Bona fide interaction-driven topological phase transition in correlated symmetry-protected topological states

It is expected that the interplay between nontrivial band topology and strong electron correlation will lead to very rich physics. Thus a controlled study of the competition between topology and correlation is of great interest. Here, employing large-scale quantum Monte Carlo simulations, we provide a concrete example of the Kane-Mele-Hubbard model on an AA-stacking bilayer honeycomb lattice with interlayer antiferromagnetic interaction. Our simulation identified several different phases: a quantum spin Hall insulator (QSH), an $xy\text{-plane}$ antiferromagnetic Mott insulator, and an interlayer dimer-singlet insulator. Most importantly, a bona fide topological phase transition between the QSH and the dimer-singlet insulators, purely driven by the interlayer antiferromagnetic interaction, is found. At the transition, the spin and charge gap of the system close while the single-particle excitations remain gapped, which means that this transition has no mean-field analog and it can be viewed as a transition between bosonic symmetry-protected topological (SPT) states. At one special point, this transition is described by a $(2+1)dO\left(4\right)$ nonlinear sigma model with exact $SO\left(4\right)$ symmetry and a topological term at exactly $\mathrm{\Theta }=\pi$. The relevance of this work towards more general interacting SPT states is discussed.

Figure 1

(a) Illustration of AA-stacked honeycomb lattice and bilayer KMH model with interlayer antiferromagnetic exchange interaction. The four-site unit cell is presented as the shaded rectangle. The gray and black lines indicate the nearest-neighbor hopping $t$ on layers 1 and 2, respectively. The spin-orbital coupling term $\lambda$, for one spin flavor, is shown by the red lines and arrows with ${\nu }_{ij}=+1$. The on-site Coulomb repulsion and interlayer AFM coupling are represented by the shaded circle and rectangle, respectively. (b) Illustration of the $xy\text{-AFM}$ Mott insulator phase. (c) Illustration of the interlayer dimer-singlet phase. Shaded ellipses are the interlayer spin singlets.

Figure 2

$U\text{−}J$ phase diagram for the AA-stacked bilayer KMH model with interlayer antiferromagnetic coupling. Shown here are the phase diagram for $\lambda =0.2t$ and $\lambda =0.3t$ cases. Solid lines (violet, green, and black) are the phase boundaries for the $\lambda =0.2t$ case. The red solid dot at $(J_c,U=0)$ and red open dots at $U=0.25$ and 0.5 and the green line going through them highlight the interaction-driven topological phase transition between QSH and the dimer-singlet insulator phase. The orange dotted line highlights the $J=2U$ path which is studied in Ref. [49]; it actually goes through a small AFM region.

Figure 3

(a) The interlayer spin-spin correlation function for the $L=6,\lambda =0.2t$ system with various $U$ values, as a function of $J$. The continuous variation of this correlation function indicates that the topological phase transition from QSH to dimer-singlet is a continuous one. At large $J$, the correlation saturates at $-3/4$ which signifies the formation of interlayer dimer singlets. (b) First-order derivative of $\langle H_J\rangle$ per bond over $J$ for $L=6,\lambda =0.2t$. The peak in every curve explicitly indicates phase transition from QSH insulator (or $xy\text{-AFM})$ phase to interlayer dimer-singlet phase. (c) First-order derivative of $\langle H_U\rangle$ per site over $J$. The peaks in these curves indicate all three possible phase transitions: QSH to dimer-singlet, QSH to $xy\text{-AFM}$, and $xy\text{-AFM}$ to dimer-singlet transitions. For $U<2t$, the peak in $d\langle H_U\rangle /dJ$ corresponds to the QSH to dimer-singlet transition; for $U>3t$, the two independent peaks as a function of $J$ correspond to the QSH to $xy\text{-AFM}$ transition at small $J$ and $xy\text{-AFM}$ to dimer-singlet transition at large $J$.

Figure 4

(a) Single-particle gap ${\mathrm{\Delta }}_{sp}\left(\mathbf{K}\right)$ of $\lambda =0.2t,U=0$ as a function of $J$. The inset shows the ${\mathrm{\Delta }}_{sp}\left(\mathbf{K}\right)$ in $J\in [3.4t,3.8t]$ region. We have checked that $\mathbf{K}$ point is indeed the minimum of single-particle gap in the whole BZ. As a function of $J$, the single-particle gap only shows a gentle dip near the topological phase transition. (b) Spin gap ${\mathrm{\Delta }}_S$ of $\lambda =0.2t,U=0$ with increasing $J$. The inset is the spin gaps in the $J\in [3.4t,3.8t]$ region. The spin gap drops very fast and closes at the topological phase transition point $J_c=3.73\left(1\right)t$.

Figure 5

Spin gap in the $J\in [3.7t,3.8t]$ region for $\lambda =0.2t,U=0$ with $L=3,6,9,12$ and the extrapolation by third-order polynomial. The inset shows the extrapolated spin gap as a function of $J$.

Figure 6

(a) The inverse amplitude of single-particle strange correlator $1/|C_{\mathbf{k}AB}^{\uparrow }|$ along the high-symmetry path for various $J=3.0t,4.0t$. (b) The spin strange correlator $S_{\mathbf{k}AA}^{\pm }$ at various $J$ values as a function of linear system size $L$.

Figure 7

(a) Finite-size extrapolation of the transverse magnetic structure factor for $L=3,6,9,12$ systems; the fits are third-order polynomial in $1/L$. The parameters are $\lambda =0.2t,U=2t$, and $J\in [2.0t,2.7t]$. Inset shows the extrapolated staggered magnetic moment $m_S$ as a function of $J$. (b) The spin gap for $L=3,6,9,12$ systems and its extrapolated thermodynamic limit (TDL) values for the same parameter set.

Figure 8

(a) Finite-size extrapolation of the transverse magnetic structure factor for $L=3,6,9,12$ systems; the fits are third-order polynomial in $1/L$. The parameter sets are $\lambda =0.2t,U=4t$, and $J\in [0.6t,2.1t]$. Inset shows the extrapolated staggered magnetic moment $m_S$ as a function of $J$. (b) The spin gap for $L=3,6,9,12$ systems and its extrapolated thermodynamic limit (TDL) values for the same parameter set.

Figure 9

Single-particle Green's function for $\lambda =0.2t,U=0,J=3.73t$ with $L=3,6,9,12$ at $\mathbf{K}$ point in (a) linear scale and (b) in semilogarithmic scale.

Figure 10

Dynamic spin-spin correlation function for $\lambda =0.2t,U=0,J=3.73t$ with $L=3,6,9,12$ at $\mathbf{\Gamma }$ point in (a) linear scale and (b) in semilogarithmic scale.

Figure 11

Extrapolation of structure factor $S^{xy}\left(\mathbf{\Gamma }\right)/N$ of $xy\text{-AFM}$ order over $1/L$ for (a) $U=0$, (b) $U=0.25t$, (c) $U=0.50t$, and (d) $U=1.00t$ over inverse system size $1/L$, at $\lambda =0.2t$. The data points with error bars in the insets are the extrapolated values in the thermodynamic limit.

Figure 12

The interlayer spin-spin correlation functions around the topological phase transitions for $U=0$ with $\lambda =0.2t$.