Correlation Effects in Quantum Spin-Hall Insulators: A Quantum Monte Carlo Study

We consider the Kane-Mele model supplemented by a Hubbard $U$ term. The phase diagram is mapped out using projective auxiliary field quantum Monte Carlo simulations. The quantum spin liquid of the Hubbard model is robust against weak spin-orbit interaction, and is not adiabatically connected to the spin-Hall insulating state. Beyond a critical value of $U>U_c$ both states are unstable toward magnetic ordering. In the quantum spin-Hall state we study the spin, charge, and single-particle dynamics of the helical Luttinger liquid by retaining the Hubbard interaction only on a ribbon edge. The Hubbard interaction greatly suppresses charge currents along the edge and promotes edge magnetism but leaves the single-particle signatures of the helical liquid intact.

Figure 1

(a) Periodic lattice structure of the KM-Hubbard model with nearest-neighbor hopping $t$, spin-orbit coupling $\lambda$, and Coulomb repulsion $U$. Arrows indicate the current direction associated with the spin-orbit term for one spin species and sublattice. (b) Effective model on a semi-infinite ribbon with periodic boundaries in the ${\mathbf{a}}_1$ direction and Coulomb repulsion $U$ only at the edge sites.

Figure 2

Phase diagram of the KM-Hubbard model from QMC simulations. Bullets correspond to computed phase boundaries. The four phases are the semimetal (SM), the quantum spin liquid (QSL), the topological or quantum spin-Hall insulator (TBI), and the antiferromagnetic Mott insulator (AFM-MI). Inset: Ground-state spin-spin correlation function [Eq. (2)] for an $L\times L$ honeycomb lattice with periodic boundary conditions and $\lambda /t=0.25$. Lines are fits to the form $a+b/L+c/L^2$. Negative values indicate that the data decay quicker than $1/L^2$. Here and in subsequent figures, error bars are omitted if smaller than the symbol size.

Figure 3

(a),(b) Single-particle gap along different cuts in Fig. 2 (${\lambda }_c/t\simeq 0.03$, $U_c/t\simeq 4.9$). (c) Raw data and size extrapolation (inset) at $U/t=4$, $\lambda /t=0.03$.

Figure 4

Transverse spin correlations along the edge of the ribbon. Results are for $L=16$, $\beta t=20$ (dashed lines) and $\beta t=40$ (solid lines). Lines are guides to the eye.

Figure 5

Dynamic spectral functions in the (a) charge sector, (b) spin-resolved one-particle sector, and (c),(d) spin sector, measured along the edge. The parameters are $U/t=2$, $L=24$, $L^{\prime }=64$, $\lambda /t=0.25$, and $\beta t=40$. Dotted lines show the velocities of the free helical liquid ($U/t=0$).

Figure 6

Same as in Fig. 5 but for $U/t=5$.