HAL Id: jpa-00247017
https://hal.archives-ouvertes.fr/jpa-00247017
Submitted on 1 Jan 1994
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Ferrimagnetism in a disordered Ising model
Giovanni Paladin, Michele Pasquini, Maurizio Serva
To cite this version:
Giovanni Paladin, Michele Pasquini, Maurizio Serva. Ferrimagnetism in a disordered Ising model.
Journal de Physique I, EDP Sciences, 1994, 4 (11), pp.1597-1617. �10.1051/jp1:1994210�. �jpa- 00247017�
Classification Phv.,ics Ahsfi.actç
05.50 02.50
Ferrimagnetism in a disordered Ising model
Giovanni Paladin ('), Michele Pasquini Ill and Maunzio Serva (2)
(1) Dipartimento di Fisica, Università dell'Aquila, 1-67100 Coppito, L'Aquila, ltaly (2) Dipartimento di Matematica, Università dell'Aquila, 1-67100 Coppito. L'Aquila, ltaly
(Received 20 May 1994, accepted 3 August 1994)
Abstract. We introduce a one dimensional l~ing model with two competing interactions
nearem neighbor random couplings ±J with equal probability and a positive infinite range
coupling Il. At low temperature T the model exhibiis a first order phase transition between a
ferromagnetic state (with magnetization mj at T 0) and a « ferrimagnetic » state (with mi 2/3 at T 0), when the diwrder Urength -Il.i is increased. For 5/12 ~J/~l
~ l, a whole spectrum of ferrimagnetic ground states with magnetization m~
= 2/(n + 1) (n 2, cc ) is
present while for.llA
~ l the ground state i~ given by a trivial one dimensional ~pin glass with
m 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic
limit. This model is exactly solved at ail temperatures and the diagram oi phase i~ calculated.
1. Introduction.
One dimensional lsing models with appropnate distributions of the disorder exhibit many
interesting features of spin glasses [Ii, such as non self-averaging overlap probability, frustration, existence of many degenerate equilibrium states [?]. It is thus interesting to look for simple models which might capture at least some aspects of the high dimensional problems.
in this paper we consider a one dimensional lsing model with nearest neighbor (n.n.) random couplings ±J which has a ferromagnetic infinite range interactions. The two limit cases are bath tnvially solvable the one dimensional disordered lsing model by a gauge transformation and the mean field model by the Curie-Weiss approach. However, the competition of the two types of interactions lead~ to a rather reach behavior. Beside the paramagnetic and ferromagnetic phases (which are present in the mean field system), and spin glass phase (which is present at zero temperature in the one dimensional system), our model exhibits a new type of ferrimagnetic order in a region of low temperatures and intermediate disorder strength.
1t is a non-trivial consequence of the disorder, and could appear in more complicated systems which share the same ingredients of our toy model.
The paper is organized as follows. In section 2 we descnbe the model and show that it can be
numencally studied by products of random transfer matrices even if there is an infinite range interaction. In section 3, we give the exact solution of the model at T
= 0 and show that it
1598 JOURNAL DE PHYSIQUE I N° 11
exhibits a ferrimagnetic order, which survives at T#0. The ferrimagnetic phases are
frustrated with zero temperature entropy which does net vanish and with a non-trivial value of the overlap between the degenerate ground states. In section 4 we study a simplified version of the model (a constrained annealed model [3() which retains the ferrimagnetic order without
taking into account the frustration. The advantage is that in this case the phase diagram can be computed in analytic way for ail T, without Montecarlo simulations. In section 5 the reader can find a summary and a critical discussion on our results.
2. The model.
The Hamiltonian of our model is
H ~ J, «, ce,
~
À ~ «, «~ (2.1)
~
N
i 1><
where ~l is a positive infinite range coupling and f are independent identically distributed random variables which assume the two values f ± J with equal probability. The partition function is a random variable depending on the (.() realization of the sy~tem,
Z~ (p. (( j = Tr e~ ~"
=
~ exp lp jj f «, «~ ~
+
(
(~«~ ~~ (2.2)
j,,, ~ ii , ,
~
and the quenched free energy iç defined as
f'= lim @ (2.3)
N-~
flN
where A is the average of an observable A over the disorder probability distribution.
in the thermodynamic limit N
- m, ail the disorder realizationç (with the exception of a çet of zero J-probability measure) have the same free energy. I.e. they self-average to the
quenched average Il 4(,
lim In Zw
= 1'. (2.4)
N-za
PN
The free energy can be numerically computed through the product of random transfer matrices
[5], since (2.2) can be transformed in an integral over an auxiliary variable ~fi. Neglecting the
factors which vanish in the thermodynamic limit, it reads
+ce @~N
Z~
=
d~fi Z~ e ~ (2.5)
-
oe
with
Z>i> = £ exP (£P-Ii WI WI
+ + (P/l1'"~
4~«1) 1?.6)
, _~~ ,
The quenched free energy is given by the saddle point estimate of the integral (2.5),
-pj=max A(~P)-~
(27)
<fi
2~Î
where
@
= lim In Z,~ = lim In fl T, (2.8)
N ce
N
N ce
N
,
is the maximum Lyapunov exponent of the product of random transfer matrices
exp(pJ, + (pA)~'~ ~fi exp(- pJ~ + (PA )"~ ~fi j
~'
exp(- pJ, (pA)~'~ ~fi) exp(pJ, (p,1)"~ ~fi) ~~'~~
corresponding to the random J~ing model (2.6). However, we can de~cribe the main features of the phase diagram without a numerical calculation by some qualitative arguments at zero
temperature, and by the analysi~ of an exactly solvable annealed model, as discussed in the next sections.
3. The ferrimagnetic phases.
Dur model obviously has some trivial phases : at high temperature T, the paramagnetic phase
of magnetization m 0 and, at low temperature and weak disorder (I small enough), the
ferromagnetic pha~e. On the other hand, when Il
=
0 and T
=
ù the system i~ in a spin gla~s phase with ni
= ù where the up or down position of a spin «, ii determined by the nature (ferro
or antiferro) of the coupling (. This phase is of antiferromagnetic type, since one con define an analogous of the staggered magnetization which is equal ta unity at T 0, even if the true
magnetization ni remains zero. in particular, the overlap cl" ~m ' jj WI «~~
N
between two equilibrium states a and f i~ cl
= ± in both ferromagnetic and glassy phases.
It is natural to expect that a third kind of phases might appear at intermediate values of
w mJlll and low temperature with a non trivial magnetic order, neither ferromagnetic nor
glassy. In other terms, as a consequence of the disorder, the ground states can have a magnetization ni # 0, # 1. Using a language borrowed from solid state physics, such a phase
of a disordered system can be called ferrimagnetic. Notice that in the following, we limit ourselves to consider states with magnetization ni mû, since the system is invariant for spin inversion, «~
- «, for ail i's.
in this section, we show that at T
= 0, the model exhibits three type of behaviours at varying
the disorder strength
1) ferromagnetic phase for OS
w ~ 5/12
2) ferrimagnetic phases for 5/l~ ~ w ~ l 3) spin glass phase for w
~
In particular, there is a first order phase transition at wj = 5/12 between ferromagnetism (mj = Il and fernmagnetism (ni~ =2/3). Moreover, there exists a whole spectrum of
ferrimagnetic phases with magnetization
ni,,=~~ fi=2.3, m (3.1)
ii+1 for disorder strength :
n(n + 3/2
~3 ?)
w e jw,,, w,~ il With mi,
"
~~~ ~ j ~~ ~ 2
1600 JOURNAL DE PHYSIQUE1 N° 11
as shown in figure1 At zero temperature, the calculation is possible because one has to
consider only the energy of the different configurations to determine the equilibrium stores.
The key idea is that the system has a ferrimagnetic state with magnetization m # which is energetically convenient if the disorder is net toc small, as a result of the competition between the antiferromagnetic nearest neighbor coupling J,
=
-J and the mean field couplings
A ~ 0.
z
Quenched
.9
Constr, ann
É ,f
~
Z
C O©
$OE
0.
~---___
i
to
Fig,
first rder tran~ition~ the and the n + 1
phase. The transition at w~ = 1 toward
the spin gla~~ phase i~ the only ontinuous one.
The ashed i~ the constrained
Here
tran~ition at w j, in
ppendix 1, the interested reader can find the fuit proof that an
ferrimagnetic hase~
existsas
ticipated by 3.1) and (3?).
For sake of suppose that initially mm J/A
T = 0 the infinite range interaction prevails and the equilibrium state i~ given by ail
spins
«~ = l for ail i s in (2.1), we see that the nergy ensity of the
E~= lim
~
=-é 133)
N-oeN 1
2
since
the
lim ~ jj ( =
N
At increasing w, some of the spins will flip in order to lower the energy due to the n-n-
interaction. TO illustrate the mechanism, figure ?a show~ a ferromagnetic state, where the first
spins to flip are «~ and «
jj~ since they have two talerai negative J,, SO that their jumps produce
the maximal energy loss. This consideration is general, as the first spins to flip are always the
spins with two talerai negative n-n- couplmgs. However, the situation is more complicated for the island of negative l's of size larger than two. For instance, as illustrated in figure 2b, in an island of 3 negative J,'s, only one of the two spins with two negative talerai n-n- couplings flips ; in an island of size 4, only two non-consecutive spins tlip to lower the energy.
a)
, ,
,
, i
i
,
2 1, 3 4 5 6 7 8 9 ',~ lO il 12
b)
, , ,
, , ,
, ,
,
2 ',~ 3 4 5 6 7 ',~ 8 9 ',~ 10 11 12
Fig. 2. De,cription of the transition tram the ferromagnetic to the first ferrimagnetic state. The fuit fine are the positive n-n- couplings.( +./ and the wrinkled fines are the negative one,./, -./. The
arrow, repre~ent the ~pin;, 2a) Disorder realization with Island; oi1w 2 negative./,',. At increa;ing
disorder ~trength w and for T=0, the first ~pins to flip are located in the middle of the
1
=
2 islands. 2b) Di;order realization with some long i,land~ oi negative.l's. The flipping spins are
distributed in an alternated way among the island, of negative couplings.
In general, immediately above a critical value wj, the antiferromagnetic order will
predominate in each island of k negative couplings ( = I, so that a number S~ É/~ of spins
are be down if k is even, and a number S~ = (É )/? if k is odd. Let us stress that the « even
»
island has only one spin configuration of minimal energy, unlike the
« odd » Island which has (É + 1)/2 equilibrium configurations of minimal energy since one negative n-n- coupling is
unsatisfied in the i,th site, with odd mi- w k.
Calling .V the total number of spins which flip in ail the islands, the resulting configuration
has magnetization
ni =
2 À
13.5)
JOURN~L DE PHYSIQUE -T 4 h'ii Ni>VEMHER iw4
1602 JOURNAL DE PHYSIQUE I N°
A moment of reflection shows that the number of islands of negative n-n- couplings is
NM +OIN"~j. Since the probability that an island is made of negative bonds is
(1/2)~, the number of islands of k n-n- coupling is
N~
=
(1/2)~ ~ (3.6)
and, if the antiferromagnetic order is present in ail the islands, the total fraction of spins down
can be easily estimated by
lim ~< jj (l/2/S~ =1/6. (3.7)
N-ce N
li
4
inserting (3.7 into 13.5), one has 2/3
w m w At T
= 0, to determine whether a ferrimagnetic
state m~ or the ferromagnetic one m
=
is of equilibrium, we should compare their respective energy densities which are given by
E(m)
= 4 J ~~ "~ m~ for 2/3
< ni < (3.8)
as shown in figure 3. E(m is a convex function of m (a parabola) so that ils minimum is given
by one of ils two extrema (either at m =1 or at m = 2/3) separated by a energy gap
AEWE~~~ max [E(2/3), E(1Ii where E~~~ is the maximum of E(ni) for me [2/3, Ii reached at ni = 2
w.
The two store have the same energy at w1 =
5/1~ where there is a first order phase transition,
since the magnetization has a discontinuous jump from mj = to m~ =
2/3.
However the first ferrimagnetic phase (n
=
~) cannot exist when the diwrder is toc strong.
Indeed, ils energy is obtained by inserting ni = ~/3 into (3.8), so that
Etern ~ J .1. (3.9)
0.49
-0.5 ~
~
i 2/3 0.7 0.7[ / 0.8 0.85 0.9 0.95
É
/ m
-0.51 /
/ /
' /
0.52
~, w= 0.35 <
w~
~w=5/12=w, .w=045>w
Fig. 3.-Energy density Ejm) » iunction of the magnetization m for the quenched model at A for different value~ of w. In the fir,t case (w 0.4
~ w 5/121 the energy minimum is reached
for m =1, in the second case (w mi ) the minimum i~ obtained bath for
m =2/3 and for
m = and in third ca~e (w 0.45
~ w for m 2/3.
On the other hand the spin glass phase with m
=
0 has energy E~ =
J, because ail the spins
are aligned with the n-n- random couplings. It follows that for w ~ 2/3, the first ferrimagnetic phase has higher energy than the spin glass phase. Actually, as shown in appendix 1, the first
ferrimagnetic phase does net minimize the energy already for w ~ w ~ = 7/12 (where a ùecond ferrimagnetic phase with m~ =
1/2 appears), while the spin glass phase prevails only for
w ~ l. In fact, a numerable infinity of different ferrimagnetic phases are exhibited by the quenched model for 5/12 ~ w ~ l with magnetization given by (3.1). This result is obtained by
a renormalization procedure which generalizes the argument used for the first ferrimagnetic phase. In the latter case, the flipping spins can be regarded as islands of length one, and in
appendix we prove that after the first transition the system is energetically equivalent to a
new one made only of islands with a number of spins n m 2, corresponding to fi positive
couplings. Note that we define the length of the island by the number of its spins and net by the number of its couplings. The renormalized system will have a transition which is due to the
spin flip of the islands of length n = 2. The mechanism can be repeated at every step,
transforming the fi-~ystem to a (n + 1)-sy~tem made of k mn + islands and the transition involves only the shortest islands, that flip as the single spins do in the first transition.
The transition from a ii-phase to the (n + 1)-phase is of a first order with an energy gap decreasing with n. The last transition toward the spin glass phase thus is continuous and one
has a critical point at w 1, T 0.
We can aise estimate the entropy and the overlap of a ferrimagnetic state at T
=
0. These two
quantities are indicative for the pre~ence of many different minima of the energy as
consequence of the frustration. In the ferromagnetic phase the system is completely ordered
and there are only two minima so that the entropy S(T=0)=0 and the overlap
cj(T
=
0)
=
±1. This is aise true for the spin glass phase. On the contrary, in the first
ferrimagnetic state there are islands of an odd number k of n-n- couplings where the spin have (É + 1)/2 possible configurations of minimal energy. Typically, in an odd island, the spins
follow an antiferromagnetic order except two neighbors which are parallel.
Using the number N~ of islands of k n-n- couplings (3.6), and considering only the
configurations with positive magnetization m =
2/3, the zero temperature entropy and overlap
can be obtained as sums which run on É', the odd values of k. The entropy is
S(T
=
OI
=
lim jj N~ In ~~ ~ = 0.034. (3. loi
,~ oe N
j ~
2
The thermal average of the overlap is defined as
(«(T)) iim '
~ ~-pi/~~-/~zi
,~ ~
N-no Z~ ~ ~T),
("" ~)
In the first ferrimagnetic phase at T
= 0, the calculation is simpler since one has to consider
only the configurations of minimal energy. The overlap has the same value for almost ail the disorder realizations, so that we get
~
(q(T
=
0))
= 1- lim jj N~ ~
~
jj ii -1'(
,~-mNj=i ((É'+')/2)~~~~j =
= jj ~~~ /,~ ~~ ~ ~
=
~
In ~=
0.895 (3. Il )
61
m~
+ 2
