Figure 1

(a) The Fermi surface at the doping level of interest is a hexagon inscribed within a hexagonal Brillouin zone (BZ), for both honeycomb and triangular lattices. The FS has three inequivalent corners, which are saddle points of the dispersion, marked by a vanishing Fermi velocity and a divergent density of states. The three inequivalent saddle points $M_i$ are connected by three inequivalent nesting vectors ${\mathbf{Q}}_i$, each of which is equal to half a reciprocal lattice vector, such that ${\mathbf{Q}}_i=-{\mathbf{Q}}_i$. (b) Spin structure for the uniaxial SDW state. The SDW order quadruples the unit cell to a unit cell with eight sites (shaded). The enlarged unit cell has a large spin moment $3\Delta$ on two sites and a small spin moment $-\Delta$ on the other six. The total spin on each unit cell is zero.Reuse & Permissions

Figure 2

The terms quartic in $\Delta$ are produced by processes represented diagrammatically by square diagrams. The diagrams for $Z_2$ and $Z_3$ correspond to patterns ${\Delta }_3$, ${\Delta }_3$, ${\Delta }_1$, ${\Delta }_1$ and ${\Delta }_3$, ${\Delta }_1$, ${\Delta }_3$, ${\Delta }_1$, respectively. The sixth order chirality sensitive term is produced by “hexagonal diagrams.” Sample square and hexagonal diagrams are shown above. The integrals are dominated by momenta that bring all the fermion propagators to the vicinity of one of the saddle points of the dispersion.Reuse & Permissions

Figure 3

(a) Excitation spectrum ${\epsilon }_k=E_k-\delta \mu$ of the $3Q$ uniaxial state. Negative $k$ are along the FS, positive $k$ are along the BZ boundary in the original BZ (along $k_x$ in the reduced zone). Placing the chemical potential at $\delta \mu =-\Delta$ ensures that four bands lie below the chemical potential (horizontal dotted line) and two lie above for all $\mathbf{k}$, irrespective of the value of $\Delta$. Thus the choice $\mu =-\Delta$ conserves electron number. Excitations with spin projection opposite to $\mathbf{\Delta }$ are in blue (solid), along $\mathbf{\Delta }$ are in red (dashed) lines. Note that gapless excitations arise in the spin-down branch only. (b) Free energy difference $\delta F=F_{\mathrm{uniaxial}}-F_{\mathrm{chiral}}$ between the $3Q$ uniaxial SDW state and the chiral state, evaluated in the mean-field approximation for the honeycomb lattice Hubbard model with $g_2=g_3=U=1.7t_1$ ($T_N\sim 0.002t_1$). The $3Q$ uniaxial state has lower Free energy over a wide range of intermediate temperatures, but at the smallest $T$ the noncoplanar, chiral state, studied in earlier works [3, 5, 10], has lower Free energy.Reuse & Permissions