Symmetry-protected fractional Chern insulators and fractional topological insulators

In this paper we construct fully symmetric wave functions for the spin-polarized fractional Chern insulators (FCIs) and time-reversal-invariant fractional topological insulators (FTIs) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction $\nu$ (and the same quantized Hall conductance ${\sigma }_{xy}$ in the FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number 1. The low-energy gauge groups of a generic ${\sigma }_{xy}=1/m·e^2/h$ FCI wave function can be either $\text{SU}\left(m\right)$ or the discrete group $Z_m$, and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground-state degeneracy. Examples of FCI/FTI wave functions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number 2 are discussed. Generic FTI wave functions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $\text{SU}\left(m\right)\times \text{SU}\left(m\right)$ or $Z_m\times Z_m$. The constructed wave functions also set up the framework for future variational Monte Carlo simulations.

Figure 1

An anyonic quasiparticle and its antiparticle in a $Z_m$ state on the honeycomb lattice. The dashed line denotes the string of $e^{\mathrm{i}2\pi k/m}$ phase shifts connecting the two plaquettes where quasiparticle $Q_1$ and its antiparticle $Q_2$ are located. On top of the ground-state mean-field ansatz, any mean-field bond crossing the string should pick up a phase shift $e^{\mathrm{i}2\pi k/m}$. Here we only demonstrate the phase shift of nearest-neighbor (NN) mean-field amplitudes by triple arrows in the figure.

Figure 2

The lattice structure and SG generators of (a) honeycomb lattice and (b) checkerboard lattice. ${\vec{a}}_1$ and ${\vec{a}}_2$ are primitive lattice vectors. Each site is labeled by coordinate $(x,y,s)$: $\vec{r}=x{\vec{a}}_1+y{\vec{a}}_2$ corresponds to the position vector of its unit cell and $s=0/1$ are the sublattice indices for $A/B$ sublattices. $C_6$ represents $\pi /3$ rotations along the $\widehat{z}$ axis around the honeycomb plaquette center. $\mathit{\sigma }$ and $P_x,P_y,P_{xy}$ are $\pi$ rotations along the axis plotted in the figure.

Figure 3

The first Brillouin zone of (a) honeycomb lattice and (b) checkerboard lattice. ${\vec{b}}_1$ and ${\vec{b}}_2$ denoted by arrows are reciprocal lattice vectors satisfying ${\vec{b}}_i·{\vec{a}}_j={\delta }_{i,j}$, where ${\vec{a}}_{1,2}$ are primitive vectors shown in Fig. 2. The solid line (red) denotes the original first Brillouin zone (BZ) of the two lattices while the bold dashed line (blue) denotes the reduced first BZ when $\pm 2\pi /3$ fluxes are inserted in each unit cell. The area of the reduced BZ is only one-third of the original BZ. (a) For the honeycomb lattice, the three circles have momenta $\Gamma ,(k_1,k_2)=(0,0)$; $K,(k_1,k_2)=(7\pi /6,\pi /3)$; and $M,(k_1,k_2)=(\pi ,0)$, where $\mathbf{k}=k_1{\vec{b}}_1+k_2{\vec{b}}_2$. (b) In the case of checkerboard lattice, the three circles have momenta $\Gamma ,(k_1,k_2)=(0,0)$; $K,(k_1,k_2)=(\pi /3,\pi )$; and $M,(k_1,k_2)=(0,\pi )$, where $\mathbf{k}=k_1{\vec{b}}_1+k_2{\vec{b}}_2$.

Figure 4

Mean-field ansatz of the parent $\text{SU}\left(3\right)$ parton states associated with FCI state CB1 on checkerboard lattice. The solid lines on NN bonds represent the complex hopping amplitude $\alpha$ along the direction of the arrow. Solid lines on NNN bonds represents real hopping amplitude ${\beta }_x$. The dashed lines on NNN bonds represent complex hopping amplitude $e^{\mathrm{i}\pi /3}{\beta }_y$ along the direction of the arrow. The triple arrow means the original hopping amplitude along its direction should be multiplied by a phase factor ${\eta }_{12}$. We only show up to NNN terms and all NNNN terms can be obtained from Eq. (12). Note that in the mean-field ansatz the lattice translation along the ${\vec{a}}_1$ direction is explicitly broken by flux insertion and the unit cell is tripled. However, the lattice translation symmetry is preserved in the corresponding electron states after projection (8).

Figure 5

The parton band structure of the parent $\text{SU}\left(3\right)$ ansatz (25) of CB1 state. Each band is threefold degenerate, corresponding to three parton flavors $f_{1,2,3}$. We plot the dispersion along $\Gamma \to K\to M\to \Gamma$, as shown in Fig. 3b. Hopping parameters are chosen as $\alpha =e^{\mathrm{i}\pi /12}$ for NN, ${\beta }_x=-0.2={\beta }_y$ for NNN, and $\gamma =0.1$ for NNNN. The Chern numbers of the six bands are $\{1,-2,-2,4,-2,1\}$ in a bottom-up order.

Figure 6

The parton band structure of a $Z_3$ mean-field ansatz (7): the CB1 state. We plot the dispersion along $\Gamma \to K\to M\to \Gamma$, as shown in Fig. 3b. Note that each threefold degenerate band in its parent $\text{SU}\left(3\right)$ ansatz (see Fig. 5) now splits into three nondegenerate parton bands. All the three lowest bands have Chern number $+1$.

Figure 7

Mean-field ansatz of the parent $\text{SU}\left(3\right)$ parton states associated with FCI states CB2 and CB3 on a checkerboard lattice. The solid lines on NN bonds represent complex hopping amplitude $\alpha$ along the direction of the arrow. Dashed lines on NNN bonds represents real hopping amplitude ${\beta }_y$. Solid lines on NNN bonds represent complex hopping amplitude $e^{\mathrm{i}\pi /3}{\beta }_x$ along the direction of the arrow. The triple arrows mean the original hopping amplitude along its direction should be multiplied by a phase factor ${\eta }_{12}$. We only show up to NNN terms and all NNNN terms can be obtained from Eq. (12). Note that in the mean-field ansatz the lattice translation along the ${\vec{a}}_1$ direction is explicitly broken by flux insertion and the unit cell is tripled. However, the lattice translation symmetry is preserved in the corresponding electron states after projection (8).

Figure 8

The parton band structure of the parent $\text{SU}\left(3\right)$ ansatz (25) of CB2 state. Each band is threefold degenerate, corresponding to three parton flavors $f_{1,2,3}$. We plot the dispersion along $\Gamma \to K\to M\to \Gamma$, as shown in Fig. 3b. Hopping parameters are chosen as $\alpha =e^{\mathrm{i}\pi /12}$ for NN, ${\beta }_x=0.1={\beta }_y$ for NNN, and $\gamma =0.09$ for NNNN. The Chern numbers of the six bands are $\{1,1,1,-5,1,1\}$ in a bottom-up order.

Figure 9

Mean-field ansatz of the parent $\text{SU}\left(3\right)$ parton states associated with FCI states HC1 and HC3 on honeycomb lattice. The solid line for NN bonds represents real hopping amplitude $\alpha$. NNN bonds represents complex hopping amplitude $\beta$ along direction of the arrow. The triple arrow means the original hopping amplitude along its direction should be multiplied by a phase factor ${\eta }_{12}$. Note in the mean-field ansatz that the lattice translation along the ${\vec{a}}_2$ direction is explicitly broken by flux insertion and the unit cell is tripled. However, the lattice translation symmetry is preserved in the corresponding electron states after projection (8).

Figure 10

The parton band structure of the parent $\text{SU}\left(3\right)$ ansatz (25) of HC1 state. Each band is threefold degenerate, corresponding to three parton flavors $f_{1,2,3}$. We plot the dispersion along $\Gamma \to K\to M\to \Gamma$, as shown in Fig. 3a. Hopping parameters are chosen as $\alpha =-1$ for NN, $\beta =-0.22\exp \left[0.2\mathrm{i}\right]$ for NNN. The Chern numbers of the six bands are $\{1,-2,-2,1,1,1\}$ in a bottom-up order.

Figure 11

The parton band structure of a $Z_3$ mean-field ansatz: HC2 state. There are 18 non-degenerate parton bands. We plot the dispersion along $\Gamma \to K\to M\to \Gamma$ as shown in FIG. 3a. The Chern numbers of the lowest 6 bands are $\{1,1,1,-2,-2,-2\}$ in a bottom-up order.

Figure 12

The parton band structure of HC3 state, a non-Abelian $\text{SU}\left(3\right)$ FCI state on honeycomb lattice. Each band is threefold degenerate, corresponding to three parton flavors $f_{1,2,3}$. We plot the dispersion along $\Gamma \to K\to M\to \Gamma$ as shown in Fig. 3a. Its mean-field ansatz is shown in Fig. 9, with ${\eta }_{12}=\exp (-\mathrm{i}2\pi /3)$. Hopping parameters are chosen as $\alpha =-1$ for NN and $\beta =-0.22\exp \left[0.2\mathrm{i}\right]$ for NNN. The Chern numbers of the six bands are $\{2,2,-1,-4,2,-1\}$ in a bottom-up order.

Figure 13

The parton band structure of CB3 state, a non-Abelian $\text{SU}\left(3\right)$ FCI state on checkerboard lattice. Each band is threefold degenerate, corresponding to three parton flavors $f_{1,2,3}$. We plot the dispersion along $\Gamma \to K\to M\to \Gamma$, as shown in Fig. 3b. Its mean-field ansatz is shown in Fig. 7, with ${\eta }_{12}=\exp (\mathrm{i}2\pi /3)$. Hopping parameters are chosen as $\alpha =e^{\mathrm{i}\pi /12}$ for NN, ${\beta }_x=0.6={\beta }_y$ for NNN, and $\gamma =0.5$ for NNNN. The Chern numbers of the six bands are $\{2,2,-1,-4,2,-1\}$ in a bottom-up order.