Frustrated Resonating Valence Bond States in Two Dimensions: Classification and Short-Range Correlations

Resonating valence bond (RVB) states are of crucial importance in our intuitive understanding of quantum spin liquids in 2D. We systematically classify short-range bosonic RVB states into symmetric or nematic spin liquids by examining their flux patterns. We further map short-range bosonic RVB states into projected BCS wave functions, on which we perform large-scale Monte Carlo simulations without the minus sign problem. Our results clearly show that both spin and dimer correlations decay exponentially in all the short-range frustrated (nonbipartite or $Z_2$) bosonic RVB states we studied, indicating that they are gapped $Z_2$ quantum spin liquids. Generically, we conjecture that all short-range frustrated bosonic RVB states in 2D have only short-range correlations.

Figure 1

(a) The flux patterns $\{{\varphi }_p\}$ of the only four symmetric NN RVB states on the kagome lattice. Here, ${\mathbf{e}}_x$ and ${\mathbf{e}}_y$ represent the unit vectors. (b) The flux patterns $\{{\varphi }_p^f\}$ in the corresponding projected BCS states.

Figure 2

The flux patterns $\{{\varphi }_p\}$ of the two symmetric NN RVB states on the triangular lattice.

Figure 3

The flux patterns $\{{\varphi }_p\}$ of (a) the two symmetric NN RVB states and (b) the four symmetric NNN RVB states on the square lattice.

Figure 4

The spin and dimer correlations as a function of distance in the NN RVB state (A) shown in Fig. 1.

Figure 5

The flux patterns $\{{\varphi }_p\}$ of (a) the four nematic NN RVB states on the kagome lattice, (b) the two nematic NN RVB states on the triangular lattice, and (c) the six nematic NNN RVB states on the square lattice.