This research was partially supported by the National Science Foundation
under Grant No. GU-2059.
CONCAVITY OF MAGNETIZATION AS A FUNCTION
OF EXTERNAL FIELD STRENGTH FOR ISING FERROMAGNETS
By
DOUGLAS G. KELLY
Departments of Mathematics and Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 635
JULY 1969
/
CONCAVITY OF MAGNETIZATION AS A FUNCTION
l
OF EXTERNAL FIELD STRENGTH FOR ISING FERROMAGNETS
by
Douglas G. Kelly
Departments of Mathematics and Statistics
University of North Carolina
ABSTRACT
For an Ising ferromagnet with
~ J .. a.a
lJ 1 j
Hamiltonian be
-
i, j).
s = a
Define
As a consequence:
i<j
+ a
l
when
J
2
l
n
n
~
-
a ,a , ... ,an' let the
spins
i=l
J.a.
1
2
l
(with
1
J'
1
j
> 0, J
-
n
= J
2
= ••• = I
= H, the magnetization
n
<s>
H.
Introduction.
The Ising model we consider here is the following:
the index set
figurations
( "up") or
N
a = (a , ... ,an)' where each
l
-1
and
J
let
{1,2, ••. ,n}; we consider the space of all
a.
2
n
N
denote
spin~-
is allowed the values
1
ij
for each pair
(i ,j )
J.
1
for each
of distinct members of
o -<
J .. = J..
lJ
<
00,
Jl -
0 < J. <
-
00.
1-
The Hamiltonian is the real-valued function on configurations whose
a
is
H
a
= - l.2,.J. l::<J.::.n
• J.. a . a
lJ 1 j-
-
n
l:: J. a. .
i=l
1
i
N, satis-
fying
value at
+1
( "down") •
Suppose that we are given extended real numbers
in
for all
+ a
+
is a concave function of the external field strength
1.
> 0
i -
1
~his research was partially supported by the National Science
Foundation under grant no. GU-2059.
2
The Gibbs probability on the space of configurations is defined by
where
(4)
k
S = (kT)-l > 0,
being Boltzmann's constant and
T
the (absolute) temperature, and
Z is defined by
where the partition function
Z= ~exp(-SH).
o
0
The expected value of a random variable
X on this space is called
its thermal average and is indicated by angular brackets:
-1
(6)
<X> = Z
8X(0)exp(-SHo)'
The average magnetization per spin is defined by
M
= n -1< s>,
where
(8)
+ ••• +0.
n
It has been conjectured that in the special case in which all
are equal to
In a paper
1
H, M is a concave function of
H; that
.
1
2
3 M
s, -2-':' O.
3H
of the author and S. Sherman, hereafter referred to as
(KS), we saw (section 7) that concavity of
M(H)
is equivalent to
323
D = <s > - 3<s><s > + 2<s> .:. 0
and also that
n
(10)
D
= i=l
~
n
~
n
~ D. 'k'
j=l k=l ;LJ
where
(11)
<0 ><0.0 >
j
1 k
- <ok><OiOj> + 2<Oi><Oj><Ok>'
In this paper we prove the following
Theorem.
In the Ising 'mode'l described by (1), (2), and (3), D"
lJ k
<0
3
for all triples
(i,j ,k)
of (not necessarily distinct) members of N.
It is of course a corollary that
J
l
M(H)
is concave in
H when
= ... = J n = H.
Note:
in section
i, J, and
when
i, j, and
2.
k
7 of
k
(KS) we observed that
are not all distinct.
D"
lJ k
is never positive
So from now on we assume that
are fixed distinct members of
N.
Notation.
First we adopt the notation of
(KS), in which the model, more
general than that above, is as follows.
A of
N a constant
J
A
is given, with
o oS.
(12 )
Suppose that for each subset
JA
.s.
00.
Let the Hamiltonian be
where
A
( 14)
o
=.lE:IIAO 1••
Again define probabilities by (3), (4), and (5).
of
(KS)
Then equation (9.2)
gives
R
Z<cr >
=2
n
00
L
k=o
1
-k1 LJ
••• J A.. '
-"k
A1
(AI"" ,A )
k
the inner sum being taken over all ordered k-tuples
~ ~
(not necessarily distinct) subsets of N which satisfy
= R.
(Here
~
denotes symmetric difference:
R
The expression we shall use for
<0 >
multiplicity functions on the subsets of
assigning a nonnegative integer
such a function
(16)
~(A)
A
~
B
J~ = AgNJ~(A) ~
0;
(A
U
... ~ ~
B) - (A n B).)
is based on the notion of
N; that is, functions
to each subset
~,define
=
of
A of
N.
For
4
lJ! = A~N(lJ(A)!);
(18)
/:,lJ =
/:, AlJ(A).
ACN
'
this last expression denotes the symmetric difference of all the subsets of
N, each subset
A being taken
lJ (A)
With this notation, define, for subsets
II
R
=
~ L
times.
R of
N,
J
/:,lJ=R lJ! lJ
the sum being over all multiplicity functions
lJ
satisfying
/:,J:! = R.
Then equations (9.7) and (9.8) of (KS) give that
II
R = <0R>. II = 2-n Z > O.
-( 20 )
II
' rp
rp
(Note: II
is an infinite series, which converges as long as all J
A
R
II
R
are finite. But
is always between 0 and 1, and is increasing
IIrp
in each of the J A' So the quotient in (20) always makes sense.)
In the model given by (1), (2), and (3), all the
except for those
A having one or two elements.
JA
are zero
So the multiplicity
functions which we will consider will be mUltiplicity functions only on
the one- and two-element subsets of
edges of the complete graph on
N; that is, on the vertices and
{1,2, ... ,n}.
It will be helpful to regard such a function
drawing
lJ({a,b})
circling vertex a
for
n = 4
bounds between each pair
lJ({a})
times.
(a,b)
lJ
of vertices and
For example, the function
by
lJ({l}) = 2, 11({2}) = 1,
lJ({1,2}) = 2, 11({1,3})
lJ(A) = 0
= 3,
for all other
would be pictured as in diagram 1.
graphically by
lJ({2,4}) = 1,
A..=. N,
lJ
defined
5
2
3
4
Diagram 1.
"Graphic" representation
of ).l in the example.
For this example, we would also have
223
= JIJ2J12J13J24;
J).l
= 2!1!2!3!1! = 24;
).l!
1I).l
3.
= {1,3,4L
Reduction of the proof.
In what follows we shall abbreviate
and
"1I).l
TIep
by
ITo'
= {i,j ,k},"
(21)
We shall also take
"1I).l
IT{.
. k}' say, by
J.,J,
= ijk"
to mean
etc.
(11) and (20) give
3
_
2
- IT TI,IT'k
IT o D,J. jk - IT 0 IT J.J
. 'k - IT oIT.IT'
J. J k
o J J.
-TI IT IT
+ 2IT.IT.IT •
o k ij
J. J k
From (19) we have, for any subsets
R, S, T
of
N,
J J J
Yv n
R S T = lI~=R lIv=S lIn=T ).l!v!n!
IT IT IT
E
E
= ~~=RlISlIT
E
Thus
where
E
1.
v_<).l
lIV=R
E
1
n_<).l-v
(
lI!1=S n!v! ').l
IT .. ,
J.J k
6
(24)
A
).l
L: F
L:
v.5.).l !1.S.jJ-v ).lVn
Av= </> An= </>
=
L:
L:
L:
v.5.).l
Av= </>
F
v.s..lJ n.s..lJ-V jJV n
Av=</> An=k
+
L:
n~j.l-v
F
L:
lJV n
An=i
L: F
VllJ n.s..j.l-V jJV n
AV= i An=j
L:
F
v.5.lJ n.5.lJ- v lJVn
Av= </> An=j
L:
+ v<lJ
Av;:;"i
L: F
n.s..lJ-v lJV n
An=k
and
(25)
F
lJVn
1
= --..,----==-----,.nlvl(lJ - v - n)1
•
Regrouping:
R
L:
~
v.5.j.l
\1.1
Av=j or k
R
(26)
A
lJ
= v'::j.l
L:~vI
Av=</>
where
(27)
R
1
1
L:
nl(lJ - v - n)1 n.s..\.l-v, nl(j.l - v - n)1
An=</>
An=i
= n.5.j.l-V
L:
j.lV
--:-...,.---"~-""I'-::-
Let
( 28)
A
for an arbitrary subset
of
N
and multiplicity function
A.
Then
proposition 1 in section 9 of (KS), which is our main lemma, provides
that
( 29)
=
lJV
R
o
t
if
B
j.l-V
(i)
#
</>
L:
1
ifB
(i) = </>
. n.5.lJ-V nl (\.l - v - n)!
j.l-V
An=</>
Thus
( 30)
A
\.l
=
1
L:
L:
v.s..\.l n.5.\.l-v v!n!(j.l - v -Ii)!
Av= </>
An=</>
B
(i )=</>
jJ-V
1
L:
L:
v.5.\.l n<lJ-V v!nI(lJ - v
Av=j or k An=</>
B
(i)=</>
\.l-V
Replacing
\.l - v
by
A, we get
-
n)!
7
E
)..'::'J.l
L\A=ij or
B)..(i)=rj>
Reversing the order of summation:
A
l.l
1
=
n.::.X.::.l.l
L1)..=ijk
B).. (i)=rj>
E
1
n<)..<l.l (l.l - )..)!().. - n)!
L1)..=ij or ik
B)..(i)=rj>
So to prove
for every
(33)
D.~J'k -<
with
\.I
E
n.9.:~Y
°
(which is our theorem), it suffices to show that
L1\.1 = ijk
and every
1
<
(l.l-)..)!().. - n)! -
L1)..=ijk
B)..(i)=rj>
Now if B (i) =f rj>, then
n
in (33) are empty sums.
)..
~
E
~-r--=l_ _~ <
L1v=ijk
B (i)=rj>
v
l.l - )..
prove that for any
l.l
with
with
)..)!().. - n)!
B (i) = rj>.
n
Because of this,
v = ).. - n, as
O~v~\.l~n v!(\.1 - n - v)!
Finally, replacing
L1n = rj>, we have
B)..(i) =f rj>, so that both sums
So we assume that
we can rewrite (33), letting
(34)
with
1
E
(l.l
n..9'::'1.1
L1)..=ij or ik
B)..(i)=rj>
implies
n
n.::. \.I
~r--=l_ _""t""':'"
E
O.::.v'::'\.l-n
V!(l.l - n- v)!
L1v=ij or ik
B (i)=rp
v
\.I,
we see that it is sufficient to
L11.1 = ijk, we have
where
L = {v: v .::. l.l, L1v = ijk, B (i) = rj>},
v
R = {v: v .::. \.I, L1v = ij or ik, BV(i) = rj>}.
8
40
Proof of (35).
Now we view multiplicity functions graphically as in section 2
above.
We will say that
circled, Le. if
]1({a,b})
]1
]1({a}) ~ 1; and that
In fact,]1
> 1.
has a singleton at vertex
has
A chain beginning at
a
]1
]1({a,b})
if
has a bond
{a,b}
bonds between
a
and ending at
finite sequence of distinct bonds
a
b
a
is
if
and
b.
(or vice versa) if? a
{a, a l }, {al , a 2 }, ... , {a. .1' a }.' {a ,b}.
nn'
n
A cycle is a chain between aanda passing through at least one other
vertex.
A subset of N is connected (with respect to a given
]1) if
there is a chain between every pair of vertices in the subset; it is a
component of
]1
if it is not a proper subset of any connected set.
The components of
]1
are of course disjoint.
B (i) :f
]1
Now the condition
A ~]1
with
~A
= i;
this means
a chain beginning at
So if
taining
~v
i
= ijk
~
means that there is a function
]1
has either a singleton at
i
or
with a singleton at one of its vertices.
B (i) =~, then
v
and
v
has a component con-
i, and this component cannot have a singleton (or else
B (i) :f ~).
v
But
tons and bonds of
i
appears an odd number of times among the singlev, so
]1
must contain a chain beginning at
i
and ending at some other vertex,which .chain cannot be lengthened.
~v
= ijk,
that it is
this other vertex must be
j
or
k.
Since
Suppose for definiteness
j.
Then this component cannot contain
there must be either a singleton at
k
k; for in order that
~v
or a chain beginning at
having a singleton at one of its vertices.
(This is because
v
k
= ijk,
and
already
9
has a chain from
i
to
which cannot be lengthened.)
j
Summing up the results of the last three paragraphs:
I:!.\)
= ijk
and either
\)
B (i) = ~,then
and
\)
or
j
\)
has a component \)
containing
\)0
call this the odd cOII!Ponent. \)
\)n' all satisf'ying
~;
I:!.\)n =
k
i
or
and a singleton; we
j
will in general have other component s
we call them null component s.
In similar fashion we see that if
\)
containing
m
k, with no singletons; we call this the main component.
also has a component
then
if
I:!.\) = ij or ik
has a main component containing
i
and
and either
B\) (i) = ~,
or
j
k, with
no singletons; all other components are nUll.
Now for any fixed
in
\)
the set of all functions in
be the set of functions in
L
L, with main component
with main component
L
I
<
AEL\) AI(~ - A)I -
is of the form
B (i)
requirement
v
m
n
i) .
contains
=~
L
m
and
I:!.n =
R
v
E \) , k ~
m
j
is
let
and let
m'
.
L
\)
be
R\)
To prove (35)
m
I
n
~
A
v
\)
m
is vacuously satisfied for
A in
\)
m'
L,
AER\) AI(~- A)I
A = \)+
n, where
m
Similarly, every
n < A- v
in
\)
Suppose for definiteness again that
L\)
\)
R with main component
it suffices to show that for every
\)
0
f th e form
.
m
Then every
and
in
I:!.n = k; the
n<A-V
A = v
m
m
(since
+ n, wh ere
~.
Thus (36) can be written
~
I
L
I
< I
L
v 1 n<A-V nl(A - v - n)1 - v ! n<A-\) n 1 (lJ - \)
m
m
m m Z"n=km
m
I:!.n=~
n) .I
•
But this follows immediately from proposition I of section 9 of
End of proof.
(KS).
10
REFERENCES
1.
D. G. Kelly and S. Sherman, J. Math. Phys. 2, 466-484 (1968).
See also R. B. Griffiths, J. Math. Phys.
484-489 (1967).
~,
478-483 and