Variational method for lattice systems: General formalism and application to the two-dimensional Ising model in an external field

A simple variational approach to the eigenvalue problem of the transfer operator is proposed. After reducing the transfer operator according to the symmetries of the Hamiltonian, the leading eigenvalues of the irreducible blocks can be evaluated by elementary variational principles. Hence the thermodynamics and a large class of correlation functions of lattice systems can be calculated. Following a natural truncation scheme the results can be improved in a systematic way. The high accuracy and the convergence of the method is demonstrated by two-dimensional Ising model. As a first application, the thermodynamics of the two-dimensional Ising ferro-and antiferromagnet in an external field is studied. We show how the same method can be used to obtain zero-temperature properties of interacting quantum lattice systems.