On Fourier and renormalization group transformations in the generalized hierarchical fermionic model

Abstract

On a two-dimensional generalized hierarchical lattice, the distance between opposite vertices of a unit cell square differs from the distance between adjacent vertices and is a new parameter of the model. At each lattice vertex, the field is defined by a set of four components that are generators of the Grassmann algebra. The Gaussian part of the model is determined by a quadratic Hamiltonian which is invariant under the renormalization group transformation. The non-Gaussian part of the model is defined by a Grassmann-valued “free measure density”, whose sets of coefficients are treated as points in a two-dimensional projective plane. The renormalization group transformation in the space of these coefficients is a homogeneous transformation of degree 4 in the projective space. The commutation relation between the Fourier transform in the space of “densities” and the renormalization group transformation is investigated.