Spin-$\frac{1}{2}$ Heisenberg model on the anisotropic triangular lattice: From magnetism to a one-dimensional spin liquid

We investigate the anisotropic triangular lattice that interpolates from decoupled one-dimensional chains to the isotropic triangular lattice and has been suggested to be relevant for various quasi-two-dimensional materials such as ${\text{Cs}}_2{\text{CuCl}}_4$ or $\kappa \text{−}{\left(ET\right)}_2{\text{Cu}}_2{\left(\text{CN}\right)}_3$. We obtain an excellent accuracy by means of a representation for the resonating valence-bond wave function with both singlet and triplet pairings. This approach allows us to establish that the magnetic order is rapidly destroyed away from the pure triangular lattice and incommensurate spin correlations are short range. A nonmagnetic spin liquid naturally emerges in a wide range of the phase diagram with strong one-dimensional character.

Figure 1

(Color online) Exact energy gap for the $6\times 6$ cluster as a function of the frustrating ratio $J^{\prime }/J$. Full and empty circles indicate singlet states with $q=\left(0,0\right)$ and different reflection symmetries (the data show a level crossing for $J^{\prime }/J\sim 0.825$). Empty and full triangles indicate triplet excitations at $q=\left(\pi ,\pi /3\sqrt{3}\right)$ and $q=\left(\pi ,\pi /\sqrt{3}\right)$, respectively. Full squares indicate triplet excitations at $q=\left(3\pi /4,0\right)$.

Figure 2

(Color online) Accuracy for the energy on the $6\times 6$ lattice, $E_0$ and $E$ denote the exact and the variational energies per site. The WF with decoupled chains (triangles) and the one with $2\times 1$ structure and $120°$ order (squares) are reported. The example of the $2\times 1$ structure for the sign of the nearest-neighbor sites is also reported: solid and dashed lines denote positive and negative pairing amplitudes, respectively. The two sites are denoted by full and empty circles.

Figure 3

(Color online) Energy per site for the $12\times 12$ (upper panel) and the $18\times 18$ (bottom panel) for the WF with three-sublattice magnetization and $2\times 1$ structure (squares) and with two decoupled chains (triangles). See text for a detailed description of the WFs.

Figure 4

(Color online) Static spin-spin correlations $S\left(q\right)$ for different lattice sizes $N=L\times L$ and frustrating ratios $J^{\prime }/J$. A darker color indicates a bigger $S\left(q\right)$.

Figure 5

(Color online) Lower panel: size scaling of the magnetic order parameter for different values of $J^{\prime }/J$. Stars indicate exact results for $N=36$ and lines are fits. Upper panel: position of the peak $q=\left(Q,0\right)$ for $J^{\prime }/J=0.8$.

Figure 6

(Color online) Phase diagram of the anisotropic triangular lattice as obtained by our variational approach.