Fermionic Hierarchical Model Research Articles

Zone Structure of the Renormalization Group Flow in a Fermionic Hierarchical Model

The Gaussian part of the Hamiltonian of the four-component fermion model on a hierarchical lattice is invariant under the block-spin transformation of the renormalization group with a given degree of normalization (the renormalization group parameter). We describe the renormalization group transformation in the space of coefficients defining the Grassmann-valued density of a free measure as a homogeneous quadratic map. We interpret this space as a two-dimensional projective space and visualize it as a disk. If the renormalization group parameter is greater than the lattice dimension, then the unique attractive fixed point of the renormalization group is given by the density of the Grassmann delta function. This fixed point has two different (left and right) invariant neighborhoods. Based on this, we classify the points of the projective plane according to how they tend to the attracting point (on the left or right) under iterations of the map. We discuss the zone structure of the obtained regions and show that the global flow of the renormalization group is described simply in terms of this zone structure.

Kondo Effect in a Fermionic Hierarchical Model

In this paper, a fermionic hierarchical model is defined, inspired by the Kondo model, which describes a 1-dimensional lattice gas of spin-1/2 electrons interacting with a spin-1/2 impurity. This model is proved to be exactly solvable, and is shown to exhibit a Kondo effect, i.e. that, if the interaction between the impurity and the electrons is antiferromagnetic, then the magnetic susceptibility of the impurity is finite in the 0-temperature limit, whereas it diverges if the interaction is ferromagnetic. Such an effect is therefore inherently non-perturbative. This difficulty is overcome by using the exact solvability of the model, which follows both from its fermionic and hierarchical nature.

Dynamics of renormalization group in the lower half-plane of the coupling constants of the fermionic hierarchical model

We consider four-component fermionic (Grassmann-valued) field on the hierarchical lattice. The Gaussian part of the Hamiltonian in the model is invariant under the block-spin renormalization group transformation with given degree of normalization factor (renormalization group parameter). The non-Gaussian part of the Hamiltonian is given by the sum of the selfinteraction forms of the 2-nd and 4-th order. The action of the renormalization group transformation in this model is reduced to the rational map in the plane of coupling constants. We investigate the global dynamics of this map in the case when the coupling constant of the 4-th order form is less than zero (lower half-plane) and the renormalization group parameter belongs to the interval [1, 3/2).

Properties of renormalization group flow in the neighborhood of Gaussian fixed point

We investigate the dynamics of the renormalization group transformation in the fermionic hierarchical model, given by the Lagrangian. We construct the invariant neighborhood of the Gaussian fixed point in the upper half-plane g > 0. We describe the subsets of the points of this neighborhood tending to the Gaussian fixed point under the iterations of the renormalization group transformation from the left and from the right.

An algorithm for studying the renormalization group dynamics in the projective space

The renormalization group dynamics is studied in the four-component fermionic hierarchical model in the space of coefficients that determine the Grassmann-valued density of the free measure. This space is treated as a two-dimensional projective space. If the renormalization group parameter is greater than 1, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann δ-function. Two different invariant neighborhoods of this fixed point are described, and an algorithm is constructed that allows one to classify the points on the plane according to the way they tend to the fixed point.

Renormalization group in a fermionic hierarchical model in projective coordinates

We study the renormalization group action in a fermionic hierarchical model in the space of coefficients determining the Grassmann-valued density of the free measure. This space is interpreted as the two-dimensional projective space. The renormalization group map is a homogeneous quadratic map and has a special geometric property that allows describing invariant sets and the global dynamics in the whole space.

Symmetries and cycles of the renormalization group in a fermionic hierarchical model

We study singular points and symmetries of the renormalization group mapping in a fermionic hierarchical model. This mapping taken at the renormalization group singular point and with the renormalization group parameter α=1 generates cycles of arbitrary lengths.

Continuum limit in the fermionic hierarchical model

We discuss the problem of rigorously constructing the continuum limit in the fermionic hierarchical model. The continuum limit constructed as the limit of fields on the refined hierarchical lattices is a field on a p-adic continuum. We investigate the problem of reconstructing the coupling constants of the continuum model from the coupling constants of the discretized model.

Critical phenomena in the fermionic hierarchical model

The dynamics of the renormalization-group transformation in the coupling-constant space of the fermionic hierarchical model are discussed. The critical behavior of this model is described in terms of the complex behavior of the Grassmann-valued mean-spin distribution density with the proper normalization. Some critical indices are calculated.

RG-invariant curves in the fermionic hierarchical model

In the fermionic hierarchical model, we investigate the renormalization group transformation given by a rational mapping in the coupling constant plane. We construct sections of stable invariant curves for all fixed points of this transformation. These curves are graphs of smooth functions, for which we give the upper and the lower estimates.