Inclusion of higher order anharmonic contributions in self-consistent phonon theory

The ansatz method of infinite summation of higher order diagrams given in Shukla and Cowley, Phys. Rev. B 58, 2596 (1998), is extended to the self-consistent phonon theory. We demonstrate the high accuracy of this approach with respect to the first-order self-consistent and improved self-consistent (ISC) phonon theories, by comparing the results from the ansatz method with their exact counterparts. The ISC theory is then extended to include the remaining diagrams of $O\left({\lambda }^4\right),$ which could not be included in its earlier formulation. This makes the ISC theory consistent, at least to $O\left({\lambda }^4\right).\mathrm{}$ This ISC theory offers a substantial improvement over the current ISC theory. The results of the equation of state for a face centered cubic nearest neighbor interaction Lennard-Jones solid from our ISC theory are shown to be in excellent agreement with the results of the classical Monte Carlo method also obtained for the same model.