Excitonic pairing and insulating transition in two-dimensional semi-Dirac semimetals

A sufficiently strong long-range Coulomb interaction can induce excitonic pairing in gapless Dirac semimetals, which generates a finite gap and drives the semimetal-insulator quantum phase transition. This phenomenon is in close analogy to dynamical chiral symmetry breaking in high-energy physics. In most realistic Dirac semimetals, including suspended graphene, the Coulomb interaction is too weak to open an excitonic gap. The Coulomb interaction plays a more important role at low energies in a two-dimensional semi-Dirac semimetal, in which the fermion spectrum is linear in one component of momenta and quadratic in the other, than a Dirac semimetal, and indeed leads to breakdown of Fermi liquid theory. We study dynamical excitonic gap generation in a two-dimensional semi-Dirac semimetal by solving the Dyson-Schwinger equation, and show that a moderately strong Coulomb interaction suffices to induce excitonic pairing. Additional short-range four-fermion coupling tends to promote excitonic pairing. Among the available semi-Dirac semimetals, we find that the ${\mathrm{TiO}}_2$/${\mathrm{VO}}_2$ nanostructure provides a promising candidate for the realization of an excitonic insulator. We also apply the renormalization group method to analyze the strong coupling between the massless semi-Dirac fermions and the quantum critical fluctuation of the excitonic order parameter at the semimetal-insulator quantum critical point, and reveal non-Fermi liquid behaviors of semi-Dirac fermions.

Figure 1

Dependence of fermion mass $m\left(0\right)$ on $\beta$ obtained under instantaneous approximation at different values of $\alpha$. (a) $N=2$; (b) $N=4$.

Figure 2

Dependence of fermion mass $m\left(0\right)$ on $\alpha$ obtained under instantaneous approximation at different values of $\beta$. (a) $N=2$; (b) $N=4$.

Figure 3

Dependence of fermion mass $m\left(0\right)$ on $N$ obtained under instantaneous approximation at different values of $\alpha$. (a) $\beta =1$; (b) $\beta =0.1$; (c) $\beta =10$.

Figure 4

Dependence of $m(p_x,p_y)$ on $p_x$ and $p_y$ obtained under instantaneous approximation at $\alpha =1$ and $N=2$. (a) $\beta =0.1$; (b) $\beta =1$; (c) $\beta =10$.

Figure 5

Dependence of $m\left(0\right)$ on $\alpha$ with $\beta =1$ in (a), $\beta =0.1$ in (b), and $\beta =10$ in (c). Method 1: Instantaneous approximation. Method 2: Khveshchenko approximation. Method 3: GGG approximation. This convention is also used in Fig. 6. Here, $N=4$.

Figure 6

Dependence of $m\left(0\right)$ as a function of flavor $N$. (a) $\alpha =1$; (b) $\alpha =\infty$. Here, $\beta =1$.

Figure 7

(a) Dependence of $g_c$ on $\beta$. (b) Dependence of $m\left(0\right)$ on $g$ at different values of $\beta$. Only the four-fermion interaction is considered in this case.

Figure 8

Dependence of $m\left(0\right)$ on $\alpha$ at different values of $g$. Both the Coulomb and four-fermion interactions are present. It is clear that larger $g$ leads to larger gap.

Figure 9

Schematic phase diagram of semi-Dirac fermion system on plane $(\alpha ,T)$ with fixed values of $N$ and $\beta$. Here, SM, EI, and NFL stand for semimetal, excitonic insulating, and non-Fermi liquid phases, respectively. The zero-temperature QCP is broadened by thermal fluctuation to a finite quantum critical region on the phase diagram at finite temperatures.

Figure 10

Flows of $Z_f,a$, and $v$ considering long-range Coulomb interaction in semimetal phase in different RG schemes. The initial conditions ${\alpha }_0=1$ and ${\beta }_0=1$ are taken.

Figure 11

Flows of $Z_f,a$, and $v$ considering both quantum fluctuation of insulating phase and long-range Coulomb interaction at QCP in RG scheme 1. The initial condition ${\beta }_0=1$ is taken.