Generalized quantum dimer model applied to the frustrated Heisenberg model on the square lattice: Emergence of a mixed columnar-plaquette phase

Aiming to describe frustrated quantum magnets with nonmagnetic singlet ground states, we have extended the Rokhsar-Kivelson (RK) loop expansion to derive a generalized quantum dimer model containing only connected terms up to arbitrary order. For the square-lattice frustrated Heisenberg antiferromagnet ($J_1\text{−}{J}_2\text{−}{J}_3$ model), an expansion up to eighth order shows that the leading correction to the original RK model comes from dimer moves on length-6 loops. This model free of the original sign problem is treated by advanced numerical techniques. The results suggest that a rotationally anisotropic plaquette phase [A. Ralko, D. Poilblanc, and R. Moessner, Phys. Rev. Lett. 100, 037201 (2008)] is the ground state of the Heisenberg model in the parameter region of maximum frustration.

Figure 1

(Color online). VBC states considered in this work. Generalized anisotropic VBC states labeled by ${\text{CP}}_a$ (mixed columnar-plaquette phases) and $T$ (trimerized phase) interpolate between the most symmetric limits shown in the figure.

Figure 2

(Color online). Typical ED low-energy spectra on a $8\times 8$ cluster (the GS energy is set to zero) for (a) $\theta =-0.3\pi$ and (b) $\theta ={\text{tan}}^{-1}\left(0.5\right)$ as a function of $\varphi$. Levels of special symmetries (see text) are highlighted as colored symbols, from bottom to top: $\left(\Gamma ,A_1\right)$ (corresponding to the GS), $\left(M,A_1\right)$, $\left(\Gamma ,B_1\right)$, $\left(K,A_1\right)$, and ${\left(M,A_1\right)}^{\ast }$. The arrow indicates the level crossing that marks the limit of the region where the latter levels correspond to the lowest excitations. Inset: Brillouin zone and its high-symmetry points $\Gamma =\left(0,0\right)$, $M=\left(\pi ,0\right)$, and $K=\left(\pi ,\pi \right)$.

Figure 3

(Color online). Extrapolations in the thermodynamic limit of the order parameters defined in the text characterizing the mixed ${\text{CP}}_1$ columnar-plaquette phase. Insets: finite-size scaling of both $M_{+}\left(\pi ,\pi \right)$ and $M_{-}\left(0,0\right)$ as a function of the inverse linear cluster size, using 36 sites $\left(6\times 6\right)$, 64 sites $\left(8\times 8\right)$, 100 sites $\left(10\times 10\right)$, 144 sites $\left(12\times 12\right)$, and 196 sites $\left(14\times 14\right)$ square clusters. The chosen value of $\theta$ corresponds to the case of the $J_1\text{−}{J}_2\text{−}{J}_3$ model studied in this work.

Figure 4

(Color online). Phase diagram in the $\left(\theta ,\varphi \right)$ plane obtained from numerical simulations. The previous knowledge of the $\varphi =0$ case (Ref. 10) has been used, in particular, to estimate the limit of the columnar phase as an approximate (black) straight line. The boundary of the pure plaquette (${\text{CP}}_1$ mixed) VBC phase obtained from the finite-size scaling of the associated order parameter is indicated by large circles (triangles). The (red) thick segment corresponds to the parameter region of the frustrated quantum antiferromagnet with $J_2+J_3=J_1/2$ according to the mapping described in the text. Crude ED estimates of the boundaries of the columnar phase and of the ${\text{CP}}_1$ phase are indicated by a thin dashed line and by small (blue) circles, respectively. The (light blue) region marked by a question mark has not been identified.

Figure 5

(Color online) (a) Variational wave functions (associated to their respective VBC states) considered in this appendix. Generalized anisotropic VWF labeled by $|{\mathrm{CP}}_a⟩$ and $|T⟩$ interpolate between the most symmetric limits shown in the figure. (b) Variational phase diagram as a function of $\theta$ ($x$ axis) and $\varphi$ ($y$ axis). The topographic map of $\alpha$ is depicted by the continuous gray lines. The line joining the Columnar phase and the trimerized one means that these regions are continuously connected in terms of $\alpha$. The thick red line shows the region of parameters of the effective model that maps to the $J_1\text{−}{J}_2\text{−}{J}_3$ model along the $J_2+J_3=J_1/2$ line for which the ground state is well described by singlet coverings.

Figure 6

(Color online). Overlap $⟨\phi |\psi ⟩$ between two VB configurations $\phi$ ad $\psi$. Closed loops that appear in the overlap diagram $g$ are represented as yellow shades.

Figure 7

(Color online). The matrix element $⟨\phi |\left(4/3\right){\mathbf{S}}_i.{\mathbf{S}}_j|\psi ⟩$ expressed as $f_{ij}⟨\phi |\psi ⟩$. Bond $\left(i,j\right)$ is represented as a solid black line. Red and blue bonds represent $|\phi ⟩$ and $|\psi ⟩$. $f_{ij}=+1$ (respectively, $f_{ij}=-1$) if $i$ and $j$ belong to the same loop at even (respectively odd) distance along the loop. $f_{ij}=0$ if $\left(i,j\right)$ connects two sites on distinct loops (see Ref. 16).