Characterization of quasiholes in two-component fractional quantum Hall states and fractional Chern insulators in $\left|C\right|$=2 flat bands

We perform an exact-diagonalization study of quasihole excitations for the two-component Halperin (221) state in the lowest Landau level and for several $\nu =1/3$ bosonic fractional Chern insulators in topological flat bands with Chern number $\left|C\right|$=2. Properties including the quasihole size, charge, and braiding statistics are evaluated. For the Halperin (221) model state, we observe isotropic quasiholes with a clear internal structure, and obtain the quasihole charge and statistics matching the theoretical values. Interestingly, we also extract the same quasihole size, charge, and braiding statistics for the continuum model states of $\left|C\right|$=2 fractional Chern insulators, although the latter possess a “color-entangled” nature that does not exist in ordinary two-component Halperin states. We also consider two real lattice models with a band having $\left|C\right|$=2. There we find that a quasihole can exhibit much stronger oscillations of the density profile, while having the same charge and statistics as those in the continuum models.

Figure 1

A single quasihole of the (221) Halperin state for $N=11,N_{\varphi }=17$, pinned by an impurity potential Eq. (2) located in the upper layer. We show the particle density in the upper layer and lower layer and the total density in (a), (b), and (c), respectively, with the pinning potential located at the center of the plot. The corresponding radial density (blue line) and excess charge (orange line) around the quasihole are shown in (d), (e), and (f), respectively. The horizontal lines in (d), (e), and (f) correspond to $2\pi {\ell }_B^2\rho =1/3,Q/(-e)=2/3$ in (d), $Q/(-e)=-1/3,2\pi {\ell }_B^2\rho =1/3$ in (e), and $Q/(-e)=1/3,2\pi {\ell }_B^2\rho =2/3$ in (f).

Figure 2

The anyonic phase of clockwise braiding one (221) Halperin quasihole around another static one along a circular path of radius $r$. The system size is $N=10,N_{\varphi }=16$. The blue (orange) curve corresponds to the case where the two impurity potentials are located in different layers (in the same layer). The dashed line corresponds to ${\overline{p}}_{\mathrm{an}}=2\pi /3$.

Figure 3

A pair of (221) Halperin quasiholes localized on top of each other in a $N=10,N_{\varphi }=16$ system. (a) The particle density in the upper layer, which is the same as that in the lower layer. (b) The radial density and excess charge in the upper layer. The dashed horizontal line corresponds to $2\pi {\ell }_B^2\rho =1/3,Q/(-e)=1/3$.

Figure 4

A single quasihole of the color-entangled $\nu =1/3$ FQH state for $N=8$, $N_s=25$, $N_x=N_y=5$, pinned by a delta potential in the upper layer. The extended twist defect connecting two layers is located at $x=\pm L/2$. In (a), (b), and (c) we show the particle density in the upper layer and lower layer and the total density, respectively, with the pinning potential located at $\mathbf{w}=(0,0)$. In (d), (e), and (f) the same quantities are shown for the pinning potential located at $\mathbf{w}=(2L/5,0)$.

Figure 5

The anyonic phase of clockwise braiding one quasihole around another static one along a circular path of radius $r$ for the color-entangled $\nu =1/3$ FQH state. The system size is $N=9$, $N_s=29$, $N_x=1$, and $N_y=29$. The aspect ratio of the sample is determined by $L_x/L_y=1$ rather than $N_x/N_y$. The braiding path crosses with the twist defect at $r\approx L/10\approx 0.95{\ell }_B$, as described in the text. The blue (orange) curve corresponds to the case where the two impurity potentials are initially located in different layers (in the same layer). The dashed line corresponds to ${\overline{p}}_{\mathrm{an}}=2\pi /3$.

Figure 6

The hopping terms in (a) the triangular lattice model and (b) the generalized Hofstadter model. The unit cell in each model is covered by a gray polygon. We show hoppings starting or ending within this unit cell. The hopping coefficients are given below the lattice configuration of each model. For a complex hopping, the given phase in the hopping coefficient is obtained when a particle moves along the arrow direction. In (b), the value of $n$ in a hopping coefficient is the $x$ position of the starting site of the hopping.

Figure 7

The lattice site occupation around one $\left|C\right|$=2 FCI quasihole at ${\nu }_{\mathrm{FCI}}=1/3$ for (a) the triangular lattice model with $N=8$ bosons on a regular $N_1\times N_2=5\times 5$ sample; and (b) the generalized Hofstadter model with $N=9$ bosons on a tilted sample. The tilted sample is generated by $n_{1,1}=1$, $n_{1,2}=9$, $n_{2,1}=-3$, $n_{2,2}=1$, with $N_s=28$ unit cells.

Figure 8

(a) The radial lattice site occupation and (b) the excess charge around a $\left|C\right|$=2 FCI quasihole at ${\nu }_{\mathrm{FCI}}=1/3$, as a function of the distance $r$ (in units of the lattice constant) from the quasihole. We use the same lattices as those in Fig. 7. The dashed line is located at $3\langle n_i\rangle =1/3$ in (a) and $Q^{\mathrm{lat}}/(-e)=1/3$ in (b).

Figure 9

Braiding one quasihole around another for $\left|C\right|$=2 FCIs at ${\nu }_{\mathrm{FCI}}=1/3$ in the generalized Hofstadter model. Here we give the braiding path on (a) a regular sample with $n_{1,1}=4$, $n_{1,2}=0$, $n_{2,1}=0$, $n_{2,2}=8$ and (b) a tilted sample with $n_{1,1}=1$, $n_{1,2}=9$, $n_{2,1}=-3$, $n_{2,2}$=2. The mobile quasihole is dragged around the static quasihole along the path indicated by the arrows, starting from the lower right corner of the sample. When the path goes outside the sample, we use periodic boundary conditions to shift the pinning potentials back into the sample. The plot of the path is superimposed on the initial lattice site occupation before the braiding.

Figure 10

Exchanging two $\left|C\right|$=2 FCI quasiholes at ${\nu }_{\mathrm{FCI}}=1/3$ in the generalized Hofstadter model. Here we give the braiding path on (a) a regular sample with $n_{1,1}=4$, $n_{1,2}=0$, $n_{2,1}=0$, $n_{2,2}=8$ and (b) a tilted sample with $n_{1,1}=1$, $n_{1,2}=9$, $n_{2,1}=-3$, $n_{2,2}$ = 2. The two quasiholes are moved following $\left(AD\right)\to \left(BD\right)\to \left(BC\right)\to \left(DC\right)\to \left(DA\right)$, as indicated by the arrows. The plot of the path is superimposed on the initial lattice site occupation before the exchange.