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

Abstract 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 ( φ ) was calculated at various values of the interaction energy in each layer ( J 1 , J 2) and between the layers ( J 3 ). The value φ ≈ 2.41 was obtained for identical interactions in the layers ( J 1 = J 2). In the remaining cases ( J 1 ≠ 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 ≠ J 2, where the transition shift exponent in the second layer takes the value 2.57.