Two-dimensional four-line models of the Ising model type

We briefly describe the theory of root transfer matrices for four-line models with the field in the new indexless form. We use theoretical and numerical methods to reveal new effects in the theory of singular points and phase transitions. A substantial part of the results is obtained using a numerical algorithm that drastically {(}at least exponentially{\rm)} reduces the computational complexity of Ising-type models by using the extremely sparse root transfer matrix.