Erratum: Anyon braiding on a fractal lattice with a local Hamiltonian [Phys. Rev. A 105, L021302 (2022)]

Figure 1

We plot $\rho \left(z_i\right)={\langle n_i\rangle }_{H+H_V,M-1}-{\langle n_i\rangle }_{H,M}$ [see Eq. (5) of the original paper] for (a) the carpet and (b) the square lattice. Here, $M=4$ and potentials with strength 100 are placed on the two sites marked by green crosses. For the carpet, almost all the charge accumulates in the vicinity of one potential. For the square lattice, the charge is more spread out, but again there is more charge close to one potential than to the other. (c) To quantify this further, we plot the total charge $-{\sum }_{j\in {\sigma }_k}\rho \left(z_j\right)$ within a circular region with radius $r$ and center at one of the sites on which a potential is applied. Specifically, ${\sigma }_k$ is the set of all $j$ for which $|z_j-w_k|\le r$, and $w_k$ is the position of the site labeled $k\in \{1,2,3,4\}$ in either (a) or (b). The vertical dotted lines show half the distance between the two sites on which the potentials are applied for the square lattice ($d_{\text{sq}}$) and the carpet ($d_{\text{c}}$). If the potentials had trapped two well-separated anyons of charge $1/2$, the curves in (c) would have had plateaus at 0.5 for $r$ large compared to the size of the anyons and short compared to the distance between the potentials. One should, however, be cautious in interpreting (c) for small lattices, as the trapped charges could have shapes far from circular. Nevertheless, considering (a) and (c) together shows that the potentials do not trap two well-separated anyons on the 64-site carpet with three particles, and (b) and (c) suggest that separated anyons are also not trapped on the 64-site square lattice with three particles. As a result, the phase computed in the paper cannot be interpreted as the Aharonov-Bohm phase obtained by moving one anyon of charge $1/2$ around the considered path.

Figure 2

(a) Particle density $\langle n_i\rangle$ of the ground state of the Hamiltonian in Eq. (1) of the original paper without potentials, $U/J\to \infty$, and four particles on a $12\times 12$ square lattice. (b) We exchange potentials along the green paths. To begin with, potentials are placed on the sites (3,10) and (10,3). Potential 1 first moves along path 1 and then path 2, while potential 2 is fixed. Potential 2 then moves along path 3 and 4, while potential 1 is fixed. Finally, potential 1 moves along path 5 and 6, while potential 2 is fixed. The paths are shown slightly displaced to make clear that the exchange happens counterclockwise. Altogether, the path involves moving a potential from one site to the neighboring site 42 times, and for each such move we use 100 steps. We hence define the position on the exchange path as a parameter within [0,42] with two decimals. (b)–(d) show $\rho \left(z_i\right)={\langle n_i\rangle }_{H+H_V,M-1}-{\langle n_i\rangle }_{H,M}$ [see Eq. (5) of the original paper] for different positions on the exchange path for the case of broad potentials and $V=5$. Sites on which potentials are applied are marked with an additional green or orange ring. The local regions defined in the text are marked by crosses (region 1) and pluses (region 2). (e) shows the charge within the local regions 1 (orange) and 2 (green), i.e., the sum of minus $\rho \left(z_i\right)$ over the sites in the region, for broad potentials and $V=5$. (f) shows the same but for narrow potentials and $V=10$.

Figure 3

Particle density $\langle n_i\rangle$ for the ground state of the Hamiltonian in Eq. (1) of the original paper without potentials, $U/J\to \infty$, and four particles for different patches of a high generation Sierpinski carpet with (a) 64 sites, (b) 128 sites, and (c) 130 sites. The highest densities are found in regions that locally look like the two-dimensional square lattice. This conclusion also holds for $M=2$ and $M=3$.