' óéóôåí Yu.Kozitsky,M.Kozlovskii,T.Krokhmalskii

A simple hierarchical scalar spin model is studied analytically and numerically in the vicinity of its critical point. The dependence of the finite size (i.e. calculated for a large but finite number of spins) susceptibility and the location of zeros of the model partition function on the number of spins at the critical point is described analytically. It is also shown analytically that the finite size correlation length in such a model diverges at the critical point slower than it is supposed in the finite size scaling theory. Certain numerical information about the critical point and ordered phase is given. In particular, the critical temperature of the model and the critical index describing the order parameter are calculated for various values of the interaction parameter.


Introduction
Hierarchical models have played a signi cant role in modern classical statistical physics.First such a model was introduced by F. Dyson 6] in 1969 as a tool in the study of one dimensional Ising{like spin models with long{range interaction potentials.Since that time these models have been studied and utilized.An excellent survey of ideas and results in this domain can be found in 3].
The main idea of F.Dyson was to substitute the one dimensional Ising-like spin model with the translation invariant interaction potential which decays as a power of the spin-spin distance by a model with the interaction decaying as the same power but of the distance de ned by special non{Euclidean metric and possessing some other symmetry instead of the translation invariance.This new symmetry was designed to allow more detailed and deep mathematically strict description which would yield also some information about properties of the initial translation invariant model.
At the other hand, during the seventieths I.R.Yukhnovskii with the collaborators have developed an approach to the study of the three dimensional Ising model based on the collective variables method (see e.g.21], 22]).The main tool of this approach was a step{by{step integration scheme, which allowed to obtain the model partition function, free energy, other thermodynamic functions in the vicinity of the critical point 17].The peculiarity of this scheme lies in the fact that the partition function is calculated as a product of the partial partition functions describing a sequence of growing boxes of spins.It was considered as a realization of known heuristic Kadano 's block{spin construction which was intended to describe the critical points of such spin models.This construction was based on the idea that the ensembles of properly renormalized total spins of cubic blocks of various linear sizes are distributed identically whenever the model is taken at its critical point.Such a property is known as a critical point self{similarity 20] or as a critical point scale invariance.The latter is considered as the main symmetry appearing at the critical point.
In fact, the level of strictness of the Yukhnovskii's approach is rather physical which means that certain approximations with no quantitatively controlled consequences were employed.Later it was understood 13] that the qualitative result of these approximations lies in the replacement of the translation invariance of the model considered by the self{similarity symmetry which was embodied in the Kadano 's construction.Moreover, it was shown in the paper 13] (see also 14] and 10]) that the ICMP{97{27E 2 step{by{step integration scheme being applied to a translation invariant spin model transforms it into the model identical to the Dyson's hierarchical model.In what follows, one can consider the Dyson's model as a realization of the Yukhnovskii's scheme of study of translation invariant Ising{like spin models.Since early seventieths, the Dyson's hierarchical model was being studied mainly analytically.At the same time, the investigations of the three dimensional Ising model within the collective variables method by I.R.Yukhnovskii and his collaborators (including also numerical ones) were performed.As a result, universal and nonuniversal aspects of the speci c heat behaviour in the vicinity of the critical point were analyzed 22], 15].The equation of state was derived, its analytic solution describing the order parameter dependence on temperature, external magnetic eld, and microscopic model parameters in the vicinity of the critical point was found 5], 16].It should be noted that a number of additional approximations was used to obtain these results.
The aim of our work is to study { analytically and numerically { a simple hierarchical model, considering it also as an emanation from the Yukhnovskii's scheme, in the vicinity of its critical point without additional approximations just mentioned.The main question here is how do the quantities, which describe the behaviour of the model in the vicinity of the critical point (such as susceptibility, order parameter, correlation length, location of zeros of the partition function, and so on), calculated for the xed large but nite number of spins, depend on thermodynamic variables, or how do they depend on the number of spins if the thermodynamic variables are xed at their critical values.Our analytic study is performed by means of rigorous methods.We discuss the asymptotic behaviour of the mentioned quantities, calculated at the critical point, when the number of spins tends to in nity.For this aim the facts which were already known are taken from the corresponding papers, whereas a number of new ones are proven here.As a result, we have suggested how to calculate numerically the order parameter for zero values of the external magnetic eld, how to nd corresponding critical index, how to calculate the critical temperature.Such an approach is partially inspired by the works on the nite size scaling, 8] and 9] in particular.We suppose to apply the nite size scaling methods to describe our model on the base of the data obtained here { it will be done in our next work.To prepare it we check the basic assumption of this method concerning the critical point correlation length asymptotics, with the size of the model tending to in nity, and nd that it should be changed in the case of our model.
As tools we use the methods developed earlier 14], 10], 11].as well as direct numerical calculations of corresponding quantities for su ciently large number of spins, based on the analysis mentioned before.Such calculations became possible also due to the special (hierarchical) structure of the model considered.

The Model
Here we deliver the description of the models and the facts from their theory which are related to our investigation.Some of these facts are proven here, another ones are taken from the papers where they have been published earlier.
Let us consider a countable set of one-dimensional spins f s 2 R; s 2 Ng which we choose to be indexed simply by positive integers.The formal Hamiltonians of the translation invariant and hierarchical models are H tr = 1 2 X s;s 0 2N J tr ss 0 s s 0 h X s2N s ; (2.1) where h is an external magnetic eld, and J tr ss 0 = J(j s s 0 j +1) 1 ; J > 0: (2.3) Here j a j stands for the absolute value of a 2 Z.The function j s s 0 j can be considered as the Euclidean distance on N, it is invariant with respect to the shifts (translations) along N. The parameter describes the decay of the potential J tr ss 0, it is a priori set being positive.As for the potential J ss 0 in the Hamiltonian (2.2), it is put to be of the form (2.3) but with the "hierarchical distance" between s and s 0 instead of j s s 0 j.This new distance can be set by means of the hierarchical structure on N. The latter is the sequence L = fL n ; n 2 Z + g of families L n = f n;r ; r 2 Ng, where n;r = fs 2 N j 2 n (r 1) + 1 s 2 n rg: (2.4)These subsets of N obey the following recursive rule n;r = s2 1;r n 1;s : (2.5) De nition 2.1 Let n, s, s 0 be chosen in N. The points s and s 0 are said to be separated on the hierarchy level n if they belong to di erent subsets from L n .Proposition 2.1 For arbitrary noncoinsiding pair of points s; s 0 2 N, there exits n 2 N such that these points are separated on the hierarchy level n 1 and are not separated on the levels n, n + 1, : : :.This number is denoted as n(s; s 0 ), that is n(s; s 0 ) = minfn 2 N j (9 2 L n )(s 2 )&(s 0 2 )g (2.6)For example, n(1; 2) = 1; n(2; 3) = 2; n(4; 5) = 3.By means of n(s; s 0 ), we can de ne the hierarchical distance on N: dist(s; s 0 ) = 2 n(s;s 0 ) 1: (2.7) Then the interaction potential in the Hamiltonian (2.2) is written as follows: J ss 0 = J(dist(s; s 0 ) + 1) 1 ; (2.8) where and J are the same as in (2.3).In order to relate such a model to the d-dimensional translation invariant model considered within the Yukhnovskii's scheme, one should put = 2=d (see 13], 14], 10]).
The following assertion can be proven directly from the de nition.
Proposition 2.2 For arbitrary pair of points s, s 0 2 N, j s s 0 j dist(s; s 0 ); (2.9) that yields J tr ss 0 J ss 0: (2.10) Thermodynamic properties of the model can be described by passing to the limit % N for the expectations < :: > , computed at nite subset with the help of Gibbs measures ;h .They are de ned as probability measures on the con guration spaces R by means of local Hamiltonians H ( ), where = f s ; s 2 g.The local Hamiltonians are constructed accordingly to the formal ones (2.1), (2.2) that is described just below.Here we consider only the Gibbs measures which correspond to zero boundary conditions.Thus where Z ;h is the normalizing constant which provides for ;h to be probabilistic, and the measure describes the single-spin (even) probability distribution.The simplest case of the latter is the measure concentrated at points 1 that corresponds to the Ising{like spins.
For the hierarchical models, the thermodynamic limit is naturally be achieved within the hierarchical structure, that is by choosing 2 L n and putting n ! 1. Hence we may set the hierarchical local Hamiltonians only for such .Thus we use the relation (2.5) and de ne the family of local Hamiltonians fH n;r j n 2 Z + ; r 2 Ng recursively by putting Clearly, these Hamiltonians are invariant with respect to those permutations of N which preserve the hierarchical structure.Such permutations form a group which is the symmetry group of the model.In particular, all Hamiltonians with the same n and di erent r 2 N are identical, therefore, can be represented by one of them, H n;1 for example.The relationship between the Hamiltonians (2.2) and (2.12) can be established as follows.We rewrite H n;1 given by (2.12) that can be used as a de nition of J (n) ss 0 .Having in mind that n;1 absorbs N when n ! 1 and using the relations (2.12){(2.14),one can prove such a statement.
Proposition 2.3 Along with the Hamiltonians (2.15) we also will consider the local Hamiltonian de ned by the Hamiltonian (2.1) where

24)
T ;h n 1 ( 2 + )T ;h n 1 ( 2 )d d : Here we have taken into account that all 1;r consist of two points, and all ( n;r ) with the same n and di erent r are distributed identically.Now let us describe in more details the family of measures which are to be chosen in this research as initial single{spin measures .The rst condition imposed on these measures is the existence of their Laplace transforms (2.25) as entire functions.Let F be a family of entire functions of one complex variable possessing the following canonical representation In other words, the family F consists of entire functions with the order of growth at most two which either have purely imaginary zeros or have them none.
De nition 2.2 A probability measure is said to possess the Lee{Yang property if its Laplace transform (2.25) belongs to the family F.
It can be shown that the measures d ( ) = 1 2 f ( 1) + ( + 1)g d ; (2.27) and (2.28) possess the Lee{Yang property.Further details can be found in the paper 12].Here we discuss only the properties of these measures which are relevant to our research.As can be seen from the de nition (2.25), the moments of each such a measure can be computed as corresponding derivatives at zero of its Laplace transform f .The following derivatives (2.29) are known as semiinvariants or cumulants of .Directly from the de nition (2.26) one obtains the following representations for these parameters u (2) = 2( + 1 ); The only one measure possessing the Lee{Yang property, the Laplace transform of which has no zeros, is the Gaussian measure.All other ones can be characterized by the location of its zeros which are nearest to the origin of the imaginary axis.These are at points z 1 = i 1=2 1 (see (2.26)).Proposition 2.4 Let for a non-Gaussian measure possessing the Lee{ Yang property, the semiinvariants be de ned by (2.29).Then the location of the nearest zeros of its Laplace transform obeys the following two sided estimate " 40 ju (4) j u (2) u (6)   # 1 4   jz 1 j " 6 u (2)   ju (4)

Critical Point and Ordered Phase
We start with the description of the ordered phase.For this purpose we will need an order parameter.In the spin models it should be a magnitization per spin in zero external eld.It can be computed by means of corresponding Gibbs measures.Consider the following expectations:  Applying standard arguments (see e.g.19]) one can extend the validity of this assertion also for the single{spin measures of the type of (2.28).Much more information about the long-range order in such models has been obtained by P.Bleher in his works 1], 2] (see also 3]).In particular, one can deduce from these papers the following facts.For our model, the most convenient parameter, for the numerical calculation, is m (n) 3 ( ) := 2 n M n ( ); (3.15) it may be taken as an approximate ( nite size) value of m 3 ( ).The fact, that the latter parameter is a lower bound for the genuine order parameter, shows that it may be used to describe the ordering phase transition in the model considered.
It should be noted here that the considerable description of the ordered phase in such hierarchical models was performed in the papers 1] and 2] where the asymptotics (for n ! 1) of the "small" partition function (2.22) was obtained.In particular, Theorem A of the paper 1] and Theorem 2.1 of the paper 2] imply the following assertion.M (2)  n ( ; 0) = O(2 2n ); for > : (3.19)At the other hand, for the disordered phase, only bulk divergences should appear M (2)  n ( ; 0) = O(2 n ); for < : (3.20)As for the critical point itself, the intermediate type of divergences is expected M (2)  n ( ; 0) = O(2 n(1+ ) ); (3.21) with some 2 (0; 1).In order to prove this conjecture and to nd , we proceed as follows.De ne the sequence of random variables f n ; n 2 Ng, where n = 2 n 2 (1+ ) ( n;r ); 0: (3.22)The probability distribution of each n can be described by means of the measure (2.18), where we put h = 0.If the sequence f 0 n g is asymptotically normal, the dependence between the spins is weak or absent as in the case described by the standard central limit theorem.It corresponds to the asymptotics (3.20).In the case of positive in (3.22), one has the abnormal normalization of the sums ( ).For weakly dependent spins, the sequence f 0 n g is asymptotically normal, then the sequences f n ; > 0g are asymptotically degenerate at zero.Therefore, the convergence of such a sequence to some nondegenerate random variable would correspond to the appearance of a strong dependence between the spins.The latter is assumed to occur at the critical point of the model.Such critical point convergences were proven.Proposition 3.4 ( 11] ) Let 2 (0; 1 2 ) and the measure possess the Lee { Yang property.Then there exists > 0 such that: (i) for = , the sequence f n g is asymptotically normal; (ii) for < the sequence f 0 n g is asymptotically normal, Proposition 3.5 ( 4] ) Let the initial measure be chosen of the form (2.28) with u = 1.Then there exist " > 0, v 0 > 0, and such that for 2 ( 1 2 ; 1 2 + "), v 2 (0; v 0 ), = , the sequence f n g converges to some non-Gaussian random variable.
For some Borel subset A R, we denote P n (A) = Prob( n 2 A): (3.23)In the case where the initial measure is absolutely continuous with respect to the Lebesgue measure on R, these are all P n .Denote )d ; where Kn = Z R 2 exp( q 2 ) (3.26) Tn 1 (2 1 2 + ) Tn 1 (2 1 2  )d d ; and q = 1 2 (1 2 1 )J (3.27)Thus, the direct corollary of Propositions 3.4 , 3.5 is the following assertion Proposition 3.6 (i) Let the conditions of Proposition 3.4 be satis ed.Then there exists such that for = , the sequence f P n ; n 2 Ng weakly converges to some Gaussian measure.
(ii) Let the conditions of Proposition 3.5 be satis ed.Then there exists such that for = , the sequence f P n ; n 2 Ng weakly converges to some non-Gaussian measure.
(iii) In both cases described above the following asymptotics holds true M (2)  n ( ; 0) = O(2 n(1+ ) ): (3.28) Remark 3.1 The restriction for to be in the small interval ( 1 2 ; 1 2 + "), mentioned in Proposition 3.5, seems to be purely technical.One can expect that the convergence stated there holds for all 2 ( 1 2 ; 1).It is con rmed by the numerical results given in the paper 4] Hence having in mind (3.18) one can deduce Proposition 3.7 Let be chosen in the interval (1=2; 1) and a n ( ) be as in (3.18), then lim n!1 a n ( ) = 0; for < ; lim n!1 a n ( ) = a 2 (0; 2); (3.29) lim n!1 a n ( ) = 2; for > : Now let us consider some consequences of the critical point asymptotics (3.28), just established.First we nd how does the correlation length n , which describes the decay of the spin{spin correlations in n:1 in such a model, diverge when n ! 1 and = .It may be used to check the assumptions of the nite size scaling method 8], 9].We follow the latter paper where the main assumption of this method is formulated as a rule for a thermodynamic quantity A L (t), calculated on a nite lattice of linear size L, to depend on this size and on t = ( )= as A L (t) = L = f A (s(L; t)); s(L; t) = L= (t): (3.30)Here a power{law critical singularity for a bulk (i.e.obtained in the thermodynamic limit L ! 1) quantity A = O(t ) is assumed, and (t) = O(t ) stands for the bulk value of .Let us choose A = , which means = , and put = , that is t = 0. Then one has in (3.30) s(L; 0) = 0, which yields in turn L (0) = Lf (0): (3.31) To check this assumption we use the de nition (3.1) and obtain M (2)  n ( ; 0) = X s;s 0 2 n;r < s s 0 > n;r : The correlation function < s s 0 > n;r depends on the hierarchical distance between s and s 0 .We set < s s 0 > n;1 = n (dist(s; s 0 )) (3.33)It is known (see 3]) that the so called "small" critical exponent equals to zero for one-dimensional and hierarchical models (see also the footnote 11] in the paper 9]).Then we can substitute in (3.32) n (x) in its asymptotic form n (x) = 0 exp( n x); n > 0; (3.34) where n stands for the inverse correlation length, i.e. n;1 := n = 1  n .In what follows, the n critical point asymptotics prescribed by the nite size scaling assumption (3.31) reads as n = O(2 n ): (3.35) To check whether it really holds, we put n = 0 2 n ; (3.36) and nd the value of which corresponds to the critical point asymptotics of M (2)  n ( ; 0) (3.28).Denote 2 n(1+ ) M (2)  n ( ; 0) = 0 2 (2 n + exp( n )S n ( ; )): Proof.One has: Here n (k) stands for the number of pairs fs; s 0 g in n;1 such that n(s; s 0 ) = k, where the latter is given by (2.6).This number depends on the number #( n;r ) of elements in n;r and can easily be computed if its following properties being utilized: 1) ; n (k) = 2 n 1 (k); k < n: Thus one obtains n (k) = 2 n+k 1 ; k = 1; 2 : : :n: (3.42) Inserting this into (3.39),one arrives at M (2)  n ( ; 0) = 0 2 (3.46)The parameter log 2 0 may belong or not to the set of integer numbers.In the rst case we denote this integer number by k 0 and conclude that the rst line of the estimates (3.46) holds for k k 0 whereas the second one holds for k k 0 +1.If the mentioned parameter is not integer, there exists an unique integer number k obeying the estimate (3.52) The corollary of (3.43) and (3.52) can be formulated as the following assertion.Proposition 3.10 Let S n ( ; ) be de ned by (3.37) with ; 2 (0; 1].Then for n !1: S n ( ; ) ! +1; for < ; S n ( ; ) ! 0; for > : (3.53)As a result, one may conclude that the nite scaling asumption (3.31) should be revised.Proposition 3.11 Let the inverse correlation length n have the form (3.36) with 6 = , then the asymptotics (3.28) does not hold.
The case of = in (3.36) is more subtle, it needs some modi cation of this dependence that is probably caused by the hierarchical nature of the model.
The last question which is to be discussed in this section is the Lee--Yang edge singularities at the critical point, similar discussion for the case of exactly soluble models may be found in 18].Due to Proposition 2.6 all P ;0 n;r possess the Lee{Yang property and may be characterized by the location of zeros of their Laplace transforms which are nearest to the origin of the imaginary axis.Let f n (z) be the Laplace transform (2.25) of P ;0 n;r , then its derivatives de ne the moments of this measure.In particular M (l)  where M (l) n are de ned by (3.1).Let z (n) 1 de ne the location of the nearest zero of f n and u (2k)  n be the semiinvariants of P ;0 n;r de ned by (2.29).Then for z (n)  1 , Proposition 2.4 gives the estimate (2.31) with just mentioned semiinvariants.In the cases described by Proposition 3.5 or by part (ii) of Proposition 3.6, that is in the cases where the critical point convergence of f P n g holds to a non-Gaussian limit, it is expected that jz (n)  1 j tends to zero.We describe this phenomenon as follows.

Numerical Results
The main advantage of hierarchical models is that they are very suitable for direct numerical calculations.By means of relatively weak computer, one can calculate recursively from (2.23), (2.24) the Radon-Nikodym derivatives T n = T ;0 n starting from some suitable measure (e.g. of the type of (2.28) ) up to su ciently high values of n.In this section, we present the results of such calculations.We choose the initial measure P ;0 0;r = P 0 = to be of the form given by (2.28) with v = 1 and u = 0:1.This is the simplest non-Gaussian measure possessing the Lee{ Yang property.Another choice of v would change only the scale of .The dependence of the results on the choice of u will be considered in a separate work, we expect it has only quantitative character.The scale of is de ned also by the choice of J in (3.27).We have put J = 2 1 2 1 2 1 ; that gives q = 1 2 : (4.1)The rst our object is the Radon-Nikodym derivative T n ( ) de ned recursively by (2.23), (2.24) with T 0 ( ) = d ( )=d .For u = 0:1, this function has only one maximum at = 0.For = 0, all these T n will have such a property.Therefore, for small , one may expect that this property of T n will hold.The appearance of the long-range order in turn, one may connect with the appearance { at some n 0 { of maxima of T n0 at = y n0 .The number n 0 must tend to in nity with approaching from above.Such a picture was proven by means of large deviation method 1], 2].We have obtained it numerically.For the value of the parameter , we have been often choosing 2=3 which corresponds to d = 3 if the hierarchical model is used to describe the Yukhnovskii's approach to the d{dimensional Ising model.Fig. 1 shows the evolution of T n ( ) with n for = 2=3 and 1 = 0:50.Here n 0 = 4. Fig. 2 shows further evolution of T n with the same values of the parameters.It can be seen that new maximum transforms subsequently into a sharp peak, that shows the corresponding P n becomes more and more close to the distribution concentrated at two point m n ( ) in full agreement with Proposition 3.3.Fig. 3 and Fig. 4 show the same behaviour of T n for = 2=3 and 1 = 0:61.This behaviour is similar to that of the previous case but here n 0 = 12.Therefore, these two values of are greater than .The behaviour of T n between the peaks m n ( ) is shown on Fig. 5.Here the dependence of log T n on x := =m n ( ) is plotted for = 0:65, 1 = 0:624, and subsequently n = 15; 20; 23; 25 (the highest curve corresponds to n = 25).This picture has been theoretically predicted by P.Bleher in his research 1].To calculate the values of the critical temperature T c = 1 for 2 ( 1 2 ; 1), we use the parameter a n ( ) described by Proposition 3.7.Hence we study evolution of a n ( ) with n for di erent xed values of , and 2 ( 1 2 ; 1).Fig. 6 shows such an evolution for = 2=3 and eight values of T = 1 .It is as described by (3.29), and one may evaluate T cr = 1 = 0:61868.
The following object of our investigation is the order parameter.In fact, it is very hard to study the genuine order parameter given by (3.5) directly.Instead of this, we have studied the lowest one m 3 ( ) (see (3.7), (3.14)).Fig. 7 shows the T := 1 -dependence of this order parameter for = 2=3 and three values of n = 10; 15; 20.Having this dependence, one may compute the value of critical index b, given by (3.13).Fig. 8 shows the dependence of b and T cr evaluated from (3.13) on the values of from the interval ( 1 2 ; 1).

Acknowledgments
This work was supported in part by the Fundamental Researches Fund of Ukraine (Grant No 2.4/173).

4 ) 9 )
Having in mind that each n;r consists of 2 n points we introduce the following Let also m tr l ( ), l = 1; 2 be de ned by the expressions(2.11),(3.1) with H tr n;r ( ) instead of H n;r ( ).In fact, only m 1 ( ) and m tr 1 ( ) are the order parameters in the hierarchical and translation invariant models respectively.The other parameters introduced above are being used to prove the positivity of the order parameters for given values of .In what follows, Propositions 2.2, 2.3 and the Gri ths inequality, which holds for all types of measures considered in this work, imply m 2 ( ) m tr 2 ( ): (3.10)At the other hand, in the case of classical (nonquantum) spin models one has 7] m 2 ( ) m 1 ( ) m tr 2 ( ) m tr 1 ( ) (3.11) Employing these estimates F.Dyson has proven the existence of the long{ range order in both models { hierarchical and translation invariant.

8
Let the correlation function (3.33) have the form (3.34) with n obeying (3.36).Then We denote this number by k 0 and conclude that the rst line of the estimates (3.46) holds for k < k 0 and the second one holds for k > k 0 .For k = k 0 , one has the following estimates 1 the solution of the inequalities (3.47).Clearly, all series in (3.50), (3.51) are convergent.Now we rewrite (3the estimates (3.46), (3.48) one can show that (3.43) holds.QEDConsider also the asymptotics of S n (1

Figure 6 .
Figure 6.Evolution of a n ( ) with n for eight subsequent values of T = 1 .

Figure 8 .
Figure 8. Critical index b de ned by m 3 ( ) = A(T cr T=T cr ) b and T cr via .
The proof of this statement can be done by showing the validity of such two sided estimate for 1 : Moreover, a = 0 if and only if is Gaussian; a = 2 if and only if is concentrated at points a, i.e. is of the type of (2.27) with some a > 0 instead of 1.The following statement is a base for the application of the Lee{ Yang property in the theory of hierarchical models.It was proven in 10] and applied in 11].