Quantum dimer model on the triangular lattice: Semiclassical and variational approaches to vison dispersion and condensation

After reviewing the concept of vison excitations in ${\mathbb{Z}}_2$ dimer liquids, we study the liquid-crystal transition of the quantum dimer model on the triangular lattice by means of a semiclassical spin-wave approximation to the dispersion of visons in the context of a “soft-dimer” version of the model. This approach captures some important qualitative features of the transition: continuous nature of the transition, linear dispersion at the critical point, and $\sqrt{12}\times \sqrt{12}$ symmetry-breaking pattern. In a second part, we present a variational calculation of the vison dispersion relation at the Rokhsar-Kivelson (RK) point, which reproduces the qualitative shape of the dispersion relation and the order of magnitude of the gap. This approach provides a simple but reliable approximation of the vison wave functions at the RK point.

Figure 1

(Color online) Some useful definitions of sets of bonds: (1) $l\left(i\right)$ is the set of the three bonds forming the edges of the plaquette $i$. (2) $l\left[a\right]$ is the set of the six (fat) bonds emanating from the triangular site $a$. (3) $l<i$ denotes the bonds forming a “zigzag” string extending (to the left) from the triangular plaquette $i$ to an edge of the lattice.

Figure 2

(Color online) Dashed lines: Horizontal paths used in Eq. (6) connecting the boundary of the system to triangles $i$ and $j$. Dotted segment: Path ${\Omega }_{ij}$. In this example, ${\Omega }_{ij}$ crosses one dimer of the reference configuration $c_0$ (ellipses) and, therefore, ${ϵ}_{ij}=-1$ [Eq. (47)].

Figure 3

(Color online) The fat (green) bonds of the hexagonal lattice (crossed by a dimer of the reference configuration on the triangular lattice) have $M_{ij}=-1$, the other bonds have $M_{ij}=1$. The four sites of the unit cell (small blue triangles) are labeled from 1 to 4, and the Bravais vectors are $x$ and $y$.

Figure 4

(Color online) Plot of the rescaled relative dimer density $D_{ij}$ in one of the 48 classical ground states of the FFIM in transverse field, for a transverse field $\Gamma$ just below ${\Gamma }_c$. Thin red bonds represent $D_{ij}=0$ (corresponding to the highest dimer density $d_{ij}=0.5$). Blue bonds are for $D_{ij}<0$, that is, a lower dimer density, with a width proportional to $\mid D_{ij}\mid$. $D_{ij}$ takes only four different values: 0, $-0.259$, $-0.518$, and $-0.776$.

Figure 5

(Color online) Spin-wave dispersion relation $S{ϵ}_k∕J$ [Eq. (34)] for the path $\mathrm{A}\to \mathrm{B}\to \mathrm{C}\to \mathrm{A}$ in the Brillouin zone (see Fig. 6) and different values of $\Gamma ∕J$: 3 (top), 2.75, 2.5, and $\sqrt{6}\simeq 2.448$ (bottom). At the critical point $\Gamma ∕J=\sqrt{6}$, the spectrum is linear around $k=(\pi ∕6,\pi ∕2)$ and $k=(5\pi ∕6,\pi ∕2)$.

Figure 6

(Color online) Dashed rectangle: Brillouin zone ($-\pi ⩽k_x⩽\pi$ and $-\pi ⩽k_y⩽\pi$) of the hexagonal lattice shown in Fig. 3. A, B, and C are the high-symmetry points used in Fig. 5. The large hexagon is the Brillouin zone of the underlying triangular lattice.

Figure 7

(Color online) Overlap $\mid ⟨i\mid j⟩\mid$ between two point-vison states [defined in Eq. (39a)] as a function the distance $d_{ij}$ between $i$ and $j$. The (blue) squares correspond to Eq. (43) and the three (red) crosses correspond to Eq. (59) with $\alpha =-0.8$. The calculations are done using an exact evaluation of the Pfaffians on a 28-site lattice with periodic boundary conditions. Dashed line: guide for the eye corresponding to an exponential decay with the dimer-dimer correlation length ${\xi }^{-1}=0.76$ (Ref. 5).

Figure 8

(Color online) Spectrum of the overlap matrix $S^{\alpha }\left(k\right)$ [Eqs. (43, 59)], for different values of $\alpha$ (0, $-0.2$, $-0.5$, and $-0.8$), along the path $\mathrm{A}\to \mathrm{B}\to \mathrm{C}\to \mathrm{A}$ in the Brillouin zone (see Fig. 6). The bottom panel is a zoom on the lowest eigenvalues of $S^{\alpha }$. In these calculations, the matrix elements $S_{ij}^{\alpha }$ are neglected for triangles $i$ and $j$ at distance $d⩾d_{\max }=10$ (102th neighbor on the hexagonal lattice). Up to this distance, the finite-size lattice used in the calculation ($28\times 28$ sites) gives practically the infinite volume limit for $S_{ij}^{\alpha }$. The value of $d_{\max }$ used here is large enough to ensure a good convergence of the spectrum since the curves obtained with a smaller truncation distance ($d_{\max }\simeq 8.47$—shown here with the same colors) are almost superposed with that for $d_{\max }=10$, except for the bottom of the spectrum at $\alpha =-0.5$ and $\alpha =-0.8$. The overlap spectrum turns out to be gapped for $\alpha =0$ and $\alpha =-0.2$ (and probably at $\alpha =-0.5$ and $\alpha =-0.8$ too), indicating the linear independence of the vison states.

Figure 9

(Color online) Variational vison [Eq. (58)] dispersion relation for different values of $\alpha$, along the path $\mathrm{A}\to \mathrm{B}\to \mathrm{C}\to \mathrm{A}$ in the Brillouin zone (see Fig. 6). The curve for $\alpha =0$ corresponds to the variational energies of point-vison states [Eq. (39a)]. The absolute minimum is $E_{\min }=0.119$, found at $(k_x,k_y)=(\pi ∕6,\pi ∕2)=B$ for a value $\simeq -0.8$ of the variational parameter $\alpha$. The exact value of the gap (0.089) (Ref. 6) is marked as a dotted horizontal line. In this calculation, the matrix elements $S_{ij}^{\alpha }$ are neglected for triangles $i$ and $j$ at distance $d⩾d_{\max }=10$. This value is large enough to ensure a perfect convergence of the spectrum below $E\lesssim 0.6$, as can be checked from the fact that the dispersion curves obtained with a smaller truncation distance ($d_{\max }\simeq 8.47$—shown here with the same colors) are practically identical. Some higher energy states, however, are not fully converged.

Figure 10

The three types of six-site plaquettes around which dimer shifts are generated at fourth order in $\Gamma ∕J$.