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Magnetization and Overlap Distributions of the Ferromagnetic Ising model on the Cayley Tree
R. Mélin
To cite this version:
R. Mélin. Magnetization and Overlap Distributions of the Ferromagnetic Ising model on the Cayley Tree. Journal de Physique I, EDP Sciences, 1996, 6 (11), pp.1435-1450. �10.1051/jp1:1996156�. �jpa- 00247256�
Magnetization and Overlap Distributions of the Ferromagnetic Ising Model on the Cayley Tree
R. Mélin (~,~,*)
(~) Centre de Recherches sur les Très Basses Températures (**), CNRS, 25 avenue des Martyrs, BP 166X, 38042 Grenoble Cedex, France
(~) NEC Research Institute, 4 Independence Way, Princeton, NJ-08540, USA
(Received 20 May 1996, revised 11 July 1996, accepted 29 July 1996)
PACS.61.43.Hv Fractals; macroscopic aggregates (including diffusion-limited aggregates)
PACS.64.60.Cn Order-disorder transformations; statistical mechanics of mortel systems
Abstract. We analyze the magnetization and the oveilap distributions on the ferromagnetic Cayley tree. Two quantities are investigated: the asymptotic scaling of ail the moments of the
magnetization and overlap distributions, as well as the computation of the fractal dimension of the magnetization and overlap probability measures.
1. Introduction
Spin mortels on trees were first introduced in trie thirties independently by Bethe and Peierls
as a way to implement mean field theories without long range interactions [1], even though it
was argued recently [2] that in some cases, Bethe-Peierls calculations are more reliable than
mean field calculations. Even though there is a mean field type transition for Ising as well as XY spins in trie Bethe-Peierls limit (see Sect. 2) when only trie properties of trie central spm
are considered and trie boundary is sent to infinity, there is no transition if one considers trie entire Cayley tree: since there are no loops, trie high temperature expansion of trie partition function in a zero magnetic field leads to trie same result as trie partition function of trie Ising chain, where it is well-known that there is only a zero temperature transition. In trie present
paper, we consider trie Ising model on trie Cayley tree
H(Isingj
~ j ~
~~~~, jl
(~Jl and also the XY model:
H~~~)
= -J ~j S~ Sj. (2)
(~,J)
The Ising model on the Cayley tree has unusual properties: it was shown in [3] that the Cayley
tree has a phase transition of continous order. It was also shown in [4] that the susceptibility (*) Address tram September 1996: International School for Advanced Studies (SISSA), Via Beirut 2-4, 34014 Trieste, Italy. (e-mail: melin@crtbt.polycnrs-gre.fr)
(** UPR 5001 CNRS
@ Les Editions de Physique 1996
1436 JOURNAL DE PHYSIQUE i N°11
Fig. 1. Trie way two 4-half-space-trees
are put together to make
a 5-half-space-tree.
is infinite below a temperature T' (that we will recover in the core of trie paper). Trie ~aim of trie present paper is a detailed investigation of trie magnetization and overlap distributions of trie Ising and XY model on trie Cayley tree.
Trie trees that are considered here are trie so-called haif-space-trees, that is recursive trees such that trie aqcestor bas two neighbors, trie bulk sites bave three neighbors and trie leaves bave only one neighbor. See Figure 1 for a picture of a 5-half-space-tree. We bave shown in [5] that below a cross-over temperature scale J/lnn, with n trie number of generations,
trie Ising spin system is magnetized and trie magnetization distribution is non Gaussian, with
trie existence of local maxima. In trie case of XY spins [6], trie spm system is magnetized
below a cross-over temperature scale J/n, but there are no local maxima in the magnetization distribution, even though there is a large, non Gaussian tail.
The magnetization distribution is investigated using two techniques. First, we calculate analytically the asymptotic scahng of the moments (M~) for large system sizes, where M is the total magnetization. We show that in the case of Ising spins and in the case of XY spins,
the moments of the magnetization have a non trivial scaling. Below a temperature T' that is
defined in the bulk of the paper, there is one scahng dimension lower than unity but larger
than 1/2. If trie temperature is larger than T' and lower than trie Bethe-Peierls temperature, there are two scaling dimensions, but one of them (even moments) is 1/2 (scaling dimension of random walks). Trie other one (odd moments) is lower than 1/2 but larger than zero. Trie
reason why trie scaling dimension of odd moments is non zero is that trie ancestor's spin is
frozen m a given direction. Above the Bethe-Peierls temperature, all the spins are uncorrelated and the scaling dimensions are trivial: 1/2 for even moments and
zero for odd moments.
Next, we study numerically the fractal properties of the magnetization distribution measure
in the Ising case. The magnetization distribution defines a measure on the interval [-Nn, Nn],
where Nn is trie number of sites:
Nn=1+2+.. +2"=2"+~-l. (3)
There exists a temperature regime where this measure is fractal in trie sense of [9]. Trie fractal dimension does not decay monotonously as trie temperature decreases.
We also analyze trie overlap distribution and reach similar conclusions.
2. Bethe Peierls Wansition
2.1. ISING CASE. It is known since the thirties [1] that the Ising model on the Bethe lattice has a mean-field like transition. For a review on spin systems in the Bethe-Peierls approach,
see [7,8j. The idea is to decimate spins at the boundary of a n-half-space-tree and to obtain an
equivalent in 1)-half-space-tree with a magnetic field at trie boundary. We note Zn(fl, h,hn)
the partition function of a n-half-space-tree with a magnetic field hn at the boundary. We first consider trie situation where two spins ai and a2 are connected to a common ancestor Z. The partition function is
~
jgj ~ ~pJEa~phna ~j
z =
a
The summation over ai and a2 bave been factored out since there are no loops. We next wnte
z(Z) under trie form z(Z)
= fifexp (flThn). Since there are two equations (one for Z
= 1 and
one for Z
= -1), trie parameters tif and Thn exist and are uniquely determined by fif~
= z(+)z(- = (4 (cosh~ (flJ) + sinh~ (flhn)))~ (5)
1~ jz(+)j l~ jcosh(fl(J+hn))j j~~
~~" 2fl~ z(-) ~fl~ cosh(fl(J-hn))'
The zero field behavior of Th is
Th = 2h tanin (fl.J) e p~~~~~~~(fl)h. (7)
There exists a finite temperature flbp such as p(flbp) = 1. If fl > flbp, the magnetic field is exponentially amplified, leading to a mean field type phase transition in the Bethe-Peierls
limit [8j. Iffl < flbp, the magnetic field is irrelevant. As far as the n-half-space-tree is-concerned,
we have
~
~ i
Zn(fl,h, hn) = (4 (cosh~ (flJ) + smh (flhn)) ~ Zn-i(fl, h, Thn). (8)
2.2. XY SPINS. We now tum to the case of XY spins. We assume that a small magnetic field h is switched on at the boundary. The partition function of two spins connected to a
common ancestor Z is
~
tj~) ~~ ~phcos81~@Jcos(8-81) (9j
~Î 1
Since the magnetic field is small, we can expand (9) up to the second order in h and reexpo- nentiate to obtain
1(Ù) = 1
+
~~~~ /~~ du e~~~°~")
~
exp (flh(cos Ù)p~~~l(fl)), (10)
2 o
with
~
p~~~~(fl) = 2 / ~
du cosuw(u), Ill)
o
where w(u) is the probability that the angular diflerence between two nearest neighbor spins
~~ "
~pJ Cos u
w(uj = ~~
(12)
du exp (fl cos u)
Î
Notice that by replacing JdÙi by JdÙi (à(Ùi) + à(Ùi r)), we recover the Ising case: w(0) is the probability x = w(0)
=
e~~~/(ePJ + e~PJ) that two nearest-neighbor spins are antiparallel;
w(r) = 1-x
=
e~PJ/(ePJ+e~PJ is the probability that two nearest neighbor spins are parallel;
p~~~l(fl) becomes p(~~~~~l(fl).
1438 JOURNAL DE PHYSIQUE I N°11
3. Scaling of trie Moments of the Magnetization
This section is organized as follows: we first derive the magnetization distribution recursion for Ising spins. The XY spin case is simply related to the Ising case by the aforementioned transformation JdÙ - JdÙ(ô(Ù) + à(Ù r)). The next step is to calculate by recursion the
scaling behavior of ail the moments.
3.1, MAGNETIzATION DISTRIBUTION RECURSION. The atm of this section is to derive the
magnetization distribution recursion for XY spins. We call PS (M) the conditional probability density of the magnetization of XY spins on a n-half-space-tree, the ancestor pointing in the direction Ù. We put two n-half-space-trees together as pictured in Figure 1 to obtain a in +1)- half-space-tree. Since the link variables are statistically mdependent, we have [6j
P~+1(M)
#
/ àMlàM2ô
(M Ml M2 ô) / àÙlàÙ2W(Ù-Ùl)W(Ù-Ù2 )P$~(Ml)P$~(M2), (13)
where w(u) is defined by (12).
3.2. LEMMA. Trie atm of this section is to derive recursion relations for trie average of trie moments. We prove trie following proposition by induction:
7ii~~i - v~ > 0 (M~)i-i
= li)l~~i ji~~~i~ii~ , (14)
where A[~~ is independent on Ù.
Trie average (...)$ means an average
over a n-half-space-tree with trie spin ancestor frozen in trie direction Ù. A by-product, we derive a recursion relation for trie AÎl's. Clearly, ~Î~l is
true, with AÎ~I = 1. We now show that ~n-1 ~ ~n. We first wnte (M~)$ under trie form
(~~)"
" ~
i
~
~
i",q>1,m(Ù), (15)
~
i
o ~~ ~
o
where
In,q,i,m(Ù) = /
dÙidÙ2w(Ù Ùi)w(Ù Ù2) ((M~)$1_~ (M~~~)$[~) ô~~~ (16)
Next, assuming ~Î~i,
we consider trie eight cases where q, and m are even or odd, and show that In,q,i,m(Ù) is either independent on if q is even
or In,~,i,m(Ù) = )In,~,i,mlô, where )In,~,i,m
is independent on Ù. It would be tedious to enumerate here trie eight cases, and we just pick-up only one case as an example: q odd, odd and m even. In this case, we bave
In,q,1,m(Ù) = /
dÙidÙ2w(Ù Ùi)w(Ù Ùi)Ab(AfIf~ô2 Il?)
Using trie fact that w(-u)
= w(u), it is easy to show that
/d92w(9 92)62 = à/ du w(u)
cosu = ~ô, (18)
2
which is of the announced form. We have wntten p instead of p~X~)(fl) because the form of this result is independent on whether
we have XY or Ising spms: one just has to choose
p(~~~~~l or p(X~l according to trie nature of trie spin variables. We can also show that In,q,i,m
is independent on q. More precisely, we note
lp/2 ifiisodd
~yj,~ = l if1 is even and m is even (19)
(p/2)~ if1 is even and m is odd.
With this notation, it is easy to show that
Îin,q,1,mÎ " JÎ~-ÎJÎ~_1 ~'Îl,m, (2Ù)
which leads us to trie following recursion for trie AÎ~~'S:
A(ql
=
jj ~ jj
~~ ~Ab(A(~/~il ôi qôm,q (1 ôl,qôm,0 + l'tq,q +'tq,0 )Af~1, (21)
~
l~ ~
~
~~
'
wbere, for furtber purposes, we bave separated tbe term A(~l~ from tbe terms Af~~, witb k < q.
3.3. ASYMPTOTIC BEHAVIOR OF THE MOMENTS. Dur aim is to calculate trie asymptotic
bebavior of tbe moments of tbe magnetization. We bave to distinguisb between two éases:
fl > fl' and fl < fl', wbere fl' is defined by p2 = 2. Notice tbat fl' corresponds to tbe inverse temperature above which tbe susceptibility per spin diverges [4j. Tbis is in fact not surprising
smce the susceptibility cari be calculated from the moments of the magnetization using the fluctuation-dissipation theorem.
3.3.i. Low Temperature Regime: fl > fl'. In the regime fl > fl', we prove by induction the
following proposition:
~(~~i - Vk,k E (0,.
,q 1), the asymptotic behavior of AÎI
m the large n limit is AÎ~
+~
akP"~, where ak is mdependent on n. (22)
~Î~l is obviously true and ao " 1. We show that ~(~Î ~ ~Î~l, and as a byproduct, we find the recursion relations for the aq's.
We first examine the case where q is odd. Then, ~yq,q + ~yq,o = p. Using (21) we can express AÎ~I m terms of tbe A)~l witb k < q. We find
AÎ " P~ +
É
(~) ~l'fi,mXn,i,m(1
ôi,qôm,q)(1 ~ qôm,01' (~~)
l=o m=o
with
n-i
x~1
~ =
£ p"-i-~A[~lA[~-~l 124)
, ,
~=o
We now assume ~(~Î in order to find the asypmtotic behavior of Xn,i,m, in the large n limit,
which is
Xn,i,m ~ aurai-m
j P"~ (25)
P P
1440 JOURNAL DE PHYSIQUE i N°11
The case q
= 1 is special due to the fact that one has to take into account the p" term in (23).
This case bas to be treated separately, and one finds là,6j: AÎ~I
+~ ai P", witb ai
= P/(P~- 1).
If q > 1, tbe p" term in (23) is sub-dominant, and we get for q > 1 AÎ~~ +~ aqp~", witb
~ ~
q
°q "
q
~ 'fq,mamaq-m. (26)
P P
~~~
mÎ
If q is even,
~yq,q+~yq,o = 2. We carry out tbe same reasoning and we find tbe following recursion relation for oq if q > 2:
We
bave
used
large n
limit.
otice tbat aq is eflectively
ndependent
q 2, weind
tbat
wbicb is exactly wbat was found in [6j.
3.3.2. Intermediate Temperature Regime: flbp < fl < fl'. In tbe regime flbp < fl < fl', we
call ~(~l tbe following proposition:
~~~~i - Vk,k E (0,..
,
j 1), tbe asymptotic bebavior of AÎI
in tbe large n limit is
A?I
+~
~f/$
n
~ ~ ~~ ~~~~
wbere flk is independent on n (29)
flk 2 p) if k is odd ' '
and we show by induction tbat ~(~l is true. We bave sbown
in [6j tbat ~Î~~ and ~Î~~ are true.
Tbe technique to show tbat ~(~( ~ ~Î~~ is tbe same as in Section 3.3.1: we use (21) and we analyze separately tbe case q even and
q odd. After straigbtforward calculations, we conclude tbat ~j~( ~ ~Î~~, and tbat
~Q 2q/1_~ ~ j~~~'ÎQ,R~ÙR~ÙQ-m (~°)
m % 0(2)
if q is even, and
fl~ "
p
q(q-Î/~
i) 1 ÎÎÎÎ'fQ,~fl~flo-~
l~~)
=Î
if q is odd.
3.4. DIscussIoN. We bave tbus sbown tbat if fl > fl', tbe dominant bebavior of tbe moments of tbe magnetization in tbe large n limit is
(M~)° « N~t+, (32)
wbere N is tbe number of sites. Using tbe analogy to multifractals [9j, tbis means tbat tbere is
one singularity of strengtb a = Inp/ In2, and of scaling dimension f = Inp/ In 2 > 1/2. Now, ifflbp < fl < fl', we bave
(MQ)° c~ NQ/~ (33)
if q is even and
(MQj° çç NFé+% j34)
if q is odd. Tbere are two singularities of scaling dimension fi(q)
= 1/2 if q is even and f2(q) "
Inp/ln2 if q is odd. If tbe temperature goes to tbe Betbe-Peierls transition temperature,
fi(q) " 1/2 and fi(q) = 0. At the Betbe-Peierls transition, tbe correlation lengtb is sborter
tban tbe lattice spacing [Si, and all the spins are thus uncorrelated. The magnetization is thus
tbe sum of N random independent spin variables. Above tbe Betbe-Peierls temperature, tbe scaling dimension of even moments is 1/2, as tbe one of random walks [loi, and tbe scaling dimension of odd moments is zero. Notice tbat tbese results are true for XY spins as well as
Ising spins. Une just has to pick-up the right p function. Tbis conclusion is to be contr>sted
with the behavior of the magnetization distribution. We have shown numerically that the
magnetization distribution has local maxima in the case of Ising spins [5j, and no local maxima
in the case of XY spins [6j. Even though the scaling of the moments of the magnetization of
XY spins is non trivial, the magnetization distribution is a smooth function, with a long tail, but obviously no local maxima, unlike the case of Ising spins where, as explained in Section 4 the magnetization distribution has a proliferation of local maxima, at low temperatures, and
we find non trivial fractal dimensions for the probability distribution measure.
4. Magnetization Distribution in the Ising Case
The aim of this section is to analyze the low temperature magnetization distribution P(M)
m the Ising case, where local maxima of the magnetization distribution are present [Si. The main result of this section is the computation of the fractal dimension of the magnetization
distribution at low temperatures. By fractal dimension, we mean the fractal dimension of the
measure P(M), as defined in [9]. We insist on the fact that these results are strongly dependent
on the discrete nature of the spin variables.
It was argued in [5] that below a temperature of the order J/ In n, the magnetization is non
Gaussian. Tbis temperature scale corresponds to tbe temperature cross-over in
(M)t ai
jpjn ~
ai j~_~~-~pJj ~35j
Nn 2 2 2 '
wbicb leads to tbe aforementioned cross-over temperature. Dur goal is to investigate tbe scaling properties of tbe low temperature magnetization distribution. Tbe magnetization distribution
is plotted in Figure 2 for tbree temperatures. We see tbat as tbe temperature decreases, tbe magnetization distribution looks more and more fractal. In order to put tbis observation on a
quantitative basis, we divide tbe segment [-Nn, Nn] mto1 pieces of size 2~~+~Nn, and we call P~ tbe probability measure of tbe 1-tb segment. We calculate [9]
xj(qj
=
fl pq, (36)
and look for tbe bebavior of xi(q) for large 1. If tbe temperature is not too low, we can identify
tbe existence of a scaling regime, namely In xi(q) is linear as a function of -1+1 for large
(sel Fig. 3). In practise, one cannot reacb tbe = +cc regime because tbe magnetization is
1442 JOURNAL DE PHYSIQUE I N°11
beta=4.9 beta=2 beta=1
_/"'h_
C / "' i i
o w
~
~
'C ''
é '
é i
C o
# ce M 4 w C cn ceE
~ w
flM
E
~
Z le-10
j
,
'
'
j
-1 5 -1 o.5 o o.5 1.5
Normalized magnetization
Fig. 2. Magnetization distribution of a n
= 15 generations tree, for fl
= 4.9, fl
= 2 and fl
= 1.
a discrete variable. Tbe next step consists in calculation tbe slope of In xi(q) as a function of In 2~~+~ in the scaling regime:
~~ ~~~~ ~~~~~ ~~ ~~~~~~~ ~~~~~~ Î~ 2 ~~l
= go f~ ~~~~
where a is the strength of the singularity, f the fractal dimension of the probability measure
and Dq the generalized dimension. In the multifractal case, both a and f depend on q. In
our case, we find that (q 1)Dq is linear as a function of q, so that tbere is only one fractal dimension in our problem. Tbe fractal dimension is plotted in Figure 4 as a function of tbe
inverse temperature for diflerent sizes. Tbe fractal dimension of tbe magnetization distribution
does Dot decay monotonously to zero as tbe temperature decreases, and we observe finite size eflects. We bave no analytic understanding of tbe existence of temperature domains wbere tbe fractal dimension of tbe magnetization increases as tbe temperature decreases. Fimte size eflects are understood as follows. Below a temperature T" of tbe order of J/lnn, tbe kinks (1.e. antiparallel nearest neigbbor spins) give rise to well-defined domains of magnetization [5].
Consider one kink on a n-balf-space-tree. Tbe probability to find a kink at generation k
(counted from tbe ancestor) is
~k
~"~~~
2(2" 11' ~~~~
