Combined spontaneous symmetry-breaking and symmetry-protected topological order from cluster charge interactions

The study of symmetry-protected topological states in the presence of electron correlations has recently aroused great interest as rich and exotic phenomena can emerge. Here we report a concrete example by employing a large-scale unbiased quantum Monte Carlo study of the Kane-Mele model with cluster charge interactions. The ground-state phase diagram for the model at half filling is established. Our simulation identifies the coexistence of a symmetry-protected topological order with a symmetry-breaking Kekulé valence bond order and shows that the spontaneous symmetry breaking is accompanied by an interaction-driven topological phase transition (TPT). This TPT features the appearance of zeros of a single-particle Green's function and gap closing in a spin channel rather than a single-particle excitation spectrum and thus has no mean-field correspondence.

Figure 1

The lattice structure and ground-state $\lambda \text{−}U$ phase diagram. (a) Illustration of the honeycomb lattice and Hamiltonian (1). A unit cell of the lattice, indicated by ${\widehat{a}}_1$, ${\widehat{a}}_2$, contains A (red dot) and B (green dot) sublattices. The black lines represent NN hopping, and the red lines show the SOC with arrows indicating ${\nu }_{\mathbf{ij}}=+1$. (b) Brillouin zone of the model and the reciprocal lattice vectors ${\widehat{b}}_1$ and ${\widehat{b}}_2$ are given corresponding to ${\widehat{a}}_1$ and ${\widehat{a}}_2$, respectively. (c) The $\lambda \text{−}U$ phase diagram from PQMC simulations. Three cases of $\lambda /t=0.05,0.2,$ and 0.4 are studied. The disordered-KVBS ordered phase transition is verified to coincide with the TPT from $C_s=+1$ to $C_s=-1$, as shown by the solid blue line. Thus, a topologically nontrivial KVBS phase is established. As a comparison, the quantum phase transition with $U_c/t=1.666\left(8\right)$ for $\lambda =0$ from a Dirac semimetal to a trivial KVBS phase from Ref. [42] is also indicated by the solid green dot.

Figure 2

(a) The total energy density derivative $\partial \langle \widehat{H}\rangle /\partial U/2L^2$ and (b) the structure factor $C_{\mathrm{VBS}}\left(\mathbf{K}\right)/L^2$ of KVBS for $\lambda /t=0.2$ with $U/t$ crossing the quantum phase transition. The results for linear system sizes with $L=9,12,15,18$, and 21 are shown.

Figure 3

(a) The correlation ratio $R_{\mathrm{corr}}$ for the KVBS order across the phase transition and (b) finite-size scaling of structure factors of KVBS order with $U/t=3.28,3.29,$ and 3.30 for $\lambda /t=0.2$.

Figure 4

Histogram of the KVBS order parameter ${\mathrm{\Delta }}_{\mathbf{K}}$ for $\lambda /t=0.2$ across the phase transition with $L=12$. $U/t=3.275$, $U/t=3.30$, $U/t=3.32$, and $U/t=3.35$ (a–d).

Figure 5

(a) The spin Chern number $C_s$ and (b) positive eigenvalue ${\eta }_{\mathbf{K}}$ of the ${\mathbf{G}}_{\sigma }(i\omega =0,\mathbf{K})$ matrix across the TPT for $\lambda /t=0.2$. In panel (a) both results of direct QMC calculations in finite-size systems and application of the interpolation technique (see Ref. [24]) are shown.

Figure 6

(a) The single-particle gap ${\mathrm{\Delta }}_{sp}\left(\mathbf{K}\right)/t$ and (b) the spin gap ${\mathrm{\Delta }}_s\left(\mathit{\Gamma }\right)/t$ for $\lambda /t=0.05$ around the phase transition $U_c/t=2.09\left(1\right)$.

Figure 7

Finite-size scaling of the single-particle gap ${\mathrm{\Delta }}_{sp}\left(\mathbf{K}\right)/t$ and the spin gap ${\mathrm{\Delta }}_s\left(\mathit{\Gamma }\right)/t$ with the dip values of different system sizes in Fig. 6 for $\lambda /t=0.05$. A quadratic and cubic fitting are applied for ${\mathrm{\Delta }}_{sp}\left(\mathbf{K}\right)/t$ and ${\mathrm{\Delta }}_s\left(\mathit{\Gamma }\right)/t$, respectively.

Figure 8

Similar calculations choosing parameter $\lambda =0.05t$ compared with the main text. (a) Structure factor $R_B\left(U,L\right)/L^2$. (b) Finite-size scaling of structure factors of KVBS order with $L=9,12,15,$ and 18.

Figure 9

Histogram of KVBS order parameter ${\mathrm{\Delta }}_{\mathbf{K}}$ for $\lambda /t=0.05$ across the phase transition with $L=12$.

Figure 10

(a) The spin Chern number $C_s$ and (b) positive eigenvalue of the ${\mathbf{G}}_{\sigma }(i\omega =0,\mathbf{K})$ matrix across the TPT for $\lambda /t=0.05$. An interpolation technique is also used here.

Figure 11

(a) Single-particle gap and (b) spin gap versus $U/t$ near the phase transition point with $\lambda /t=0.2$.

Figure 12

Finite-size scaling of the single-particle gap ${\mathrm{\Delta }}_{sp}\left(\mathbf{K}\right)/t$ and the spin gap ${\mathrm{\Delta }}_s\left(\mathit{\Gamma }\right)/t$ with the dip values of different system sizes in Fig. 11 for $\lambda /t=0.2$. Quadratic fittings are applied.

Figure 13

The lattice structure and the Brillouin zone of the mean-field Hamiltonian. (a) Each site in an unit cell is marked with an integer, and the lattice vectors are given as ${\widehat{a}}_1$ and ${\widehat{a}}_2$. The bold lines stand for the electron hoppings that couple to the order parameter $m_{\mathrm{KVBS}}$. We also draw the three nearest unit cells $t_{{\mathbf{R}}_i,{\mathbf{R}}_j}$ which contribute the hopping matrices $T^{{\mathbf{R}}_i,{\mathbf{R}}_j}$ to the Bloch Hamiltonian $H_{\mathbf{k}}$. (b) The reciprocal lattice vectors ${\widehat{b}}_1$, ${\widehat{b}}_2$ are given corresponding to the lattice vectors ${\widehat{a}}_1$ and ${\widehat{a}}_2$, respectively. The first Brillouin zone is marked by the solid green line. By comparison, we also draw the reciprocal lattice vector and the Brillouin zone of Hamiltonian (1) with ${\widehat{B}}_1$, ${\widehat{B}}_2$ and the black dashed line, respectively.

Figure 14

Numerical results for Hamiltonian (B1) with $\lambda /t=0.05$. (a) Energy band in the first Brillouin zone at the gapless point ($\mathrm{\Gamma }$) with $m_{\mathrm{VBS}}/t\sim 0.26$. (b) Spin Chern number $C_s$ jumps from $+1$ to 0 before and after the gapless point with $L=60,120,240,$ and 600, where $L=L_x=L_y$ equals the number of unit cells along both the $x$ and $y$ directions.

Figure 15

(a) Imaginary-time single-particle Green's function and (b) spin-spin correlation function at the quantum phase transition point $U/t=2$ for $\lambda =0.05t$ with $L=9,12,15,18,$ and 21.