SU(3) trimer resonating-valence-bond state on the square lattice

We propose and study an SU(3) trimer resonating-valence-bond (tRVB) state with $C_{4v}$ point-group symmetry on the square lattice. By devising a projected entangled-pair state representation, we show that all (connected) correlation functions between local operators in this SU(3) tRVB state decay exponentially, indicating its gapped nature. We further calculate the modular $S$ and $T$ matrices by constructing all nine topological sectors on a torus and establish the existence of ${\mathbb{Z}}_3$ topological order in this SU(3) tRVB state.

Figure 1

(a) Schematics of the SU(3) trimers. (b) Typical trimer covering configuration on the square lattice. (c) Projected entangled-pair state representation of the SU(3) tRVB state. (d) Two kinds of projectors mapping virtual states to physical states.

Figure 2

(a) Each of the local tensor $\mathcal{A}$ of infinite PEPS is contracted with its complex conjugate by the physical index to form the bilayer tensor network. (b) Schematic of the contraction of bilayer tensor network based on boundary iMPS.

Figure 3

The spectrum of the transfer matrix (only the absolute values are shown). The eigenvalues are normalized such that the largest eigenvalue is 1. The inset defines the transfer matrix for a two-site unit cell and shows the upper bound of the correlation length $\xi$ versus the truncation bond dimension $\chi$.

Figure 4

The correlation functions (a) $|\langle {\lambda }^3\left(0\right){\lambda }^3\left(d\right)\rangle |$, (b) $|\langle {\lambda }^8\left(0\right){\lambda }^8\left(d\right)\rangle |$, (c) $C_1\left(d\right)=|\langle B_1\left(0\right)B_1\left(d\right)\rangle -\langle B_1\left(0\right)\rangle \langle B_1\left(d\right)\rangle |$, and (d) $C_2\left(d\right)=|\langle B_1\left(0\right)B_2\left(d\right)\rangle -\langle B_1\left(0\right)\rangle \langle B_2\left(d\right)\rangle |$ versus distance $d$, respectively. The black dashed lines correspond to the plot of the exponentially decaying function $\propto exp(-d/\xi )$ with correlation length $\xi =1.23$ estimated from the gap of the transfer matrix.

Figure 5

(a) ${\mathbb{Z}}_3$ gauge symmetry of the PEPS local tensor. (b) Constructing nine states by inserting gauge flux along two non-contractible loops on a torus. (c) and (d) A $3\times 1$ torus formed by double tensors and the ${\mathbb{Z}}_3$ gauge symmetry elements is used to compute modular $S$ and $T$ matrices.

Figure 6

Modular matrices from TRG. With increasing RG steps, the modular matrices converge to their fixed-point results, as seen from their traces.