Mott multicriticality of Dirac electrons in graphene

We study the multicritical behavior for the semimetal-insulator transitions on graphene's honeycomb lattice using the Gross-Neveu-Yukawa effective theory with two order parameters: the SO(3) (Heisenberg) order parameter describes the antiferromagnetic transition, and the ${\mathbb{Z}}_2$ (Ising) order parameter describes the transition to a staggered density state. Their coupling induces multicritical behavior which determines the structure of the phase diagram close to the multicritical point. Depending on the number of fermion flavors $N_f$ and working in the perturbative regime in the vicinity of three (spatial) dimensions, we observe first-order or continuous phase transitions at the multicritical point. For the graphene case of $N_f=2$ and within our low-order approximation, the phase diagram displays a tetracritical structure.

Figure 1

(a) Schematic phase diagram of the extended Hubbard model on the honeycomb lattice with on-site interaction $U$ and nearest-neighbor interaction $V$. Neutral suspended graphene is found to be in the semimetallic state indicated by the star. The neighborhood of the multicritical point (gray shaded area) may be governed by a (b) tetracritical or (c) bicritical structure or by a (d) first-order multicritical point. Solid lines denote second-order and dashed lines first-order phase transitions.

Figure 2

RG flow and fixed-point structure in the Yukawa-coupling subsector for different values of $N_f$ below, at, and above the critical value of $N_f=4$. For $N_f<4$, fixed point B is stable within this subsector, while fixed point D is located in the unphysical domain $g_{\chi }^2<0$ (top left). At $N_f=4$, D becomes physical and collides with B (top right), and subsequently exchanges stability with the latter for $N_f>4$ (bottom left). For $N_f\to \infty$, the stable fixed point D is located at $g_{\varphi }^{*2}=g_{\chi }^{*2}$ (bottom right). Note that A, B, C, and D may be true (and possibly stable) fixed points of the full system only if suitable corresponding zeros of the bosonic $\beta$ functions can be found; see text.

Figure 3

Upper panel: Coordinates of the bosonic couplings at the fixed point D as a function of $N_f$. The gray areas indicate the ranges of $N_f$ where fixed point D does not exist and no other stable fixed point exists. Lower panel: Third critical exponent as a function of $N_f$ for the two possible stable fixed points B and D.

Figure 4

Value of the parameter $\mathrm{\Delta }$ as a function of $N_f$ for the stable fixed point, which governs the multicritical behavior in the phase diagram of Fig. 1. For $N_f\le 4$, the multicritical point is given by the chiral Heisenberg plus Ising fixed point (B) and it has positive $\mathrm{\Delta }>0$, corresponding to tetracritical behavior with a mixed CDW and SDW phase, as indicated in Fig. 1. For $4<N_f<4.64$, the multicritical point is still tetracritical, but its long-range behavior defines another universality class governed by fixed point D. For $4.64<N_f<4.7$, the multicritical point is bicritical [see Fig. 1]. For $4.7<N_f<16.6$, there is no physically admissible stable fixed point in the system and we expect the multicritical point to become discontinuous. For $N_f>16.6$, the multicritical behavior is again bicritical.