Renormalization group and critical phenomena

Review of Phys.Rev.B4(1971)9 3174,
by C Sochichiu
Renormalization Group and Critical Phenomena
by K.G. Wilson
Monday, June 17, 13

The paper
There are in fact two papers...
Monday, June 17, 13

The paper
6We shall use the notation of the Anderson model in
this paper; the results obtained are identical with those
based on the Wolff model.
M. T. Baal-Monod and D. L. Mills, Phys. Rev. Let-
ters 24, 225 (1970).
A. A. Abrikosov, Physics 2, 5 (1965); B. Roulet,
J. Gavoret, and P. Nozieres, Phys. Rev. 178, 1072
(1969).
9R. A. Weiner, Phys. Rev. Letters 24, 1071 (1970).
~06. Baym and L. P. Kadanoff, Phys. Bev. 124, 287
(1961).
~~The parquet diagrams are only a formal method of
a narrow low-frequency mode.
'4The solution to the Suhl vertex equation is roughly
proportional to 6~,p at high temperatures (see M. J.
Levine and H. Suhl, Ref. 4).
~This shifts the effective location of the d-state reso-
nance to & =0 (see Bef. 5).
~
A similar calculation of the susceptibility in a local-
moment model has been reported by J. A. Appelbaum and
D. B. Penn, Phys. Rev. 8 3, 942 (1971).
J. B. Schrieffer and P. A. Wolff, Phys. Bev. 149,
491 (1966).
PHYSICAL REVIEW B VOLUME 4, NUMBER 9 1 NOVK MBER 1971
Renormalization Group and Critical Phenomena.
I. Renormalization Group and the Kadanoff Scaling Picture*
Kenneth G. 'Wilson
Laboratory of Nuclear Studies, Cornell University, Ithaca, Net York 14850
(Received 2 June 1971)
The Kadanoff theory of scaling near the critical point fox an Ising ferromagnet is cast in
differential form. The resulting differential equations are an example of the differential
equations of the renormalization group. It is shown that the Widom-Kadanoff scaling laws
arise naturally from these differential equations if the coefficients in the equations are ana-
lytic at the critical point. A generalization of the. Kadanoff scaling picture involving an "ir-
relevant" variable is considered; in this case the scaling laws result from the renormaliza-
tion-group equations only if the solution of the equations goes asymptotically to a fixed point.
The problem of critical behavior in ferromagnets
(and other systems) has.long been a puzzle. ' Con-
sider the Ising model of a ferromagnet; the parti-
tion function is
zoic a)=E exp zZEs, s,.;+esp),(s)
where E= —J/kT, Jisa coupling constant, sl is the
spin at lattice site n, P& is a sum over nearest-
neighbor sites, and h is a magnetic field variable,
The spin s; is restricted to be + 1; 5, ~,&
means a
sum over all possible configurations of the spins.
T is the temperature, and k is Boltzmann's con-
stant. The partition function is a sum of exponen-
tials each of which is analytic in K and k. There-
fore one would expect the partition function itself
to be analytic in K and h. In fact, however, the
partition function is singular for K=0, and h=O,
where K, is the critical value of K. To be precise,
the singularity occurs only in the infinite-volume
limit, in which case one calculates the free-energy
density
J'(Ã, k) =lim —lnZ(K, k),
1
(2)
y V
Monday, June 17, 13

The paper
6We shall use the notation of the Anderson model in
this paper; the results obtained are identical with those
based on the Wolff model.
M. T. Baal-Monod and D. L. Mills, Phys. Rev. Let-
ters 24, 225 (1970).
A. A. Abrikosov, Physics 2, 5 (1965); B. Roulet,
J. Gavoret, and P. Nozieres, Phys. Rev. 178, 1072
(1969).
9R. A. Weiner, Phys. Rev. Letters 24, 1071 (1970).
~06. Baym and L. P. Kadanoff, Phys. Bev. 124, 287
(1961).
~~The parquet diagrams are only a formal method of
a narrow low-frequency mode.
'4The solution to the Suhl vertex equation is roughly
proportional to 6~,p at high temperatures (see M. J.
Levine and H. Suhl, Ref. 4).
~This shifts the effective location of the d-state reso-
nance to & =0 (see Bef. 5).
~
A similar calculation of the susceptibility in a local-
moment model has been reported by J. A. Appelbaum and
D. B. Penn, Phys. Rev. 8 3, 942 (1971).
J. B. Schrieffer and P. A. Wolff, Phys. Bev. 149,
491 (1966).
PHYSICAL REVIEW B VOLUME 4, NUMBER 9 1 NOVK MBER 1971
I. Renormalization Group and the Kadanoff Scaling Picture*
Kenneth G. 'Wilson
Laboratory of Nuclear Studies, Cornell University, Ithaca, Net York 14850
(Received 2 June 1971)
The Kadanoff theory of scaling near the critical point fox an Ising ferromagnet is cast in
tion function is
means a
T is the temperature, and k is Boltzmann's con-
stant. The partition function is a sum of exponen-
density
1
(2)
y V
PHYSICAL REVIEW 8 VOLUME 4, NUMBER 9 NOVE MBER 1971
II. Phase-Space Cell Analysis of Critical Behavior*
Kenneth G. Wilson
L aborato~ of Nuclear Studies, Come/l University, Ithaca, Near York 14850
Qeceived 2 June 1971)
.A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the
generalization the spin s~ at a lattice site n can take on any value from — to ~. The interac-
tion contains a quartic term in order not to be pure Gaussian. The interaction is investigated
by making a change of variable sa=P g (n)s, where the functions g (n) are localized wave-
packet functions, There are a set of orthogonal wave-packet functions for each order-of-mag-
nitude range of the momentum k . An effective interaction is defined by integrating out the
wave-packet variables with momentum of order 1, leaving unintegrated the variables with mo-
mentum &0.5. Then the variables with momentum between 0.25 and 0.5 are integrated, etc.
The integrals are computed qualitatively. The result is to give a recursion formula for a se-
quence of effective Landau-Ginsberg-type interactions. Solution of the recursion formula gives
the following exponents: g=0, y=1.22, p=0. 61 for three dimensions. In five dimensions or
higher one gets q=0, y=1, and v=2, as in the Gaussian model (at least for a small quartic
term). Small corrections neglected in the analysis may make changes (probably small) in the
exponents for three dimensions.
I. INTRODUCTION
In Paper I of this series' the Kadanoff picture of
scaling for the Ising model was discussed. Kada-
noff considered the problem of the critical behavior
of the Ising model. 3
He proposed that the critical
behavior could be understood in terms of the effec-
ponents within the Kadanoff picture; the best one
can do is to derive the scaling laws' which relate
all the critical exponents to two unknown param-
eters.
Is there some generalization of the Kadanoff
picture which can be derived from the exact parti-
tion function of the Ising model? This problem millMonday, June 17, 13

The main idea
Scaling symmetry if treated inﬁnitesimally,
can reveal the properties of the system at
criticality
Based on a “wrong” Kadanoff’s idea…
...which still is useful and makes sense
Monday, June 17, 13

The system
Ising model of a ferromagnet partition
function
coupling constant (hopping
parameter) is the spin at the lattice site
is the magnetic ﬁeld
Every contribution to Z(K,h) is analytic, so
one expects Z to be analytic too…
In fact, however...
arise naturally from these differential equations if the coefficients in th
lytic at the critical point. A generalization of the. Kadanoff scaling pictu
relevant" variable is considered; in this case the scaling laws result fro
tion-group equations only if the solution of the equations goes asymptotic
tion function is
means a
T is the temperature,
stant. The partition
tials each of which is
fore one would expec
to be analytic in K an
partition function is
where K, is the critic
the singularity occur
limit, in which case
density
J'(Ã, k) =lim —l
1
y V
relevant" variable is considered; in this case the scaling laws result
tion-group equations only if the solution of the equations goes asymptot
tion function is
means a
T is the temperature
stant. The partition
tials each of which
fore one would exp
to be analytic in K
partition function i
where K, is the cri
the singularity occ
limit, in which cas
density
J'(Ã, k) =lim —1
y V
oblem of critical behavior in ferromagnets
systems) has.long been a puzzle. ' Con-
Ising model of a ferromagnet; the parti-
on is
a)=E exp zZEs, s,.;+esp),(s)
= —J/kT, Jisa coupling constant, sl is the
ttice site n, P& is a sum over nearest-
sites, and h is a magnetic field variable,
s; is restricted to be + 1; 5, ~,&
means a
all possible configurations of the spins.
T is the temperature, and k is Boltzmann's con
stant. The partition function is a sum of expone
tials each of which is analytic in K and k. Ther
fore one would expect the partition function itsel
where K, is the critical value of K. To be preci
limit, in which case one calculates the free-ener
density
1
y V
~n
lem of critical behavior in ferromagnets
systems) has.long been a puzzle. ' Con-
sing model of a ferromagnet; the parti-
n is
)=E exp zZEs, s,.;+esp),(s)
—J/kT, Jisa coupling constant, sl is the
ice site n, P& is a sum over nearest-
ites, and h is a magnetic field variable,
; is restricted to be + 1; 5, ~,&
means a
all possible configurations of the spins.
T is the temperature, an
stant. The partition fun
tials each of which is an
fore one would expect th
to be analytic in K and h
partition function is sing
where K, is the critical
the singularity occurs o
limit, in which case one
density
J'(Ã, k) =lim —lnZ(
1
y V
Monday, June 17, 13

Critical point
There is a singularity at and
for the free energy
Problem: The methods of evaluation of F(K,h),
or Z(K,h) can be used only far away from
the critical point
idea: Renormalization group can be used to
modify the parameters in order to make
physical quantities calculable
K = Kc h = 0
the parti-
),
sl is the
arest-
variable,
means a
he spins.
density
1
(2)
y V
Monday, June 17, 13

(Vague) idea beyond the RG
Regular equations can have singular
solutions…
Consider the equation
“Motion of a ball rolling on a hill…”
ormalization group is a nonlinear trans-
group of the kind that occurs in classical
. The equations of motion of a classical
th time-independent potentials define
tions on phase space which form a
he finite transformations of the group are
ormations induced by a finite translation
he infinitesimal transformation is de-
he equations of motion themselves. It
own how a translation group can arise
lysis of critical behavior. This group is
renormalization group for historical
the connection with renormalization will
ed at the conclusion of paper II4). The
al transformation of the renormalization
analogous to an equation of motion, and
use the language of differential equations
n the language of group theory in the re-
of this paper.
antage of a reformulation of Eq. (1) in
the differential equations of the renormali-
up is that it allows the singularities of
al point to occur naturally. Before set-
ese differential equations we shall show
mple classical example how singularities
nerated from an equation of motion. Con-
equation
——(x)
dx
x) is the function shown in Fig. 1. One
of this equation as describing the motion
olling on a hill with height given by V(x).
(3) is not strictly speaking the equation of
r said ball, but qualitatively the solution
uation is similar to the solution of the
der equation one should write down (this
l
A xc
t
xB
FIG. 1. Potential V(x) with minima at x& and x& and
a maximum at pc.
continuity in x(~, xp) as a function of xp whereas
x(t, xp) for finite f is continuous. It is assumed
here that the potential V(x) is analytic in x, as in-
dicated by Fig. 1, 'so the discontinuity in x(~, xp) at
xp= xc cannot be blamed on any singularity in V(x)
itself.
The basic proposal of this paper is that the sin-
gularities at the critical point of a ferromagnet can
be understood as arising from the t=~ limit of the
solution of a differential equation. In order to de-
velop an understanding of how one relates critical
behavior to a differential equation, we shall set up
Kadanoff's scaling picture in differential form.
Kadanoff 's original hypothesis which led to the
Widom-Kadanoff scaling laws was that near the
critical point one could imagine blocks of spins
acting as a unit, i.e., all spins in a block would
be up or down simultaneously. Kadanoff then
argued that one could treat all spins in a block as a
single effective spin, agd one could write an effec-
tive Hamiltonian in the Ising form for these effec-
tive spins. He then showed how these assumptions
we shall use the language of differential equations
rather than the language of group theory in the re-
mainder of this paper.
The advantage of a reformulation of Eq. (1) in
terms of the differential equations of the renormali-
zation group is that it allows the singularities of
the critical point to occur naturally. Before set-
ting up these differential equations we shall show
with a simple classical example how singularities
can be generated from an equation of motion. Con-
sider the equation
dx—= ——(x)
dx
where V(x) is the function shown in Fig. 1. One
can think of this equation as describing the motion
of a ball rolling on a hill with height given by V(x).
Equation (3) is not strictly speaking the equation of
motion for said ball, but qualitatively the solution
of this equation is similar to the solution of the
second-order equation one should write down (this
a
b
a
s
t
t
Monday, June 17, 13

Singularities in rolling balls
Released from a point the ball will roll to
either or
The ﬁnal result is a discontinuous function of
even if is an analytic function
Kadanoff: Can we use a description of the
system which remains analytic even when at
the singular point?
xA xB
x0
x0
V (x)
Monday, June 17, 13

Widom-Kadanoff scaling laws
Kadanoff: ...one could treat all spins in a
block as a single effective spin, and one
could write an effective Hamiltonian in the
Ising form for these effective spins.
These assumptions lead to scaling laws
Wilson: wrong assumptions → right
conclusions; used as a basics for
generalizations
Monday, June 17, 13

Kadanoff’s picture
Imagine an inﬁnite cubic lattice divided into
cubic blocks L lattice sites on a side. Each
block contains L3 lattice sites...
The total spin is sum of L3 spins, takes
values from -L3 to +L3
Normalize to
counts the blocks
s0
~m
s0
~m s0
~m = ±1
~m
Monday, June 17, 13

Block system
Interaction of blocks should be still
described by the Ising model, but with
modiﬁed couplings
Assuming the same free energy per block,
Assuming the correlation lengths agree
KL, hL
F(K& lz) = L F(Kz& hz) . (4)
In the Kadanoff picture one can also compute the
correlation length using the block Hamiltonian.
Let $(K, fz) be the correlation length for the original
Hamiltonian in units of the lattice spacing. Then
Kadanoff proposes in particular that the total free
energy of the original Ising model is the same as
the free energy of the blocks calculated using the
block parameters. In practice this equivalence is
expressed in terms of the free-energy density
rather than the total free energy. Let F(K, h) be
the free energy per lattice site of the original Ising
Hamiltonian of Eq. {1). The free energy per block
of the block Hamiltonian is simply F(Kz,, hz, ). If
the total free energy is the same for both, then
Vfe ex
metry
the old
Hamilt
old blo
change
lattice
h» m
This
SL, L
case,
L to (
can de
TH G. %II SON
se
-
ng
e
$(Kz„hz, ) is the correlation length of the block
Hamiltonian, in units of the block spacing. For the
two to agree, one must have
$ (K, Iz) = L )(Kz„lzz, ). (5)
The Kadanoff picture is, in summary, that there
exists effective coupling parameters KI, and h~ such
that Eqs. (4) and (5) hold, for any L. Kadanoff al-
so requires that correlation functions for large
distances be calculable through the block Hamil-Monday, June 17, 13

Implications of Kadanoff scaling
Assumption that there exist effective
couplings which make these relations
possible
Wilson: existence of such relations is
following trivially from the deﬁnition of the
partition function or free energy
Non-trivial thing: this relations can be made
regular even at the critical point (see the
rolling ball)
Wilson: analytically extend L to continuous
KL, hL
Monday, June 17, 13

Scaling properties
Kadanoff’s scaling relations bring to
valid only for
To obtain the diff. eqn’s, note that e.g.
can be functions of but not of L
separately
d
ugh
which
nd the
study-
nto cubic
k con-
lock is
rding to
the
n has
f spins
e lattice
of in-
'-
s@=+1,
he
earest-
uples to
the in-
ssed in
ubstitute
The differential equations of the renormalization
group will be, in the Kadanoff picture, equations
for EI, and h~. So far nothing has been said about
how to compute KI, and h~. Kadanoff proposed
definite forms for the dependence of KI. and h~ on
L, namely,
Kz =Kc &L
h, =aL',
where
&=K, —K (6)
and Eqs. (6) and (V) are valid only for L «$( ,K)I.z
Here we shall first derive differential equations for
KI, and h~ and show later that the solution of the
differential equations has Kadanoff 's form.
To obtain the general form of the differential
equations for KI, and h&, we note the following. The
constants K» and h» are functions of K~ and AI.
but not of L separately. The change from KI. and
hr, to K» and h» is equivalent to making new
blocks of size 2L out of old blocks of size L. Each
new block is a cube containing eight old blocks.
ferential equations of the renormalization
l be, in the Kadanoff picture, equations
d h~. So far nothing has been said about
mpute KI, and h~. Kadanoff proposed
orms for the dependence of KI. and h~ on
y,
Kc &L
aL',
—K (6)
(6) and (V) are valid only for L «$( ,K)I.z
shall first derive differential equations for
~ and show later that the solution of the
l equations has Kadanoff 's form.
ain the general form of the differential
for KI, and h&, we note the following. The
K» and h» are functions of K~ and AI.
f L separately. The change from KI. and
and h» is equivalent to making new
size 2L out of old blocks of size L. Each
K2L, h2L
KL, hL
Monday, June 17, 13

The Scaling argument
The change from , to
is equivalent to making new blocks of size 2L out
of old blocks of size L. Each new block is a cube
containing eight old blocks. But in writing an
effective Hamiltonian with constants
one has substituted a lattice for
the old blocks; having made this substitution the
Hamiltonian does not know what the size L of the
old blocks was. Regardless of the value of L, the
change to 2L is simply a matter of combining
eight lattice sites to make the new block, so
must be the same function of for
any L.
KL, and hL K2L, and h2L
K2L, and h2L
K2L, and h2L
KL, and hL
Monday, June 17, 13

Differential equations
This should remain true when going to any L
e.g.
In fact, these relations follow from the
deﬁnition of free energy, but using it we
should encounter singularity
Kadanoff: block-spin transformation suggests
non-singular relation;
L ! (1 + )L(4)
e
inal
n
e
sing
ock
f
Vfe expect u to depend only on hr, owing to the sym-
metry of the Ising Hamiltonian for hr, --h~. The
h» must be the same function of K& and h& for any L.
This continues to be true if one goes from L to
SL, L to 4L, etc. Generalizing to the continuous
case, we assume this is true also for going from
L to (1+5)L, for small 5. This means 6LdK~/dL
can depend on K~ and h~ but not L separately:
CRITI
analogous equation for h& is
-~ = I.-'a, v(K„I,') .dJ (10)
Equations (9) and (10) are the renormalization-
group equations suggested by the Kadanoff block
picture. Because of the questionable validity of
this picture one would expect the differential equa-
tions to be equally questionable. Actually this is
not so; there is another way to derive the differen-
tial equations which involves only minimal assump-
tions, such that the differential equations become
essentially a, tautology. Namely, let us define K~
and h~ to be the solutions of Eqs. (4) and (5). That
back to
tion V(x
be discon
u(K, h )
point.
es u(K,
which a
that u is
tions of
such as
which ap
tions.
about th
The KMonday, June 17, 13

The Advantage of RG equations
One can ﬁnd if you know the
solution for the free energy exactly
Finding the differential equations for
is easier than ﬁnding the solution
To understand critical behavior: integrate
until , then
are far from their critical
values ⇒ one can compute
then go back close to the critical point
KL, and hL
KL, and hL
L ⇠ ⇠ ⇠ ⇠ 1
KL, and hL
⇠(KL, hL) and F(KL, hL)
Monday, June 17, 13

The Use of Scaling Equations
Start with T slightly above the Tc, K slightly
smaller than Kc
As L increases KL decreases away from Kc, say
to Kc/2 ﬁnd that L
Integrate
For exactly critical values
So one should start from a point slightly away
from the critical point
⇠(KL, hL) and F(KL, hL)
KL = Kc and hL = 0
u(K„O) = 0 . (i4)
Comparing this result to the classical analog of
the ball on the hill, the point K= K, is analogous
to one of the points of equilibrium for the ball
(x=x„, or xs, or xc).
Now let K and h be near the critical values K,
and 0. For small values of L, namely for L
than K,. As L increases KL must decrease, so as
to go away from K,. This ensures that $(K~, h~)
decreases as I. increases, as required by Eq. (5).
Pick a value for K~ well away from K„say K,/2;
let us integrate the renormalization-group equations
until a value of L is reached for which K~ = K,/2,
then stop and compute F(K, 0) and f(K, I) from Eqs.
(4) ~d (5).
If K and h have exactly the critical values K, and
0, respectively, then one must have KL = K, and
h~ =0 for all I,. The reason for this is that $(K„O)
is infinite; therefore &(K~, h~) must be infinite for
all L. For $(K~, h~) to be infinite, K~ and h~ must
have the critical values. Hence KL =-K, and hL =0
must be a solution of the renormalization group
equations, which is true only if
which is a
For this v
h, = h(K
Hence one
F(K, a)
$(K, h)
These form
F(K, h) and
been reduce
multiplying
variable h&
see Ref. l
To be ac
nonlinear
What can o
this case'P
K =Q(
h~ =P(
be the solu
over the raMonday, June 17, 13

Linearized equations
will be large and KL and hL will also be near the
critical values. For this range of L one can use
a linearized form of the renormalization group
equations to compute KL and AL. The linearized
equations for KL and hL are
' =—(K, -K.)y, (15)
Lx & (16)
Q(
$(
that is
critica
near t
hL sho
giving
where x and y are constants:
p(
g(
y= —, (K„o),
x=v(K,, O) . (is)
In writing these equations we have assumed that
u and v are differentiable at the critical point;
this is how one uses in practice the arialyticity
for I,
ing on
equatio
in term
does.
K
Monday, June 17, 13

The Scaling law
For
therefore
KL = Kc/2
and
nd
out
N(K, h)
ne
ng
eed
e
maller
o as
~)
(5).
/2;
ations
2,
Eqs.
, and
d
K„O)
so
K, /2=&L'
L = (K,/2e)'~',
(i9)
(2o)
which is a scaling law for L as a function of c.
For this value of L, AL is
h, = h(K,/2~)"" .
Hence one can compute F(K, 8) and $(K, h) to be
F(K, a) = (K, /2e) '"F[K,/2, @(K,/2~)""], (22)
$(K, h) = (K, /2&)' ' &[K,/2, &(K,/2&) ~'l . (23)
These formulas are scaling laws; the functions
F(K, h) and $(K, h) depending on two variables have
7 —
The solutions of Eqs. (15) and (16) are the form-
ulas (6) and (V) proposed by Kadanoff. Assume
that this approximation is valid until K~ = K,/2.
Then one can solve for the value of L giving KL
= K, /2:
d out
N(K, h)
one
ng
ceed
re
maller
so as
h~)
(5).
,/2;
uations
/2,
Eqs.
K, and
nd
(K„O)
for
must
=0
so
K, /2=&L'
L = (K,/2e)'~',
(i9)
(2o)
h, = h(K,/2~)"" .
F(K, a) = (K, /2e) '"F[K,/2, @(K,/2~)""], (22)
$(K, h) = (K, /2&)' ' &[K,/2, &(K,/2&) ~'l . (23)
been reduced to explicit powers of e (i.e., 7 —T,)
multiplying functions depending only on the single
variable h&
" '. For consequences of these laws
= K, /2:
d out
N(K, h)
one
ng
ceed
re
maller
so as
h~)
. (5).
,/2;
uations
/2,
Eqs.
K, and
nd
(K„O)
for
must
so
K, /2=&L'
L = (K,/2e)'~',
(i9)
(2o)
h, = h(K,/2~)"" .
F(K, a) = (K, /2e) '"F[K,/2, @(K,/2~)""], (22)
$(K, h) = (K, /2&)' ' &[K,/2, &(K,/2&) ~'l . (23)
been reduced to explicit powers of e (i.e., 7 —T,)
multiplying functions depending only on the single
variable h&
" '. For consequences of these laws
= K, /2:
Monday, June 17, 13

If we would know the exact solution...
The difference would not be much: we would
get the same scaling laws with slightly
modiﬁed proportionality coefﬁcients
Monday, June 17, 13

Problems of the Kadanoff’s picture
)
)
ly
),
)
)
singularity at the critical point is a consequence of
the infinite time required to move away from a point
of unstable equilibrium.
The problem with the simple renormalization-
group equations discussed earlier is that there is
at present no hope of showing that the functions
u(K, h2) and v(K, h ) are analytic at the critical
point. Without the analyticity, the renormaliza-
tion-group equations become a tautology, as ex-
plained earlier. Instead of trying to prove that
u and v are analytic, one can try to generalize the
renormalization-group equations in the hope that
analyticity will be easier to establish for the gen-
eralization. The generalizations which the author
has been able to construct are rather complicated,
involving an infinite number of L-dependent coupling
constants. To prepare for these generalizations
Monday, June 17, 13

Wilson’s generalization...
Introduce an additional “irrelevant” coupling
e.g. the coefﬁcient of the next-to-neighbor
coupling
qL
(0) is a con-
top of the hill
ent from K,
and this is
m the top of
K, is analo-
in classical
he correla-
K, 0) is.pro-
ake K~ = K, /2.
e t»~ for
As the initial
op of the hill,
inite as xo
by computing
ear K,. This
dKI,
I s qI & ~I ) (40)
IV(KI, q gI,~ llI, ) i
dq~ 1
(41)
dhl,
kJ V(KJ p qgp kI ) (42)
The initial values of KI, qI, , and kI, (for L = 1) are
denoted K, q, and h. We assume the same rules
for computing the free energy and the correlation
q prove only
respects that will be explained later.
Imagine that the renormalization-group equations
involve three L-dependent variables K~, ql, , and
hl, . %e think of K~ and @ as coefficients of first-
and second-nearest-neighbor couplings which are
even to the exchange s-'- —s-' while h~ still multi-
plies the spins s-' themselves which are odd to this
exchange. Hence one expects the form of the re-
normalization-group equations to be
length as before, namely,
P(E, q, I ) = I.-'P(E„q„a,),
t(E,q, a) =r, t (E„q„I,).
KENNETH G.
1
Vf IL 8GN
~'N
y = —Bq
~d
x =v (K„
We now expect there wiQ be a line of critical
points; for each value of q there should be a criti-
cal value K,(q) for K. If the initial values of K,
q, and h lie on the critical line (K=K,(q), h = 0)
There will,
solutions of
power of I,
E~ -E,Monday, June 17, 13

“The phase diagram”
We expect a line of critical points
If you are on the line you remain on it…
Assume an equilibrium point ...
Solution to linearized equations gives two
scalings at : one attracting and one
repelling...
Kc(q)
qc
qc
Monday, June 17, 13

Behavior near critical point
q~ = q, —& r„L' +q
hl, =hI",
where z and q depend o
r,(K K,) —(q —q
r„(K-K,)-(q-q
PIG. 2. Potential VN, q) plotted by contours of con-
stant V. The minimum of V is at the origin (P&); the
maximum is at Pz and Pc, is a saddle point. The ridge
line is the trajectory going to Pz,. the gully line is the
trajectory from Pc to P~. Unmarked lines are contours.
The dashed line marks an arbitrary value (qo) for q.
One is on the critical li
then Kl. K„q& -q, w
the critical line away fr
k is small. Then for I
and since z is negative,
and q, (i.e., the ball is
the saddle point}. For
doIQlDates RDd Kg RDd q
q, . The g term continu
neglected. Since g me
along the critical line,
independence of the init
line. One can now com
which K~ =K,/2, giving
ol
K, /2 = &I'
Monday, June 17, 13

Irrelevance of q
For KL = Kc/2
of con-
); the
ridge
is the
contours.
r q.
q, . The g term continues to decrease and can be
neglected. Since g measures the initial location
along the critical line, independence of q is
independence of the initial location on the cx'itical
line. One can now compute the value of L, for
ol
K, /2 = &I' (04)
line
s to the
can be
baQ
one
of the
line,
one
levant
nt and
whether
es not
start-
line,
e saddle
as before. For this value of L,
The important property of these formulas is that
q~ is a constant independent of h, e, or q, while
hI, again depends only on the ratio h& "~'. So
from Egs. (43) and (44) one gets the same form of
scaling laws for E(K, h, q) and $ (K, h, q) as was ob-
tained earlier when there was no irrelevant vari-
able q. Since K=K, (q) for e =0, it follows from
Eq. (62) that & is proportional to K-K,(q), which
—
of con-
); the
ridge
is the
contours.
r q.
ol
K, /2 = &I' (04)
line
s to the
can be
baQ
one
of the
line,
one
levant
nt and
whether
s not
start-
line,
e saddle
is proportional to T —T,(q). Hence no matter what
rs of con-
(P&); the
The ridge
ne is the
re contours.
) for q.
q, . The g term continues to decrease and can be
ol
K, /2 = &I' (04)
cal line
gous to the
on can be
the baQ
if one
ath of the
uQy line,
ne one
irrelevant
evant and
but whether
does not
,qr, start-
ical line,
the saddle
be close to
ch is slow).
I becomes
), at which
is proportional to T —T,(q). Hence no matter what
q is, e is the customary temperature variable
T -T, apart from a meaningless normalization
factol o
So far nothing has been said about initial condi-Monday, June 17, 13

Summary
of the renormalization-group equations, but the
conclusions will not be changed by using the non-
linear equations.
To summarize the results of this analysis, one
starts with values of E and q near the critical line
K=K, (q). The functions A~ and q~ have an initial
transient behavior in which El. and q& adjust to the
critical values E, and q, . In other words, while
the initial Ising Hamiltonian can have an arbitrary
fraction of second-nearest-neighbor coupling, by the
time one has gone to a block Hamiltonian with rea-
sonably large block size the couplings have become
essentially fixed at E, and q, . The Hamiltonians
for larger I. are therefore independent of the initial
fraction of second-nearest-neighbor coupling, and
the critical behavior of the theory is likewise in-
dependent of the initial fraction.
In fact the parameters of the critical behavior
such as the exponents x and y are determined by the
renormalization-group differential equation, rather
of t
this
gro
con
the
solu
whe
are
cha
pen
equ
var
are
cer
sad
grou
cyc
osc
whi
areMonday, June 17, 13

Summary, cont’d
essentially fixed at E, and q, . The Hamiltonians
for larger I. are therefore independent of the initial
fraction of second-nearest-neighbor coupling, and
the critical behavior of the theory is likewise in-
dependent of the initial fraction.
In fact the parameters of the critical behavior
such as the exponents x and y are determined by the
renormalization-group differential equation, rather
than the initial Hamiltonian. If one knows the form
of the functions u, v, and se, one can calculate the
exponents x and y by finding the saddle point E„q,
of the differential equation and then solving the
linearized equations about the saddle point. In
principle one then has two choices for how to find
the exponents x and y, one choice being as just
a
c
sa
g
c
o
w
a
th
ch
P
fe
fe
o
*Supported by the National Science Foundation.
~For reviews of the theory of critical phenomena, see
P
SMonday, June 17, 13

Renormalization group and critical phenomena

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