Many-body breakdown of indirect gap in topological Kondo insulators

We show that the inclusion of nonlocal correlation effects in a variational wave function for the ground state of a topological Anderson lattice Hamiltonian is capable of describing both topologically trivial insulating phases and nontrivial ones characterized by an indirect gap, as well as its closure at the transition into a metallic phase. The method, though applied to an oversimplified model, thus captures the metallic and insulating states that are indeed observed in a variety of Kondo semiconductors, while accounting for topologically nontrivial band structures.

Figure 1

(a) $U$ vs ${\varepsilon }_f$ phase diagram of the model (1) at half filling, with the value of the emergent indirect gap marked on a color scale. The phase diagram comprises three topologically distinct phases, a trivial topological insulator (TTI) and two nontrivial ones, TKI($\mathrm{\Gamma }$) and TKI($M$). Along the two red dashed lines, the $f$ occupancy is constant, with $n_f=0.9$ and $n_f=1.1$. (b) The continuation of the phase diagram for larger $U$ where the TKI can undergo a transition to a metallic state. The region where the $f$ occupancy locks at $n_f=1$ with negligible fluctuations, $d_f\to 0$, is denoted as an orbital selective (OS) Mott regime. (c)–(e) Exemplary low-energy band structures for the selected phases: (c) TKI($\mathrm{\Gamma }$) for $U=7$ and ${\varepsilon }_f=-5$; (d) TKI($M$) for $U=9$ and ${\varepsilon }_f=-2.5$; (e) metal for $U=20$ and ${\varepsilon }_f=-0.5$. The valence and the conduction bands are marked with different colors.

Figure 2

Evolution of the indirect gap ${\mathrm{\Delta }}_g$ with increasing $U$ across the topological quantum phase transition (tQPT) between TKI($\mathrm{\Gamma }$) and TKI($M$) at ${\varepsilon }_f=-2$. The inset shows the low-energy band structure at the topological phase transition where both direct and indirect gaps close, which indicates that the TKI($\mathrm{\Gamma }$) and TKI($M$) are phases not adiabatically connected.

Figure 3

Evolution vs $U$ of (a) the indirect gap ${\mathrm{\Delta }}_g$; (b) the variationally generated $f\text{−}f$ hopping amplitudes $t^f$. Three regimes are identified: (i) initial rise of the gap; (ii) its collapse associated with the sign change of the nearest neighbor $[\mathbf{r}=(1,0\left)\right]$ $t^f$; (iii) the metallic state characterized by the negative value of ${\mathrm{\Delta }}_g$, namely, by overlapping valence and conduction bands, after a quantum phase transition (QPT). We also mark the topological quantum phase transition (tQPT) for small $U\simeq 0.5$.