Resonating valence bond states with trimer motifs

A trimer resonating valence bond (tRVB) state consisting of an equal-weight superposition of trimer coverings on a square lattice is proposed. A model Hamiltonian of the Rokhsar-Kivelson type for which the tRVB becomes the exact ground state is written. The state is shown to have $9^g$ topological degeneracy on a genus $g$ surface and support ${\mathbb{Z}}_3$ vortex excitations. Correlation functions show exponential behavior with a very short correlation length consistent with the gapped spectrum. The classical problem of the degeneracy of trimer configurations is investigated by the transfer matrix method.

Figure 1

(a)–(j) Ten possible configurations of a local tensor to realize the trimer covering. Known trimer coverings consisting of linear and bent types only arise as special limits. The numbers 0, 1, and 2 denote the indices of the site tensor. (k) Red and blue bars represent all possible L- and B-type trimers, respectively.

Figure 2

(a) Assignment of the weight $\omega =e^{2\pi i/3}$ and its conjugate ${\omega }^{*}$ for the passage through the dual lattice with the center of the trimer (blue dot) on the right and left side of the path, respectively. (b) For any elementary loop surrounding a site, the total weight is always $\mathrm{\Gamma }=\omega$. For any loop surrounding a single trimer, the total weight is $\mathrm{\Gamma }={\omega }^3=1$.

Figure 3

(a) Staggered trimer configuration. A nonflippable six-trimer block inside the blue boundary can be connected to (b) a flippable configuration through six-trimer resonance moves.

Figure 4

Correlations between (a) L-type, (b) B-type, (c) L- and B-type, and (d) (LL)-trimers are measured on a $120\times 120$ lattice with the open boundary condition. Here, $R_{ij}$ is the distance between two trimers at the site $i$ and $j$, and the estimated correlation length $\xi$ is shown for each plot.

Figure 5

Schematic phase diagram of the quantum trimer Hamiltonian.