A CONDITION EQUIVALENT TO FERROMAGNETISM FOR A GENERALIZED ISING MODEL

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1 This research was partially supported by the National Science Foundation under grant No. GU-2059 and the U.S. ir Force Office of Scientific Research under grant No. FOSR-68-l4l5. CONDITION EQUIVLENT TO FERROMGNETISM FOR GENERLIZED ISING MODEL by DOUGLS G. KELLY Department of Statistics University of North Carolina at Chapel Hill Institute of Statistics Mimeo Series No. 668 FEBRURY 1970

2 CONDITION EQUIVLENT TO FERROMGNETISM FOR GENERLIZED ISING MODEL OOUGLS G. KELLY JBSTRCT In the real algebra R(G) over a certain finite group, the operator exp is defined. condition is stated on exp J which is necessary and sufficient for J in R(G) to be nonnegative (where J is viewed as a function G + R). Physically, this amounts to a condition on the correlations of a generalized system of Ising spins which is necessary and sufficient for the ferromagnetism of the system. KEv WORm ND PHRSES Ising ferromagnet, correlations, real group algebra.

3 CONDITION EQUIVLENT TO FERROMGNETISM FOR GENERLIZED ISING MODEL DOUGLS G. KEl.LY INTRODUCTION. Let N {l,,n} and let {o1,,on} be a generalized system of Ising spins with Hamiltonian H (1) where, for any subset of N, (2) and J() R (the real numbers) is the many-body potential. The system is called ferromagnetic if J() ~ 0 for each nonempty S N. In reference [2], hereafter referred to as (KS), the following problem is stated: Find conditions on the correlations sufficient for ferromagnetism. ~() = lo\ which are necessary and \ I In that paper, a condition equivalent to ferromagnetism is given in terms of the Fourier transform of the function ~; Theorem 1 below restates this condition. In this note, we translate that into a condition on ~ itself, stated in terms of a sum of products ~(B1) ~(Bk)' each product bearing a coefficient equal to the permanent of a certain matrix. (The result is given as Theorem 2 below.) This research was partially supported by the National Science Foundation under grant No. GU-2059 and the U.S. ir Force Office of Scientific Research under grant No. FOSR-68-l4l5.

4 2,ltiSTRCT FORfIlJLTlOO. In (KS) it is shown that the numbers J() and ~() = (a ) are naturally associated with a certain real group algebra, as follows. Let G denote the group (2 N,6) of subsets of N under symmetric difference. (The identity is the empty set ~, and each member of G is its own inverse.) The real group algebra R(G) will be viewed here as the vector space of all functions G ~ R, with multiplication given by convolution: (f*g) () - I f(b)g(6b). Be:G (3) The multiplicative identity is the function 0 for which o(~) = 1 and o() = 0 when ~ ~. Define the operator exp on R(G) by exp f = o + f + f*f + f*f*f (4) (It was shown in Section 11 of (KS) that exp is well-defined; that is, that the series in (4) converges when applied to any e: G.) Now if the "many-body potential" J() is viewed as a member J of R(G), then it happens that (exp J)() (exp J)«(I) (5) «KS), equation (11.13»; moreover, judicious choice of J«(I) (whose value does not affect the physical nature of the system) will guarantee (exp J) (~) 1. Thus the problem stated in the introduction is reduced to the following: Given J e: R(G), find conditions on exp J which are necessary and sufficient for J() to be nonnegative for each nonempty e: G.

5 3 STTEM::NT OF THE RESLLT I For any two subsets and B of N, define B a = (-1) InBI, (6) where IRI denotes the cardinality of R. The identity = (7) is easily checked. Define, for subsets and B of N, C + = {C~N: a = l}, C = {C~N: a = -l}, +B+ and similarly for +B-, -B+, B. C 1 and C B = {C~N: a = a = l}, (ii) If and B are unequal, nonempty subsets of N, then each of the n-2 2 members. (This lemma appears in Section 3 of (KS).) Note that + and +B+ are subgroups of G and each of the other families is a coset. Thus we have The characters of G are the functions and so the Fourier transform of f R(G) is given by ~() = l a: f(b). B G (8)

6 4 Now let J e R(G) and define, for e G, ~() = (exp J)(). (9) From Theorem 12.1 of (KS), from the Proof of 12.1, and from the statement following the Proof, can be gleaned THEOREM I. For R;:", J (R) ~ 0 if and only if II + " ~(E) EeR II FeR "~(F) (10) To interpret (10) in terms of the function ~, we adopt the following notations: Fix ReG, R;:,,; By Lemma 1, n-1 k = 2 ; let R- = {F 1,,F k }. For any unordered k-tup1e ~ = {B 1 } of (not necessarily distinct) members of G, define and let ~I be the number of permutations ~ of {l,,k} for which {B~l,,B~k} = {B 1,.,B k } For example, if k = 8 and ~ = {,,,B,B,C,D,E}, then ~~ = ~C~D~E and ~I = lso define the k x k matrix of ±l's M = ~,R (12) THEOREM 2, J(R) ~ 0 if and only if l ~ per (M~,Rh(Bl) ~(Bk) ~ 0, ~~=R (13)

7 5 the sum extending over unordered k-tuples ~ = {B 1 } of members of G satisfying 6~ = R. PROOF OF THEOREM 2. With the notation we have established, (10) is equivk E E = i R II L 0B 1T(B ) i i=l B G i 1 (14) L L BE 1 E k 1T(B1 ) 1T (B k ) = B B G 1 k 1 Bk G Similarly = B B G 1 k 1 Bk G OR L L BF 1 F k 1T(B1 )... 1T (B k ) (15) Now (14) can be rewritten E = R (16) the sum extending over all unordered k-tup1es ~ = {B 1 }, where (17) Here ~ ranges over all distinguishable permutations of {B 1 }. Similarly, (18) where (19)

8 6 Now Lemma 2 says that for any i = l,,k, (20) Thus for each i = l,.,k, = ~ E1!1F i B l. o,j,b jj <p 'I' 1... (21) That is, we have B jj... (22) From (22) it follows that (i) If!1jJ = ~, (ii) If!1jJ = R, then B = jj jj' then B = -. jj jj If!1jJ = where ~ ~ and ~ R, then Lemma 1 implies that exactly Fl Fk half of the numbers 0,,o are +1 and the other half are -1; so (iii) If jj ~ ~ or R, then B jj = jj = O. Combining (16), (18), and (i) - (iii) above, we see that ER-O R ~ 0 if and only if L 1T (B ) 1T (B ) ~ O.!1jJ=R jj 1 k (23) Finally we note that if instead of (17) we form, for a fixed jj,... (24)

9 7 the sum extending over all permutations of {l,..,k} rather than over distinguishable permutations of {B 1 }, we obtain ~I~. But (24) is exactly per(m R); thus Theorem 2 is proved. ~, REMRKs. For the case n 2, condition (13) reduces to the Griffiths inequalities (see [1] and (KS». That is, if we denote {1} by and {2} by B, then Theorem 2 gives J() ~ 0 iff ~(~)~() ~ ~(B)~(B) J(B) ~ 0 iff ~(~)~(B) ~ ~()~(B) J(N) ~ 0 iff ~(~)~(N) ~ ~()~(B). For n ~ 3, of course, (13) is not only unlike the Griffiths inequalities, it is also quite unwieldy. The evaluation of permanents of matrices of ±l's is a subject that seems to have received scant attention. REFERENCES 1. R. B. Griffiths: Correlations in Ising Ferromagnets. I and II. J. Math. Phys. 8 (1967) D. G. Kelly and S. Sherman: General Griffiths' Inequalities on Correlations in Ising Ferromagnets. J. Math. Phys. 9 (1968)