Spin-liquid and magnetic phases in the anisotropic triangular lattice: The case of $\kappa \text{−}{\left(\text{ET}\right)}_2\text{X}$

The two-dimensional Hubbard model on the anisotropic triangular lattice, with two different hopping amplitudes $t$ and $t^{\prime }$, is relevant to describe the low-energy physics of $\kappa \text{−}{\left(\text{ET}\right)}_2\text{X}$, a family of organic salts. The ground-state properties of this model are studied by using Monte Carlo techniques, on the basis of a recent definition of backflow correlations for strongly correlated lattice systems. The results show that there is no magnetic order for reasonably large values of the electron-electron interaction $U$ and frustrating ratio $t^{\prime }/t=0.85$, suitable to describe the nonmagnetic compound with $\text{X}={\text{Cu}}_2{\left(\text{CN}\right)}_3$. On the contrary, Néel order takes place for weaker frustrations, i.e., $t^{\prime }/t\sim 0.4\text{–}0.6$, suitable for materials with $\text{X}={\text{Cu}}_2{\left(\text{SCN}\right)}_2$, $\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Cl}$, or $\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{Br}$.

Figure 1

(Color online) Illustration of the lattice in the square topology (a) used in this work and in the equivalent triangular one. (b) Solid and dashed lines indicate hopping amplitudes $t$ and $t^{\prime }$, respectively.

Figure 2

(Color online) Results for 18 electrons on 18 sites as a function of $U/t$. Upper panels: accuracy of energy $\Delta E=\left(E_0-E_v\right)$, $E_v$ and $E_0$ being the variational and the exact values, respectively. Lower panels: overlap between the exact ground state and the variational BCS wave functions. The state without (with) backflow correlations is denoted by diamonds (squares).

Figure 3

(Color online) Variational energies per site (in unit of $J=4t^2/U$) for the BCS state with a density Jastrow factor (diamonds) and for the AF wave function with both density and spin Jastrow terms (circles); the correlated Fermi gas with Jastrow factor is also reported for $t^{\prime }/t=0.85$ (squares). All states have backflow correlations and results are for 100 sites.

Figure 4

(Color online) Variational results for the density-density correlations $N_q$ divided by $\left|q\right|$, along the (1,0) direction, for 100 (red symbols) and 196 (black symbols) sites. Full (empty) symbols refer to the BCS (AF) wave function. Upper panel: from top to bottom, $U/t=4$, 5, 6, 8, and 10. Lower panel: from top to bottom, $U/t=6$, 7, 8, 10, and 20.

Figure 5

(Color online) Size scaling of the spin-spin correlations $S_Q/N$ for $Q=\left(\pi ,\pi \right)$. Data are for $t^{\prime }/t=0.6$ with $U/t=10$ (squares) and $U/t=20$ (circles), and $t^{\prime }/t=0.85$ with $U/t=10$ (triangles) and $U/t=20$ (diamonds). Variational and fixed-node results are denoted by empty and full symbols, respectively. The variational results do not depend substantially upon $U$ and $t^{\prime }$. The fixed-node results indicate long-range order for $t^{\prime }/t=0.6$ but not for $t^{\prime }/t=0.85$.

Figure 6

(Color online) Proposed phase diagram for the two discussed hopping ratios, $t^{\prime }/t=0.6$ and $t^{\prime }/t=0.85$. In the first case, variational (VMC) and fixed-node (FN) results indicate both a direct transition between a metal and an insulator with AF Néel order at a critical value of the electron-electron repulsion $U_c$. For $t^{\prime }/t=0.85$, the variational results predict the existence of three different phases at increasing $U/t$: a metal, an AF insulator with Néel order and a spin liquid, while, within the fixed-node approximation, the nonmagnetic ground state extends down to the metal-insulator transition.