This research was partially supported by the National Science Foundation under grant No. GU-2059 and the U.S. Air Force Office of Scientific
Research under grant No. AFOSR-68-l4l5.
A CONDITION EQUIVALENT TO FERROMAGNETISM
FOR A GENERALIZED ISING MODEL
by
DOUGLAS G. KELLY
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 668
FEBRUARY 1970
A CONDITION EQUIVALENT TO FERROMAGNETISM
FOR A GENERALIZED ISING MODEL
OOUGLAS G. KELLY
JBSTRACT
In the real algebra
exp
is defined.
sufficient for
function
G+
R(G)
over a certain finite group, the operator
A condition is stated on
J
in
R(G)
exp J
which is necessary and
to be nonnegative (where
J
is viewed as a
R). Physically, this amounts to a condition on the corre-
lations of a generalized system of Ising spins which is necessary and
sufficient for the ferromagnetism of the system.
KEv
WORm AND PHRASES
Ising ferromagnet, correlations, real group algebra.
A CONDITION EQUIVALENT TO FERROMAGNETISM
FOR A GENERALIZED ISING MODEL
DOUGLAS G. KEl.LY
INTRODUCTION. Let N· {l, ••• ,n} and let {o1, ••• ,on} be a generalized
system of Ising spins with Hamiltonian
(1)
H •
where, for any subset
A of N,
(2)
and
J(A)
€
R (the real numbers) is the many-body potential. The system
is called ferromagnetic if
J(A)
~
ence [2], hereafter referred to as
Find conditions on the correlations
sufficient for ferromagnetism.
0
for each nonempty
(KS),
A S N.
In refer-
the following problem is stated:
~(A) = loA\
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.
that into a condition on
~(B1) ••• ~(Bk)'
~
In this note, we translate
itself, stated in terms of a sum of products
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. Air Force Office of
Scientific Research under grant No. AFOSR-68-l4l5.
2
,ltiSTRACT FORfIlJLATlOO.
~(A) =
(a A)
In
(KS)
i t is shown that the numbers
J(A)
and
are naturally associated with a certain real group algebra,
as follows.
Let
N
(2 ,6)
G denote the group
difference.
of subsets of
~,
(The identity is the empty set
its own inverse.)
The real group algebra
G ~ R,
vector space of all functions
R(G)
N under symmetric
and each member of
G is
will be viewed here as the
with multiplication given by convo-
lution:
(f*g) (A)
I
-
f(B)g(A6B).
(3)
Be:G
The multiplicative identity is the function
o(A)
=0
when
A
for which
=
exp
on
=1
and
R(G)
by
o + f + f*f + f*f*f +
21
(It was shown in Section 11 of
(4)
31
(KS)
that
exp
is well-defined; that is,
that the series in (4) converges when applied to any
Now
o(~)
~ ~.
Define the operator
exp f
0
if the "many-body potential"
J(A)
A e: G.)
is viewed as a member
J
of
R(G), then it happens that
(exp J)(A)
(exp J)«(I)
«KS),
(5)
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),
and sufficient for
J(A)
find conditions on
exp J
which are necessary
to be nonnegative for each nonempty
A e: G.
3
STATEM::NT OF THE RESLLT I
a
where
IRI
B
For any two subsets
=
A
A and
B of
N,
define
(-1) IAnBI,
(6)
R.
denotes the cardinality of
The identity
=
is easily checked.
(7)
(ii)
If
A and
C
A+ =
{C~N:
a
A
=
{C~N:
a
A = -l},
=
{C~N:
a
A=1
A+B+
and similarly for
A and B of N,
Define, for subsets
A+B- , A-B+, A B
A
=
l},
C
C
and
C
B=
l},
.
B are unequal, nonempty subsets of
n-2
2
(This lemma appears in Section 3 of (KS).)
are subgroups of
a
N,
then each of the
members.
Note that
A+ and
G and each of the other families is a coset.
A+B+
Thus we
have
The characters of
G are the functions
Fourier transform of
f
~(A) =
l
B€G
€
R(G)
and so the
is given by
a: f(B).
(8)
4
Now let
J e R(G)
=
~(A)
and define, for
A e G,
(exp J)(A).
(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;:",
+
II
EeR
"~(E)
J (R) ~ 0
II
FeR
if and only if
"
~(F)
(10)
•
To interpret (10) in terms of the function
~,
we adopt the following
notations:
let
Fix
ReG,
R-
{F , ••• ,F }.
1
k
=
R;: ,,;
By Lemma 1,
(not necessarily distinct) members of
and let
~I
be the number of permutations
{B~l, ••• ,B~k}
~
k
x
k
M~,R
THEOREM
2,
of
define
~
of
{l, ••• ,k}
for which
= {B 1 ,.··,Bk }· For example, if k = 8 and
= {A,A,A,B,B,C,D,E},
define the
G,
= 2n-1 ;
= {B 1 , ••• ,Bk }
~
For any unordered k-tup1e
k
matrix of
~~=R
~~
= A~C~D~E
and
~I
= 3121111111.
Also
±l's
=
J(R) ~ 0
l
then
(12)
if and only if
~ per (M~,Rh(Bl) ••• ~(Bk) ~
0,
(13)
5
the sum extending over unordered k-tuples
G satisfying
6~
PROOF OF THEOREM
=
= {B , ••• ,B } of members of
1
k
= R.
2.
E =
R
~
With the notation we have established, (10) is equiv-
k
E
II
L 0B i 1T(Bi )
i=l B €G
i
1
L
L
B €G
1
Bk€G
E
1
°B
1
(14)
E
k 1T(B ) • •• 1T (B ) •
°B
1
k
k
Similarly
OR
=
L
B €G
1
L
Bk€G
F
1
°B
1
Fk
1T(B ) ... 1T (B ) •
°B
1
k
k
(15)
Now (14) can be rewritten
ER
=
the sum extending over all unordered k-tup1es
(16)
~
= {B 1 , ••• ,Bk },
where
(17)
Here
~
ranges over all distinguishable permutations of
{B , ••• ,B }.
1
k
Similarly,
(18)
where
(19)
6
Now Lemma 2 says that for any
= l, ••• ,k,
i
(20)
Thus for each
= l, .•• ,k,
i
~
=
B
jJ
...
E1 !1Fi
l. o,j,B
<p
'I'
1
(21)
That is, we have
B
...
jJ
(22)
From (22) it follows that
If
(i)
If
!1jJ =
~,
then
B = A •
jJ
jJ'
(ii)
If
!1jJ = R,
then
B
!1jJ
=A
where
A
If
AjJ
and
Fl
half of the numbers
(iii)
~ ~
= -AjJ .
jJ
A
~
Fk
0A , ••• ,oA
~ ~
or
R,
R,
are
then
BjJ
then
+1
Lemma 1
implies that exactly
and the other half are
-1; so
= AjJ = O.
Combining (16), (18), and (i) - (iii) above, we see that
ER-O
R
~
0
if and
only i f
L
!1jJ=R
A
1T
(B )
jJ
1
•••
1T (B
k
) ~ O.
Finally we note that if instead of (17) we form, for a fixed
...
(23)
jJ,
(24)
7
the sum extending over all permutations of
distinguishable permutations of
is exactly
REMARKs.
per(M
~,
R);
{B , ••• ,B },
1
k
For the case
B,
by
For n
~
3,
n· 2,
~IA~.
we obtain
But (24)
condition (13) reduces to the Griffiths
That is, if we denote
{1}
by
A and
then Theorem 2 gives
J(A)
~
0
iff
~(~)~(A)
~
~(B)~(AAB)
J(B)
~
0
iff
~(~)~(B)
~
~(A)~(AAB)
J(N)
~
0
iff
~(~)~(N)
~
~(A)~(B).
of course, (13) is not only unlike the Griffiths inequalities,
it is also quite unwieldy.
±l's
rather than over
thus Theorem 2 is proved.
inequalities (see [1] and (KS».
{2}
{l, .•. ,k}
The evaluation of permanents of matrices of
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) 478-489.
2.
D. G. Kelly and S. Sherman: General Griffiths' Inequalities on
Correlations in Ising Ferromagnets. J. Math. Phys. 9 (1968)
466-484.