Quasielectrons in lattice Moore-Read models

Moore-Read states can be expressed as conformal blocks of the underlying rational conformal field theory, which provides a well explored description for the insertion of quasiholes. It is known, however, that quasielectrons are more difficult to describe in continuous systems, since the natural guess for how to construct them leads to a singularity. In this work, we show that the singularity problem does not arise for lattice Moore-Read states. This allows us to construct Moore-Read Pfaffian states on lattices for filling fraction $5/2$ with both quasiholes and quasielectrons in a simple way. We investigate the density profile, charge, size, and braiding properties of the anyons by means of Monte Carlo simulations. Further we derive an exact few-body parent Hamiltonian for the states. Finally, we compare our results to the density profile, charge, and shape of anyons in the Kapit-Mueller model by means of exact diagonalization.

Figure 1

(a),(c),(e),(g) Circles, stars, and squares represent lattice sites, quasiholes, and quasielectrons, respectively. We fix $N=112$. The density profile $\rho \left(z_i\right)$, defined as the particle density difference between the states with and without anyons, is plotted with color bar for the cases of (a) two quasielectrons, (c) one quasihole and one quasielectron, (e) four quasielectrons, and (g) two quasiholes and two quasielectrons in the states, respectively. (b),(d),(f),(h) The excess charge distributions are computed. The anyon charges approach $\sim \pm 0.25$ with an error of magnitude ${10}^{-4}$. Note that the density profiles are similar for quasiholes and quasielectrons except for a sign as evident from the plot of the excess charge, ${\sum }_k{Q}_k$, of the anyons shown in (d) and (h), respectively.

Figure 2

(a) Lattice sites and anyons are denoted by circles and diamonds, respectively. Proper choices of the charges of the anyons lead to the configurations shown in Fig. 1. The anyons are placed in the bulk and sufficiently separated from each other. Subsequently one anyon is moved around one lattice site through a closed loop along the path midway in the lattice plaquette while keeping the other anyon positions fixed. The lattice size is set to $N=96$. (b),(c) We take the setup as described in (a). We plot the overlap quantity $P$ as a function of different moves, i.e., the position $l$ of the circulating anyon for the cases of two quasielectrons (2 qe) and one quasihole and one quasielectron (1qh-1qe) as shown in (b) and (c), respectively. For both cases, we note that the quantities vary with the period of the lattice up to finite-size effects.

Figure 3

We keep the anyons (pluses) fixed in the bulk and sufficiently separated from each other and increase the lattice size $N$ by adding more sites as shown in (a). We denote different lattice sizes with symbols as $N=52$ (circles), $N=60$ (squares added), $N=68$ (hexagons added), $N=76$ (down pointing triangles added), $N=88$ (diamonds added), and $N=96$ (right pointing triangles added). Proper choices of the charges of anyons lead to the configurations as shown in Figs. 1 and 1, respectively. In (b) we plot the overlaps $\mathcal{O}$ (circles) and $\mathcal{N}$ (squares) as defined in (21) and (22), respectively, in the semilog scale as a function of the lattice size for the cases of four quasielectrons (4 qe) and two quasiholes and two quasielectrons (2qh-2qe) present in the system. We note that the quantities of interest in both plots follow an exponential decay for sufficiently large lattice sizes. A linear fitting indeed proves the exponential decay implying that the states ${\mathrm{\Psi }}_I$ and ${\mathrm{\Psi }}_{\psi }$ are orthogonal in the thermodynamic limit.

Figure 4

We take the setup as shown in Fig. 2 and a proper choice of anyon charges leads to the configurations in Figs. 1 and 1. We plot the overlap quantity $P^{\alpha }$ with $\alpha \in \{I,\psi \}$ as a function of different moves, i.e., $l$ of the circulating anyon. In (a) and (b) we plot respectively $P^I\text{and}{P}^{\psi }$ for the case of four quasielectrons (4 qe) and in (c) and (d) concurrently we plot $P^I\text{and}{P}^{\psi }$ for the case of two quasiholes and two quasielectrons (2qh-2qe). It is seen that the quantities vary with the period of the lattice up to finite-size effects.

Figure 5

In (a), (c), (e), and (g) we symbolize lattice sites $(N=24)$ by the circles. Quasiholes and quasielectrons are depicted by stars and squares, respectively. We place a quasihole potential $V$ and a quasielectron potential $-V$ on two different lattice sites in (a). Similarly we place two quasihole potentials (each with $V)$ and two quasielectron potentials (each with $-V)$ in (c). The density profile $\rho \left(z_i\right)$ that is defined as the particle density difference between the states with and without anyons, is plotted with color bar. We plot the excess charge distribution $Q_k$ as defined in (13) as a function of the radial distance from the anyons in (b) and (d) as marked by the circles respectively in (a) and (c). In (e) and (g) we place one quasihole and one quasielectron [hence exploit (6) and (9)] and two quasiholes and two quasielectrons in the system [hence use (6) and (10)]. The density profile $\rho \left(z_i\right)$ is plotted with color bar. We also plot the excess charge distribution $Q_k$ as a function of the radial distance in (f) and (h). We observe that in all cases the quasiparticles, both quasiholes and quasielectrons, are trapped with the charge of $\sim 0.5$ and $\sim -0.5$, respectively. Note that (b) and (f) are similar, although not exactly the same, and the same applies to (d) and (h).