Corollary to a GHS inequality

Volume 49A, number 2

PHYSICS LETTERS

26 August 1974

COROLLARY TO A GHS INEQUALITY H. FALK* Inst jruut voor Theoretische Fysica, Ri/ksuniversiteit Utrecht, Utrecht, The Netherlands Received 25 July 1974 For two Ising spin systems linked by a single ferromagnetic bond and satisfying the hypothesis ofthe GriffithsHurst-Sherman inequality, increasing any member of a subset of bonds in system 2 leads to a decrease of the pairspin covariance in system 1.

Introduction. In the process of establishing the concavity of the magnetization asa function of magnetic field for Ising spin systems with purely ferromagnetic, binary interactions, Griffiths, Hurst and Sherman [1] developed a powerful inequality which implies that the covariance between two spins is a non-increasing function ofthe magnetic field on any spin. An alternative proof of the inequality has recently been given by Lebowitz [2]. A corollary to the GHS result is the following: consider two, independent Ising spin systems satisfying the hypothesis of the GHS inequality. Connect the two systems by a single ferromagnetic bond from, say, spin ~ in system ito spin a~,in system 2. Then the effect0qofinincreasing Jpq between a~,and system 2 any is tobond decrease (not increase) any spin the covariance cov(or, a 5) between any two spins tsr’ a~in system 1. When combined with a Griffiths inequality [3], this corollary shows that~1cov(ar, a 5) is independent of Jpq for zero external field and temperatures above which the averages
*1

Long-term address: Physics Department, City College CUNY, NY, NY, 10031, USA. If the spins in system 2 are in zero field, it may be verified directly that the canonical ensemble average of any function of the configurations of the spins in system 1, only, is independent of all bond strengths relating to pairs of spins in system 2.

We expect (but have not proved) that the corollary is true for any bond in system 2; i.e., it is not restricted to bonds emanating from Corollary and proof Consider two sets N~(n = 1,2) of positive integers: N1

= { 1~2~ ,p—1}, N2 {p,p+l, ,M} (1) where 3 ~ p
...

—

i
—

~

—

iEN

h~a1

(2)

where ~ h1 ~ 0, a~= ±1.The Hamiltonians 1C1, ~2 refer toferromagnetic two independent Ising spinand systems eachfield with binary, interactions external parameters Now linkh1.the systems with a single, ferromagnetic bond j(l2) ~ 0 connecting spins 0p—1 and as,. The interaction Hamiltonian ~

~(l2)

=

3

0p

12 the Hamiltonian ~‘p_1 of the combined system is and —

~C=JC1 +~C +~C 2 12

(4)

.

The canonical ensemble average
~ Q exp(—~3~C)}/~exp(—jThC) a

$2

(5)

a

In commenting on the preprint of this letter, J. Groeneveld and, subsequently, Masuo Suzuki [4j kindly provided elegant proofs of this conjecture.

93

Volume 49A, number 2

PHYSICS LETTERS

where the absolute temperature

1,

~

T—(kB~3Y

=


(6)

—

J, 1

For i ~/=j the covariance of any two spins a,, a1 is definedby cov(a1, a1)

26 August 1974

(p+l ~
=h1

(p=i
~ =~~1 (/=p +1

M)

Corollary. Forr,sEN1,qEN2,q>p:

(12)

D cov(a~,a5)IaJpq ~ 0.

(7)

Before considering the proof recall that Griffiths’ inequality [3] requires that a 0 and h1 0. But for T> T~and all h~= 0,
~‘

a cov(ar, as)Iafpq

=

=h~

0

(8)

(j=p).

The transformed Hamiltonian ~iCis thus of the form for which the GHS inequality obtains and the J~,1,for / p +1 ,M, play the role of external field parameters. Furthermore, for r,sEN1 cov(a~,a5) = consequently

(13)

COV(tr~ts) —

3Jpq= a cov(ç, t 8hq ~ 0. cov(ar, G~)I 5)I

(14)

forProof all T>InT~ ~ 0, ofh1new = 0, EN1, tk,q where EN2. terms spinr,s variables tk = ± 1 for k = 1, 2, M write the transformation

a

a,

Th.W. Ruijgrok for their kind and constructive interest in this work and for the hospitality of the Instituux.

=

(1 = 1,

t,

2,

...

,

...

,

p)

I would like to thank J. Groeneveld and

t 1t,,

(i’p+1,p+2,...,M).

(9)

The 1C1 and ~12 are unchanged except for the replacement of each a, by t,; whereas,

References [1] R.B. Griffiths, C.A. Hurst and S. Sherman, J. Math. Phys.

~I 2

where

94

i
j..t~t.— ~

-~

~

j~N2

h.p.

(10)

11 (1970)790. 12] J.L. Lebowitz, Commun. math Phys. 35 (1974) 87. 13] R.B. Griffiths, J. Math. Phys. 8 (1967) 484, Theorem 5. See also, M. Suzuki, J. Math. Phys. 14 (1973) 837 and 1088. [4] 1. Groeneveld and Masuo Suzuki, private communication.