${\mathbb{Z}}_2$ topological liquid of hard-core bosons on a kagome lattice at $1/3$ filling

We consider hard-core bosons on the kagome lattice in the presence of short-range repulsive interactions and focus particularly on the filling factor $1/3$. In the strongly interacting limit, the low-energy excitations can be described by the quantum fully packed loop coverings on the triangular lattice. Using a combination of tensor product state based methods and exact diagonalization techniques, we show that the system has an extended ${\mathbb{Z}}_2$ topological liquid phase as well as a latti ce nematic phase. The latter breaks lattice rotational symmetry. By tuning appropriate parameters in the model, we study the quantum phase transition between the topological and the symmetry broken phases. We construct the critical theory for this transition using a mapping to an Ising gauge theory that predicts the transition to belong to the $O\left(3\right)$ universality class.

Figure 1

(a) Mapping from the $1/3$ filled bosonic problem on kagome to FPL on the triangular lattice. Each particle on the kagome lattice is mapped to a loop segment on the triangular lattice that is obtained by connecting the centers of the kagome hexagons. The strength of the 1st, 2nd, and 3rd repulsive interactions are denoted as $V_1$, $V_2$, and $V_3$, respectively. The limit of $V_1=V_2=V_3=2V$ is the focus of this paper. Panel (b) demonstrates the allowed lowest-order processes in $t/V$ (see the text). (c) Shown is the sublattice used for the gauge transformation that changes the sign of $g$ in Eq. (5) as described in the text. The sublattice is constructed by the $b$ and $c$ sites of the shaded unit cells.

Figure 2

The triangular lattice is put on a torus by setting periodic boundary conditions along two independent directions given by the lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. $A_0$ denotes the intersection point (a link on the lattice) of the two noncontractible loops on the torus. There are four topological sectors characterized by the doublets ($0,0$), ($0,1$), ($1,0$), and ($1,1$) [19, 46]. The local Hamiltonian in Eq. (7) cannot mix configurations from different topological sectors, hence block diagonal in the full configuration basis.

Figure 3

The phase diagram of the dimer models at different fractional fillings $f$. The important details about each of them are given in the text. The filling fraction $1/6$ corresponds to a triangular lattice QDM [19, 46] while the half-filled case maps to an easy-axis kagome antiferromagnet or a 3-dimer model [16, 17]. The focus of the present paper is at $f=1/3$.

Figure 4

The plot shows the finite-size dependence of the topological entanglement entropy $\gamma$ as a function of $L$, the perimeter of the subsystem $A$. The green dashed line corresponds to the saturated value of $ln2$ in the thermodynamic limit. The lattice is constructed on an infinite cylinder assuming periodic boundary condition in one direction. The black dashed line indicates the bipartition of the cylinder separating the subsystem $A$ from the rest $B$. The inset shows the linear growth of the entanglement entropy $S$ with $L$.

Figure 5

(a) The real-space dimer-dimer correlation $C_{ij}$ is plotted on a triangular lattice cluster at $V_{\text{RK}}=-3.0$ which indicates the parallel loop pattern deep into the LN phase. (b) Same is plotted for $V_{\text{RK}}=1.0$ showing the exponential decay of the dimer-dimer correlation function in the liquid phase at the RK point. The reference link, with respect to which correlations at all the other links are measured, is denoted by $R_0$. Red and blue links stand for positive and negative correlation, respectively, while the width measures the correlation strength. (c) Plotted is the same correlator as a function of $|r_i-r_j|$ at different RK potential. The red dashed line represents an approximate exponential fit $e^{-r/\xi }$ for $\xi \sim 1.0$.

Figure 6

In the left panel, the gap ($\mathrm{\Delta }$) to the first excited state tends to vanish at $V_{\text{RK}}\sim -0.3$ for different symmetric and stripped clusters. The ground state always lies in the zero-flux sector $(0,0)$ for all of them. In the right panel, the derivative of the gap is plotted as a function of $V_{\text{RK}}$ to locate the transition point at $V_{\text{RK}}\sim -0.3$.

Figure 7

The ground state susceptibility $-{\delta }^2{E}_g/\delta V_{\text{RK}}^2$ in the $(0,0)$ sector indicates a continuous phase transition between two phases for different clusters. The transition point is close to $V_{\text{RK}}\sim -0.3$ as evident from the right panel of Fig. 6.

Figure 8

The two-vison correlator $v_{IJ}$ is plotted as a function of $|r_I-r_J|$ at different values of $V_{\text{RK}}$ for a symmetric cluster of $L=4$. The red dashed line represents an exponential decay: $e^{-|r_I-r_J|/\xi }$ for $\xi \sim 0.7$.

Figure 9

The medial honeycomb lattice is shown along with the original kagome lattice. Removal of the blue-colored boson creates two spinon excitations sitting at the centers of the adjacent shaded hexagons (the “defect hexagons”) which violate the constraint of having two bosons. By virtue of the single-particle hopping term, these two defects can separate farther apart; however their creation is energetically suppressed in the strong-coupling limit.

Figure 10

(a) A flippable bow tie. (b) The first Brillouin zone of the honeycomb lattice showing $M$ points. (c) Loop ordering at $M_1$ point. A loop segment is placed on the link of the triangular lattice whenever it crosses a honeycomb bond joining antiparallel spins which represent the vison degrees of freedom. (d) The yellow region qualitatively shows the allowed ranges of $t_1$ and $t_2$ for which the minima of $J\left(k\right)$ lie in the vicinity of the $M$ points within an energy window of ${10}^{-3}$. The vertical phase boundary indicates the case when the minima are exactly on the $M$ points corresponding to the LN phase for $V_{\text{RK}}\ll 0$.