Chern-Simons composite fermion theory of fractional Chern insulators

We formulate a Chern-Simons composite fermion theory for fractional Chern insulators (FCIs), whereby bare fermions are mapped into composite fermions coupled to a lattice Chern-Simons gauge theory. We apply this construction to a Chern insulator model on the kagome lattice and identify a rich structure of gapped topological phases characterized by fractionalized excitations including states with unequal filling and Hall conductance. Gapped states with the same Hall conductance at different filling fractions are characterized as realizing distinct symmetry fractionalization classes.

Figure 1

Kagome lattice unit cell with phases arising from a magnetic flux indicated by arrows. The net flux is zero.

Figure 2

(a) Kagome lattice unit cell with spatial components of the statistical gauge field $A_i$ and fluxes ${\mathrm{\Phi }}^{\alpha }=J_i^{\alpha }{A}_i$. (b) Dice lattice unit cell with the hydrodynamic gauge field $B_i$ and fluxes ${\mathrm{\Phi }}^{*\alpha }=J_i^{*\alpha }{B}_i$. (c) Orientation of the dual (dice) lattice relative to the direct (kagome) lattice.

Figure 3

(a) Composite fermion Hofstadter spectrum on the kagome lattice with $k=1$ and ${\varphi }_{\pm }=\pi /2$. The blue line is the Fermi energy. Some examples of gapped states are labeled with their filling and Hall conductance. Vertical red (purple) lines are drawn at fillings corresponding to the principal particle (hole) Jain sequence. The ratio of the mean-field sublattice shift $\mathrm{\Delta }$ to the filling and the mean-field current per link $k$ are plotted in (b) and (c), respectively. All quantities are in units of $J=1$.

Figure 4

Band structure of the model given by Eq. (2.1) in the absence of interactions with ${\varphi }_{\pm }=\pi /2$ and $J=1$. The lower, middle, and upper bands have $C_0=+1,0,-1$, respectively

Figure 5

Composite fermion Hofstadter spectrum on the kagome lattice with $k=1$ and ${\varphi }_{\pm }=\pi /2$ assuming a uniform density of composite fermions and equal fluxes through all plaquettes. The blue line is the Fermi energy. Some examples of gapped states are labeled with their filling and Hall conductance. Vertical red (purple) lines are drawn at fillings corresponding to the principal particle (hole) Jain sequence.

Figure 6

Plots of the sublattice imbalance and band gap for (a) and (b) $n_L=1/3$, and (c) and (d) $n_L=2/3$ as a function of interaction strength $g$. Note that for small $g$ the imbalance $\mathrm{\Delta }$ and the band gap vary smoothly with the latter never vanishing.

Figure 7

Spectrum of the Hermitian $6\times 6$ matrix $i\mathcal{M}\left(\mathbf{q}\right)$ as a function of $q_1$ for the choice $q_2=\pi$.