Double-semion topological order from exactly solvable quantum dimer models

We construct a generalized quantum dimer model on two-dimensional nonbipartite lattices, including the triangular lattice, the star lattice, and the kagome lattice. At the Rokhsar-Kivelson (RK) point, we obtain its exact ground states that are shown to be a fully gapped quantum spin liquid with the double-semion topological order. The ground-state wave function of such a model at the RK point is a superposition of dimer configurations with a nonlocal sign structure determined by counting the number of loops in the transition graph. We explicitly demonstrate the double-semion topological order in the ground states by showing the semionic statistics of monomer excitations. We also discuss possible implications of such double-semion resonating valence bond states to candidate quantum spin-liquid systems discovered experimentally and numerically in the past few years.

Figure 1

Sign rule of dimer configurations. (a) The reference dimer configuration, which is a columnar VBS state. (b) An arbitrary dimer configuration. (c) The transition graph constructed by superposing the configurations in (a) and (b). In this transition graph, there are four loops, including two trivial length-two loops at the bottom-left and the bottom-right corners, and therefore the count $N_c=4$ in Eq. (6).

Figure 2

Different types of resonance terms and their effects on transition graphs. The shaded rhombus shows the location of the resonance term, and the thick lines in (b) and (c) are transition loops. (a) Two types of resonance terms. The red dimers show the reference dimer configuration as in Fig. 1. (b) Effect of a type-A resonance term. A type-A term reconnects two transition loops and changes the number of loops by one. (c) Effect of a type-B resonance term. A type-B resonance term reconnects the loop locally and does not change the number of loops.

Figure 3

A monomer string and a half-vison string. (a) The transition graph of a dimer configuration with two monomer excitations. Besides the transition loops, the graph also contains a string connecting two monomer sites, for which we coin the name “monomer string.” (b) A half-vison string connecting the two monomer sites. The half-vison string is represented by a dashed line. Both the monomer string and the half-vison string are oriented from site $i$ to $j$. The arrows on the dimers and the dashed line show the direction of monomer and half-vison strings. Note that the half-vison string runs through the dual lattice, so it starts and ends on an adjacent dual lattice site to the monomer excitations. According to the definition of the half-vison string, the two crossings at the left and the right give phase factors of $-i$ and $i$, respectively. This is explained in Appendix pp2.

Figure 4

Effects of different types of resonance terms on the defect string connecting two monomers. Similar to Fig. 2, the shaded area shows the location of the resonance term and the thick line shows the defect string that connects two monomer excitations. The dashed line shows the half-vison string. (a) A type-B resonance term changes the shape of the string locally without changing the topology of the string. (b) A type-A resonance term absorbs a nearby loop into the string. (c) A type-A resonance term reconnects the monomer string. In the left configuration, the half-vison string does not intersect the monomer string, and in the right configuration, the half-vison string intersects the monomer string twice with the same phase factors $i$, so the total phase is $-1$. Therefore, the half-vison string gives an extra phase factor $-1$ after the resonance.

Figure 5

Monomer excitations and fusion rules. The solid lines with arrows are the monomer strings, and dashed lines with arrows are the half-vison strings. The dashed line without an arrow is a vison string. (a) A monomer excitation with one outgoing monomer string and one outgoing half-vison string. (b) An antimonomer with one outgoing monomer string and one incoming half-vison string. (c) Two monomers are joined by one monomer string and one half-vison string running in parallel directions and fuse into vacuum. (d) Two antimonomers are joined by one monomer string and one half-vison string running in opposite directions and fuse into vacuum. (e) One monomer and one antimonomer are joined by one monomer string, but the two half-vison strings meet in the middle and fuse into a vison string. Hence one monomer and one antimonomer fuse into a vison.

Figure 6

Length-six resonance term.

Figure 7

Exchanging two monomers.

Figure 8

Moving one monomer along exactly the same path as in Fig. 7.

Figure 9

Fully symmetric dimer configuration on the star lattice.

Figure 10

General definition of a half-vison string operator. The solid lines represent loops and monomer strings in the transition graph, and the dashed line represents a half-vison string.