A Renormalization Group Analysis of the Hierarchical Model by Pierre Collet

By Pierre Collet

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Extra resources for A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics

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2) then I ~(~,-) = ~ - I ( S ~ c r i t ( ~ o ) temperature trajectory of eo (or fo). )~/~ crit 1 ~ near ~crit,the It is generally assumed in RG theory that the temperature can be used as one coordinate on the unstable m a n i f o l d ~ u . , c5) we can show that this is indeed the case. 2 . ) is differentlable in L and it is stable manifold. Therefore the (inverse) temperature is one of the "relevant" directions and can be used as a coordinate . ) > ~crit ~ 6 6crit Figure 6. The temperature as a coordinate We now derive the equations for the moments of the sum of the spins.

2) Choose a number ~crit > 0 which will be the critical the model we are going to define. 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.

24) and the relation between "M" and "P". 25) X2 -N ~ + ~crit ' ~ ~ 0 , we get (cf. 18) <~>N X , L = X 2N,~N,~(~ N , . ) /-~ ^ 1,6 N, JTL(~N) x , a 2 ~o. 26) 1,6N,a282 + 0(i) Therefore P log X N - A ^ '6N'-~c~itlog S6crit(%) 2 lim N ~ ~ log i~N - 6crit I log < ~ >N L XI, 6N, a2~ 2 lim = N ~ ~ log(c/2) log k 2 -N log k 2 Anticipating again the existence of the thermodynamic limit, above the critical temperature P l P = ~,f lim N -, ~ l N,O,f we get P log × log( c(~)/ 6, f /2 ) as l°gI6 - 6criti log k2(~) ~ ~ ~crit " 54 An analogous result will be seen to hold below the critical tempera- ture.

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