Importance of anisotropy in the spin-liquid candidate Me${}_3$EtSb[Pd(dmit)${}_2$]${}_2$

Organic charge-transfer salts based on the molecule Pd(dmit)${}_2$ display strong electronic correlations and geometrical frustration, leading to spin-liquid, valence bond solid, and superconducting states, among other interesting phases. The low-energy electronic degrees of freedom of these materials are often described by a single band model: a triangular lattice with a molecular orbital representing a Pd(dmit)${}_2$ dimer on each site. We use ab initio electronic structure calculations to construct and parametrize low-energy effective model Hamiltonians for a class of Me${}_{4-n}$ Et${}_{n}X$[Pd(dmit)${}_2$]${}_2$ ($X=$ As, P, N, Sb) salts and investigate how best to model these systems by using variational Monte Carlo simulations. Our findings suggest that the prevailing model of these systems as a $t-t^{\prime }$ triangular lattice is incomplete and that a fully anisotropic triangular lattice description produces importantly different results, including a significant lowering of the critical $U$ of the spin-liquid phase.

Figure 1

(a) Structure of the spin-liquid candidate Me${}_3$EtSb[Pd(dmit)${}_2$]${}_2$ (Sb-1). (b) Dimerization of the Pd(dmit)${}_2$ molecules, where dimers form 2D triangular lattice layers in the $a$-$b$ plane, separated along $c$ by cation layers (in this case, Me${}_3$EtSb). We follow Ref. 21 in labeling the hopping integrals in the $a$-$b$ plane $t_a$, $t_b$, and $t_c$.

Figure 2

Band structure and density of states of Sb-1 in a wide energy window around the Fermi energy. The four pairs of bands highlighted in green are those identified as being of Pd(dmit)${}_2$ HOMO or LUMO origin. The pair that crosses the Fermi energy is of HOMO-ab origin. These are the bands used in a minimal one-orbital model of the system. The direct energy gap between the HOMO-ab bands and the others is more than 0.1 eV, although the indirect gap is smaller.

Figure 3

Schematic of the hybridization of the orbital of the two Pd(dmit)${}_2$ in each dimer and the resultant crossing of energy levels leading to the unusually ordered frontier bands of Pd(dmit)${}_2$ complexes.

Figure 4

Left panel: Wannier orbital for the HOMO-ab bands of Sb-1. The antibonding HOMO character of the orbital is clearly visible. Right panel: The 2D tight-binding lattice generated in the $a$-$b$ plane from the Wannier orbital of the left panel. Each gray point of the lattice represents a dimer. The widths of the various cylinders in the lattice are linearly proportional to the magnitude of the corresponding $t$ (see Table ).

Figure 5

Upper panel: Variational energies per site for the FATL model with the parameters for Sb-1, by using the paramagnetic state (red squares), $|{\Psi }_{\text{BCS}}\rangle$, and the magnetic spiral state (blue circles), $|{\Psi }_{\text{SP}}\rangle$, as a function of $U/t_a$ and in units of $J=4t_a^2/U$. Both wave functions were computed on a $L=324$ lattice, for which the optimal pitch angles ($\theta =7\pi /9,{\theta }^{\prime }=3\pi /9$) are commensurate. Lower panel: $N\left(q\right)/q$ at $U/t_a=6.4$ by using $|{\Psi }_{\text{BCS}}\rangle$ (empty triangles) and $|{\Psi }_{\text{SP}}\rangle$ (empty diamonds) and at $U/t_a=7.1$ by using $|{\Psi }_{\text{BCS}}\rangle$ (full triangles) and $|{\Psi }_{\text{SP}}\rangle$ (full diamonds). Data show the metal to insulator transition in $|{\Psi }_{\text{BCS}}\rangle$, while the magnetic state is always insulating. Plots are presented along the line connecting the point $Q=(\pi ,\pi /\sqrt{3})$ to the point $\Gamma =(0,0)$ in reciprocal space.

Figure 6

Phase diagram vs $U/t_{\max }$, where $t_{\max }$ is the biggest hopping parameter in the model, for the isotropic, $t-t^{\prime }$, and FATL models for Sb-1, showing the metallic (green), spiral ordered magnetic insulator (orange), and spin-liquid (cyan) phases. The spin-liquid phase is strongly enhanced by including the full anisotropy of the system, while the metal-insulator transition is only slightly changed.