

New Trends in Mathematical and Theoretical Physics
October 3, 2016 17:00–17:30, Moscow, MIAN, Gubkina, 8






Renormalization group dynamics in the plane of the coupling constants of the fermionic hierarchical model
Mukadas Missarov^{}, A. F. Shamsutdinov^{} ^{} Kazan Federal University

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Abstract:
We consider fourcomponent fermionic (Grassmannvalued) field on the hierarchical lattice. The Gaussian part of the Hamiltonian in the model is invariant under the blockspin renormalization group transformation with given degree of normalization factor ( renormalization group parameter). The nonGaussian part of the Hamiltonian is given by the selfinteraction forms of the 2nd and 4th order with coupling constants $r$ and $g$. The action of the renormalization group transformation in the model is reduced to the rational map in the plane of coupling constants $(r,g)$. The upper halfplane $\{(r,g): g > 0\}$ and the lower halfplane are invariant under the renormalization group transformation We investigate the dynamics of this map in the lower and upper halfplanes for different values of renormalization group parameter.
To describe the global picture of the renormalization group flow we use also the space of the coefficients of the expansion of free measure density wich is denoted as the cspace. The renormalization group action in cspace is given as a homogeneous quadratic map. This space is treated as a twodimensional projective space and is visualized as a unit disk. If the renormalization group parameter is greater than the dimension of the lattice, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann deltafunction . We describe two different (left and right) invariant neighborhoods of this fixed point and classify the points on the plane according to the way they tend to this fixed point (from the left or from the right).
We describe explicitly the zone structure of the classified domains and show that the global renormalization group flow has a nice description in terms of this zone structure. We discuss also the global behavior of all RGinvariant curves.
Language: English

