Classical lattice models with single-node interactions on hierarchical lattices: The two-layer Ising model

A general approach is proposed for renormalization group transformations at arbitrary hierarchical lattices with two root nodes and the presence of single-node interactions (interactions between layers, magnetic field, chemical potential, etc.). The effectiveness of the proposed approach was shown for the two-layer Ising model in a zero magnetic field on the simplest representative of folded square hierarchical lattices. The phase diagram was investigated and the shift exponent (phi) was calculated at various values of the interaction energy in each layer (J(1), J(2)) and between the layers (J(3)). The value phi approximate to 2.41 was obtained for identical interactions in the layers (J(1) = J(2)). In the remaining cases (J(1) not equal J(2)) the shift exponent turned out to be close to 0.5, which is consistent with the data for the square lattice. The exceptional case is J(1) > 0, J(2) > 0, and J(1) not equal J(2), where the transition shift exponent in the second layer takes the value 2.57. (C) 2020 Elsevier B.V. All rights reserved.