Interaction-driven phases in the half-filled spinless honeycomb lattice from exact diagonalization

We investigate the fate of interaction-driven phases in the half-filled honeycomb lattice for finite systems via exact diagonalization with nearest- and next-nearest-neighbor interactions. We find evidence for a charge density wave phase, a Kekulé bond order, and a sublattice charge-modulated phase in agreement with previously reported mean-field phase diagrams. No clear sign of an interaction-driven Chern insulator phase (Haldane phase) is found despite being predicted by the same mean-field analysis. We characterize these phases by their ground-state degeneracy and by calculating charge-order and bond-order correlation functions.

Figure 1

The extended Hubbard model on the honeycomb lattice. A and B sublattices are represented by green and red circles. The basis vectors and nearest-neighbors vectors defined in the text are ${\mathit{a}}_{1,2}$ and ${\mathit{\delta }}_{1,2,3}$, respectively. The NN and NNN interactions $V_1$ and $V_2$ are represented by gray ellipses. The gray dotted lines enclose the $\Omega =3\times 3$ cluster with periodic boundary conditions.

Figure 2

Contour plot of the noninteracting band structure obtained from (1) with $V_1=V_2=0$. Superimposed is the discretized BZ (dashed line) for a lattice with (a) $\Omega =3\times 3$ and (b) $\Omega =3\times 4$ unit cells. The inset text shows the momentum label $\mathit{Q}=(Q_1,Q_2)$ (see text).

Figure 3

(i) ED phase diagram at $n=\frac{1}{2}$ for a $\Omega =3\times 3$ system. The brightness of each color is proportional to the size of the many-body gap $\Delta /t$. The right-hand side shows the energy spectrum against total momentum $Q=Q_1+L_1{Q}_2$ for the (b) CMs phase, (c) Kekulé phase, (d) SM phase, and (e) CDW phase. The small numbers indicate the degeneracy of each state. The zero of energies is chosen to be the ground-state energy. The phases are identified by the number of ground states over which there is the highest gap. (ii) Mean-field phase diagram calculated following Ref. [10].

Figure 4

CDW pattern at $V_1/t\to \infty$ with $V_2=0$.

Figure 5

(i) The six different types of Kekulé orders. From these six patterns one can produce the second row out of the first row only by including an overall homogeneous hopping decrease (d) + (e) + (f). Thus, only four patterns are independent. (ii) Four independent Kekulé super-positions obtained from (5) with $V_1=3t$ and $V_2=2t$. Different colors correspond to different hopping magnitudes.

Figure 6

CMs patterns with their corresponding degeneracies due to a sixfold-rotational symmetry.