Wave functions for fractional Chern insulators

We provide a parton construction of wave functions and effective field theories for fractional Chern insulators. We also analyze a strong-coupling expansion in lattice gauge theory that enables us to reliably map the parton gauge theory onto a microscopic electron Hamiltonian. We show that this strong-coupling expansion is useful because of a special hierarchy of energy scales in fractional quantum Hall physics. Our procedure is illustrated using the Hofstadter model and then applied to bosons at half filling and fermions at one-third filling in a checkerboard lattice model recently studied numerically. Because our construction provides a more or less unique mapping from microscopic model to effective parton description, we obtain wave functions in the same phase as the observed fractional Chern insulators without tuning any continuous parameters.

Figure 1

A rendition of the checkerboard model showing the next-next-nearest-neighbor (NNNN) hoppings interpreted as “bridges” out of the plane connecting more distant sites. The red and blue dots are the two sublattices. Our lattice gauge theory version of this model associates a link variable with every elementary hopping path shown here. Included in the sum over loops are loops running along these out-of-plane bridges representing the NNNN hoppings.

Figure 2

The two most important processes in the strong-coupling expansion. Process (A) generates an electron kinetic term of order ${\left(t^f\right)}^3/h^2$. Process (B) generates repulsive electron interactions of order ${\left(t^f\right)}^{4}K/h^4$.

Figure 3

A sketch of the phase diagram of the gauge theory based on the strong-coupling expansion and the energy scales of the electron model. The phase boundaries are determined by the condition that the electron-electron interaction either grossly exceeds the electron band gap (upper curve) or falls well below the bandwidth of the lowest electron band (lower curve).

Figure 4

The checkerboard flat band model and its cube root. The original unit cell is shaded in green; the $3\times$-enlarged unit cell is less shaded. The arrows indicate the direction in which the hopping amplitude is $e^{i\varphi }$. $t_1^{\prime }$ ($t_2^{\prime }$) is the NNN amplitude associated with the solid (dashed) lines.

Figure 5

Left: the topological flat bands found in the checkerboard model by Ref. 31, at the optimal hopping amplitudes. Energy is measured in units of the NN amplitude $t$, and wave vectors are measured in units of the inverse NNN lattice spacing. The Hamiltonian is the one used in the numerical work,[9, 10, 11, 12, 13] which is minus that studied in Ref. 31. Right: the band structure for a square root of the checkerboard flat band model. Each parton hopping amplitude is the square root of the optimal-flatness values chosen in Ref. 31: $t=1,t_1^{\prime }=\sqrt{\frac{1}{2+\sqrt{2}}},t_2^{\prime }=t_1^{\prime }{e}^{\frac{i\pi }{2}},t^{\prime \prime }=i\sqrt{\frac{1}{2+2\sqrt{2}}},\varphi =5\pi /8,\vec{\alpha }=(1,1,0),\vec{\theta }=(0,1,1,1)$. The discrete parameter $\vec{\theta }$ is explained in Appendix C. The Chern number of the bottom band is $-1$.

Figure 6

The band structure for a cube root of the checkerboard flat band model. Each parton hopping amplitude is the square root of the optimal values chosen in Ref. 31: $t=1,t_1^{\prime }=\sqrt[3]{\frac{1}{2+\sqrt{2}}},t_2^{\prime }=t_1^{\prime }{e}^{-\frac{\pi i}{3}},t^{\prime \prime }=\sqrt[3]{\frac{1}{2+2\sqrt{2}}},\varphi =\pi /12,\vec{\alpha }=(1,1,0)$. The Chern numbers of these bands, from bottom to top, are $(1,0,-1,0,0,0)$.