By Pierre Collet
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Additional resources for A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics
1) e ~o ~ D s (the stable manifold of T This function should satisfy furthermore cl) ~o > 0 , e2) ~o ~ 03) z az % ( z ) c4) log % ( z ) C5) llaz ~o - az ~= II~ is small , cf. 5). the following conditions. 1. ci)-c5) z az % ( z ) / (%(z)) ~ ~ L There are o n ~ s , . functions ~o ~ ~ satisfying (Proof : Section 16). The choice of ~crit and ~o determines a Hamiltonian ~ = ~N, fo through the formula ! 2) Ir 46 For any such Hamiltonian we shall calculate the critical indices. They do not depend on the particular choice of ~o and this fact is called universality.
32) is also equal to log M P 2N' ~N' hN# f lim N ~ l o g hN ~ l o g Z2 / l o g ~'l i and k i = 2c -~ . This describes the magnetization perature as a function of the magnetic Finally assume that M~,f at the critical tem- field. = llm exists and is different h ~ 0 M6'h'f from zero. This will only be the case below the critical (6 > 6crit ), as we shall see later. 35) near the critical in the two phase region. Summarizing, near the critical we see that the behaviour of the various quantities temperature completely controlled or near the Critical by the llnearization field (h = O) is of the tangent m a p ~ j ~ ( ~ c ).
Flows , Mathematical notes, Princeton. Princeton University Press (1969). J. W E G N E R : Corrections to scaling laws, Phys. Rev. %) of Sternberg . 5. Discussion of the Critical Indices One of the triumphs in the RG approach has been the correct prediction of experimentally measured critical indices. The critical indices are defined as follows : Let Q(6) be some physical quantity de- pending on the inverse temperature 6 = I/kT, where k is the Boltzmann constant. e. (or diverge) as ~ ~c " Then the critical index of Q a t ~c (from above or below) is the limit (if it exists) VQ = lim ~ Note that in particular if ges as 6 ~ 6c log Q(~) / log IB-~c I ~c VQ ~ 0 , then this means that Q(~) diver- and ~Q measures in some sense how fast this diver- gence is.