Strong correlation effects on topological quantum phase transitions in three dimensions

We investigate the role of short-ranged electron-electron interactions in a paradigmatic model of three-dimensional topological insulators, using dynamical mean-field theory and focusing on nonmagnetically ordered solutions. The noninteracting band structure is controlled by a mass term $M$, whose value discriminates between three different insulating phases, a trivial band insulator and two distinct topologically nontrivial phases. We characterize the evolution of the transitions between the different phases as a function of the local Coulomb repulsion $U$ and find a remarkable dependence of the $U-M$ phase diagram on the value of the local Hund's exchange coupling $J$. However, regardless of the value of $J$, following the evolution of the topological transition line between a trivial band insulator and a topological insulator, we find a critical value of $U$ separating a continuous transition from a first-order one. When the Hund's coupling is significant, a Mott insulator is stabilized at large $U$. In proximity of the Mott transition we observe the emergence of an anomalous “Mott-like” strong topological insulator state.

Figure 1

Unit cell of the cubic reciprocal lattice. The arrows indicate the three-dimensional path in the reciprocal space used to depict the bands in the next Fig. 2.

Figure 2

Evolution of the noninteracting band structure (left) as a function of $M$ and consequent “phase diagram” (right) of the model (3). The coloured intervals on the right indicates the different topological phases of the system. The colors of the band structures correspond to the orbital character. The plot also illustrates the difference between the strong topological insulator realized at $M>0$, i.e., ${\mathrm{STI}}_{\mathrm{\Gamma }}$, and the one realized at $M<0$, i.e., ${\mathrm{STI}}_R$.

Figure 3

Interacting phase diagram of the three-dimensional model (1) as a function of $M$ and $U$, with $J=0$ (left panel) or $J=U/4>0$ (right panel). Data are for $\lambda =0.3$. The solid (red) line ends in the critical point and is continued beyond this value by the dashed (orange) line. The union of these two lines indicates the boundary between the trivial insulator and the strong topological insulator $(1,000)$. The dotted (blue) line indicates the transition from the STI to the weak topological insulator $(0,111)$. In the right panel the solid (yellow) line is the boundary of the Mott insulating region. The filled (gray) area near the Mott phase shows the re-entrant anomalous STI $(1,111)$ phase, precursor of the Mott transition. The dotted area in the right panel indicates the AFM phase region. The solid (black) line corresponds to the boundary of the AFM phase, which is separated from the nontrivial region by a first-order transition.

Figure 4

Real part of the scalar Matsubara self-energy $\mathrm{Re}\mathrm{\Sigma }\left(i{\omega }_n\right)$ for an increasing interaction strength $U$. Data are shown for $M=0.3$ with $J=0$ (left panel), or $M=4.5$ with $J=U/4$ (right panel). The figure illustrates the opposite algebraic sign of the self-energy in the two investigated cases. It also points out the pronounced frequency dependence at low-energy.

Figure 5

Evolution of the correlation strength measured by $\mathrm{\Xi }$ as a function of $M$ and for increasing values of $U$. All data are for $J=U/4$. The dotted line indicate the transition points.

Figure 6

Evolution of the effective Mass term $M_{\mathrm{eff}}$ as a function of the bare mass $M$. Panel (a)-(b) illustrates the transition from Trivial Band Insulator to Strong Topological Insulator for $J=0$ (a) and $J>0)$ (b). Inset: Hysteresis loop of $M_{\mathrm{eff}}$ for $U=9$ and $J=U/4$.

Figure 7

Absence of bulk gap closing: Evolution of the Green's functions poles, i.e. the band dispersions. We find a continuous transition at weak interaction ($U=2$), contrasted by the absence of bulk gap closure at strong interaction ($U=11$).

Figure 8

(a) Renormalization constant $Z$ as a function of $M$. Data are for increasing values of $U$ and for $J=U/4$. Formation of the Mott insulating state is signaled by the vanishing of $Z$. The transition is generically of the first order. The discontinuity reduces approaching the triple point. (b) Orbital occupation across the WTI-Mott Insulator transition through the ${\mathrm{STI}}_R$. First-order jump takes place at WTI-${\mathrm{STI}}_R$ transition. (c) Real part of the scalar Matsubara self-energy $\mathrm{Re}\mathrm{\Sigma }\left(i{\omega }_n\right)$ for increasing values of $M$. Data are for $U=6$ and $J=U/4$. The plot illustrates how large self-energy lead to the formation of the anomalous ${\mathrm{STI}}_R$ phase.