Loops, sign structures, and emergent Fermi statistics in three-dimensional quantum dimer models

We introduce and study three-dimensional quantum dimer models with positive resonance terms. We demonstrate that their ground state wave functions exhibit a nonlocal sign structure that can be exactly formulated in terms of loops, and as a direct consequence, monomer excitations obey Fermi statistics. The sign structure and Fermi statistics in these “signful” quantum dimer models can be naturally described by a parton construction, which becomes exact at the solvable point.

Figure 1

(a) Eight cubic unit cells of the CSO lattice. (b) The two types of length-4 loops on the CSO lattice. Left: Square loop, right: bent-square loop.

Figure 2

Arrow pattern for CSO lattice.

Figure 3

Two different ways to move a monomer. The first and second rows show two different ways to move a monomer, and the last row shows the transition graph obtained by compositing the two dimer coverings in the same column. The curvy segment shows the part of the transition loop through the rest of the lattice, which is identical before and after the monomer move.

Figure 4

(a) The two holes are exchanged using nearest-neighbor hops defined by numbered arrows. Possible hole positions are marked in italics. Dimers are flipped accordingly. (b) We move a single hole along the same path to find the flux, changing our reference tiling using dimer flips along bent-square loops when necessary. When such a dimer flip is performed, the resulting new dimer tiling is marked in red. In total, 4 bent square loops are used, giving a +1 sign.

Figure 5

Parton model. (a) One dimer degree of freedom is represented by two partons on the two ends of the dimer. (b) The partons live on a decorated CSO lattice in which every original CSO site is replaced by a eight-site cluster that represents the eight ways to draw a dimer from that site.

Figure 6

Overlapping boundary of two transition graphs. The black bonds belong to dimer covering $D_b$, green and orange bonds belong to dimer coverings $D_a$ and $D_c$, respectively, and the bond painted in both green and orange belongs to both $D_a$ and $D_c$.

Figure 7

Two loops with one overlapping boundary. The orientations of the two loops are chosen such that the boundaries are oriented in opposite directions on the two loops, and the merged loop can naturally inherit the orientations.

Figure 8

One loop with one overlapping boundary. The arrow shows the orientation of the loop. (a) A planar loop without twisting. (b) A nonplanar loop with a twist. The loop is the edge of a Möbius strip. (c) The result of merging of the loop in (b). Comparing to the orientation in (b), the direction of the bonds labeled by the dashed line are flipped.

Figure 9

(a) The two holes are exchanged on the square sublattice using next-nearest-neighbor hops, which requires only one square loop. (b) We move a single hole along the same path to find the background flux. Two square loops are needed, giving $\varphi$ = 0.

Figure 10

(a) A Levin-Wen exchange is performed on two holes. (b) We move a single hole along the same path to find that background flux $\varphi$ = 0.

Figure 11

(a) A Levin-Wen exchange is performed on two holes. Two reference changes occur after the second and third hops. (b) The sequence of hops on two holes is repeated, this time without exchange. Only a single reference change is necessary, after the first hop.

Figure 12

(a) Two holes are exchanged using two nearest-neighbor hops, and one next-nearest neighbor hop. Red arrows show corresponding dimer moves. (b) A single hole is moved along the same path, generating nonzero flux.