The effective half-filled band model is inappropriate for the dimerized 2D organic superconductors

The antiferromagnetism in $\kappa$-(ET)$_{2}$X can be understood within the effective $\frac{1}{2}$-filled band anisotropic triangular lattice Hubbard Hamiltonian for strong anisotropy. DMFT theories have claimed antiferromagnetic-to-superconductor transition within the same model, as the anistropy is reduced. In previous work we have shown the absence of superconductivity within the triangular lattice $\frac{1}{2}$-filled band Hubbard model for any Hubbard $U$ and any anisotropy. Other DMFT approaches theories have claimed superconductivity within the so-called Hubbard-Heisenberg model, which incorporates an additional antiferromagnetic spin-exchange over and above that due to the Hubbard $U$. Very recent work has also claimed a valence-bond solid (VBS) phase within the Hubbard-Heisenberg model, which would seemingly explain the observed VBS phase in EtMe$_{3}$P[Pd(dmit)$_{2}$]$_{2}$. We report exact calculations that show that neither the VBS nor the superconducting phase occur within the Hubbard-Heisenberg model, showing clearly that the effective $\frac{1}{2}$-filled band model is unsuitable for describing the complete phase space of the $\kappa$-(ET)$_{2}$X. Our work raises serious doubts about the DMFT theories of superconductivity of metal intercalated C$_{60}$ and picene.