Multi-Loop Calculations of Anomalous Exponents in the Models of Critical Dynamics

L Ts Adzhemyan,E V Ivanova,M Dančo,M V Kompaniets,Gh Adam,J Buša,M Hnatič

doi:10.1051/epjconf/201610802004

Abstract

The Renormalization group method (RG) is applied to the investigation of the E model of critical dynamics, which describes the transition from the normal to the superfluid phase in He4 . The technique “Sector decomposition” with R’ operation is used for the calculation of the Feynman diagrams. The RG functions, critical exponents and critical dynamical exponent z , which determines the growth of the relaxation time near the critical point, have been calculated in the two-loop approximation in the framework of e -expansion. The relevance of a fixed point for helium, where the dynamic scaling is weakly violated, is briefly discussed.

Highlights

IntroductionThe renormalization group method (RG) for the study of phase transitions and critical phenomena [1,2,3] allows one to justify the critical scaling and gives a recipe for calculating critical exponents as expansions in a small parameter ε = (dc − d)/2, which is the deviation from the critical dimension dc = 4
The Renormalization group method (RG) functions, critical exponents and critical dynamical exponent z, which determines the growth of the relaxation time near the critical point, have been calculated in the two-loop approximation in the framework of ε-expansion
The renormalization group method (RG) for the study of phase transitions and critical phenomena [1,2,3] allows one to justify the critical scaling and gives a recipe for calculating critical exponents as expansions in a small parameter ε =/2, which is the deviation from the critical dimension dc = 4

Summary

Introduction

The renormalization group method (RG) for the study of phase transitions and critical phenomena [1,2,3] allows one to justify the critical scaling and gives a recipe for calculating critical exponents as expansions in a small parameter ε = (dc − d)/2, which is the deviation from the critical dimension dc = 4. The calculation of the renormalization-group functions is the main technical problem. It is solved by determining the renormalization constants from calculations of the corresponding Feynman diagrams. In this case, the analytical calculation of loop diagrams in higher order is quite complicated, so it is convenient to use numerical methods that allow subsequently to automate the process of search of the renormalization-group functions.

Formulation of the problem

The method of sector decompostion

RG functions and scaling regimes

Conclusion