Interaction-driven instabilities of a Dirac semimetal

We explore the possible particle-hole instabilities that can arise in a system of massless Dirac fermions on both the honeycomb and $\pi$-flux square lattices with short range interactions. Through analytical and numerical studies we show that these instabilities can result in a number of interesting phases. In addition to the previously identified charge and spin density wave phases and the exotic “quantum anomalous Hall” (Haldane) phase, we establish the existence of the dimerized “Kekulé” phase over a significant portion of the phase diagram and discuss the possibility of its spinful counterpart, the “spin Kekulé” phase. On the $\pi$-flux square lattice we also find various stripe phases, which do not occur on the honeycomb lattice. The Kekulé phase is described by a ${\text{Z}}_3$ order parameter whose singly quantized vortices carry fractional charge $\pm e/2$. On the $\pi$-flux lattice the analogous dimerized phase is described by a ${\text{Z}}_4$ order parameter. We perform a fully self-consistent calculation of the vortex structure inside the dimerized phase and find that close to the core the vortex resembles a familiar superconducting U(1) vortex, but at longer length scales a clear $Z_4$ structure emerges with domain walls along the lattice diagonals.

Figure 1

(Color online) The model: (a) square lattice with $\frac{1}{2}{\Phi }_0$ magnetic flux per plaquette and dimerized hopping amplitudes. The $\pm$ on the left for each row show the choice of gauge for the Peierls phase factors. The thick (thin) bonds indicate an increased (decreased) hopping amplitude in the $\widehat{x}$ and $\widehat{y}$ directions, the ●/○ dots on the lattice sites represent an increase/decrease in local charge density, and the dotted red line indicates the NNN hopping (showing only inside a single plaquette). The four sites of the unit cell are marked 1–4. (b) Honeycomb lattice set up in analogous fashion with dimerized “Kekulé” phase, without magnetic field.

Figure 2

(Color online) Phase diagram for $\pi$-flux model with spinless fermions. (a) $V_3=0$. (b) $V_3=1.5t$. $\left(\text{SM}=\text{Semimetal}\right)$.

Figure 3

(Color online) Phase diagram for honeycomb lattice. At the mean-field level all transitions from the SM phase are second order whereas transitions between all the gapped phases are first order. The $\times$’s represent the line along which the ratio $V_2/V_1$ would be expected to fall in graphene based on a crude estimate of the bare Coulomb repulsion (Ref. 19).

Figure 4

(Color online) Self-consistent vortex structure in the dimerized phase. (a) Phase and (b) magnitude of the on-site dimer order parameter $g_i$ defined in Eq. (31). (c) Charge density profile showing excess charge density in the vortex core. The total accumulated charge is $e/2$. (d) Anisotropy parameter $\delta$ defined in Eq. (35). For figures (a)–(c), $\Delta /t\approx 0.24$.