Monte Carlo study of Dirac semimetals phase diagram

In this paper the phase diagram of Dirac semimetals is studied within a lattice Monte Carlo simulation. In particular, we concentrate on the dynamical chiral symmetry breaking which results in a semimetal-insulator transition. Using numerical simulation, we determine the values of the critical coupling constant of the semimetal-insulator transition for different values of the anisotropy of the Fermi velocity. This measurement allows us to draw a tentative phase diagram for Dirac semimetals. It turns out that within the Dirac model with Coulomb interaction both ${\mathrm{Na}}_3\mathrm{Bi}$ and ${\mathrm{Cd}}_3{\mathrm{As}}_2$, known experimentally to be Dirac semimetals, would lie deep in the insulating region of the phase diagram. This result probably shows a decisive role of screening of the interelectron interaction in real materials, similar to the situation in graphene.

Figure 1

The chiral condensate $\langle \overline{\mathrm{\Psi }}\mathrm{\Psi }\rangle$ as a function of $\beta$ for different values of mass $m$. Fermi velocity is isotropic ${\xi }_1={\xi }_2={\xi }_3=1$. The black line corresponds to chiral limit $m\to 0$ taken with the help of EOS (8).

Figure 2

Susceptibility $\chi$ of the chiral condensate as a function of $\beta$ for different values of mass $m$. Fermi velocity is isotropic, ${\xi }_1={\xi }_2={\xi }_3=1$.

Figure 3

Logarithmic derivative of $R$ of the chiral condensate as a function of mass $m$ for different values of the coupling constant $\beta$. Fermi velocity is isotropic, ${\xi }_1={\xi }_2={\xi }_3=1$.

Figure 4

The square chiral condensate $\sigma$ as a function of $\frac{m}{\sigma }$. Fermi velocity is isotropic, ${\xi }_1={\xi }_2={\xi }_3=1$. Straight lines correspond to the fit of all points with EOS (8).

Figure 5

The dependence of the critical coupling constant ${\alpha }_{\mathrm{eff}}^c$ on the Fermi velocity anisotropy $\xi$. For ${\alpha }_{\mathrm{eff}}>{\alpha }_{\mathrm{eff}}^c\left(\xi \right)$ the system is in the insulator phase. Smaller values of ${\alpha }_{\mathrm{eff}}<{\alpha }_{\mathrm{eff}}^c\left(\xi \right)$ correspond to the semimetal phase. Statistical errors are smaller than data points. Lines are to guide the eyes.