Interacting type-II semi-Dirac quasiparticles

Abstract

Type-II semi-Dirac fermions in two dimensions have been proposed to describe topologically nontrivial low energy excitations in titanium/vanadium oxide heterostructures. These quasiparticles appear at the merger of three Dirac cones, resulting in a non-zero Berry phase. We find, by employing Hartree-Fock, renormalization group and RPA techniques, that the spectrum is very sensitive to long-range electron-electron interactions and can undergo a profound transformation. Specifically the quasiparticle spectrum evolves, driven by interactions, from anisotropic Dirac dispersion at the lowest energies, towards the characteristic type-II semi-Dirac boomerang shape as the energy increases. The corresponding density of states varies between linear and power one third ($ρ(\varepsilon) \sim |\varepsilon| \rightarrow |\varepsilon|^{1/3}$). The crossover scale is controlled by the interaction strength $α= e^2/(\hbar v)$, and the specifics of the effective interacting Hamiltonian. Our results imply that various physical characteristics exhibit critical behavior with continuously varying 'critical exponents'; for example Landau levels in a magnetic field vary with the energy scale: $|\varepsilon_n(B)|\sim (nB)^{1/2} \rightarrow (nB)^{3/4}, n=0,1,2,...$, and similarly for other observables.

Figures (6)

• Figure 1: Low energy dispersion corresponding to the Hamiltonian in Eq.(\ref{['ham']}) under a small perturbation $\Delta$, such that $\mathcal{H}=\left(g_1 k_{x}^{2} - v k_y +\Delta\right)\hat{\sigma}_{x}+g_2 k_{x}k_{y}\hat{\sigma}_{y}$. (a) $\Delta<0$, leading to the presence of three Dirac cones located at $(0,\Delta/v),(\pm\sqrt{-\Delta g_1},0)$. (b) $\Delta=0$: The three cones coalesce at the origin, giving rise to a type-II semi-Dirac point with a finite Berry phase. (c) $\Delta>0$: The band crossing is shifted to $k_y=(g_1^2/vg_2)\Delta$. At the smallest momenta the spectrum is linear, but exhibits type-II semi-Dirac behavior at larger energies.
• Figure 2: Contour plot of the critical low energy spectrum Eq.(\ref{['spec_rescaled']}), illustrated schematically in Figure \ref{['fig:intro']}b). Energies are measured in units of $\varepsilon_0=v^2/g_1$ and momenta in units of $v/g_1$. We take $g=1/2$, similar to the cross anisotropy in the candidate material. The isoenergy curves have a characteristic boomerang shape.
• Figure 3: Type-II semi-Dirac spectrum renormalized by the bare Coulomb interaction at the one-loop Hartree-Fock level for $\alpha=0.2$ and $g=1/2$. The convex lowest energy contours indicate an anisotropic Dirac cone, with concave semi-Dirac behavior appearing at higher energies corresponding to Eq.(\ref{['crossover']}). An ultraviolet momentum cutoff $\vert k_{\Lambda}^{x,y}\vert=1.1$ on a square region, in units of $v/g_1$, is used to find the values $\Delta/\varepsilon_0=1.88/4\pi$ and $c/v=1.39/4\pi$.
• Figure 4: Evolution of the power $\eta(E)$ in the density of states $\rho(E)\sim E^{\eta(E)}$ for $g=1/2$ under different interaction strengths $\alpha=0.05,0.10,0.20$ at the Hartree-Fock level. The smooth transition from linear Dirac behavior $\eta=1$ at the smallest energies, approaching type-II semi-Dirac behavior $\eta=1/3$ (dashed line) with increasing energy, captures the variation in the interacting spectrum shown in Figure \ref{['fig:isosurfHF']}.
• Figure 5: Static polarization function for type-II semi-Dirac fermions. The ravine residing on $p_y=(c_1/c_2)^3 p_x^{3/2}$ demarcates two regions where the polarization is dominated by $\sqrt{p_x}$ and $p_y^{1/3}$ dependence, and flat in the other direction. The dashed and solid mesh lines at constant $p_x,\,p_y$ exhibit this structure. Values of $c_1=0.196\pi$ and $c_2=0.154\pi$ were calculated, and $w=50$ was used for illustration.