Spontaneous parity breaking in spin-orbital coupled systems

Effects of spontaneous parity breaking by charge, spin, and orbital orders are investigated in a two-band Hubbard model on a honeycomb lattice. This is a minimal model in which the interorbital hopping, atomic spin-orbit coupling, and strong electron correlation give rise to fascinating properties, such as the magnetoelectric effects, quantum spin Hall effect, and spin or valley splitting in the band structure. We perform the symmetry analysis of possible broken-parity states and the mean-field analysis of their competition. We find that the model at 1/4 filling exhibits a spin-orbital composite ordered state and a charge ordered state, in addition to a paramagnetic quantum spin Hall insulator. We show that the composite ordered phase exhibits two types of magnetoelectric responses. The charge ordered state shows spin splitting in the band structure, while the topological nature varies depending on electron correlations.

Figure 1

(a) Schematic picture of a honeycomb lattice; the primitive translation vectors are ${\mathit{a}}_1=(\sqrt{3}/2,1/2)$ and ${\mathit{a}}_2=(-\sqrt{3}/2,1/2)$. Open circles (triangles) indicate the inversion centers (the parity-broken sites). (b) Schematic picture of the energy levels of the two-band model Hamiltonian in Eq. (1). (c) Ground-state phase diagram of the model in Eqs. (1) and (4) by the mean-field calculations. We take $t_0=t_1=\lambda =0.5$ and $J_{\mathrm{H}}=0.1U$. NM, CO, and SOO stand for the nonmagnetic, charge ordered, and spin-orbital composite ordered states, respectively.

Figure 2

(a) Temperature dependence of the ME coefficients $K_{yx}^{\mathrm{u}}$, $K_{xy}^{\mathrm{u}}$, and $K_{zy}^{\mathrm{s}}$, and the order parameters $m_{\mathrm{SOO}}^{xy}$ and $m_{\mathrm{SOO}}^{yx}$ in the SOO region in Fig. 1. The data are calculated at $U=2.5$ and $V=0$. The vertical dashed line shows the transition temperature. (b) Electronic band structure for the same parameters at $T=0$. The results are shown along the high-symmetry lines in the Brillouin zone [see the inset of Fig. 3]. The Fermi level is set to zero.

Figure 3

Electronic band structures: (a) for the NM state at $U=V=0$ and (b) for the CO state at $U=0$ and $V=0.6$. In (b), the dashed blue (dotted red) lines show the bands with the up-(down-)spin polarization. The inset of (b) shows the energy contour below the Fermi level by $0.05$.