Correlation Inequalities for Spin Glass in one Dimension

0.1 MB·English

by  Pierluigi Contucci

correlation inequalities for spin glass
in one dimension
Pierluigi Contucci †, Francesco Unguendoli ∗
† Dipartimento di Matematica, Universita` di Bologna, e-mail: [email protected]
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2 ∗ Dipartimento di Matematica, Universita` Modena, e-mail: [email protected]

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Abstract
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We prove two inequalities for the direct and truncated correlation for the nearest-
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h
t neighboor one-dimensionalEdwards-Andersonmodelwith symmetricquencheddis-
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order. The second inequality has the opposite sign of the GKS inequality of type
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II. In the non symmetric case with positive average we show that while the direct
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correlation keeps its sign the truncated one changes sign when crossing a suitable
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line in the parameter space. That line separates the regions satisfying the GKS
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second inequality and the one proved here.
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1 Introduction and Results
:
v
i
X
r In a recent paper [CL] a correlation inequality was proved for spin systems with quenched
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symmetric random interaction in arbitrary dimension, extending a previous result for the
Gaussian case [CG]. That inequality yield results for spin glasses similar to those ob-
tained for ferromagnetic systems from the first GKS inequality [Gr, Gr2, KS] e.g. it gives
monotonicity of the pressure in the volume and bounds on the surface pressure. Other
inequalities were considered: in particular the extension to non-symmetric interactions
and possible versions of a second type GKS inequality.
In this work we study the d = 1 case with nearest neighboor interaction. In the
same spirit of the GKS systems no assumption of translation invariance is made on the
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interaction distributions and by consequence our results cannot be obtained by an exact
solution. We prove that both the inequality of the first type does extend to the non
symmetric case and that an inequality of the second type holds indeed in the symmetric
case. A similar result with a complete proof of inequalities of type I and II has been
obtained so far only in the Nishimori line [CMN, MNC].
Let us consider a chain with periodic boundary condition
N
H(σ,J) = − J σ σ
i i i+1
i=1
X
with σ = σ . The random variables J have independent distributions p(J ). Those
N+1 1 i i
fulfills one of the three following hypothesis, which will be called system I, II and III in
the remaining part of the paper:
I)
p(|J |) ≥ p(−|J |), ∀i and ∀|J | ∈ R+
i i i
II) the J are symmetric around a positive mean µ > 0:
i i
p(µ +|J |) = p(µ −|J |), ∀i and ∀|J | ∈ R+
i i i i i
In the case of discrete variables: J = µ ±J(i), p(µ +J(i)) = p(µ −J(i)) = 1/2, we assume
i i i i
that J(i) > µ (see below for further explanations) and we introduce the notations:
i
a = µ +J(i)
i i
−b = µ −J(i)
i i
a ,b > 0
i i
III) the J are discrete variables taking on values ±J(i) with J(i) > 0 such that:
i
where p = p(J(i)), q = p(−J(i)) ,
i i
and
α := (p −q ) ≥ 0 .
i i
i
Y
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Let ω the thermal average of the quantity σ σ , ω that of the quantity σ σ σ σ
h h h+1 h,k h h+1 k k+1
and Av[·] the average over the quenched disorder.
Our main results are:
Proposition 1.1 For all three systems:
Av[J ω ] > 0, ∀h = 1...N (1.1)
h h
Proposition 1.2 For systems I and III with α = 0:
Av[J J (ω −ω ω )] < 0, ∀h,k = 1...N, h 6= k (1.2)
h k hk h k
Proposition 1.3 For system III, with α > 0, the following properties hold:
∀l, there exists in the (J(l),α) quadrant, a curve α(J(l)) such that the quantity
Av[J J (ω −ω ω )] (1.3)
h k hk h k
changes its sign from negative to positive when crossing the curve α(J(l)) by increas-
ing α and such that on the curve α(J(l))
Av[J J (ω −ω ω )] = 0, ∀h,k = 1...N, h 6= k . (1.4)
h k hk h k
Moreover Av[J J (ω −ω ω )] is increasing in α along the J(l) = const lines.
h k hk h k
2 Proofs
We start by proving the following lemmata.
Lemma 2.1 System III can be rewritten as:
N−1
H(τ,K) = −K τ τ − J(i)τ τ (2.5)
N N N−1 i i+1
i=1
X
with:
N
K = J(N) sgn(J ) = ±J(N) . (2.6)
N i
i=1
Y
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Setting P = prob(K = J(N)) and Q = prob(K = −J(N)) we have:
N N
1+ (p −q )
P = i i i (2.7)
2
Q
1− (p −q )
Q = i i i (2.8)
2
Q
Lemma 2.2 Consider system II with discrete variables and assume that µ = 0 for at
h
least one h. Such a system can be rewritten as:
N
H(τ,K) = − K τ τ
i i i+1
i=1
X
where:
K = J = ±a , a > 0 (2.9)
h h h h
a > 0
i
K = (2.10)
i

b > 0
 i
the two cases having probability 1/2. 
Proof of Lemma 2.1
The proof is based on the Gauge transformation α = sgn(J ), for 2 ≤ j ≤ N
j 1≤i<j i
α1 = 1. Set τi = αiσi H is given by (2.5) with KN = J(NQ) Ni=1sgn(Ji). We have now to
N
compute the new probability measure for i=1sgn(Ji). CleQarly the expectation
N N Q N
Av sgn(J ) = Av[sgn(J )] = (p −q ) = P −Q .
i i i i
" #
i=1 i=1 i=1
Y Y Y
Proof of Lemma 2.2
Group the bond configurations in couples that only differ for the sign of J and Gauge
h
transform them using the same transformation of Lemma 3.1 What we obtain is:
K = |J | > 0
i i
K = J = ±J(h)
h h
Moreover, since p(K(h)) = p(J(h)) or p(K(h)) = p(−J(h)), p(K(l)) = 1/2 for all l.
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Introduce, for system III, the following shorthand notations:
C := cosh(K(i)) = cosh(J(i)); S := sinh(K(i)) = sinh(J(i))
i i
Proof of Proposition 1.1
First we prove the thesis for discrete variables (system II and III). The partition function
and correlation of an N spins chain with periodic boundary conditions can be written as:
Z = C + S (2.11)
i i
i i
Y Y
1
ω = S C +C S (2.12)
h h i h i
Z
" #
i6=h i6=h
Y Y
System III Using Lemma 2.1 one has:
Av [J ω ] = Av [K ω ] = K(h) Pω| −Qω| =
{J} h h (Kh) h h kh=k(h) kh=−k(h)
(cid:8) (cid:9)
= K(h) Q ω| −ω| +(P −Q)ω| ≥ 0
kh=k(h) kh=−k(h) kh=k(h)
(cid:8) (cid:2) (cid:3) (cid:9)
due to the first Griffith’s inequality for ferromagnetic systems.
System II (discrete variables) Since the pressure is a convex function of the µ ’s we can
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prove our theorem for µ = 0. If for some i J(i) ≤ µ the variable J takes positives values
h i i
and it doesnt influence the sign of the average. Now, using lemma 2.2 and observing that
P = Q = 1/2 and the J average is a linear combination of K average with all positive
h
remaining K , we have the thesis with the same steps as before. The extension to the
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continuos case is obtained by the usual method of integrating over the positive parts of
the J distributions.
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Proof of Proposition 1.2
For discrete variables (system III), using the standard hyperbolic expansion:
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ω = S S C +C C S ⇒
hk h k i h k i
Z
" #
i6=h,k i6=h,k
Y Y
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1
ω −ω ω = S S C +C C S C + S +
hk h k h k i h k i i i
Z2
( ! !
i6=h,k i6=h,k i i
Y Y Y Y
− S C +C S S C +C S =
h i h i k i k i
! !)
i6=h i6=h i6=k i6=k
Y Y Y Y
1
= (C S )· C2C2 +S2S2 −C2S2 −S2C2 =
Z2 i i h k h k h k h k
( )
i6=h,k
Y (cid:0) (cid:1)
C S
i i
i6=h,k
= Y
2
C + S
i i
!
i i
Y Y
If at least one of the random variables is symmetric we have: P = Q = 1/2; using lemma
2.1 one has:
K · (C S )
h i i
Av[K K (ω −ω ω )] = J(k) ·Av  i6=Yh,k  =
h k hk h k (KN) ( C + S )2
i i i i
 
 Q Q 
 1 1 1
= J(k)J(h) (C S )· − =
i i
2 ( C + S )2 ( C − S )2
i6=h,k (cid:26) i i i i i i i i (cid:27)
Y
= −2J(k)J(h) (C S )· Q i(CiQSi) < 0Q Q
i i ( C2 − S2)2
i6=h,k iQi i i
Y
The extension to the continuous case is as above. Q Q
Proof of Proposition 1.3
Let α > 0 or equivalently P = 1+α > 1 > Q = 1−α.
2 2 2
We obtain analogously as before:
P( C − S )2 −Q( C + S )2
Av[K K (ω −ω ω )] = J(k)J(h) (C S ) i i i i =
h k hk h k i i ( C2 − S2)2
i6=h,k Q Q i Qi Q
Y
J(k)J(h) (C S ) Q Q
= i6=h,k i i ·{(P −Q)( C2 + S2)−2 (C S )}
( C2 − S2)2 i i i i i i i
i iQ i i
Q Q Q
The sign of the previous expression is, by inspection, the same as that of the curly
Q Q
parentheses:
g(α;{J}) := α( C2 + S2)−2 (C S )
i i i i
i i i
Y Y Y
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One obtains:
- α = 0 (zero mean spin glass) ⇒ g(α;{J}) < 0;
- α = 1 (ferromagnetic) ⇒ g(α;{J}) = ( C − S )2 > 0;
i i i i
Q Q
- for all J(l), g(α;{J}) is increasing function of α;
- Av [K K (ω − ω ω )] = 0 on the (J(l),α) plane curve with J(l) > 0 and
(Kh) h k hk h k
0 ≤ α ≤ 1 defined by:
2C S (C S )
α(J(l)) = l l i6=l i i (2.13)
C2 C2 +S2 S2
l i6=l Qi l i6=l i
Q Q
Theproofoftheinequalitiesforonedimensionalsystemswithfreeboundaryconditions
or for tree-like lattices is trivial since, due to the absence of loops the partition function
factorizes
Z = 2N cosh(λ J )
i i
i
Y
and by consequence the first inequality is fulfilled even without taking the average and
the second inequality reduces obviously to the equality to zero.
3 Comments
Weproved that aonedimensional spinglasssystem fulfillsafamilyofcorrelationinequali-
ties without the assumption of translation invariance for the interaction distribution. The
first inequality extends a similar one proved in [CL] for any lattice and any interaction
with zero mean value. Here we have shown that the inequality is stable by suitable defor-
mations of the zero mean hypotheses. The inequality of type II proved here shows that in
the zero mean case the truncated correlation function has the opposite sign of the stan-
dard GKS inequality i.e. the case of interactions with zero variance and positive mean.
We have moreover identifyed the line crossing which the truncated correlation changes its
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sign. It would be interesting to establish if an inequality of type (1.2) is fulfilled also in
higher dimensions (see [KNA]). In fact, as a straightforward computation shows in the
Gaussian case, if such an inequality holds then the overlap expectation would be mono-
tonic in the volume and several nice regularity properties would follow [CG2]. We also
mention that the inequality (1.2) doesn’t hold in general topologies as it was shown to us
by Hal Tasaki for a Bernoulli spin chain with an extra bond connecting two non adjacent
sites. Moreover a similar violation for the inequality (1.2) can be obtained in the case in
which the disorder, still having zero average, is non symmetric.
Acknowledgments. The authors thank Joel Lebowitz for many fruitful suggestions.
P.C. thank Cristian Giardina, Sandro Graffi, Frank Den Hollander and Hidetoshi Nishi-
mori for useful discussions.
References
[CL] P.Contucci, J.Lebowitz, Correlation Inequalities for Spin Glasses, Annales Henri
Poincare, to appear
[CG] P.Contucci, S.Graffi, Monotonicity and thermodynamic limit, Jou. Stat. Phys., Vol.
115, Nos. 1/2, 581-589, (2004)
[Gr] R. B. Griffiths, Correlation in Ising Ferromagnets, Jou. Math. Phys, Vol. 8, 478-483,
(1967)
[Gr2] R. B. Griffiths, A proof that the free energy of a spin system is extensive, Jou.
Math. Phys, Vol. 5, 1215-1222, (1964)
[KS] D.G.Kelly, S. Sherman: General Griffiths’ Inequalities on Correlations in Ising Fer-
romagnets, Jou. Math. Phys. 9, 466, (1968)
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[CMN] P.Contucci, S.Morita, H.Nishimori, Surface Terms on the Nishimori Line of the
Gaussian Edwards-Anderson Model Journal of Statistical Physics, Vol. 122, N. 2,
303-312, (2006)
[MNC] S.Morita, H.NishimoriandP.Contucci, GriffithsInequalities fortheGaussianSpin
Glass, Journal of Physics A: Mathematical and General, Vol 37, L203-L209, (2004)
[KNA] H.Kitatani, H.Nishimori, A.AokiInequalities forthelocalEnergyofRandomIsing
Models. Jou. of the Physical Society of Japan, Vol. 76, Issue 7, pp. 074711 (2007).
[CG2] P.Contucci, S.Graffi, On the surface pressure for the Edwards-Anderson Model,
Comm. Math. Phys. Stat. Phys., Vol. 248, 207-220, (2004)
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