Topological antiferromagnetic phase in a correlated Bernevig-Hughes-Zhang model

Topological properties of antiferromagnetic phases are studied for a correlated topological band insulator by applying the dynamical mean-field theory to an extended Bernevig-Hughes-Zhang model including the Hubbard interaction. The calculation of the magnetic moment and the spin Chern number confirms the existence of a nontrivial antiferromagnetic (AF) phase beyond the Hartree-Fock theory. In particular, we uncover the intriguing fact that the topologically nontrivial AF phase is essentially stabilized by correlation effects but not by the Hartree shifts alone. This counterintuitive effect is demonstrated, through a comparison with the Hartree-Fock results, and should apply for generic topological insulators with strong correlations.

Figure 1

The results obtained with the Hartree-Fock approximation: (a) spin configuration of the AF phase, (b) sketch of the reduced Brillouin zone, (c) magnetic moment at orbital 1, (d) spin-Hall conductivity, (e) energy level of each orbital modified by the Hartree shift. For $3.5<U<3.78$, the coexistence region inherent in the first-order transition can be found. The energy of each phase crosses at $U\sim 3.6$, which determines the phase transition point for the thermodynamically stable phase diagram. Note that in this figure, no nontrivial AF phase is found.

Figure 2

(a) The AF moment of orbital 1 as a function of interaction strength. (b) The SChN as a function of interaction strength. Inset: The SChN (red) and the magnetic moment of orbital 1 (blue) are plotted. In figure (a), the coexistence region is also observed in the DMFT results for $4.2<U<4.5$. Here, the first-order phase transition point, where the energy of each phase crosses, is estimated as $U\sim 4.3$.

Figure 3

Spectral functions for an up-spin dominant sublattice [$A_{\alpha ,\sigma }\left(\omega \right)$] for several values of interaction strength. At $U=3$ $(4.4,4.8)$, the system is in the TBI (nontrivial AF, trivial AF) phase, respectively. Solid red (solid blue, dashed red, dashed blue) line represents that of the state $(\alpha ,\sigma )=(1,\uparrow )$ [$(1,\downarrow ),(2,\uparrow ),(2,\downarrow )$], respectively. Inset (left): the total spectral functions (${\Sigma }_{\alpha ,\sigma }{A}_{\alpha ,\sigma }$) near the Fermi energy ($\omega =0$). Inset (right): the spectral function near the Fermi energy around $\omega =0$.

Figure 4

Momentum-resolved spectral functions [$A_{\alpha ,\sigma }(\mathbf{k},\omega )$] for $(\alpha ,\sigma )=(1,\downarrow ),(2,\uparrow )$: (a), (b) for $U=4.4$, (c), (d) for $U=4.8$.

Figure 5

The renormalized energy level of each orbital (${\varepsilon }_{\alpha ,\sigma }^{*}$) as a function of interaction.