ for a length-$x$ subregion of a periodic two-chain system with length $L=60$. Data were obtained at the critical point using parameters $t_1/t_3=t_2/t_3=1$. By fitting to the CFT prediction we extract a central charge $c=\frac{4}{5}$, consistent with analytical arguments.

Figure 9

Full phase diagram of the two-chain system in units where $t_3=1$. The trivial phase (green circles) surrounds the $t_1>0$ axis while the topological phase (blue triangles) surrounds the $t_2>0$ axis. The remainder of the phase diagram shown is gapless with central charge $c=1$, which we verified numerically for points indicated with red squares. Transition lines separating the gapless state from the two gapped phases are approximate. The critical point along the line $t_1=t_2$ is exact, however, and represents a precursor to the Fibonacci phase appearing in the 2D limit.

Figure 10

Relative energy splitting between quasidegenerate ground states of cylinders with $N_y=4–10$ as a function of $t_{\perp }$ (in units where $t_3=1$). Notice that for $N_y\ge 8$ the states remain nearly degenerate even up to the isotropic-triangular-lattice point $t_{\perp }=1$. This is the first evidence suggesting that in the 2D limit the Fibonacci phase enjoys a wide stability window extending from weakly coupled chains past the isotropic-triangular-lattice limit.

Figure 11

Entanglement “energies” of the quasi-ground states $|1\rangle$ (left column) and $|\varepsilon \rangle$ (right column) for the $N_y=4, t_{\perp }/t_3=0.2$ cylinder after shifting and rescaling the spectra to match the lowest two levels of the ${\mathbb{Z}}_3$ parafermion CFT prediction. The numbers in parentheses indicate the predicted degeneracies. The same rescaling was used for both spectra, but with different shifts; thus, the fit requires only three fitting parameters.

Figure 12

Entanglement entropy fits for $N_y=4,6,8$ cylinders with two different interchain couplings $t_{\perp }$ (with $t_3=1$). All entanglement-entropy values are shifted downward by $ln\sqrt{3}$ as explained in the text. The $y$ intercepts closely match the topological entanglement entropies ${\gamma }_n$ predicted for the Fibonacci phase.

Figure 13

Entanglement entropy difference between the quasidegenerate ground states of $N_y=4,6,8,10$ cylinders as a function of interchain coupling $t_{\perp }$ (with $t_3=1$). The horizontal dashed line denotes the thermodynamic-limit prediction $ln\phi \approx 0.481$. Our $N_y=8$ and 10 data further indicate that the Fibonacci phase survives even beyond the isotropic-triangular-lattice point in the 2D limit, corroborating the evidence presented in Fig. 10.

Figure 14

Maxima in $N_y=8$ cylinder entanglement entropies as a function of $t_2$ for various fixed $t_1$ ($t_3=1$ throughout). To estimate the location of each maximum, which in the 2D limit coincide with the transition out of the Fibonacci phase, data points near each maximum were fitted to a quadratic form. Fits are shown as thick gray curves; the resulting locations of the maxima are indicated by vertical dashed lines. The locations of the maxima give the four $N_y=8$ data points in Fig. 15. The $t_1=0.4,0.5,0.6$ data were computed using DMRG with a fixed number of states $m$. The $t_1=0.3$ curve was found by extrapolating fixed $t_2$ entropies as a power law in $m$.

Figure 15

(a) Estimated extent of the Fibonacci phase near the weakly-coupled-chain limit. Data points correspond to the locations of peaks in the entanglement entropy observed when tuning $0<t_2<t_1$ for fixed $t_1$ in $N_y=4,6,8$ cylinders ($t_3=1$ throughout). The solid curves are fits of the data to the function $(t_2-t_1)=C{(t_2+t_1)}^{8/5}$ with $C$ an $N_y$-dependent fitting parameter. Dashed curves show the fits reflected across the line $t_1=t_2$ since in the thermodynamic limit the Hamiltonian is symmetric about this line. (b) $N_y=8$ fit reflected under all $t_1–t_2–t_3$ permutations, estimating the full extent of the Fibonacci phase. The black squares and triangle mark the location of the isotropic square- and triangular-lattice models.

Figure 16

Global phase diagram (some assembly required) of the Hamiltonian in Eq. (5) with couplings constrained to $|t_1|+|t_2|+|t_3|=1$. Note that Fig. 15 shows the face with $t_1,t_2,t_3\ge 0$ containing the Fibonacci phase. The six corners of the octahedron correspond to various exactly solvable decoupled-chain limits, and directions (a)–(e) mark weakly-coupled-chain limits where DMRG shows strong evidence of the semion phase. Direction (f) indicates anisotropic-square-lattice systems studied in Fig. 18 and small squares mark locations of the isotropic-square-lattice model in the 2D limit. The bottom right corner shows the phase diagram after assembly.

Figure 17

(a) States of the $\mathrm{U}{\left(1\right)}_6$ chiral CFT, organized by charge $n$. The highest weight states have energies $\frac{n^2}{12}$ and thus form a parabola. Their descendants have energies $\frac{n^2}{12}+m$ for integers $m=0,1,2,...$ with multiplicities given by the partitions of the integers $1,1,2,3,...$. Combining all the states for even $n$ yields the edge spectra for the $|1_s\rangle$ ground state, shown on the left axis. The states for odd $n$ constitute the edge spectra for the $|s\rangle$ ground state, shown on the right axis. (b) Numerical entanglement spectra of the quasi-ground states $|1_s\rangle$ (blue, left column) and $|s\rangle$ (red, right column) at $\vec{t}=(-0.1,-0.1,-1)$ for the $N_y=8$ cylinder after shifting and rescaling the spectra to match the lowest two levels of the $\mathrm{U}{\left(1\right)}_6$ CFT prediction. We used the same rescaling for both spectra but different shifts. Numbers in parentheses denote the predicted degeneracies. The excellent agreement with analytics strongly supports the onset of the semion phase.

Figure 18

Relative energy splitting between quasidegenerate ground states of cylinders with $N_y=4–10$ as a function of $t_1$ (in units where $t_3=1$ with $t_2=0$ throughout). The horizontal axis corresponds to direction (f) in Fig. 16.

Figure 19

Possible ordering of parafermion operators in an $N_y=4$ cylinder. For the Hamiltonian in Eq. (5), this choice yields no interactions that cross the parafermion ordering path. Parafermion “hopping” processes that pass through regions labeled $R$ or $L$ are expressed in terms of the corresponding operator representation, e.g., ${\alpha }_{3R}^{†}{\alpha }_{6R}$ or ${\alpha }_{5L}^{†}{\alpha }_{8L}$.

Figure 20

Ordering of parafermion operators used in DMRG for this work (shown for a cylinder with $N_y=4$). Note that in terms of the underlying clock degrees of freedom seen by DMRG, this path requires a circumference of only two sites in the $y$ direction.

Figure 21

“Braiding” parafermion sites 3 and 4 clockwise two times transforms the pairwise interaction between sites 3 and 7, defined originally in the $L$ representation, into an interaction that crosses the parafermion ordering path. This scheme allows us to simulate the model of Eq. (5) using the numerically very convenient ordering shown in Fig. 20.

Figure 22

Parafermion ordering path used to discuss the self-duality of the two-leg ladder model studied in Sec. 3. Solid lines and numbering of the parafermion operators indicate the ordering path, which coincides with the $t_1$ and $t_2$ interactions. The $t_3$ interactions, shown as dashed lines, involve processes that take place above or below the path.