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Journal of Physics A Mathematical and General

Wilson-Feynman graph technique and universality

Monique Combescot, Michel Droz

Published Nov 1, 1976• 0 citations
Publisher

Abstract

The critical exponents and scaling function are calculated with the Wilson-Feynman graph technique to second order in epsilon =4-d for an n-component system with anisotropic Brillouin zone, using a general cut-off for the bare propagator G(p,r 0 , lambda )=(r 0 +p 2 ) -1 f(p/ lambda ,r 0 / lambda 2 ). Universality holds provided that f(v,x) is analytic in v and x around (O,O) and the first derivative f v 1 (O,x)=0.

DOI
10.1088/0305-4470/9/11/016

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The critical exponents and scaling function are calculated with the Wilson-Feynman graph technique to second order in epsilon =4-d for an n-component system with anisotropic Brillouin zone, using a general cut-off for the bare propagator G(p,r 0 , lambda )=(r 0 +p 2 ) -1 f(p/ lambda ,r 0 / lambda 2 ). Universality holds provided that f(v,x) is analytic in v and x around (O,O) and the first derivative f v 1 (O,x)=0.
Feynman graph methods are used to analyse crossover behaviour in an n-component Heisenberg system, with a spin anisotropy that results asymptotically in m-component critical behaviour. An expression for the universal n- and m-dependent amplitude locating the singularity of the crossover scaling function is derived to second order in epsilon =4-d and found to give fair agreement with the results of recently derived series expansions. The same techniques are used to obtain, to 0( epsilon 2 ), a universal amplitude ratio characterizing the shape of the phase boundary in the vicinity of spin-flop bicritical point.
It is shown that the critical indices corresponding to (0, lll perturbations can be calculated by a Feynman-diagram method. At order ${\ensuremath{\epsilon}}^{2}$, the results are consistent with the special cases l=1,2,and4l=1,2, \mathrm{and} 4l=1,2,and4 obtained previously. There is also general agreement to order $\ensuremath{\epsilon}$ with the exponents calculated from Wilson's recurrence relation.
The application of renormalisation group recursion relations to the calculation of critical exponents in d dimensions to second order in epsilon =4-d is discussed. It is shown that in the limit of a large scale factor b, the fixed point value, u* of the four-spin coupling constant is correctly given to second order by a renormalisation group equation in which all irrelevant variables, mass renormalisation graphs, and mass terms in the propagators are ignored. Related approximations in the calculation of the critical exponents are justified. The unexpected equality to order epsilon 2 of u* and u 0c , the Feynman graph coupling constant, is discussed.
Details of the calculation to O(1n2)O(\frac{1}{{n}^{2}})O(n21​) of the polarization part and of the vertex function with one ${\ensuremath{\phi}}^{2}$ insertion are given for an nnn-component, three-dimensional system with O(n)\mathrm{O}(n)O(n) symmetric, short-ranged interaction. The critical indices $\ensuremath{\lambda}$ and $\ensuremath{\mu}$ are deduced and are found to conform to other indices available to the same order. An analysis of how the requirements of exponentiation and universality are fulfilled shows that they are consequences of the fulfillment of a scaling law at the previous order. Cutoff-dependent contributions are seen to split off automatically. With possible higher-order calculations in mind considerable stress is put on the technical side: An apparently novel type of regularization technique allows one to pick up the relevant terms directly and is shown to be connected to a kind of analytic regularization. Graphs with diagonal insertions are calculated from generalized Feynman amplitudes associated with first-order diagrams. Those with vertex insertions, as well as irreducible graphs are handled by an integration technique taken over from conformal invariant field theory. The numerical performance of the 1n\frac{1}{n}n1​ expansion is discussed, and an attempt is made to obtain more meaningful numbers from the expansion through various Pad\'e- and Pad\'e---Borel-type resummations. The results are compared with information available from other sources.
A class of renormalization groups characterized as the general Gaussian renormalization group is defined. It includes the sharp cutoff renormalization group of Wilson, the linear renormalization group of Bell and Wilson, and the incomplete‐integration renormalization group of Wilson. The renormalization group approach to critical phenomena and the expansion around four dimensions is reviewed. The fixed point of the general Gaussian renormalization group is determined to order ε2, where ε=4−d (d is the dimensionality of the system), and the critical exponents are determined to the same order. The main conclusion is that the exponents do not depend on the cutoff functions of the renormalization group.