Fermionic quantum dimer and fully-packed loop models on the square lattice

We consider fermionic fully packed loop and quantum dimer models which serve as effective low-energy models for strongly correlated fermions on a checkerboard lattice at half- and quarter-filling, respectively. We identify a large number of fluctuationless states specific to each case and which are due to fermionic statistics. We discuss the symmetries and conserved quantities of the system and show that, for a class of fluctuating states in the half-filling case, the fermionic sign problem can be gauged away. This claim is supported by a numerical evaluation of the low-lying states and can be understood by means of an algebraic construction. The elimination of the sign problem then allows us to analyze excitations at the Rokhsar-Kivelson point of the models using the relation to the height model and its excitations, within the single-mode approximation. We then discuss a mapping to a U(1) lattice gauge theory which relates the considered low-energy model to the compact quantum electrodynamics in $2+1$ dimensions. Furthermore, we point out consequences and open questions in the light of these results.

Figure 1

Panels (a) and (b) show a configuration of fermions satisfying the “two-per-tetrahedron” rule at half-filling and its representation in terms of FPLs. Panels (c) and (d) show configurations of fermions satisfying the “one-per-tetrahedron” rule at quarter-filling and their representations in terms of quantum dimers. In both cases, “flippable plaquettes” corresponding to two- and three-particle ring exchanges are shaded (see text for details).

Figure 2

Recipe of how to map dimer and FPL coverings to a height model. First the links of the bipartite square lattice are oriented in such a way that the arrows are always pointing from the black to the white sublattice. Next we start from one arbitrary plaquette and set its value to zero. We then set the height value of the other plaquettes in the following ways: (a) Dimer model: If the traversed link is empty, we change the height by $\Delta h=1$ if the arrow is from left to right and by $\Delta h=-1$ if the arrow is from right to left. If the traversed link is occupied by a dimer, we change the height by $\Delta h=-3$ if the arrow is from left to right and by $\Delta h=3$ if the arrow is from right to left. The dashed line shows one possible path assigning the heights to the plaquettes. (b) FPL model: The rules are nearly identical to those for dimers except that $\left|\Delta h\right|=1$ for both occupied and empty links. Since an FPL state is always an overlap of two dimer states, one can also obtain the same heights by simply adding those corresponding to the dimer states and dividing the result by two.

Figure 3

Panels (a) and (b) show the representation of the configuration as fully packed directed loops. (c) Action of a “non-sign-changing” process of the effective Hamiltonian. (d) Three different actions of “sign changing” processes on the topology of loop configurations.

Figure 4

(a) GS energy and energies of lowest excited states of the effective Hamiltonian in sectors with different slopes (${\kappa }_x$, $0$). Left panel: Data from an exact diagonalization of $H_{\text{eff}}$ on a 72-site cluster where the Fermi sign is taken into account. Right panel: Same data from a calculation with excluded Fermi signs. (b) Counter example of a fifth-order process which changes the sign of the wave function but not the topology of the covering loops.

Figure 5

Two distinct “plaquette flips” and corresponding colorings of the plane. The two moves are different by the relative “$-$” sign. The dashed diagonal lines represent either of the two equivalent (for the purpose of our argument) locations of a dimer along one of the adjacent sides of a square. The red crosses indicate the double plaquette around which the loop segments resonate.

Figure 6

Fragments of possible “frozen” states: (a,b) are fluctuationless under any local ring exchanges at quarter-filling (a) and half-filling (b); (c,d) are maximally flippable for bosonic, but fluctuationless for fermionic ring exchanges; (e,f) are the examples of generic fluctuationless configurations of fermions at both quarter-filling (e) and half-filling (f); (g,h) are other examples of fluctuationless configurations of fermions at half-filling.

Figure 7

Fragments of states stabilized by quantum fluctuations. (a) The state maximizing the number of three-fermion ring exchanges at quarter-filling. (b) The “squiggle” state that maximizes the number of three-fermion ring exchanges at half-filling. The flippable rectangles involving the central plaquette are shaded.

Figure 8

Structure factor $S_{\mathrm{x̂}\mathrm{x̂}}$ for the (a) FPL and (b) hard-core dimers on the square lattice.

Figure 9

Labeling of links on flippable double-plaquettes at position $\mathbf{x}$ with unit vectors ${\mathrm{ê}}_1$ and ${\mathrm{ê}}_2$.