Bona fide interaction-driven topological phase transition in correlated SPT states

Abstract

It is expected that the interplay between non-trivial 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 (QMC) simulations, we provide a concrete example of the Kane-Mele-Hubbard (KMH) model on an AA stacking bilayer honeycomb lattice with inter-layer antiferromagnetic interaction. Our simulation identified several different phases: a quantum spin-Hall insulator (QSH), a $xy$-plane antiferromagnetic Mott insulator ($xy$-AFM) and an inter-layer dimer-singlet insulator (dimer-singlet). Most importantly, a bona fide topological phase transition between the QSH and the dimer-singlet insulators, purely driven by the inter-layer 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 analogue and it can be viewed as a transition between bosonic SPT states. At one special point, this transition is described by a $(2+1)d$ $O(4)$ nonlinear sigma model (NLSM) with {\it exact} $SO(4)$ symmetry, and a topological term at {\it exactly} $Θ= π$. Relevance of this work towards more general interacting SPT states is discussed.

Figures (12)

• Figure 1: (color online) (a) Illustration of AA-stacked honeycomb lattice and bilayer KMH model with inter-layer antiferromagnetic exchange interaction. The four-site unit cell is presented as the shaded rectangle. The gray and black lines indicates the nearest-neighbor hopping $t$ on layer 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 inter-layer AFM coupling are represented by the shaded circle and rectangle, respectively. (b) Illustration of the $xy$-AFM Mott insulator phase. (c) Illustration of the inter-layer dimer-singlet phase. Shaded ellipses are the inter-layer spin singlets.
• Figure 2: (color online) $U$-$J$ phase diagram for the AA-stacked bilayer KMH model with inter-layer antiferromagnetic coupling. Showing 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 goes through them highlights 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. Slagle2015, it actually goes through a small AFM region.
• Figure 3: (color online) (a) The inter-layer spin-spin correlation function for $L=6, \lambda=0.2t$ system with various $U$ values, as a function of $J$. The continuous variation of this correlation function indicates 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 inter-layer 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 $xy$-AFM) phase to inter-layer 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 $xy$-AFM and $xy$-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 $xy$-AFM transition at small $J$ and $xy$-AFM to dimer-singlet transition at large $J$.
• Figure 4: (color online) (a) Single-particle gap $\Delta_{sp}(\mathbf{K})$ of $\lambda=0.2t, U=0$ as a function $J$. The inset shows the $\Delta_{sp}(\mathbf{K})$ 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 $\Delta_S$ of $\lambda=0.2t, U=0$ with increasing $J$. The inset is the spin gaps in $J \in [3.4t,3.8t]$ region. The spin gap drops very fast and closes at the topological phase transition point $J_c=3.73(1)t$.
• Figure 5: (color online) Spin gap in $J\in[3.7t,3.8t]$ 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 $J$.