Tentative Structural Features of a Gapped RVB State in the Anisotropic Triangular Lattice

The self-consistency equations for the independent order parameters as well as the free energy expression for the mean-field RVB model of the spin-1/2 Heisenberg Hamiltonian on the anisotropic triangular lattice is considered in the quasi-one-dimensional approximation. The solutions of the self-consistency equations in the zero-temperature limit are in fair agreement with the previous numerical analysis of the same model by other authors. In particular, the transition from the ungapped 1D-RVB state to the gapped 2D-RVB state occurs at an arbitrarily weak transversal exchange ($J_{2}\rightarrow0)$ although the amount of the gap is exponentially small: $\frac{12J_{1}}{\pi}\exp(-\frac{2J_{1}}{J_{2}})$, where $J_{1}$ is the longitudinal exchange parameter. The structural consequences of the formation of the 2D-RVB state are formulated by extending the famous bond order \emph{vs}. bond length relation known for polyenes (one-dimensional Hubbard chains). Analytical estimates of this effect are given.