Short-ranged interaction effects on $Z_2$ topological phase transitions

We develop an analytical approach based on a combined perturbative and self-consistent mean-field treatment of interactions that is capable of capturing topological phase transitions beyond either method when used independently. As an illustration of the method, we study the effects of short-range interactions on the $Z_2$ topological insulator phase, also known as the quantum spin Hall phase, in two generalized versions of the Kane-Mele model at half-filling on the honeycomb lattice. The results are in excellent agreement with quantum Monte Carlo calculations on the same model and cannot be reproduced by either a perturbative treatment or a self-consistent mean-field treatment of the interactions. Our analytical approach helps to clarify how the symmetries of the one-body terms of the Hamiltonian determine whether interactions tend to stabilize or destabilize a topological phase. Moreover, our method should be applicable to a wide class of models where topological transitions due to interactions are in principle possible but are not correctly predicted by either perturbative or self-consistent treatments.

Figure 1

Schematic of the honeycomb lattice with two sublattices labeled $A$ and $B$. The vectors ${\mathbf{e}}_{1/2}=(\pm 1/2,\sqrt{3}/2)$ connect the same sublattice in different unit cells. The lattice constant $a$ is set to 1.

Figure 2

Self-consistent mean-field data for QSH boundary shift in the (a) GKM model and (b) DKM model within the perturbation theory plus mean-field picture. (a) The amount of the boundary shift, red open squares, is linearly proportional to $U^2/t^2$. The inset is the illustration of the GKM model on the honeycomb lattice. The red lines represent the $t_3$ hoppings. (b) The open blue diamonds represent the data of the shift amount. The inset represents the DKM model on the honeycomb lattice with anisotropic hoppings that break $C_3$ rotation. The red lines represent the $t_d$ hoppings with $t_d>t$. A positive shift indicates that the TI phase is stabilized; for a negative shift it is destabilized.

Figure 3

QMC data for the QSH boundary shift in (a) the GKM model and (b) the DKM model. (a) The shift is positive, which means QSH is more stable against $t_3$ hoppings. More interestingly, the shift amount is linearly proportional to ${(U/t)}^2$, consistent with our mean-field picture. (b) The shift is negative and linearly proportional to ${(U/t)}^2$. The short-range interaction makes the QSH phase more destabilized by the dimerization $t_d$. Statistical errors are denoted by the error bars.

Figure 4

QMC data for QSH boundary shift in the attractive Kane-Mele-Hubbard model in the presence of staggered potential $M$. The shift amount of the critical $M,\Delta M^c$ is negative and linearly proportional to $U/t$. Note that we choose ${\lambda }_{so}=0.2t,U<0$, and PHS is explicitly broken due to the staggered potentials. QMC is only fermion-sign free in the attractive $U<0$ case.