Effects of interaction in the Hofstadter regime of the honeycomb lattice

We investigate phases of spinless fermions on the honeycomb lattice with nearest-neighbor interaction in the Hofstadter regime. The interaction induces incompressible nematic and ferrielectric phases with broken translation symmetry. Some of the transitions are accompanied by changes in the Hall conductivity. We study pair correlations and show that the quantum metric, averaged over the Brillouin zone, characterizes the shape of the pair correlation function.

Figure 1

Hall conductivity vs Fermi energy plot for $q=30$. The Fermi regime, van Hove filling, and Dirac regime are shown in this figure.

Figure 2

Honeycomb lattice in a magnetic field with flux $\varphi =\frac{h}{3e}$ passing through each plaquette. $A$ and $B$ are the two sublattices. We consider two possible choices of unit cell: linear choice (I) shown as a gray rectangle and hexagonal choice (II) shown as a yellow hexagon. $(A_1,B_1,A_2,B_2,A_3,B_3)$ are sublattices of these magnetic unit cells. ${\widehat{e}}_1$ and ${\widehat{e}}_2$ represent the basis vectors of the original lattice.

Figure 3

Phase diagram for $m=1,2,3$ filling as $V$ is varied. The charge distribution in each phase is shown by the unit cell whose repetition gives the full lattice picture of the ground state. The Chern number distribution of the filled bands is given in bracket. For example, in the $m=2$ case, the left number in bracket is for the lowest band and the next number is for the second band. The green circle is the transition point. The blue circle is the point where the Chern numbers of the filled bands change without any change in the Hall conductivity.

Figure 4

The quadrupole and dipole moments as a function of $V$ in the three phases for $m=2$. The blue line represents the polarization (i.e., dipole moment per original unit cell) and the red line represents the magnitude of the eigenvalue of the quadrupole moment matrix per original unit cell, $\left|r\right|$.

Figure 5

Flux in the plaquette and current flowing on the links on the lattice for $m=2$ in (a) nematic phase (b) ferrielectric phase. The flux distribution is shown after subtracting out the background flux $2\pi /3$. In ferrielectric phase, $\delta {\varphi }_1=\delta {\varphi }_3+\delta {\varphi }_2$. The bonds with arrows are the links on which there is nonzero current. Same color represents same magnitude of current.

Figure 6

(a) Bond order parameter strengths for $m=2$ broken symmetry phases. The red bonds are the strongest and the dotted violet bonds the weakest. (b) Bond order parameters for the $m=1$ broken symmetry phase. The red bonds are stronger than the dotted blue bonds.

Figure 7

$l_g^2=\sqrt{g_1{g}_2}$ and $\gamma$ vs $V$ for $m=2$. The blue line represents $l_g$ and the red line represents $\gamma$. The discontinuity in the plot shows the first order nature of the phase transition.

Figure 8

Pair correlation function on lattice in (a) symmetric phase showing $2\pi /3$ rotation preserved, (b) nematic phase for $m=2$ showing $2\pi /3$ rotation broken.

Figure 9

Energy band diagram with respect to $k_2$ for (a) $q=3$ in symmetric phase for $m=2$ at $V=2.744$, (b) $q=3$ in nematic phase for $m=2$ at $V=2.744$. The energy band diagrams are plotted by diagonalizing the mean-field Hamiltonian using the order parameters obtained by solving the self-consistency equations for $m=2$ and $q=3$.