I L NUOVO CIMENTO
VOL. 25 B, N. 2
11 Febbraio 1975
The Lavoisier Law (1) and the Critical Point.
M. CASSANDRO
Istituto di 2'isica d e l l ' U n i v e r s i t h - R o m a
Istituto N a z i o n a l e d i 2'isica .Yucleare - Sezione di R o m a
G. GALLAVOTTI (*)
l n s t i t u u t voor Theoretisd, e .Fysica K a t h o l i e k e Universiteit N i ] m e g e n - Ni~megen
(ricevuto il 3 Giugno 1974)
Summary. - - This paper contains mainly an exposition of Wflson's theory
of the Kadanoff renormalization group.
l.
-
Introduction.
BLEHER a n d SINAI in a r e c e n t w o r k on t h e hierarchical models (2) shed
some light on t h e feasibility of a rigorous a p p r o a c h t o t h e n o r m a l i z a t i o n g r o u p
t h e o r y of critical p h e n o m e n a . T h e a i m of this p a p e r is t o discuss, in t h e f r a m e
of a quite general model, t h e f o r m u l a t i o n of this t h e o r y in order t o t r y t o single
o u t a set of a s s u m p t i o n s on which t h e general v a l i d i t y of this p r o g r a m should rely.
T h e p a p e r is divided in t w o p a r t s : in t h e first we discuss for a general
m o d e l t h e basic ideas of W i l s o n ' s t h e o r y of t h e K a d a n o f f r e n o r m a l i z a t i o n g r o u p
a n d t h e associated scaling laws. W e p u t some effort in t r y i n g to isolate four
a s s u m p t i o n s which, if s u i t a b l y f o r m u l a t e d , should be c h e c k e d in models.
(1) (~We must lay it down as an incontestable axiom, that in all operations of art
and nature, nothing is created: an equal quantity of mattez exists both before and after
the experiment ... and nothing takes place beyond changes and modifications of these
elements ,~. A. L. LAVOISI]~R: Traitd gl~mentaire de chimie prdsentg d a n s u n ordre n o u v e a u
(Paris, 1789) (new edition, Paris, 1937).
(*) On leave of absence from Istituto di Fisica Teorica dell'UniversitY, Napoli.
(2) P. H. BLv,m ~ and JA. G. SINAI: C o m m . . M a t h . . P h y s . , 33, 23 (1973).
691
692
M. CASSANDRO a n d G. GALLAVOTTI
I n the second part we discuss the s-expansion and the related calculations
of the critical exponents in the case of the hierarchical model where the approximations can be controlled.
We shall only deal with Ising models on a d-dimensional square lattice Z ~
at zero external field and above the critical temperature. The t r e a t m e n t for
the general case (h, T arbitrary) can be followed along the same lines with
some minor changes and it is left out mainly for the sake of a simpler formalism.
As the reader will soon realize we are inspired b y the very clear exposition
of ~V]~G~ER (s) and b y the rigorous results of BLEHER and STZiAI (2) and, in some
sense, we are only p u t t i n g together the qualitative aspects of the two papers
without being able to be as general as the first or as rigorous as the second,
b u t we suggest a n u m b e r of conjectures, still u n p r o v e n even in simple models,
which m a y help the m a t h e m a t i c a l physicist to the understanding of the mathemetical aspects of the problem.
2. -
The
Ising
model.
~u shall call here Ising model a somewhat more extended class of models.
On each site S E Z a sits a spin r
cxD, q- oo). The energy of a configuration contained in a box A is given b y
(2.])
H ( ' % ; ) = -- ~:
_
~
l ~,..~(%,,
(~
..-, a~),
where the functions t(k)
I~,. ,r
1, ..., a~,) are defined for all k's and ~ , ..., ~ Z a
and are even in the a's and periodic over the simultaneous lattice translations
of all the labels ~1, ".., ~, (with a period which m a y depend on k). The functions ](*) are assumed to vanish if one of their arguments vanishes. F o r convenience we shall restrict the side L of the box A to be a power of 2, L = 2 N,
and the periods of the potentials will also be restricted to be powers of 2.
I f L = 2 ~ we shall write H(~)(_a) instead of H(A)(cr).
A model (H, z) is characterized b y an H of the above type and b y w h a t
we shall call the (( free-spin probability measure ~> (;r(a)da) which would be
the Gibbs distribution of each spin if H were zero. The measure ~r(a) da must
be even and normalized so t h a t f~r(a)da ~ 1.
The partition function for the model (H, z) will be, if A is a box with side 2 ~,
(2.2)
ZAH, ~) :fexp
[-- H(N'(_a)]1-[ z(a~) da~.
(3) F. J. WIgGlER: Phys. Rev. B, 5, 4529 (1972).
~eA
TIIE LAVOISIER
LAW AND THE
693
CRITICAL POINT
The a b o v e generality is necessary for t h e coming discussion e v e n ~hough we
m a y e v e n t u a l l y be interested only in the nearest-neighbour spin -1 Ising model
which is given b y #~)
= 0 unless k----2 and l ~ 1 - - ~2[ : 1
(2.3)
~(a) =
O(a+ z ) + O(a--z)
2
'
(2)
F o r m u l a e (2.1), (2.2) and the e x a m p l e (2.3) should clarify the meaning of the
above s o m e w h a t a b s t r a c t setting. We shall assume t h a t the functions
] (~) (o~, ..., a~) and ~(a) h a v e enough (~good ~ properties (like not too long range,
stability, etc .... ) so t h a t the t h e r m o d y n a m i c limit for t h e free energy and cor
relation functions of the model (H, ~) exist ~nd are b o u n d a r y condition inde
pendent.
I n other words, we assume t h a t the model (H, ~) defines a unique Gibbs
r a n d o m field.
I t is this last a s s u m p t i o n t h a t will force us to w o r k only for T > To.
_
3. - B l o c k spins.
Renormalization
group.
The block spins were introduced, in the context of the t h e o r y of the critical
point, b y KADA~OrF (4). This concept also arises natura~lly in p r o b a b i l i t y theory
(and precisely in t h e limit theorems) ~nd t h e connection between t h e limit
theorems of p r o b a b i l i t y t h e o r y and t h e t h e o r y of the critical point is p r o b a b l y
rather deep (2.s.6). L e t X--~(x~, ..., xa) be d integers and t h i n k of Z a us the union
of adjacent boxes (say squares) w i t h side 2"; we denote b y {x, n} the b o x
(3.~)
( ~ e z ~, 2 " x , < ~, < 2"(x, + 1), i = 1, ..., d},
where ~, are t h e co-ordinates of $ (integers).
Define
(3.2)
r(~)
~,@
~e{,.~}
2nclql2
where 0 < ~ < 2 is a n a r b i t r a r y (( scaling ~) p a r a m e t e r . W h e n
or n or x is deducible f r o m the c o n t e x t we shall simply ~ ' i t e
for v~.Q("). Consider a model (H, ~r) a n d the associated (unique)
Gibbs equilibrium state. Then t h e @ a n d the ~.e(~) are r u n d o m
well-defined p r o b a b i l i t y distributions in t h e equilibrium state
will be the (~~-field ~> and the (~v(~)-field ~) of (H, 7~).
the value of @
v~.Q or v~ or v(~)
infinite-volume
variables with
of (H, ~): t h e y
(4) L. [P. ]~ADA~OFF: Physics, 2. 263 (1966).
G. JONA LASINIO: The ~'enormalization groups: a probabilistic view, Padova prcprint (1974).
(6) G. GALLAVOm~I,H. J. F. K~OPS and G. C. MARTIN LSS: pre,print (t974).
(5)
694
M. CASS~NDRO and G. GkLLAVOTTI
W e ask t h e q u e s t i o n : does t h e r e exist u m o d e l ( H ' , g ' ) whose a-field is dist r i b u t e d as t h e vr
of (H, ~) ? I t is easy t o give a f o r m a l answer t o t h e a b o v e
question. L e t us first define
I~
(3.3)
a~
)
-"
Z,(tt, ~)
w h e r e {x, 1} is a u n i t cell (containing 2 d points) a n d H (t) is t h e H a m i l t o n i a n H
r e s t r i c t e d to t h e b o x (x, 1}. I n o t h e r words z ' is t h e d i s t r i b u t i o n t h a t t h e block
spins v,(~)qw o u l d h a v e if all t h e interactions b e t w e e n spins in different blocks
were t u r n e d off. T h e n t h e p a r t i t i o n f u n c t i o n Z~(H, :~) can be w r i t t e n as
(3.4)
zAH,
= ( I I ~'(v.)dv.ZN(H , ~, v)
J
~
-
,
w h e r e t h e p r o d u c t over x r u n s over t h e blocks of 2 d spins into w h i c h t h e b o x A
w i t h side 2 ~ is divided a n d Z~,(H, :~, ~_)is t h e p a r t i t i o n f u n c t i o n over t h e v o l u m e A
associated to t h e a-spin d i s t r i b u t i o n
(3.5)
i.e. to t h e r a n d o m field g e n e r a t e d b y t h e H a m i l t o n i a n (H, :~) with a c o n s t r a i n t :
t h e allowed configurations are those w h i c h precisely p r o d u c e t h e set of block
spins w h i c h appears in t h e label. W e can n o w define t h e effective H a m i l t o n i a n as
(3.6)
1 Zu(H, ~, v_)
fI'(~) = -
n ~
~ , o) '
t h e ~ indicating t h a t / t ' depends on N.
T h e a b o v e /t'(_v) c a n be w r i t t e n as
(3.7)
/~,(_~) = _ ~
~ 7q~)t
:
~'~ ~ . . . ~ V~k)
t~ ~a, ,~,~
w i t h t h e functions ]' v a n i s h i n g if one of their a r g u m e n t s vanishes. F o r i n s t a n c e
(3.s)
1'(')(~=) = in z~(//, ~, ~, v=)
z~,(//, :~, o) '
where ~), v~ is the set of b l o c k spins w h i c h are 0 for all boxes except t h e one
w i t h label x. Similar f o r m u l a e can be recursively f o u n d for t h e o t h e r f u n c t i o n s .
W e o b s e r v e t h a t t h e r~tio of p a r t i t i o n f u n c t i o n s in (3.8) can be i n t e r p r e t e d as
THE
LAVOISIEICr L/kW 2iND T H E
CRITICIkL F 0 1 N T
~
the average of a local q u a n t i t y in t h e Gibbs r a n d o m field, generated b y the
t I a m i l t o n i a n (H, =) (restricted to t h e box A), with a constraint on the allowed
configurations a: the allowed configurations are those which produce sets of
block spins w i t h zero magnetization. I f we assume t h a t also this modified
(H, =)-model has a unique equilibrium state, t h e n we should expect t h a t the
following limits exist:
(3.9)
/'(*)(v~,, ..., v~,) = lim ]'(*)(v~,, ..., vt,)"
.W--+~
We shall denote the constrained (H, ~)-model b y (H, ~)o.
The limiting values ](*) together with =' define a new model (H', u') and the
t r a n s f o r m a t i o n f r o m (H, ~) to ( H ' , =') will be denoted b y
(3.10)
(H', =') = KQ(H, =).
We assume t h a t also (H', ~') is a (~good model ~, i.e. it has a unique Gibbs
r a n d o m field which is precisely t h e same as the v(.~)Q-field.
The conditions on (H, =) under which the a b o v e assumptions hold
1) (H, ~)o has a unique Gibbs raltdom field,
2) (H', ~') has a unique Gibbs r a n d o m field which is precisely the v(.~Q-field,
are not known.
These two s t a t e m e n t s are not unreasonable and m a y be true for v e r y
general (H, ~) which h a v e a unique Gibbs distribution.
I n the following we are going to assume 1) and 2) for all (H, ~) of interest
and, also, for t h e associated models of the form K~(H, ~), n ~ 1, 2, .... W e
shall also assume t h a t the interchange of the limits involved in assumption 2) allows one to deduce f r o m t h e obvious relation
z.(f/, =) = z~(=, =, o)z~_fl~', =')
(3.11)
the nontrivial one
(3.12)
:1
t
!
/(H,=) = F ( = , = ) § ~ / ( = ,= ),
where
(3.13)
1
/(H, :~) = lim 2~r~ In Z~(H, :~)
and
(3.14)
1
/~(H, ~) = lim 2~~ l n Z ~ ( H , =, 0) = / ( ( H , =)._).
696
~ . CASSANDRO &nd G. GALLAVOTTI
Hence, b y iteration, if (H~, ~ ) =
(a.~)
t(~, ~) =
K~(H, ~z),
_L /((//~
~r~k_l)_0)
~- 2 - , . / ( = , , =,) = ,u, + 2-,,,/(=,, =,),
~=~
2 ~(~-~)
where /~ is defined here.
4. - The basic assumption for the scaling laws.
I n this Section we list a n d briefly discuss the basic assumptions (da) t h a t
are needed for the derivation of the scaling laws in zero field. These assumptions seem quite n a t u r a l and, therefore, i m p l y some understanding of the
mechanism behind the sealing laws.
I) The transformation Ke in (3.11), which depends on ~ only, is smooth
in the neighbourhood of a n y (H, ~) to which u unique Gibbs state is associated.
This assumption is r a t h e r loosely stated here and needs some comments.
First the neighbourhood of (H, =) t h a t appears in I) must consist of models
(/t~ ~) with (/t, ~) (( close ~ to (H, =) but still such t h a t the Gibbs equilibrium
distribution for (/t,~) is unique (s).
The second comment is a more explicit definition of what is m e a n t b y
smoothness ; we regard (H, a) as a sequence of <1coupling functions>> (~, {]~I ..~})
and denote b y (H, ~)d-z(3H, ~:~) the model corresponding to the sequence
z oz, t:~,. ,~-4-z ~, ,~}). Then we wish t h a t the model
K~((R, =) + z(a/f, ~=)) = K~(~, =) + z~(a~ ', ~=')
where w is a suitable nonnegative constant (i.e. independent of (SH, 8~))
and (SH', 8~') is an increment of <(order 1 ~> when z - ~ 0. This requirement
should hold for increments (SH, 8z) which are not (i too wild ~>. We do not
specify w h a t we really m e a n b y (<order 1 >) and prefer to leave it to t h e reader
to figure out the several possible interpretations of this sentence, which would
allow the formal manipulations performed below.
I n conclusion we can say t h a t <(smooth ~) has to be interpreted in the sense
t h a t K~ satisfies a ttSlder condition with constant w which is ~<direction >>-independent at ]east for the interesting directions. One would be t e m p t e d to say
(7) K. G. WILSON: Phys. Rev. B, 4, 3174, 3184 (1971).
(s) Hence if we consider the /sing model (flcHo,~o) at the critical temperature, the
models (flHo, ~o) will be in some <1neighbourhood ,> only if fl< tic"
THE
LAVOISIER
LAW
AND
THE
CRITICAL
697
POINT
that this statement should follow from the fact that the new coupling functions
can be expressed as t h e r m o d y n a m i c local averages in the constrained model
(H, x)o (cf. (3.9)) which should not ha~'e long-range correlations simultaneously
with (H, ~) (because t h e y are just different models) and therefore the local
averages should depend smoothly on the parameters of (H, x)o, i.e. of (H, ~).
Actually one would even say t h a t ~ = 1.
This is, however, a very dangerous statement (9) and it is n o t unlikely t h a t
there are some interesting instances in which (H, u) and (H, ~)o are simultaneously
critical. E v e n in this ease, however, nothing is against the H61der continuity
of the local t h e r m o d y n a m i c averages as functions of the variation of (H, ~)
and the only questionable thing is the direction independence of w (we remember,
however, t h a t we are here discussing only the h = 0 ease and so directions
which lead to uneven interactions and ~'s are not allowed).
I I ) Consider the models (H, ~) which give rise to a unique equilibrium
state and such t h a t there is a value ~ for which
(~.1)
0 < lira/(v(~)~ ~\
= A < oo
n---~ co
Then the limit
(4.2)
lim K~(H, ~) = (H*, ~*)
exists and is nontrivial (i.e. we exclude ~ * ( v ) = 5(v)).
One should specify the sense of (4.2) and there are several possible alternatives. The simplest would be to interpret (4.2) as saying t h a t the Gibbs
r a n d o m field of K~(H, 7~) has a weak limit as ~ -+ cx~ which is, also, a r a n d o m
field. The precise meaning of this statement is the following: let P(A*>(a~,..., aA).
9da~ ... da A be t h e probability distribution, in the Gibbs r a n d o m field of K~(H, :~),
of the a-spins of the box A, t h e n for all test functions with compact support
(4.3)
l i m fP~n)(a~, ..., aA)q~(a ~. . . . , aA) d a l ... d a A
=
=f_P~'(...)~(a,
..., a,,)dal ... da.~,
where P(A~)(al, ..., aA)da 1 ... da A is a probability measure which satisfies the
(9) There is a very nice counterexample due to KAST]~LEYN to the general validity
of such a statement: consider the two-dimensional Ising model with the constraint that
the block spins {x, 1} are all zero and furthermore the block configurations + + and + +
are forbidden; this model is exactly solvable and has the same critical temperature as
the original unconstrained Ising model! (KAsT]~L]~YN:private communication.)
~
M. CASS&NDRO
a n d G. G A L L ~ V O T T I
necessary a n d obvious c o m p a t i b i l i t y requirements which allow one to t h i n k of
it as a reduced p r o b a b i l i t y of a r a n d o m field (~0).
The existence of (4.2) in this w e a k sense is not a (<too strong )> r e q u i r e m e n t :
in fact (4.1) means (if assumptions 1) and 2) of Sect. 3 are accepted)
(~.4)
A (ffl' ""'
f p(n}
aA)a~dal "" daA .~---A-g~A < oo,
which allows, via simple compactness arguments, possibly b y passing to a
subsequence, to deduce t h a t t h e P(A~)(~a, ..., aA) da~.., d~A exist and satisfy
the c o m p a t i b i l i t y requirements.
So t h e real assumption, if one interprets (4.2) in t h e a b o v e minimal sense,
is t h a t t h e limit exists over t h e whole sequence and not only on subsequences.
N o t e v e r y t h i n g of w h a t follows would crumble if one only assumes (4.2) for
subsequenees: one would h a v e to introduce the set of a c c u m u l a t i o n points of
t h e sequence K o ( H , Jr) ~nd use t h e m instead of t h e simple point (H*, 3r*) (limit
cycles). We cMI t h e a b o v e compactness a r g u m e n t s t h e Lavoisier law (~); we
leave to t h e reader to e x p o u n d w h y (hint: (4.4) says t h a t the <(mass )> of t h e
p r o b a b i l i t y distributions P~)(...) is forced to s t a y in a finite region while n
varies) (~).
One m a y wonder w h y t h e exponent 2 in (4.1) plays such a special role.
Actually one hopes t h a t at least for short-range models with a ~ with c o m p a c t
support t h e same ~ defined b y (4.1) guarantees t h a t
0 < lim
<~) ~
for all choices of r.
M a t h e m a t i c a l l y however this is not a consequence of (4.1) and therefore
the special role of the e x p o n e n t o is an essential p a r t of a s s u m p t i o n I I ) . B u t
again, as in the above discussed case of the (( limit cycles ~, the theories t h a t
follows seem, at least partially, a d a p t a b l e to m o r e pathological situations.
U n f o r t u n a t e l y such a m i n i m a l i n t e r p r e t a t i o n of (4.2) is not enough for our
purposes: we shall also h a v e to interpret (4.2) as saying t h a t the limit r a n d o m
field, discussed above, can be generated as the unique equilibrium r a n d o m
field of a model (H*, ~r*) and the coupling functions of K~(H, ~r) approach the
coupling functions of (H*, ~r*) so t h a t t h e f o r m a l m a n i p u l a t i o n s which we are
going to m a k e on this basis are allowed.
Again I I ) is a r a t h e r well-posed m a t h e m a t i c a l p r o b l e m and here it does
(lo) I . I . CrHICHMANand A. V. SCOI~OCHOD:Theory o] Random Processes (Moscow, 1971);
DOBRUSHIN: Theory o/ prob. and applications. 13. 197 (1968); 25. 458 (1970).
(11) The existence of the above limit in a sense stronger than the weak one just described
can be rigorously established for noncritical cases el. ref. (6).
I ). L .
THE
LAVOISIER
LAW
AND
THE
CRITICAL POINT
699
n o t m a k e m u c h sense t o s p e c i f y i t m o r e b y e x p l i c i t l y s t a t i n g t h e c o n v e r g e n c e
c o n d i t i o n s (~).
I~et us n o w c o n s i d e r ~ m o d e l of t h e f o r m (flHo, no) w i t h fi a r e a l p a r a m e t e r fl>fl~.
A s s u m e t h a t flo is t h e i n v e r s e c r i t i c a l t e m p e r a t u r e f o r t h e m o d e l w i t h H a m i l t o n i a n Ho a n d f r e e - s p i n d i s t r i b u t i o n ~r0. B y t h i s w e m e a n t h a t t h e G i b b s
e q u i l i b r i u m s t a t e of (flHo, z0) is u n i q u e for fl<flr a n d n o t u n i q u e for f l > rio.
T h e n b y I ) , I I ) we c a n s a y t h a t
(4.5)
"
H ,o ~o) + (fl-- fi~)"A ,,
K~(flHo, ~ro)= Ke(fl~
w i t h A . of (~o r d e r i ,) for n f i x e d a n d
K~(~oHo, ~o) = (R*, ~*) + ~.
w i t h e. ~
0 (in s o m e sense t o b e specified).
I I I ) T h e m o d e l s K~(flHo, no) a n d K~(/3cHo, ~o) w i l l s t a r t b e i n g d i f f e r e n t
as soon as 2" e x c e e d s t h e c o r r e l a t i o n l e n g t h , which, t h e r e f o r e , is d e t e r m i n e d b y
t h e co n d i t i o n t h a t ( f i - - f i c ) ~ A . b e of (( o r d e r 1 )~ a t fl fixed. T h i s is a r a t h e r
s t r i c t i n t e r p r e t a t i o n of K a d a n o f f ' s i d e a t h a t , close t o t h e c r i t i c a l p o i n t , t h e
d i s t r i b u t i o n of t h e s p i n s i n s i d e a b o x w i t h side less t h a n t h e c o r r e l a t i o n l e n g t h
is e s s e n t i a l l y e q u a l t o t h e one t h e y w o u l d h a v e if t h e y w e r e a t t h e c r i t i c a l
p o i n t (').
I V ) T h e d i f f e r e n c e A , c a n b e t h o u g h t , if/~ ~ fi~, as b e l o n g i n g to a l i n e a r
s p a c e ~(H*, :~*) a n d t h e o p e r a t o r Kq l i n e a r i z e d a r o u n d (H*, ~*), a l o n g 5(H*, ~*),
h a s a s p e c t r u m w i t h o n l y one e i g e n v a l u e 2 > ].
Since it is w e l l -known t h a t t h e s p e c t r u m of a u o p e r a t o r d e p e n d s on t h e
s p a c e on w h i c h i t acts, we h a v e t o s p e c i f y t h a t ~(H*, zt*) m u s t b e t h e (~m i n i m a l ~
s p a c e w i t h t h e a b o v e p r o p e r t y , i.e. t h e m i n i m a l s p a c e c o n t a i n i n g tt~e r e l e v a n t
i n c r e m e n t s z].. I n g e n e r a l , b y t a k i n g a s p a c e ~(H*, s*) t o o large, one m a y
cause t h e l i n e a r i z e d o p e r a t o r t o h ~ v e a v e r y b a d s p e c t r u m w i t h as m a n y p o i n t s
as w a n t e d .
So t h i s ~ s s u m p t i o n seems q u i t e u n c o n s t r u c t i v e b e c a u s e its v e r i f i c a t i o n m i g h t
m e a n a g o o d k n o w l e d g e of t h e b e h a v i o u r of t h e m o d e l n e a r t h e c r i t i c a l
(12) The existence of a ~ such t h a t (4.1) holds is essentially equivalent to the requirement t h a t the p a i r correlation function in (H, n) behaves as a pure powt, r law for large
distances: if (a0an>___R-(a-z+") , then Q= 1 + (2--~t)/d. If this is not true, i.e. there
are, say, logarithmic corrections, then one has to choose ~ as a function of n but, usually,
lim q, will exist and one could t r y to give arguments similar to the ones used in the case
of constant Q. We do not enter here in this discussion.
700
M. CASSANDRO
and
G. G A L L K V O T T I
point. I t seems t h a t one could hope to a p p l y it to some explicit ease where
t h e r e are reasons to p u t severe limitations on t h e space (H*, ~*) (for instance
in t h e simplest a p p r o x i m a t i o n scheme, which is t h e lowest-order s-expansion (~a),
one guesses t h a t ~(H*, ~r*) is a certain 2-dimensional space (see end of Sect. 7
for a m o r e precise comment)).
I t is therefore quite i m p o r t a n t to h a v e some rigorous restrictions on ~(H*, ~*) :
such restrictions could, ior instance, be p r o v i d e d b y conservation laws. S o m e
examples on this point can be found in (~4).
5. - The sealing laws.
H e r e we rapidly derive t h e scaling laws at h--~ 0. W e show t h a t ~11 the
critical exponents can be expressed in t e r m s of t h e two constants @= 1 q- [ - ( 2 - - r l ) / d (see (6)) and (ln~2)/w.
The following a r g u m e n t s are m a i n l y heuristic since t h e n u m b e r of a s s u m p tions m ~ d e so far is so large t h a t , anyhow, we h a v e lost control of t h e errors.
We a p p l y to (4.5) K~ and we obtain
(5.])
where d~ is the operator Kq linearized around K~(fioHo, ~ro) ~_ (H*, zr*). Therefore using I I I ) we interpret the value i, such t h a t (fi-- fi,)wt~ = ], as linked
to the correlation length L(fi) b y
(5.2)
s(fl) oc e~ ~: ( f i - fl~)-w,o,~ ,
where t h e symbol cc is supposed to m e a n a s y m p t o t i c a l l y proportional in the
limit fi-~fi~. So the e x p o n e n t k n o w n as v is
(5.3)
~ _
ln~ 2 "
We h a v e already r e m a r k e d t h a t the exponent k n o w n as ~ is given b y @-~ ] qq- ( 2 - - ~ ) / 6 (el. footnote (12)). L e t us compute one m o r e exponent: consider
the e x p o n e n t called ~ in the literature (4). The starting point is (3.15) w r i t t e n
(13) K. G. WILSON and 5[. E. FISHER: Phys. Rev. Zett., 28, 240 (1972); M. E. FISHER:
Phys. Rev. Lett., 29, 917 (1972).
(14) G. GALLAVOTTI,H. J. F. KNOPS and It. VAN BEYEREN: preprint (1974).
THE
LAVOISIER
LAW
AND
THE
CRITICAL
701
POINT
for l = l - - ~ w i t h $ an a r b i t r a r y large c o n s t a n t a n d 1 g i v e n b y (5.2). ~Ve find
(5.4)
= ~. i ( ( ~ r ~ _ ~ ,
k=l
~ ~)o_)-I((/3o~_~, ~,~_~)~)+
~=1
~
U s i n g (5.1) a n d a s s u m i n g t h a t t h e free e n e r g y of t h e c o n s t r a i n e d m o d e l
(flH~_~, ~-~)o_ is a HSlder c o n t i n u o u s f u n c t i o n of t h e H a m i l t o n i a n p a r a m e t e r s
in t h e n e i g h b o u r h o o d of (fl~H~_l, ~-1)o a n d if we define its H61der e x p o n e n t
as 2 - - a 0 , we find t h a t (see (5.2), (5.3))
(5.5)
/(~, ~)~o (~- ~o)~F + ~ k~~ l [(~-fl~2 d k ~]"-~' ?
where P a n d G are some constants.
I f one assumes 2 ~ 2 d a n d either w = 1~ ~o = I (1~) or s o m e suitable rela%ions b e t w e e n 2~, w~ ~o, it follows t h a t
(5.6)
t ( ? H , ~) ~- F(fi -
~)~,
which in t e r m s of t h e usual definitions in t h e l i t e r a t u r e reads
dv=2--~.
H o w e v e r , if one does n o t m a k e t h e a b o v e assumptions, m a n y other results
are possible including a n explicit d e p e n d e n c e of e on t h e p a r a m e t e r w, w h i c h
would then become a third independent parameter.
6. -
The hierarchical
models.
This is a class of models (1~,1~) w h e r e t h e a s s u m p t i o n s 1), 2), I ) - I V ) c a n
be rigorously c h e c k e d (*). H o w e v e r these models h a v e l o n g - r a n g e periodic interactions. R o u g h l y t h e hierarchical models are c h a r a c t e r i z e d b y t h e p r o p e r t y
t h a t t h e r e is one value of r r such t h a t
(6.1)
Kq,(fiH, ~) ---- (fill, ~') ,
(ts) This would be more natural if the constrained Hamiltonian (Hz, z~)o were never
critical or had, in some sense, an oo-order transition (cf. Sect. 4, discussion of
assumption I)).
(16) F. J. DYsoN: Comm. Math. Phys., 12, 91, 212 (1969); 21, 269 (1971).
702
M. CASSANDROand G. GALLAVOTTI
i.e. t h e , interaction ~ p a r t of the H a m i l t o n i a n is i n v a r i a n t under Kq. (2). W e
shall not need in the sequel an explicit expression of H, which the interested
reader can find for instance in ref. (2.1e.17).
A m o n g the hierarchical models which give rise to the same ~' in (6.1)
there are particular models (i.e. particular z's) which h a v e a critical point fl~
and, at fi~, a s s u m p t i o n I I ) is verified with Q ~- Q'.
L e t us denote b y H e, a hierarchical H a m i l t o n i a n which in (6.1) gives rise
to a certain value ~'. BLEttER and SI~A~ h a v e b e e n able to s t u d y the critical
b e h a v i o u r of classes of models of the form (flH~, ~) in a range of the p a r a m e t e r Q
b e t w e e n 1 < ~ < ~ ~- s, where s is some positive n u m b e r (~). T h e y were n o t
only able to prove 1), 2), I ) - I V ) in precise m a t h e m a t i c a l t e r m s b u t also to go
often b e y o n d these assumptions and obtain a v e r y detailed picture of t h e neighbourhood of the critical point for such models (~8).
W e n o t e t h a t , for a hierarchical model with index Q, the recursion f o r m u l a
for t h e block spins is simply an integral operator on the (( free ~) p a r t of t h e
Hamiltonian
(6.2)
normalization
we call /~q.~ the operator defined b y (6.2). This e q u a t i o n under mild conditions on H (~ always has a Gaussian fixed point which is usually called t h e
(~ simplest ~ or (~trivial ~) fixed point.
7. -
The s-expansion.
The t r a n s f o r m a t i o n KQ seems to be hopelessly complicated in t h e general
case, p a r t i c u l a r l y when one comes to the discussion of t h e fixed point (H*, ~*)
a n d its t a n g e n t space ~(H*, ~z*) which is relevant for a particular initial m o d e l
(fill, ~). To deal with such question in, say, t h e I s i n g model one needs some
drastic a p p r o x i m a t i o n s whose physical meaning is not always clear.
There is, however, an interesting class of models, t h e hierarchical models,
for which it is possible to develop an a p p r o x i m a t i o n scheme which seems to
be u n d e r control.
The a p p r o x i m a t i o n scheme is the famous e-expansion, which we discuss
heuristically in the following pages and which has r e c e n t l y p r o v e n to be rigorous
for a certain class of hierarchical models (18).
I t is assumed t h a t the hierarchical model (flHe, ~), see preceding Section,
(1:) G. A. BAKER jr.: Phys.
(18) JA. G. SINAI: preprint.
T2ev. B, 5, 2622 (1972).
THE
LAVOISIER
.LAW A N D T H E
CRITICAL
POINT
703
has a critical point which is described as long as possible b y the simplest Gaussian fixed point ~* of eq. (6.2), together with the space ~(H*, u*) consisting
of the functions of the form z~*(v)](v), /~ L ,(~* (v)dv) or a subspace of it prescribed b y a priori arguments (~). <(As long as possible ~> means as long as
p r o p e r t y IV) holds.
I t can be checked t h a t the spectrum of the operator obtained b y linearizing
/{~ around ~* has the spectrum (independent of H (~) and fl)
(7.1)
).~ = 2'~(2'tcQ-z))" ,
n ~-- 1, 2, ...,
and the associated eigenfunctions are polynomials of degree 2n.
So, unless some conservation laws force ~(H*, ~*) to be orthogonal to the
eigenspace of ~ , the critical point cannot be described b y Wilson's t h e o r y
around the above fixed point if
(7.2)
~ > ~.
When ~ > ~ one assumes t h a t there is another fixed point of the form
(v) = ~, (v) exp [av~ Jr by~ + O(e~?s)],
(7.3)
where a, b are assumed to be of order ~----~--~, which describes, together
with the linear space of the functions of the form ~*(v)](v), /(v)eL,(~*(v)dv),
the t h e o r y of the hierarchical model (unless a priori reasons forbid it, as in
the Gaussian case; see also ref. (~4)).
I t is easy to show, consistently to order ~, t h a t such a fixed point indeed
exists and, if Q -- ~ = ~ > 0, has the right spectral properties.
Applying _~ to (7.3) one gets
(7.4)
|,~C=.1}
z'(v,) ---- e x p [ - - B ( L _~)]~ \ ~
cf
,
~, II,=~(~*(v,) exp [av~ d- by,]dr,),
where B(_v, _~) is the Hamiltonian restricted to a box of side 2 ~. Introducing
orthogonal co-ordinates ~, Yl, " ' , Y~'-I such t h a t
24--I
(7.5)
J=l
70~
M.
and
CASSANDRO
G.
GALLAVOTTI
f u r t h e r m o r e , we a s s u m e t h a t B is n e g a t i v e definite a n d t h a t (zg)
(7.6)
B(_~, _~) ---- -- ~o ~ +/~(Y, Y).
I f one retains o n l y t e r m s u p t o order C-, one o b t a i n s t h a t t h e c o n s t a n t s a a n d b
b e c o m e a', b' such t h a t
a ' = 2 a $2a-~ 6~"<y~'),b-~ b"-~z-~- O(ea),
(7.7)
b' = 2ag4b + 1854(<y '} -- <y-~}~)b 2 + O(ea),
where
2d--1 ~.@
--
_
2 a _ 1 2n
] ~ ( ~ (y,) exp [ay~] dy,) exp [fl(y, y)] ~ y,
(7.8)
(y2,,> =
i=1
f
i=1
2a--1
1-[ (z*(y~)exp[ay~] dye) exp[fl(y, y)]
a n d ~ is a k n o w n f u n c t i o n of m o r e c o m p l i c a t e d averages of t h e f o r m (7.8).
This leads to t h e fixed p o i n t
(7.9)
2
2
1894((Y4)~=o - - <Y )~=o)
where, indeed, ~, b are 0(e) w i t h e = 2~$4 - 1. T h e o p e r a t o r /~Q linearized
a r o u n d t h e fixed point ~z*(v) can be studied t o first order in e on t h e 2 - p a r a m e t e r
space (a, b) a n d one easily finds t h a t t h e eigenvalues of t h e linearized o p e r a t o r are
[
(7.10)
;tl = 2~"(1 + e/6) + O(e2),
2~ =
1--
E§
O(e2).
T h e linearization a r o u n d t h e trivial fixed point, w h i c h to order s can be identi-
(19) Notice that if B(v,_~) is a bilinear form in the v,'s a sufficient condition for (7.6)
2a
is given by ~ B ~ j = % (independent of i!), which is a condition of symmetry saying
J=l
that all the spins in the same block enter symmetrically in the Hamfltonian. The conditions on the signs of the eigenvalues of B is a kind of attractiveness condition which
is needed also to avoid divergences in (7.8).
THE
LAVOISIER
LA.W
AND
THE
fled with t h e choice a :
(7.11)
CRITICA.L
705
POINT
b : 0, gives of course
2g~- -- - 1 + e
o
So the e-expansion provides a tool for estimating t h e change of 21 as a function of e in t h e hierarchical model.
8. -
Conclusions.
W e h a v e tried to explore the basic assumptions of Wilson's t h e o r y of the
Kadanoff renormalization group in order to point out explicitly a n u m b e r of
conjectures which seem v e r y interesting for the m a t h e m a t i c a l physicist.
We hope t h a t the above v e r y heuristic and, sometimes, puzzling discussion
m a y help in understanding the relations between t h e beautiful intuitions of
KADAi~OFF a n d ~VI~SO~ and the m o d e r n t h e o r y of the limit theorems in probability theory.
$$*
I t is a pleasure to t h a n k H. vA~ BEYEREN and H. K ~ o P s for illuminating
suggestions.
9
RIASSUNT0
Questo lavoro contiene essenzialmente un'esposizione della teoria di Wilson del gruppo
di rinormalizzazione di Kadanoff.
3aROH glaBya3be H RpHTHqecKaH TO~Ca.
Pe3mMe (*). - - B 3 r o ~ p a 6 o r e paccMaTpnaaeTca T e o p n a B n n b c o n a Ztan rpylmbi nepertopMapoaKn Ka~laHosa.
(*) I-Iepeeec)eno peOatzque~t.
45 - II N u o v o
Ctmento
B.