Charge and statistics of lattice quasiholes from density measurements: A tree tensor network study

We numerically investigate the properties of the quasihole excitations above the bosonic fractional Chern insulator state at filling $\nu =1/2$, in the specific case of the Harper-Hofstadter Hamiltonian with hard-core interactions. For this purpose, we employ a tree tensor network technique, which allows us to study systems with up to $N=18$ particles on a $16\times 16$ lattice and experiencing an additional harmonic confinement. First, we observe the quantization of the quasihole charge at fractional values and its robustness against the shape and strength of the impurity potentials used to create and localize such excitations. Then, we numerically characterize quasihole anyonic statistics by applying a discretized version of the relation connecting the statistics of quasiholes in the lowest Landau level to the depletions they create in the density profile [E. Macaluso et al., Phys. Rev. Lett. 123, 266801 (2019)]. Our results give a direct proof of the anyonic statistics for quasiholes of fractional Chern insulators, starting from a realistic Hamiltonian. Moreover, they provide strong indications that this property can be experimentally probed through local density measurements, making our scheme readily applicable in state-of-the-art experiments with ultracold atoms and superconducting qubits.

Figure 1

Schematic representation of three bosons (red circles) on a $5\times 5$ lattice with $\varphi =\alpha {\mathrm{\Phi }}_0$ flux per plaquette.

Figure 2

Upper panel: variation in the occupation of the different lattice sites with respect to the unperturbed state $\delta n_j={\langle {\widehat{n}}_j\rangle }_{V_i}-{\langle {\widehat{n}}_j\rangle }_{V_i=0}$, induced by a $2\times 2$ pinning potential of strength $V_i$ ($V_i/t=0.1,1.0$, and 2.5, from left to right) placed in the center of the lattice. The system parameters considered here are those of case I; see Table 2. Lower panel: Charge $Q_l$ induced by a square-shaped pinning potential with intensity $V_i$ on the central plaquette (i.e., on the central $2\times 2$ lattice sites). For large enough $l$, the charge saturates to the expected values for one and two QHs, namely $Q_l=-1/2$ and $Q_l=-1$. The other parameters are as in case I; see Table 2. The charge is computed as in Eq. (12), for the values of $l$ indicated in the legend. The horizontal dashed line corresponds to $-4\times (\alpha /2)=-0.3$, that is, minus four times the bulk density of the unperturbed FCI state. The TTN bond dimension is $D=350$ for all data points.

Figure 3

Two-dimensional density profiles for case I [panels (a), (b), and (c)] and case II [panels (d), (e), and (f)], obtained by TTN calculations. Panels from left to right represent states with zero, one, or two QHs pinned at site (8,8). For case I, we used a single-site pinning potential of strength $V_i/t=2.0$ located at (8,8), to pin the single QH [panel (b)], and a plus-shaped potential of strength $V_i/t=1.0$ acting on the sites (7,8), (8,7), (8,8), (8,8), and (9,8), to pin the two overlapping QHs [panel (c)]. For case II, we used a single-site pinning potential of strength $V_i/t=4.0$, still at (8,8), for the single QH [panel (e)], and the same plus-shaped potential as before, for the two overlapping QHs [panel (f)]. The TTN bond dimension is $D=350$ for case I and $D=500$ for case II.

Figure 4

(a) Comparison between the density depletions created by a single QH (blue dots) and two overlapping QHs (orange dots) in the interacting HH model and in the continuum $\nu =1/2$ Laughlin state (blue and orange dashed lines). (b) Comparison between the quasihole braiding phase, as a function of the cutoff radius $R_{\text{max}}$, for the interacting HH model (red dots), for the discretized quasihole wave functions in Eq. (3) with $a\simeq 0.97l_{\text{B}}$ (black crosses), and for their continuum version (black dashed line). The Hamiltonian parameters are those of case I and the properties—shape and strength—of the pinning potentials considered in the TTN calculations are reported in Table 3. The TTN bond dimension is $D=350$ for all data sets.

Figure 5

Panels (a) and (b): Comparison between the density depletions created by a single QH (blue markers) and two overlapping QHs (orange markers) in the interacting HH model and in the continuum $\nu =1/2$ Laughlin state (blue and orange dashed lines). Different markers indicate the use of different pinning potentials (see Table 3) to create the quasiholes in a system with case II parameters. Panels (c) and (d): Comparison between the quasihole braiding phase, as a function of the cutoff radius $R_{\text{max}}$, obtained for the interacting HH model with different pinning potentials (red markers), for the discretized quasihole wave functions in Eq. (3) with $a\simeq 1.25l_{\text{B}}$ (black crosses), and for their continuum version (black dashed line). In panels (a) and (c), quasiholes are centered on a lattice site, while in panels (b) and (d) they are located at the center of a plaquette. The TTN bond dimension is $D=500$ for all data sets.

Figure 6

(a) Density profiles for the two states ${\mathrm{\Psi }}_0$ and ${\mathrm{\Psi }}_1$ (see text), computed via ED and TTN. (b) Density profiles averaged over ${\mathrm{\Psi }}_0$ and ${\mathrm{\Psi }}_1$, both for TTN and ED. In both panels, the dashed black line is the result for a Laughlin QH state in continuum space.

Figure 7

Finite-$D$ deviations in the depletion profile of a double QH in case II; see Eq. (B1). The reference value $\mathrm{\Delta }{d}_{2\mathrm{QH}}\left(\rho \right)=0$ corresponds to $D=500$ [see orange diamonds in Fig. 5].