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PROFESSOR: OK, let's start.

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00:00:25,710 --> 00:00:30,460
So we are going to change
perspective again and think

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00:00:30,460 --> 00:00:32,070
in terms of lattice models.

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00:00:38,750 --> 00:00:42,810
So for the first
part of this course,

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00:00:42,810 --> 00:00:45,780
I was trying to change
your perspective

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00:00:45,780 --> 00:00:49,400
from thinking in terms of
microscopic degrees of freedom

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00:00:49,400 --> 00:00:51,640
to a statistical field.

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00:00:51,640 --> 00:00:55,930
Now we are going to go back
and try to build pictures

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00:00:55,930 --> 00:00:59,310
around things that
look more microscopic.

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00:00:59,310 --> 00:01:03,860
Typically, in many solid
state configurations,

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00:01:03,860 --> 00:01:09,860
we are dealing with transitions
that take place on a lattice.

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00:01:09,860 --> 00:01:14,440
For example, imagine a square
lattice, which is easy to draw,

20
00:01:14,440 --> 00:01:19,570
but there could be all kinds
of cubic and more complex

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00:01:19,570 --> 00:01:20,970
lattices.

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00:01:20,970 --> 00:01:24,730
And then, at each
side of this lattice,

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00:01:24,730 --> 00:01:28,380
you may have one microscopic
degrees of freedom

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00:01:28,380 --> 00:01:32,640
that is ultimately participating
in the ordering and the phase

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00:01:32,640 --> 00:01:34,930
transition that
you have in mind.

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00:01:34,930 --> 00:01:37,130
Could, for example,
be a spin, or it

27
00:01:37,130 --> 00:01:42,670
could be one atom
in a binary mixture.

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00:01:42,670 --> 00:01:48,010
And what you would like to do
is to construct a partition

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00:01:48,010 --> 00:01:54,330
function, again, by summing
over all degrees of freedom.

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00:01:54,330 --> 00:01:58,220
And we need some
kind of Hamiltonian.

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00:01:58,220 --> 00:02:01,510
And what we are going
to assume governs

32
00:02:01,510 --> 00:02:06,950
that Hamiltonian is the analog
of this locality assumption

33
00:02:06,950 --> 00:02:09,759
that we have in
statistical field theories.

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00:02:09,759 --> 00:02:11,590
That is, we are going
to assume that one

35
00:02:11,590 --> 00:02:13,960
of our degrees of
freedom essentially

36
00:02:13,960 --> 00:02:18,100
talks to a small
neighborhood around it.

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00:02:18,100 --> 00:02:19,900
And the simplest
neighborhood would

38
00:02:19,900 --> 00:02:24,970
be to basically talk to
the nearest neighbor.

39
00:02:24,970 --> 00:02:29,730
So if I, for example,
assign index i

40
00:02:29,730 --> 00:02:32,420
to each side of
the lattice, let's

41
00:02:32,420 --> 00:02:36,880
say I have some variable at each
side that could be something.

42
00:02:36,880 --> 00:02:40,430
Let's call it SI.

43
00:02:40,430 --> 00:02:42,360
Then my partition
function would be

44
00:02:42,360 --> 00:02:47,800
obtained by summing
over all configurations.

45
00:02:47,800 --> 00:02:53,090
And the weight I'm going to
assume in terms of this lattice

46
00:02:53,090 --> 00:02:58,220
picture to be a sum
over interactions

47
00:02:58,220 --> 00:03:01,360
that exist between
pairs of sides.

48
00:03:01,360 --> 00:03:05,680
So that's already an assumption
that it's a pairwise thing.

49
00:03:05,680 --> 00:03:11,720
And I'm going to use this
symbol ij with an angular

50
00:03:11,720 --> 00:03:15,390
bracket along it,
around it, to indicate

51
00:03:15,390 --> 00:03:19,605
the sum over nearest neighbors.

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00:03:25,230 --> 00:03:30,870
And there is some
function of the variables

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00:03:30,870 --> 00:03:32,518
that live on these neighbors.

54
00:03:36,790 --> 00:03:40,490
So basically, in the
picture that I have drawn,

55
00:03:40,490 --> 00:03:47,250
interactions exist only between
the places that you see lines.

56
00:03:47,250 --> 00:03:49,630
So that this pin
does not interact

57
00:03:49,630 --> 00:03:52,930
with this pin, this
pin, or this pin,

58
00:03:52,930 --> 00:03:55,710
but it interacts
with these four pins,

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00:03:55,710 --> 00:03:57,610
to which it is near neighbor.

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00:04:00,940 --> 00:04:03,120
Now clearly, the form
of the interaction

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00:04:03,120 --> 00:04:07,410
has to be dictated by
your degrees of freedom.

62
00:04:07,410 --> 00:04:12,900
And the idea of
this representation,

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00:04:12,900 --> 00:04:16,040
as opposed to the previous
statistical field theory

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00:04:16,040 --> 00:04:21,519
that we had, is that in
several important instances,

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00:04:21,519 --> 00:04:27,190
you may want to know not only
what these universal properties

66
00:04:27,190 --> 00:04:32,170
are, but also, let's say, the
explicit temperature or phase

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00:04:32,170 --> 00:04:36,630
diagram of the system as a
function of external parameters

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00:04:36,630 --> 00:04:38,140
as well as temperature.

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00:04:38,140 --> 00:04:43,610
And if you have some idea of
how your microscopic degrees

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00:04:43,610 --> 00:04:45,940
of freedom interact
with each other,

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00:04:45,940 --> 00:04:50,630
you should be able to solve
this kind of partition function,

72
00:04:50,630 --> 00:04:52,470
get the singularities,
et cetera.

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00:04:55,390 --> 00:05:01,690
So let's look at some simple
versions of this construction

74
00:05:01,690 --> 00:05:03,930
and gradually discuss
the kinds of things

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00:05:03,930 --> 00:05:05,338
that we could do with it.

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00:05:07,970 --> 00:05:09,695
So some simple models.

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00:05:12,570 --> 00:05:22,680
The simplest one
is the Ising model

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00:05:22,680 --> 00:05:27,090
where the variable that
you have at each side

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00:05:27,090 --> 00:05:29,750
has two possibilities.

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00:05:29,750 --> 00:05:37,320
So it's a binary variable in
the context of a binary alloy.

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00:05:37,320 --> 00:05:40,780
It could be, let's
say, atom A or atom

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00:05:40,780 --> 00:05:44,800
B that is occupying
a particular site.

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00:05:44,800 --> 00:05:48,080
There are also cases
where there will be some--

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00:05:48,080 --> 00:05:51,450
if this is a surface and
you're absorbing particles

85
00:05:51,450 --> 00:05:55,120
on top of it, there could be
a particle sitting here or not

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00:05:55,120 --> 00:05:55,790
sitting here.

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00:05:55,790 --> 00:05:59,690
So that would be also another
example of a binary variable.

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00:05:59,690 --> 00:06:06,010
So you could indicate that by
empty or occupied, zero or one.

89
00:06:06,010 --> 00:06:10,470
But let's indicate it by
plus or minus one as the two

90
00:06:10,470 --> 00:06:11,995
possible values
that you can have.

91
00:06:15,640 --> 00:06:20,330
Now, if I look at the
analog of the interaction

92
00:06:20,330 --> 00:06:24,120
that I have between two
sites that are neighboring

93
00:06:24,120 --> 00:06:25,980
each other, what
can I write down?

94
00:06:29,684 --> 00:06:33,560
Well, the most general
thing that I can write down

95
00:06:33,560 --> 00:06:38,630
is, first of all, a
constant I can put,

96
00:06:38,630 --> 00:06:41,250
such as shift of energy.

97
00:06:41,250 --> 00:06:46,280
There could be a linear
term in-- let's put

98
00:06:46,280 --> 00:06:48,540
all of these with minus signs.

99
00:06:48,540 --> 00:06:51,520
Sigma i and sigma j.

100
00:06:51,520 --> 00:06:57,080
I assume that it is symmetric
with respect to the two sides.

101
00:06:57,080 --> 00:06:59,730
For reasons to become
apparent shortly,

102
00:06:59,730 --> 00:07:04,130
I will divide by z, which
is the coordination number.

103
00:07:07,350 --> 00:07:11,100
How many bonds per side?

104
00:07:15,920 --> 00:07:17,380
And then the next
term that I can

105
00:07:17,380 --> 00:07:23,480
put is something like
j sigma i sigma j.

106
00:07:23,480 --> 00:07:27,340
And actually, I can't
put anything else.

107
00:07:27,340 --> 00:07:31,270
Because if you s of this as
a power series in sigma i

108
00:07:31,270 --> 00:07:34,940
and sigma j, and sigma
has only two values,

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00:07:34,940 --> 00:07:35,950
it will terminate here.

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00:07:35,950 --> 00:07:39,890
Because any higher power of
sigma is either one or sigma

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00:07:39,890 --> 00:07:40,390
itself.

112
00:07:43,010 --> 00:07:46,930
So another way of writing
this is that the partition

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00:07:46,930 --> 00:07:53,670
function of the Ising model is
obtained by summing over all 2

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00:07:53,670 --> 00:07:56,043
to the n configurations
that I have.

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00:07:56,043 --> 00:07:59,230
If I have a lattice
of n sides, each side

116
00:07:59,230 --> 00:08:11,270
can have two possibilities of
a kind of Hamiltonian, which

117
00:08:11,270 --> 00:08:20,180
is basically some constant
plus h sum over i sigma i kind

118
00:08:20,180 --> 00:08:24,250
of field that prefers one
side or the other side.

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00:08:24,250 --> 00:08:29,010
So it is an analog of a magnetic
field in this binary system.

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00:08:29,010 --> 00:08:34,010
And basically, I convert it from
a description that is overall

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00:08:34,010 --> 00:08:37,039
balanced to a description
overall sides.

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00:08:37,039 --> 00:08:40,690
And that's why I had put the
coordination number there.

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00:08:40,690 --> 00:08:44,630
It's kind of a
matter of convention.

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00:08:44,630 --> 00:08:53,010
And then a term that
prefers neighboring sites

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00:08:53,010 --> 00:08:55,480
to be aligned as
long as k positive.

126
00:09:00,560 --> 00:09:03,338
So that's one
example of a model.

127
00:09:06,760 --> 00:09:12,050
Another model that
we will also look at

128
00:09:12,050 --> 00:09:15,540
is what I started to
draw at the beginning.

129
00:09:15,540 --> 00:09:19,820
That is, at each site,
you have a vector.

130
00:09:19,820 --> 00:09:26,340
And again, going in terms of
the pictures that we had before,

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00:09:26,340 --> 00:09:31,600
let's imagine that we have a
vector that has n components.

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00:09:31,600 --> 00:09:39,980
So SI is something
that is in RN.

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00:09:39,980 --> 00:09:45,380
And I will assume that the
magnitude of this vector

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00:09:45,380 --> 00:09:47,260
is one.

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00:09:47,260 --> 00:09:49,130
So essentially,
imagine that you have

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00:09:49,130 --> 00:09:55,050
a unit vector, and each
site that can rotate.

137
00:09:55,050 --> 00:10:03,210
So if n equals to one, you
have essentially one component

138
00:10:03,210 --> 00:10:04,970
vector.

139
00:10:04,970 --> 00:10:07,610
Its square has to
be one, so the two

140
00:10:07,610 --> 00:10:11,170
values that it can have
are plus one and minus one.

141
00:10:11,170 --> 00:10:18,140
So the n equals to one case
is the Ising model again.

142
00:10:18,140 --> 00:10:22,200
So this ON is a
generalization of the Ising

143
00:10:22,200 --> 00:10:25,630
model to multiple components.

144
00:10:25,630 --> 00:10:30,320
n equals to two corresponds
to a unit vector

145
00:10:30,320 --> 00:10:34,620
that can take any angle
in two dimensions.

146
00:10:34,620 --> 00:10:39,130
And that is usually
given the name xy model.

147
00:10:42,310 --> 00:10:44,970
n equals to three is
something that maybe you

148
00:10:44,970 --> 00:10:52,540
want to use to describe
magnetic ions in this lattice.

149
00:10:52,540 --> 00:10:57,120
And classically, the direction
of this ion could be anywhere.

150
00:10:57,120 --> 00:11:01,130
The spin of the
ion can be anywhere

151
00:11:01,130 --> 00:11:03,870
on the surface of a sphere.

152
00:11:03,870 --> 00:11:06,470
So that's three
components, and this model

153
00:11:06,470 --> 00:11:08,392
is sometimes called
the Heisenberg model.

154
00:11:17,420 --> 00:11:19,051
Yes.

155
00:11:19,051 --> 00:11:21,012
AUDIENCE: In the
Ising model, what's

156
00:11:21,012 --> 00:11:27,520
the correspondence between g
hat h hat j hat and g, h, and k?

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00:11:27,520 --> 00:11:28,580
PROFESSOR: Minus beta.

158
00:11:28,580 --> 00:11:29,080
Yeah.

159
00:11:29,080 --> 00:11:31,570
So maybe I should have written.

160
00:11:31,570 --> 00:11:38,910
In order to take this-- if I
think of this as energy pair

161
00:11:38,910 --> 00:11:44,310
bond, then in order to
get the Boltzmann weight,

162
00:11:44,310 --> 00:11:50,860
I have to put a minus beta.

163
00:11:50,860 --> 00:11:54,120
So I would have said
that this g, for example,

164
00:11:54,120 --> 00:11:56,850
is minus beta g hat.

165
00:11:56,850 --> 00:12:00,790
k is minus beta j hat.

166
00:12:00,790 --> 00:12:03,790
And h is minus beta h hat.

167
00:12:11,390 --> 00:12:13,970
So I should have
emphasized that.

168
00:12:13,970 --> 00:12:18,920
What I meant by b--
actually, I wrote

169
00:12:18,920 --> 00:12:24,680
b-- so what I should have
done here to be consistent,

170
00:12:24,680 --> 00:12:26,470
let's write this as minus beta.

171
00:12:34,340 --> 00:12:34,840
Thank you.

172
00:12:34,840 --> 00:12:37,930
That was important.

173
00:12:40,820 --> 00:12:44,690
If I described these b's
as being energies, then

174
00:12:44,690 --> 00:12:46,817
minus beta times
that will be what

175
00:12:46,817 --> 00:12:48,150
will go in the Boltzmann factor.

176
00:12:53,530 --> 00:12:56,160
OK

177
00:12:56,160 --> 00:13:01,130
So whereas these were examples
that we had more or less seen

178
00:13:01,130 --> 00:13:05,920
their continuum version in
the Landau-Ginzburg model,

179
00:13:05,920 --> 00:13:09,580
there are other symmetries
that get broken.

180
00:13:09,580 --> 00:13:11,900
And things for which
we didn't discuss

181
00:13:11,900 --> 00:13:15,630
what the corresponding
statistical field theory is.

182
00:13:15,630 --> 00:13:19,220
A commonly used case
pertains to something

183
00:13:19,220 --> 00:13:20,470
that's called a Potts model.

184
00:13:23,770 --> 00:13:28,470
Where at each side
you have a variable,

185
00:13:28,470 --> 00:13:33,200
let's call it SI,
that takes q values,

186
00:13:33,200 --> 00:13:35,870
one, two, three,
all the way to q.

187
00:13:39,580 --> 00:13:42,200
And I can write
what would appear

188
00:13:42,200 --> 00:13:47,080
in the exponent
minus beta h to be

189
00:13:47,080 --> 00:13:50,770
a sum over nearest neighbors.

190
00:13:53,520 --> 00:14:00,480
And I can give some kind of
an interaction parameter here,

191
00:14:00,480 --> 00:14:04,870
but a delta si sj.

192
00:14:04,870 --> 00:14:08,900
So basically, what it says is
that on each side of a lattice,

193
00:14:08,900 --> 00:14:10,920
you put a number.

194
00:14:10,920 --> 00:14:13,820
Could be one, two,
three, up to q.

195
00:14:13,820 --> 00:14:18,710
And if two neighbors are
the same, they like it,

196
00:14:18,710 --> 00:14:21,980
and they gain some
kind of a weight.

197
00:14:21,980 --> 00:14:26,740
That is, if k is positive,
encourages that to happen.

198
00:14:26,740 --> 00:14:29,660
If the two neighbors
are different,

199
00:14:29,660 --> 00:14:30,970
then it doesn't matter.

200
00:14:30,970 --> 00:14:33,050
You don't gain energy,
but you don't really

201
00:14:33,050 --> 00:14:36,510
care as to which one it is.

202
00:14:36,510 --> 00:14:42,620
The underlying symmetry that
this has is permutation.

203
00:14:42,620 --> 00:14:47,400
Basically, if you were to
permute all of these indices

204
00:14:47,400 --> 00:14:50,420
consistently across the
lattice in any particular way,

205
00:14:50,420 --> 00:14:53,010
the energy would not change.

206
00:14:53,010 --> 00:15:04,930
So permutation symmetry
is what underlies this.

207
00:15:04,930 --> 00:15:12,640
And again, if I look at the
case of two, then at each side,

208
00:15:12,640 --> 00:15:15,190
let's say I have one or two.

209
00:15:15,190 --> 00:15:19,390
And one one and two two are
things that gain energy.

210
00:15:19,390 --> 00:15:21,530
One two or two one don't.

211
00:15:21,530 --> 00:15:25,110
Clearly it is the same
as the Ising model.

212
00:15:25,110 --> 00:15:29,310
So q equals to two is another
way of writing the Ising model.

213
00:15:32,680 --> 00:15:36,810
Q equals to three is something
that we haven't seen.

214
00:15:36,810 --> 00:15:40,760
So at each site, there
are three possibilities.

215
00:15:40,760 --> 00:15:47,040
Actually, when I started at MIT
as a graduate student in 1979,

216
00:15:47,040 --> 00:15:50,770
the project that I had
to do related to the q

217
00:15:50,770 --> 00:15:53,660
equals to three Potts model.

218
00:15:53,660 --> 00:15:55,280
Where did it come from?

219
00:15:55,280 --> 00:16:00,620
Well, at that time,
people were looking

220
00:16:00,620 --> 00:16:03,846
at surface of graphite,
which as you know,

221
00:16:03,846 --> 00:16:08,910
has this hexagonal structure.

222
00:16:08,910 --> 00:16:12,230
And then you can
absorb molecules

223
00:16:12,230 --> 00:16:15,910
on top of that, such as,
for example, krypton.

224
00:16:15,910 --> 00:16:19,190
And krypton would
want to come and sit

225
00:16:19,190 --> 00:16:22,620
in the center of one
of these hexagons.

226
00:16:22,620 --> 00:16:26,320
But its size was such
that once it sat there,

227
00:16:26,320 --> 00:16:29,800
you couldn't occupy
any of these sides.

228
00:16:29,800 --> 00:16:35,340
So the next one would have
to go, let's say, over here.

229
00:16:35,340 --> 00:16:41,600
Now, it is possible to
subdivide this set of hexagons

230
00:16:41,600 --> 00:16:43,820
into three sub lattices.

231
00:16:43,820 --> 00:16:49,300
One, two, three, one,
two, three, et cetera.

232
00:16:49,300 --> 00:16:51,750
Actually, I drew
this one incorrectly.

233
00:16:51,750 --> 00:16:54,690
It would be sitting here.

234
00:16:54,690 --> 00:17:01,230
And what happens is that
basically the agile particles

235
00:17:01,230 --> 00:17:06,550
would order by occupying one of
three equivalent sublattices.

236
00:17:06,550 --> 00:17:10,630
So the way that that order
got destroyed was then

237
00:17:10,630 --> 00:17:17,020
described by the q equals to
three Potts universality class.

238
00:17:17,020 --> 00:17:18,890
You can think of
something like q

239
00:17:18,890 --> 00:17:26,339
equals to four that would
have a symmetry of a tetragon.

240
00:17:26,339 --> 00:17:30,820
And so some structure that
is like a tetragon getting,

241
00:17:30,820 --> 00:17:34,250
let's say, distorted in
some particular direction

242
00:17:34,250 --> 00:17:38,250
would then have four equivalent
directions, et cetera.

243
00:17:38,250 --> 00:17:42,510
So there's a whole set of other
types of universality classes

244
00:17:42,510 --> 00:17:46,460
and symmetry breakings that
we did not discuss before.

245
00:17:46,460 --> 00:17:49,110
And I just want
to emphasize that

246
00:17:49,110 --> 00:17:53,420
what we discussed before does
not cover all possible symmetry

247
00:17:53,420 --> 00:17:54,510
breakings.

248
00:17:54,510 --> 00:17:58,100
It was just supposed to
show you an important class

249
00:17:58,100 --> 00:18:01,090
and the technology
to deal with that.

250
00:18:01,090 --> 00:18:03,570
But again, in this
particular system,

251
00:18:03,570 --> 00:18:07,460
let's say you really wanted
to know at what temperature

252
00:18:07,460 --> 00:18:11,890
the phase transition occurs,
as well as what potential phase

253
00:18:11,890 --> 00:18:14,340
diagrams and
critical behavior is.

254
00:18:14,340 --> 00:18:16,570
And then you would
say, well, even

255
00:18:16,570 --> 00:18:19,490
if I could construct a
statistical field theory

256
00:18:19,490 --> 00:18:21,460
and analyze it in
two dimensions,

257
00:18:21,460 --> 00:18:24,170
and we've seen how
hard it is to go

258
00:18:24,170 --> 00:18:27,000
below some other
critical dimension,

259
00:18:27,000 --> 00:18:30,810
it doesn't tell me things about
phase diagrams, et cetera.

260
00:18:30,810 --> 00:18:33,960
So maybe trying to understand
and deal with this lattice

261
00:18:33,960 --> 00:18:38,120
model itself would tell
us more information.

262
00:18:38,120 --> 00:18:43,280
Although about quantities that
are not necessarily inverse.

263
00:18:43,280 --> 00:18:46,770
Depending on your
microscopic model,

264
00:18:46,770 --> 00:18:49,840
you may try to introduce
more complicated systems,

265
00:18:49,840 --> 00:18:53,870
such as inspired by quantum
mechanics, you can think

266
00:18:53,870 --> 00:18:56,730
of something that
I'll call a spin S

267
00:18:56,730 --> 00:19:04,980
model in which your SI
takes values from minus s,

268
00:19:04,980 --> 00:19:11,260
minus s plus 1, all
the way to plus s.

269
00:19:11,260 --> 00:19:14,330
There's 2s plus 1 possibilities.

270
00:19:14,330 --> 00:19:18,110
And you can think of this
as components of, say,

271
00:19:18,110 --> 00:19:23,640
a quantum spin of s
along the zed axis.

272
00:19:23,640 --> 00:19:28,410
Write some kind of
Hamiltonian for this.

273
00:19:28,410 --> 00:19:31,235
But as long as you deal
with things classically,

274
00:19:31,235 --> 00:19:33,980
it turns out that
this kind of system

275
00:19:33,980 --> 00:19:37,942
will not really have different
universality from the Ising

276
00:19:37,942 --> 00:19:38,442
model.

277
00:19:42,110 --> 00:19:46,830
So let's say we have
this lattice model.

278
00:19:46,830 --> 00:19:48,420
Then what can we do?

279
00:19:52,360 --> 00:19:56,280
So in the next
set of lectures, I

280
00:19:56,280 --> 00:20:02,380
will describe some tools for
dealing with these models.

281
00:20:02,380 --> 00:20:08,110
One set of approaches, the
one that we will start today,

282
00:20:08,110 --> 00:20:20,080
has to do with the position
space renormalization group.

283
00:20:20,080 --> 00:20:22,340
That is the approach
that we were following

284
00:20:22,340 --> 00:20:24,900
for renormalization previously.

285
00:20:24,900 --> 00:20:27,500
Dealt with going
to Fourier space.

286
00:20:27,500 --> 00:20:30,110
We had this sphere,
hyper sphere.

287
00:20:30,110 --> 00:20:33,010
And then we were basically
eliminating modes

288
00:20:33,010 --> 00:20:37,340
at the edge of this
sphere in Fourier space.

289
00:20:37,340 --> 00:20:40,100
We started actually by
describing the process

290
00:20:40,100 --> 00:20:41,130
in real space.

291
00:20:41,130 --> 00:20:44,040
So we will see
that in some cases,

292
00:20:44,040 --> 00:20:48,520
it is possible to do a
renormalization group directly

293
00:20:48,520 --> 00:20:49,610
on these lattice models.

294
00:20:53,760 --> 00:21:00,860
Second thing is, it turns out
that as combinatorial problems,

295
00:21:00,860 --> 00:21:05,600
some, but a very small
subset of these models,

296
00:21:05,600 --> 00:21:07,730
are susceptible to
exact solutions.

297
00:21:10,460 --> 00:21:14,020
Turns out that practically
all models in one dimension,

298
00:21:14,020 --> 00:21:18,990
as we will start today,
one can solve exactly.

299
00:21:18,990 --> 00:21:21,810
But there's one
prominent case, which

300
00:21:21,810 --> 00:21:25,450
is the two dimensionalizing
model that one can also

301
00:21:25,450 --> 00:21:26,680
solve exactly.

302
00:21:26,680 --> 00:21:29,450
And it's a very
interesting solution

303
00:21:29,450 --> 00:21:36,030
that we will also examine in, I
don't know, a couple of weeks.

304
00:21:36,030 --> 00:21:38,310
Finally, there are
approximate schemes

305
00:21:38,310 --> 00:21:44,680
that people have evolved for
studying these problems, where

306
00:21:44,680 --> 00:21:52,980
you have series expansions
starting from limits, where

307
00:21:52,980 --> 00:21:55,320
you know what is happening.

308
00:21:55,320 --> 00:21:58,570
And one simple
example would be to go

309
00:21:58,570 --> 00:22:01,320
to very high temperatures.

310
00:22:01,320 --> 00:22:04,910
And at high temperatures,
essentially every degree

311
00:22:04,910 --> 00:22:07,290
of freedom does what it wants.

312
00:22:07,290 --> 00:22:09,810
So it's essentially a
zero dimensional problem

313
00:22:09,810 --> 00:22:11,080
that you can solve.

314
00:22:11,080 --> 00:22:14,960
And then you can start treating
interactions perturbatively.

315
00:22:14,960 --> 00:22:18,780
So this is kind of
similar to the expansions

316
00:22:18,780 --> 00:22:23,720
that we had developed in 8
333, the virial expansions,

317
00:22:23,720 --> 00:22:27,530
et cetera, about
the ideal gas limit.

318
00:22:27,530 --> 00:22:30,940
But now done on a system
that is a lattice,

319
00:22:30,940 --> 00:22:33,240
and going to
sufficiently high order

320
00:22:33,240 --> 00:22:38,070
that you can say something
about the phase transition.

321
00:22:38,070 --> 00:22:41,570
There is another extreme.

322
00:22:41,570 --> 00:22:45,740
In these systems, typically
the zero temperature state

323
00:22:45,740 --> 00:22:46,600
is trivial.

324
00:22:46,600 --> 00:22:48,050
It is perfectly ordered.

325
00:22:48,050 --> 00:22:50,480
Let's say all the
spins are aligned.

326
00:22:50,480 --> 00:22:53,760
And then you can start
expanding in excitations

327
00:22:53,760 --> 00:22:57,220
around that state and
see whether eventually,

328
00:22:57,220 --> 00:23:00,030
by including more
and more excitations,

329
00:23:00,030 --> 00:23:04,730
you can see the phase transition
out of the ordered state.

330
00:23:04,730 --> 00:23:07,790
And something that is actually
probably the most common use

331
00:23:07,790 --> 00:23:12,090
of these models, but I
won't cover in class,

332
00:23:12,090 --> 00:23:16,470
is to put them on the computer
and do some kind of a Monte

333
00:23:16,470 --> 00:23:23,320
Carlo simulation,
which is essentially

334
00:23:23,320 --> 00:23:27,270
a numerical way of trying to
generate configurations that

335
00:23:27,270 --> 00:23:29,260
are governed by this weight.

336
00:23:29,260 --> 00:23:33,570
And by changing the temperature
as it appears in that weight,

337
00:23:33,570 --> 00:23:36,134
whether or not one can,
in the simulation, see

338
00:23:36,134 --> 00:23:37,300
the phase transition happen.

339
00:23:41,210 --> 00:23:44,330
So that's the change
in perspective

340
00:23:44,330 --> 00:23:47,060
that I want you to have.

341
00:23:47,060 --> 00:23:49,860
So the first thing
that we're going to do

342
00:23:49,860 --> 00:23:56,100
is to do the number one here,
to do the position space

343
00:23:56,100 --> 00:24:05,290
renormalization group of
one dimensional Ising model.

344
00:24:05,290 --> 00:24:09,030
And the procedure that
I describe for you

345
00:24:09,030 --> 00:24:11,190
is sufficiently general
that in fact you

346
00:24:11,190 --> 00:24:15,070
can apply to any other
one dimensional model,

347
00:24:15,070 --> 00:24:19,660
as long as you only have these
nearest neighbor interactions.

348
00:24:19,660 --> 00:24:26,890
So here you have a lattice
that is one dimensional.

349
00:24:26,890 --> 00:24:31,720
So you have a set
of sites one, two.

350
00:24:31,720 --> 00:24:38,820
At some point, you have i,
i minus one, i plus one.

351
00:24:38,820 --> 00:24:41,200
Let's say we call
the last one n.

352
00:24:43,910 --> 00:24:46,060
So there are n sites.

353
00:24:46,060 --> 00:24:51,940
There are going to be 2 to
the n possible configurations.

354
00:24:51,940 --> 00:24:55,300
And your task is given
that at each site,

355
00:24:55,300 --> 00:24:59,440
there's a variable
that is binary.

356
00:24:59,440 --> 00:25:03,020
You want to calculate a
partition function, which

357
00:25:03,020 --> 00:25:08,700
is a sum over all these 2
to the n configurations.

358
00:25:08,700 --> 00:25:14,380
Of a weight that is
this e to the sum

359
00:25:14,380 --> 00:25:22,220
over i B, the interaction
that couples SI and SI plus 1.

360
00:25:22,220 --> 00:25:29,400
Maybe I should have
called this e hat.

361
00:25:29,400 --> 00:25:34,132
And B is the thing that has
minus beta absorbed in it.

362
00:25:39,130 --> 00:25:43,350
So notice that basically, the
way that I have written it,

363
00:25:43,350 --> 00:25:46,380
one is interacting with two.

364
00:25:46,380 --> 00:25:48,740
i minus one is
interacting with i.

365
00:25:48,740 --> 00:25:50,850
i is interacting
with i plus one.

366
00:25:50,850 --> 00:25:53,750
So I wrote the nearest
neighbor interaction

367
00:25:53,750 --> 00:25:57,170
in this particular fashion.

368
00:25:57,170 --> 00:26:00,890
We may or may not worry
about the last spin,

369
00:26:00,890 --> 00:26:04,980
whether I want to
finish the series here,

370
00:26:04,980 --> 00:26:08,810
or sometimes I will use
periodic boundary condition

371
00:26:08,810 --> 00:26:12,280
and bring it back and couple
it to the first one, where

372
00:26:12,280 --> 00:26:13,300
I have a ring.

373
00:26:13,300 --> 00:26:14,550
So that's another possibility.

374
00:26:17,320 --> 00:26:21,880
Doesn't really matter all
that much at this stage.

375
00:26:21,880 --> 00:26:30,330
So this runs for i
going from one to n.

376
00:26:30,330 --> 00:26:34,320
There are n degrees of freedom.

377
00:26:34,320 --> 00:26:37,830
Now, renormalization
group is a procedure

378
00:26:37,830 --> 00:26:43,150
by which I get rid of
some degrees of freedom.

379
00:26:43,150 --> 00:26:47,270
So previously, I have
emphasized that what we did

380
00:26:47,270 --> 00:26:50,200
was some kind of an averaging.

381
00:26:50,200 --> 00:26:51,900
So we said that
let's say I could

382
00:26:51,900 --> 00:26:55,060
do some averaging of
three sites and call

383
00:26:55,060 --> 00:26:58,930
some kind of a representative
of those three.

384
00:26:58,930 --> 00:27:04,210
Let's say that we
want to do a RG

385
00:27:04,210 --> 00:27:06,780
by a factor of b equals to two.

386
00:27:09,790 --> 00:27:17,230
So then maybe you say that
I will pick sigma i prime

387
00:27:17,230 --> 00:27:24,330
and u sigma i to be sigma i
plus sigma i plus 1 over 2,

388
00:27:24,330 --> 00:27:27,230
doing some kind of an average.

389
00:27:27,230 --> 00:27:31,120
The problem with this choice
is that if the two spins are

390
00:27:31,120 --> 00:27:33,290
both pluses, I will get plus.

391
00:27:33,290 --> 00:27:35,375
If they're both minuses,
I will get minus.

392
00:27:35,375 --> 00:27:39,680
If there is one plus and
one minus, I will get zero.

393
00:27:39,680 --> 00:27:42,860
Why that is not
nice is that that

394
00:27:42,860 --> 00:27:46,370
changes the structure
of the theory.

395
00:27:46,370 --> 00:27:48,810
So I started with
binary variables.

396
00:27:48,810 --> 00:27:50,110
I do this rescaling.

397
00:27:50,110 --> 00:27:56,390
If I choose this scheme, I will
have three variables per site.

398
00:27:56,390 --> 00:28:01,340
But I can insist upon keeping
two variables per site, as long

399
00:28:01,340 --> 00:28:05,460
as I do everything
consistently and precisely.

400
00:28:05,460 --> 00:28:11,140
So maybe I can say that
when this occurs, where

401
00:28:11,140 --> 00:28:17,400
the two sites are different,
and the average would be zero,

402
00:28:17,400 --> 00:28:20,340
I choose as tiebreaker
the left one.

403
00:28:29,240 --> 00:28:32,940
So then I will
have plus or minus.

404
00:28:32,940 --> 00:28:35,770
Now you can convince
yourself that if I do this,

405
00:28:35,770 --> 00:28:39,400
and I choose always the
left one as tiebreaker,

406
00:28:39,400 --> 00:28:43,130
the story is the same as
just keeping the left one.

407
00:28:46,610 --> 00:28:52,500
So essentially, this kind of
averaging with a tiebreaker

408
00:28:52,500 --> 00:28:58,555
is equivalent to getting
rid of every other spin.

409
00:29:01,190 --> 00:29:06,460
And so essentially
what I can do is

410
00:29:06,460 --> 00:29:11,640
to say that I call
a sigma i prime.

411
00:29:11,640 --> 00:29:16,989
Now, in the new listing that I
have, this thing is no longer,

412
00:29:16,989 --> 00:29:18,030
let's say, the tenth one.

413
00:29:18,030 --> 00:29:21,980
It becomes the fifth one because
I removed half of things.

414
00:29:21,980 --> 00:29:28,200
So sigma i prime is
really sigma 2i minus 1.

415
00:29:28,200 --> 00:29:39,590
So basically, all the odd ones
I will call to be my new spins.

416
00:29:39,590 --> 00:29:43,475
All the even ones I want to
get rid of, I'll call them SI.

417
00:29:47,200 --> 00:29:50,740
So this is just a
renaming of the variables.

418
00:29:50,740 --> 00:29:53,540
I did some handwaving
to justify.

419
00:29:53,540 --> 00:29:57,970
Effectivity, all I did
was I broke this sum

420
00:29:57,970 --> 00:30:04,050
into two sets of sums, but
I call sigma i prime and SI.

421
00:30:04,050 --> 00:30:06,740
And each one of
them the index i,

422
00:30:06,740 --> 00:30:10,500
rather than running from
1 to n in this new set,

423
00:30:10,500 --> 00:30:15,691
the index i runs
from 1 to n over 2.

424
00:30:21,350 --> 00:30:24,450
So what I have said,
agian, is very trivial.

425
00:30:24,450 --> 00:30:27,610
I've said that the
original sum, I

426
00:30:27,610 --> 00:30:31,790
bring over as a sum over
the odd spin, whose names

427
00:30:31,790 --> 00:30:36,530
I have changed, and a
sum over even spins,

428
00:30:36,530 --> 00:30:40,180
whose names I have called SI.

429
00:30:40,180 --> 00:30:48,240
And I have an interaction,
which I can write as sum over i,

430
00:30:48,240 --> 00:30:53,750
essentially running
from 1 to n over 2.

431
00:30:53,750 --> 00:30:58,310
I start with the interaction
that involves sigma i

432
00:30:58,310 --> 00:31:03,840
prime with si because
now each sigma i

433
00:31:03,840 --> 00:31:09,280
prime is acting with
an s on one side.

434
00:31:09,280 --> 00:31:11,160
And then there's
another interaction,

435
00:31:11,160 --> 00:31:13,650
which is SI, and the next.

436
00:31:19,290 --> 00:31:24,810
So essentially, I rename
things, and I regrouped bonds.

437
00:31:24,810 --> 00:31:31,590
And the sum that was n terms
now n over 2 pairs of terms.

438
00:31:31,590 --> 00:31:34,530
Nothing significant.

439
00:31:34,530 --> 00:31:37,920
But the point is
that over here, I

440
00:31:37,920 --> 00:31:43,400
can rewrite this as a
sum over sigma i prime.

441
00:31:43,400 --> 00:31:54,430
And this is a product over
terms where within each term,

442
00:31:54,430 --> 00:32:02,260
I can sum over the spin that
is sitting between two spins

443
00:32:02,260 --> 00:32:03,140
that I'm keeping.

444
00:32:03,140 --> 00:32:12,190
So I'm getting rid of this spin
that sits between spin sigma

445
00:32:12,190 --> 00:32:16,840
i prime and sigma
i plus 1 prime.

446
00:32:27,090 --> 00:32:33,800
Now, once I perform
this sum over SI here,

447
00:32:33,800 --> 00:32:36,790
then what I will
get is some function

448
00:32:36,790 --> 00:32:42,470
that depends on sigma i prime
and sigma i plus 1 prime.

449
00:32:42,470 --> 00:32:46,360
And I can choose to
write that function as e

450
00:32:46,360 --> 00:32:52,840
to the b prime sigma i
prime sigma i plus 1 prime.

451
00:32:52,840 --> 00:32:56,850
And hence, the
partition function

452
00:32:56,850 --> 00:33:02,210
after removing every
other spin is the same

453
00:33:02,210 --> 00:33:07,970
as the partition function that
I have for the remaining spins

454
00:33:07,970 --> 00:33:11,815
weighted with this b prime.

455
00:33:20,110 --> 00:33:25,030
So you can see that I took the
original partition function

456
00:33:25,030 --> 00:33:29,340
and recast it in
precisely the same form

457
00:33:29,340 --> 00:33:32,800
after removing half of
the degrees of freedom.

458
00:33:32,800 --> 00:33:36,340
Now, the original b
for the Ising model

459
00:33:36,340 --> 00:33:40,980
is going to be parameterized
by g, h, and k.

460
00:33:40,980 --> 00:33:46,540
So the b prime I
did parameterize.

461
00:33:46,540 --> 00:33:52,750
So this, let's say, emphasizes
parameterized by g, h, k.

462
00:33:52,750 --> 00:33:56,110
I can similarly
parameterize this

463
00:33:56,110 --> 00:34:00,140
by g prime, h prime, k prime.

464
00:34:00,140 --> 00:34:02,980
And how do I know that?

465
00:34:02,980 --> 00:34:04,890
Because when I was
writing this, I

466
00:34:04,890 --> 00:34:07,510
emphasized that this is
the most general form

467
00:34:07,510 --> 00:34:10,860
that I can write down.

468
00:34:10,860 --> 00:34:14,800
There is nothing else
other than this form

469
00:34:14,800 --> 00:34:17,719
that I can write down for this.

470
00:34:17,719 --> 00:34:24,850
So what I have essentially is
that this e to the b prime,

471
00:34:24,850 --> 00:34:30,454
which is e to the g prime plus
h prime sigma i prime plus sigma

472
00:34:30,454 --> 00:34:36,460
i plus 1 prime plus k
prime sigma i prime sigma

473
00:34:36,460 --> 00:34:42,210
i plus 1 prime involves
these three parameters,

474
00:34:42,210 --> 00:34:46,190
is obtained by summing over SI.

475
00:34:46,190 --> 00:34:54,659
Let me just call it s being
minus plus 1 of e to the g plus

476
00:34:54,659 --> 00:35:06,630
h sigma 1 sigma i prime plus
SI plus k sigma i prime SI

477
00:35:06,630 --> 00:35:20,610
plus g plus kh sigma SI plus
sigma i plus 1 prime plus k SI

478
00:35:20,610 --> 00:35:21,653
sigma i plus 1.

479
00:35:26,970 --> 00:35:30,270
So it's an implicit
equation that

480
00:35:30,270 --> 00:35:33,740
relates g prime, h prime,
k prime, to g, h, and k.

481
00:35:36,440 --> 00:35:40,620
And in particular, just
to make the writing

482
00:35:40,620 --> 00:35:46,170
of this thing explicit more
clearly, I will give names.

483
00:35:46,170 --> 00:35:53,990
I will call e to the k to
the x, e to the h to by,

484
00:35:53,990 --> 00:35:57,430
e to the g to bz.

485
00:35:57,430 --> 00:36:01,720
And here, similarly,
I will write x prime e

486
00:36:01,720 --> 00:36:05,396
to the k prime, y
prime e to the h prime,

487
00:36:05,396 --> 00:36:08,800
and z prime is e to the g prime.

488
00:36:15,230 --> 00:36:19,590
So now I just have
to make a table.

489
00:36:19,590 --> 00:36:23,830
I have sigma i prime
sigma i plus 1 prime.

490
00:36:26,860 --> 00:36:34,400
And here also I can
have values of s.

491
00:36:34,400 --> 00:36:43,430
And the simplest possibility
here is I have plus plus.

492
00:36:43,430 --> 00:36:48,580
Actually, let's put
this number further out.

493
00:36:48,580 --> 00:36:52,390
So if both the sigma
primes are plus,

494
00:36:52,390 --> 00:36:54,800
what do I have on
the left hand side?

495
00:36:54,800 --> 00:37:00,430
I have e to the g
prime, which is z prime.

496
00:37:00,430 --> 00:37:06,150
e to the 2h prime, which
is y prime squared.

497
00:37:06,150 --> 00:37:08,500
e to the k prime,
which is x prime.

498
00:37:11,300 --> 00:37:14,120
What do I have on
the left hand side?

499
00:37:14,120 --> 00:37:16,780
On the left hand side, I
have two possible things

500
00:37:16,780 --> 00:37:18,190
that I can put.

501
00:37:18,190 --> 00:37:21,420
I can put s to be either
plus or s to be minus,

502
00:37:21,420 --> 00:37:25,900
and I have to sum over
those two possibilities.

503
00:37:25,900 --> 00:37:30,380
You can see that in all
cases, I have e to the 2g.

504
00:37:30,380 --> 00:37:32,060
That's a trivial thing.

505
00:37:32,060 --> 00:37:35,650
So I will always
have a z squared.

506
00:37:35,650 --> 00:37:38,780
Irrespective of s, I
have two factors of e

507
00:37:38,780 --> 00:37:40,710
to the h sigma prime.

508
00:37:49,080 --> 00:37:52,260
OK, you know what happened?

509
00:37:52,260 --> 00:37:55,080
I should've used--
since I was using

510
00:37:55,080 --> 00:37:58,600
b, this factor of h over 2.

511
00:37:58,600 --> 00:38:06,420
So I really should have
put here an h prime over 2,

512
00:38:06,420 --> 00:38:09,940
and I should have
put here an h over 2

513
00:38:09,940 --> 00:38:17,100
and h over 2 because the field
that was residing on the sites,

514
00:38:17,100 --> 00:38:21,170
I am dividing half of it
to the right and half of it

515
00:38:21,170 --> 00:38:23,120
to the left bond.

516
00:38:23,120 --> 00:38:25,620
And since I'm writing things
in terms of the bonds,

517
00:38:25,620 --> 00:38:27,940
that's how I should go.

518
00:38:27,940 --> 00:38:35,644
So what I have here actually
is now one factor of i prime.

519
00:38:35,644 --> 00:38:37,880
That's better.

520
00:38:37,880 --> 00:38:42,400
Now, what I have on
the right is similarly

521
00:38:42,400 --> 00:38:46,810
one factor of y
from the h's that

522
00:38:46,810 --> 00:38:50,390
go with sigma 1
prime, sigma 2 prime.

523
00:38:50,390 --> 00:38:56,460
And then if s is
plus, then you can

524
00:38:56,460 --> 00:39:02,510
see that I will get two
factors of e to the k

525
00:39:02,510 --> 00:39:04,830
because both bonds
will be satisfied.

526
00:39:04,830 --> 00:39:07,540
Both bonds will be plus plus.

527
00:39:07,540 --> 00:39:12,020
So I will get x squared,
and the contribution

528
00:39:12,020 --> 00:39:14,740
to the h of the
intermediate bond s

529
00:39:14,740 --> 00:39:17,130
is going to be 1e to the h.

530
00:39:17,130 --> 00:39:21,310
So I will get x squared y.

531
00:39:21,310 --> 00:39:26,940
Whereas if I put it the
intermediate sign for s

532
00:39:26,940 --> 00:39:30,890
to be minus 1, then I have
two pluses at the end.

533
00:39:30,890 --> 00:39:32,410
The one in the middle is minus.

534
00:39:32,410 --> 00:39:36,020
So I would have two
dissatisfied factors of e

535
00:39:36,020 --> 00:39:39,270
to the k becoming
e to the minus k.

536
00:39:39,270 --> 00:39:41,360
So it's x to the minus 2.

537
00:39:41,360 --> 00:39:45,160
And the field also
will give a factor of e

538
00:39:45,160 --> 00:39:50,860
to the minus h or y inverse.

539
00:39:50,860 --> 00:39:53,130
So there are four
possibilities here.

540
00:39:53,130 --> 00:39:56,320
The next one is minus minus.

541
00:39:56,320 --> 00:40:00,070
z prime will stay the way it is.

542
00:40:00,070 --> 00:40:04,500
The field has switched sign.

543
00:40:04,500 --> 00:40:08,510
So this becomes y prime inverse.

544
00:40:08,510 --> 00:40:11,030
But since both of them
are minus-- minus,

545
00:40:11,030 --> 00:40:14,900
minus-- the k factor
is satisfied and happy.

546
00:40:14,900 --> 00:40:20,300
Gives me x prime because
it's e to the plus k.

547
00:40:20,300 --> 00:40:24,260
On the right hand side, I
will always get the z squared.

548
00:40:24,260 --> 00:40:29,590
The first factor
becomes the inverse.

549
00:40:29,590 --> 00:40:34,080
Now, s plus 1 is a
plus site, plus spin,

550
00:40:34,080 --> 00:40:36,830
that is sandwiched
between two minus spins.

551
00:40:36,830 --> 00:40:39,240
So there are two unhappy bonds.

552
00:40:39,240 --> 00:40:42,760
This gives me x to the minus 2.

553
00:40:42,760 --> 00:40:47,260
Why the spin is pointing in
the direction of the field.

554
00:40:47,260 --> 00:40:49,440
So there's the y here.

555
00:40:49,440 --> 00:40:52,670
And here I will get x
squared and y inverse

556
00:40:52,670 --> 00:40:56,470
because now I have
three negative signs.

557
00:40:56,470 --> 00:40:58,230
So all the k's are happy.

558
00:41:01,320 --> 00:41:03,960
There's the next one,
which is plus and minus.

559
00:41:03,960 --> 00:41:08,320
Plus and minus, we can see that
the contribution to the field

560
00:41:08,320 --> 00:41:10,540
vanishes.

561
00:41:10,540 --> 00:41:15,070
Because sigma i prime and plus
sigma i plus 1 prime are zero.

562
00:41:15,070 --> 00:41:19,170
I will still get z,
the contribution,

563
00:41:19,170 --> 00:41:22,180
because plus and minus,
the bond between them

564
00:41:22,180 --> 00:41:25,740
gives me e to the minus k prime.

565
00:41:25,740 --> 00:41:27,345
I have here z squared.

566
00:41:30,070 --> 00:41:34,910
There is no overall contribution
of y for the same reason

567
00:41:34,910 --> 00:41:37,640
that there was nothing here.

568
00:41:37,640 --> 00:41:47,160
But from s, I will get a
factor of y plus y inverse.

569
00:41:47,160 --> 00:41:49,560
And there's no
contribution for x

570
00:41:49,560 --> 00:41:52,100
because I have a
plus and a minus.

571
00:41:52,100 --> 00:41:56,230
And if the spin is either
plus or minus in the middle,

572
00:41:56,230 --> 00:41:59,820
there will be one satisfied
and one dissatisfied bond.

573
00:41:59,820 --> 00:42:03,740
Again, by symmetry,
the other configuration

574
00:42:03,740 --> 00:42:04,810
is exactly the same.

575
00:42:09,230 --> 00:42:14,650
So while I had four
configuration, and hence

576
00:42:14,650 --> 00:42:22,360
four things to match, the
last two are the same.

577
00:42:22,360 --> 00:42:25,280
And that's consistent
with my having

578
00:42:25,280 --> 00:42:29,020
three variables, x prime, y
prime, and z prime, to solve.

579
00:42:29,020 --> 00:42:31,700
So there are three equations
and three variables.

580
00:42:36,140 --> 00:42:41,620
Now, to solve these equations,
we can do several things.

581
00:42:46,660 --> 00:42:53,980
Let's, for example, multiply
all four equations together.

582
00:42:53,980 --> 00:42:56,420
What do we get on
the left hand side?

583
00:42:56,420 --> 00:43:01,110
There are two x's, two
inverse x's, y, and inverse y,

584
00:43:01,110 --> 00:43:02,930
but four factors of z.

585
00:43:02,930 --> 00:43:07,130
So I get z prime to
the fourth factor.

586
00:43:07,130 --> 00:43:10,610
On the other side, I
will get z to the eight,

587
00:43:10,610 --> 00:43:12,775
and then the product
of those four factors.

588
00:43:33,110 --> 00:43:39,820
I can divide one, equation
one, by equation two.

589
00:43:39,820 --> 00:43:41,580
What do I get?

590
00:43:41,580 --> 00:43:43,240
The x's cancel.

591
00:43:43,240 --> 00:43:44,350
The x's cancel.

592
00:43:44,350 --> 00:43:46,440
So I will get y prime squared.

593
00:43:49,300 --> 00:43:52,110
On the other side,
divide these two.

594
00:43:52,110 --> 00:43:53,350
I will get y squared.

595
00:43:56,910 --> 00:44:01,580
x squared y plus x
squared y inverse,

596
00:44:01,580 --> 00:44:11,768
x minus 2y plus x2 minus--
yeah, x2 y inverse.

597
00:44:15,600 --> 00:44:21,040
And finally, to get the equation
for x prime, what I can do

598
00:44:21,040 --> 00:44:29,002
is I can multiply 1 and
2, divide by 3 and 4.

599
00:44:29,002 --> 00:44:32,740
And if I do that, on
the left hand side

600
00:44:32,740 --> 00:44:38,075
I will get x prime
to the fourth.

601
00:44:38,075 --> 00:44:41,410
And on the right
hand side, I will

602
00:44:41,410 --> 00:44:49,450
get x squared y plus x
to the minus 2 y inverse

603
00:44:49,450 --> 00:44:57,660
x minus 2y x squared y inverse
divided by y plus y inverse

604
00:44:57,660 --> 00:44:58,160
squared.

605
00:45:07,070 --> 00:45:09,840
So I can take the log of
these equations, if you like,

606
00:45:09,840 --> 00:45:13,950
to get the recurrence
relations for the parameters.

607
00:45:13,950 --> 00:45:16,610
For example, taking the
log of that equation,

608
00:45:16,610 --> 00:45:19,540
you can see that I
will get g prime.

609
00:45:19,540 --> 00:45:26,990
From here, I would get
2g plus some function.

610
00:45:26,990 --> 00:45:35,420
There's some delta g
that depends on k and h.

611
00:45:35,420 --> 00:45:37,440
If I do the same
thing here, you can

612
00:45:37,440 --> 00:45:39,540
see that I will
get an h prime that

613
00:45:39,540 --> 00:45:43,910
is h plus some function
from the log of this

614
00:45:43,910 --> 00:45:46,400
that depends on k and h.

615
00:45:46,400 --> 00:45:51,231
And finally, I will get k
prime some function of k and h.

616
00:45:58,570 --> 00:46:02,840
This parameter g is
not that important.

617
00:46:02,840 --> 00:46:05,150
It's basically an
overall constant.

618
00:46:05,150 --> 00:46:07,370
The way that we started
it, we certainly

619
00:46:07,370 --> 00:46:11,810
don't expect it to modify
phase diagrams, et cetera.

620
00:46:11,810 --> 00:46:16,380
And you can see that it never
affects the equations that

621
00:46:16,380 --> 00:46:20,410
govern h and k, the
two parameters that

622
00:46:20,410 --> 00:46:23,820
give the relative weight of
different configurations.

623
00:46:23,820 --> 00:46:27,340
Whether you are in a ferromagnet
or a disordered state

624
00:46:27,340 --> 00:46:30,030
is governed by these
two parameters.

625
00:46:30,030 --> 00:46:32,510
And indeed, we can ignore this.

626
00:46:32,510 --> 00:46:37,120
Essentially, what it amounts to
is as follows, that every time

627
00:46:37,120 --> 00:46:41,230
I remove some of the
spins, I gradually

628
00:46:41,230 --> 00:46:44,410
build a contribution
to my free energy.

629
00:46:44,410 --> 00:46:47,070
Because clearly, once
I have integrated out

630
00:46:47,070 --> 00:46:51,070
over all of the energies, what I
will have will be the partition

631
00:46:51,070 --> 00:46:51,700
function.

632
00:46:51,700 --> 00:46:54,230
Its log would be
the free energy.

633
00:46:54,230 --> 00:46:56,720
And actually, we encountered
exactly the same thing

634
00:46:56,720 --> 00:46:59,530
when we were doing
momentum space RG.

635
00:46:59,530 --> 00:47:02,460
There was always, as a
result of the integration,

636
00:47:02,460 --> 00:47:07,020
some contribution that I
call delta f that I never

637
00:47:07,020 --> 00:47:10,230
looked at because, OK, it's
part of the free energy

638
00:47:10,230 --> 00:47:13,130
but does not govern
the relative weights

639
00:47:13,130 --> 00:47:15,440
of the different configurations.

640
00:47:15,440 --> 00:47:17,440
So these are really
the two things

641
00:47:17,440 --> 00:47:20,980
that we need to focus on.

642
00:47:20,980 --> 00:47:27,220
Now, if we did things
correctly and these

643
00:47:27,220 --> 00:47:32,190
are correct equations, had
I started in the subspace

644
00:47:32,190 --> 00:47:37,040
where h equals to zero,
which had up-down symmetries,

645
00:47:37,040 --> 00:47:40,700
I could have changed all of
my sigmas to minus sigma,

646
00:47:40,700 --> 00:47:44,350
and for h equals to zero, the
energy would not have changed.

647
00:47:44,350 --> 00:47:47,070
So a check that we
did things correctly

648
00:47:47,070 --> 00:47:51,390
is that if h equals to zero,
then h prime has to be zero.

649
00:47:51,390 --> 00:47:52,960
So let's see.

650
00:47:52,960 --> 00:47:59,530
If h equals to zero
or y is equal to one,

651
00:47:59,530 --> 00:48:01,690
well if y is equal
to one, you can

652
00:48:01,690 --> 00:48:05,660
see that those two factors in
the numerator and denominator

653
00:48:05,660 --> 00:48:07,250
are exactly the same.

654
00:48:07,250 --> 00:48:11,630
And so y prime stays to one.

655
00:48:11,630 --> 00:48:17,060
So this check has
been performed.

656
00:48:17,060 --> 00:48:22,000
So if I am on h equals to
zero on the subspace that

657
00:48:22,000 --> 00:48:28,030
has symmetry, then I have only
one parameter, this k or x.

658
00:48:28,030 --> 00:48:32,810
And so on this space, the
recursion relation that I have

659
00:48:32,810 --> 00:48:36,060
is that x prime
to the fourth is x

660
00:48:36,060 --> 00:48:39,280
squared plus x to
the minus 2 squared.

661
00:48:44,520 --> 00:48:46,110
I think I made a mistake here.

662
00:49:09,400 --> 00:49:12,390
No, it's correct.

663
00:49:12,390 --> 00:49:14,850
y plus y inverse.

664
00:49:14,850 --> 00:49:16,970
OK, but y is one.

665
00:49:16,970 --> 00:49:23,490
So this is divided by
2 squared, which is 4.

666
00:49:23,490 --> 00:49:40,090
But let me double check
so that I don't go-- yeah.

667
00:49:42,750 --> 00:49:45,690
So this is correct.

668
00:49:45,690 --> 00:49:54,480
So I can write it in terms
of x being e to the k.

669
00:49:54,480 --> 00:49:58,610
So what I have here
is e to the 4k prime.

670
00:49:58,610 --> 00:50:03,810
I have here e to the 2k plus e
to the minus 2k divided by 2,

671
00:50:03,810 --> 00:50:09,310
which is hyperbolic
cosine of 2k, squared.

672
00:50:09,310 --> 00:50:18,650
So what I will get is k
prime is 1/2 log hyperbolic

673
00:50:18,650 --> 00:50:19,890
cosine of 2k.

674
00:50:30,030 --> 00:50:39,490
Now, k is a parameter
that we are changing.

675
00:50:39,490 --> 00:50:43,980
As we make k stronger,
presumably things

676
00:50:43,980 --> 00:50:46,540
are more coupled to each
other and should go more

677
00:50:46,540 --> 00:50:48,110
towards order.

678
00:50:48,110 --> 00:50:52,900
As k goes towards zero,
basically the spins

679
00:50:52,900 --> 00:50:53,730
are independent.

680
00:50:53,730 --> 00:50:56,650
They can do whatever they like.

681
00:50:56,650 --> 00:50:59,530
So it kind of makes
sense that there

682
00:50:59,530 --> 00:51:06,180
should be a fixed point at
zero corresponding to zero

683
00:51:06,180 --> 00:51:09,110
correlation length.

684
00:51:09,110 --> 00:51:10,910
And let's check that.

685
00:51:10,910 --> 00:51:19,240
So if k is going to zero--
it's very small-- then

686
00:51:19,240 --> 00:51:28,400
k prime is 1/2 log hyperbolic
cosine of a small number,

687
00:51:28,400 --> 00:51:34,330
is roughly 1 plus that
small number squared over 2.

688
00:51:37,170 --> 00:51:39,670
There's a series like that.

689
00:51:39,670 --> 00:51:43,890
Log of 1 plus a small
number is a small number.

690
00:51:43,890 --> 00:51:49,130
So this becomes 4k
squared over 2 over 2.

691
00:51:49,130 --> 00:51:49,970
It's k squared.

692
00:51:52,880 --> 00:51:58,200
So it says that if I, let's say,
start with a k that is 1/10,

693
00:51:58,200 --> 00:52:02,650
my k prime would be
1/100 and then 1/10,000.

694
00:52:02,650 --> 00:52:07,600
So basically, this
is a fixed point

695
00:52:07,600 --> 00:52:11,220
that attracts everything to it.

696
00:52:11,220 --> 00:52:13,670
Essentially, what
it says is you may

697
00:52:13,670 --> 00:52:17,130
have some weak
amount of coupling.

698
00:52:17,130 --> 00:52:19,930
As you go and look at the spins
that are further and further

699
00:52:19,930 --> 00:52:24,080
apart, the effective coupling
between them is going to zero.

700
00:52:24,080 --> 00:52:25,810
And the spins that
are further apart

701
00:52:25,810 --> 00:52:27,190
don't care about each other.

702
00:52:27,190 --> 00:52:29,820
They do whatever they like.

703
00:52:29,820 --> 00:52:33,129
So that all makes
physical sense.

704
00:52:33,129 --> 00:52:35,170
Well, we are really
interested in this other end.

705
00:52:35,170 --> 00:52:37,750
What happens at k
going to infinity?

706
00:52:43,820 --> 00:52:47,850
We can kind of look
at this equation.

707
00:52:47,850 --> 00:52:50,690
Or actually, look at this
equation, also doesn't matter.

708
00:52:50,690 --> 00:52:52,630
But here, maybe it's clearer.

709
00:52:52,630 --> 00:52:56,390
So I have e to the 4k prime.

710
00:52:56,390 --> 00:52:58,755
And this is going to
be dominated by this.

711
00:52:58,755 --> 00:53:00,440
This is e to the 2k.

712
00:53:00,440 --> 00:53:02,840
e to the minus 2k is very small.

713
00:53:02,840 --> 00:53:04,910
So it's going to
be approximately

714
00:53:04,910 --> 00:53:09,110
e to the 4k divided by 4.

715
00:53:13,790 --> 00:53:16,710
So what I have out here
is that k is very large.

716
00:53:16,710 --> 00:53:20,790
k prime is roughly
k, but then smaller

717
00:53:20,790 --> 00:53:25,300
by a factor that
is 1/2 of log 2.

718
00:53:29,560 --> 00:53:34,090
So we try to take two spins that
are next to each other, couple

719
00:53:34,090 --> 00:53:35,810
them with a very strong k.

720
00:53:35,810 --> 00:53:39,550
Let's say a million,
10 million, whatever.

721
00:53:39,550 --> 00:53:43,880
And then I look at spins
that are too further apart.

722
00:53:43,880 --> 00:53:47,710
They're still very strongly
coupled, but slightly less.

723
00:53:47,710 --> 00:53:49,690
It's a million minus log 2.

724
00:53:49,690 --> 00:53:56,150
So it very, very gradually
starts to move away from here.

725
00:53:56,150 --> 00:54:01,850
But then as it kind of goes
further, it kind of speeds up.

726
00:54:01,850 --> 00:54:04,550
What it says is that it is true.

727
00:54:04,550 --> 00:54:08,100
At infinity, you
have a fixed point.

728
00:54:08,100 --> 00:54:11,320
But that fixed
point is unstable.

729
00:54:11,320 --> 00:54:17,210
And even if you have very
strong but finite coupling,

730
00:54:17,210 --> 00:54:21,970
because things are finite, as
you go further and further out,

731
00:54:21,970 --> 00:54:24,810
the spins become less
and less correlated.

732
00:54:24,810 --> 00:54:28,540
This, again, is another
indication of the statement

733
00:54:28,540 --> 00:54:31,140
that a one dimensional
system will not order.

734
00:54:35,030 --> 00:54:39,030
So one thing that you
can do to sort of make

735
00:54:39,030 --> 00:54:43,160
this look slightly better, since
we have the interval from zero

736
00:54:43,160 --> 00:54:47,040
to infinity, is to
change variables.

737
00:54:47,040 --> 00:54:52,380
I can ask what does
tang of k prime do?

738
00:54:52,380 --> 00:54:56,790
So tang k prime is e to the
2k prime minus 1 divided by

739
00:54:56,790 --> 00:55:00,160
e to the 2k prime plus 1.

740
00:55:00,160 --> 00:55:07,750
You write e to the 2k's
in terms of this variable.

741
00:55:07,750 --> 00:55:11,750
At the end of the day,
a little bit of algebra,

742
00:55:11,750 --> 00:55:15,320
you can convince yourself
that the recursion relation

743
00:55:15,320 --> 00:55:18,650
for tang k has a
very simple form.

744
00:55:18,650 --> 00:55:21,660
The new tang k is simply the
square of the old tang k.

745
00:55:28,870 --> 00:55:34,380
If I plot things as
a function of tang k

746
00:55:34,380 --> 00:55:41,890
that runs between zero
and one, that fixed point

747
00:55:41,890 --> 00:55:46,200
that was at infinity
gets mapped into one,

748
00:55:46,200 --> 00:55:50,970
and the flow is
always towards here.

749
00:55:50,970 --> 00:55:55,080
There is no other
fixed point in between.

750
00:55:55,080 --> 00:55:57,670
Previously, it wasn't
clear from the way

751
00:55:57,670 --> 00:56:00,790
that I had written whether
potentially there's

752
00:56:00,790 --> 00:56:03,380
other fixed points
along the k-axis.

753
00:56:03,380 --> 00:56:08,280
If you write it as t
prime tang is t squared,

754
00:56:08,280 --> 00:56:11,380
where t stands for tang,
clearly the only fixed points

755
00:56:11,380 --> 00:56:14,750
are at zero and one.

756
00:56:14,750 --> 00:56:16,860
But now this also
allows us to ask

757
00:56:16,860 --> 00:56:19,590
what happens if
you also introduce

758
00:56:19,590 --> 00:56:25,410
the field direction,
h, in this space.

759
00:56:28,070 --> 00:56:33,710
Now, one thing to
check is that if you

760
00:56:33,710 --> 00:56:39,225
start with k equals to zero,
that is independent spins,

761
00:56:39,225 --> 00:56:42,700
you look at the equation here.

762
00:56:42,700 --> 00:56:47,450
k equals to 0 corresponds
to x equals to one.

763
00:56:47,450 --> 00:56:48,870
This factor drops out.

764
00:56:48,870 --> 00:56:52,010
You see y prime is equal to y.

765
00:56:52,010 --> 00:56:54,950
So if you have no
coupling, there

766
00:56:54,950 --> 00:56:58,700
is no reason why the
magnetic field should change.

767
00:56:58,700 --> 00:57:05,230
So essentially, this entire line
corresponds to fixed points.

768
00:57:05,230 --> 00:57:10,386
Every point here
is a fixed point.

769
00:57:10,386 --> 00:57:12,760
And we can show
that, essentially,

770
00:57:12,760 --> 00:57:16,460
if you start along
field zero, you go here.

771
00:57:16,460 --> 00:57:21,210
If you start along different
values of the field,

772
00:57:21,210 --> 00:57:25,690
you basically go to
someplace along this line.

773
00:57:25,690 --> 00:57:30,700
And all of these flows
originate, if you go and plot

774
00:57:30,700 --> 00:57:34,450
them, from back here
at this fixed point

775
00:57:34,450 --> 00:57:38,800
that we identified before
at h equals to zero.

776
00:57:38,800 --> 00:57:43,500
So in order to find out what is
happening along that direction,

777
00:57:43,500 --> 00:57:49,730
all we need to do is to go and
look at x going to infinity.

778
00:57:49,730 --> 00:57:51,650
With x going to
infinity, you can

779
00:57:51,650 --> 00:57:55,730
see that y prime squared, the
equation that we have for y,

780
00:57:55,730 --> 00:57:58,760
y prime squared is y squared.

781
00:57:58,760 --> 00:58:01,020
And then there's this fraction.

782
00:58:01,020 --> 00:58:04,050
Clearly, the terms that are
proportional to x squared

783
00:58:04,050 --> 00:58:06,170
will overwhelm
those proportional

784
00:58:06,170 --> 00:58:08,350
to x to the minus 2.

785
00:58:08,350 --> 00:58:10,910
So I will get a
y and a y inverse

786
00:58:10,910 --> 00:58:13,280
from the numerator
and denominator.

787
00:58:13,280 --> 00:58:18,270
And so this goes
like y to the fourth.

788
00:58:18,270 --> 00:58:22,990
Which means that in the vicinity
of this point, what you have is

789
00:58:22,990 --> 00:58:24,775
that h prime is 2h.

790
00:58:28,770 --> 00:58:32,490
So this, again, is
irrelevant direction.

791
00:58:32,490 --> 00:58:38,420
And here, you are flowing
in this direction.

792
00:58:38,420 --> 00:58:40,650
And the combination
of these two really

793
00:58:40,650 --> 00:58:44,930
justifies the kind of flows
that I had shown you before.

794
00:58:48,790 --> 00:58:52,050
So essentially, in the
one dimensional model,

795
00:58:52,050 --> 00:58:55,560
you can start with any
microscopic set of k and h's

796
00:58:55,560 --> 00:58:57,000
that you want.

797
00:58:57,000 --> 00:59:00,895
As you rescale, you essentially
go to independent spins

798
00:59:00,895 --> 00:59:02,270
with an effective
magnetic field.

799
00:59:06,690 --> 00:59:13,200
So let's say we start with
very close to this fixed point.

800
00:59:13,200 --> 00:59:16,510
So we had a very
large value of k.

801
00:59:16,510 --> 00:59:19,510
I expect if I have
a large value of k,

802
00:59:19,510 --> 00:59:23,860
neighboring spins are
really close to each other.

803
00:59:23,860 --> 00:59:29,290
I have to go very far before,
if I have a patch of pluses,

804
00:59:29,290 --> 00:59:32,080
I can go over to a
patch of minuses.

805
00:59:32,080 --> 00:59:34,550
So there's a large
correlation length

806
00:59:34,550 --> 00:59:36,290
in the vicinity of this point.

807
00:59:39,160 --> 00:59:43,620
And that correlation
length is a function

808
00:59:43,620 --> 00:59:48,020
of our parameters k and h.

809
00:59:48,020 --> 00:59:51,240
Now, the point is that
the recursion relation

810
00:59:51,240 --> 00:59:57,150
that I have for k does not look
like the type of recursions

811
00:59:57,150 --> 00:59:59,020
that I had before.

812
00:59:59,020 --> 01:00:03,780
This one is fine because I can
think of this as my usual h

813
01:00:03,780 --> 01:00:08,830
prime is b to the yh h, where
here I'm clearly dealing

814
01:00:08,830 --> 01:00:15,890
with a b equals to two, and so
I can read off my yh to be one.

815
01:00:15,890 --> 01:00:19,640
But the recursion
relation that I have for k

816
01:00:19,640 --> 01:00:22,730
is not of that form.

817
01:00:22,730 --> 01:00:26,700
But who said that I
choose k as the variable

818
01:00:26,700 --> 01:00:28,470
that I put over here?

819
01:00:28,470 --> 01:00:31,290
I have been doing all of
these manipulations going

820
01:00:31,290 --> 01:00:34,380
from k to x, et cetera.

821
01:00:34,380 --> 01:00:36,870
One thing that behaves
nicely, you can see,

822
01:00:36,870 --> 01:00:40,700
is a variable if I
call e to the minus k

823
01:00:40,700 --> 01:00:48,660
prime is square root
of 2 e to the minus k.

824
01:00:52,460 --> 01:00:55,080
At k equals to infinity,
this is something

825
01:00:55,080 --> 01:00:57,450
that goes to zero on both ends.

826
01:00:57,450 --> 01:01:00,350
But if its k is
large but not finite,

827
01:01:00,350 --> 01:01:03,070
it says that on
the rescaling, it

828
01:01:03,070 --> 01:01:06,770
changes by a factor of root 2.

829
01:01:06,770 --> 01:01:10,150
So if I say, well,
what is c as a function

830
01:01:10,150 --> 01:01:12,930
of the magnetic field,
rather than writing k,

831
01:01:12,930 --> 01:01:16,010
I will write it as
e to the minus k.

832
01:01:16,010 --> 01:01:20,080
It's just another way
of writing things.

833
01:01:20,080 --> 01:01:24,450
Well, I know that under
one step of RG, all

834
01:01:24,450 --> 01:01:27,730
my length scales have
shrunk by a factor of two.

835
01:01:27,730 --> 01:01:34,920
So this is twice the c that
I should write down with 2h

836
01:01:34,920 --> 01:01:37,180
and root 2 e to the minus k.

837
01:01:41,050 --> 01:01:44,950
So I just related the
correlation length

838
01:01:44,950 --> 01:01:49,760
before and after one
step of RG, starting

839
01:01:49,760 --> 01:01:51,935
from very close to this point.

840
01:01:54,830 --> 01:01:58,250
And for these two
factors, 2 and root 2,

841
01:01:58,250 --> 01:02:02,320
I use the results that
I have over there.

842
01:02:02,320 --> 01:02:05,130
And this I can keep doing.

843
01:02:05,130 --> 01:02:06,900
I can keep iterating this.

844
01:02:06,900 --> 01:02:09,720
After l steps of
RG, this becomes 2

845
01:02:09,720 --> 01:02:15,630
to the l, c of 2
to the lh, and 2

846
01:02:15,630 --> 01:02:18,070
to the l over 2
e to the minus k.

847
01:02:25,530 --> 01:02:28,770
I iterate this sufficiently
so that I have moved away

848
01:02:28,770 --> 01:02:33,280
from this fixed point, where
everything is very strongly

849
01:02:33,280 --> 01:02:35,110
correlated.

850
01:02:35,110 --> 01:02:37,730
And so that means
I move, let's say,

851
01:02:37,730 --> 01:02:41,110
until this number is something
that is of the order of one.

852
01:02:41,110 --> 01:02:41,830
Could be 2.

853
01:02:41,830 --> 01:02:42,840
Could be 1/2.

854
01:02:42,840 --> 01:02:45,090
I really don't care.

855
01:02:45,090 --> 01:02:47,230
But the number of
iterations, the number

856
01:02:47,230 --> 01:02:49,340
of rescalings by a
factor of two that I

857
01:02:49,340 --> 01:02:53,085
have to do in order to achieve
this, is when 2 to the l

858
01:02:53,085 --> 01:02:55,498
is of the order of e to the k.

859
01:02:58,920 --> 01:03:00,200
e to the 2k, sorry.

860
01:03:11,770 --> 01:03:15,320
And if I do that, I
find that c should

861
01:03:15,320 --> 01:03:25,600
be e to the 2k some
function of h e to the 2k.

862
01:03:37,660 --> 01:03:42,590
So let's say I am
at h equals to zero.

863
01:03:42,590 --> 01:03:46,530
And I make the strength of the
interaction between neighbors

864
01:03:46,530 --> 01:03:48,430
stronger and stronger.

865
01:03:48,430 --> 01:03:51,140
If I asked you, how does
the correlation length,

866
01:03:51,140 --> 01:03:55,350
how does the size of the patches
that are all plus and minus

867
01:03:55,350 --> 01:03:58,440
change as k becomes
stronger and stronger?

868
01:03:58,440 --> 01:04:02,740
Well, RG tells you that
it goes in this fashion.

869
01:04:02,740 --> 01:04:04,620
In the problem
set that you have,

870
01:04:04,620 --> 01:04:08,357
you will solve the problem
exactly by a different method

871
01:04:08,357 --> 01:04:09,440
and get exactly this form.

872
01:04:15,610 --> 01:04:16,155
What else?

873
01:04:16,155 --> 01:04:21,370
Well, we said that one of the
characteristics of a system

874
01:04:21,370 --> 01:04:23,620
is that the free
energy has a singular

875
01:04:23,620 --> 01:04:27,010
part as a function
of the variables

876
01:04:27,010 --> 01:04:31,910
that we have that is related
to the correlation length

877
01:04:31,910 --> 01:04:33,690
to the power d.

878
01:04:33,690 --> 01:04:37,100
In this case, we
have d equals to one.

879
01:04:37,100 --> 01:04:40,020
So the statement is
that the singular part

880
01:04:40,020 --> 01:04:44,170
of the free energy, as you
approach infinite coupling

881
01:04:44,170 --> 01:04:47,580
strength, behaves
as e to the minus 2k

882
01:04:47,580 --> 01:04:52,404
some other function
of h, e to the 2k.

883
01:04:58,020 --> 01:05:01,440
Once you have a
function such as this,

884
01:05:01,440 --> 01:05:03,930
you can take two
derivatives with respect

885
01:05:03,930 --> 01:05:07,330
to the field to get
the susceptibility.

886
01:05:07,330 --> 01:05:11,530
So the susceptibility would
go like two derivatives

887
01:05:11,530 --> 01:05:14,850
of the free energy with
respect to the field.

888
01:05:14,850 --> 01:05:18,410
You can see that each derivative
brings forth a factor of e

889
01:05:18,410 --> 01:05:19,930
to the 2k.

890
01:05:19,930 --> 01:05:23,850
So two derivatives will bring
a factor of e to the 2k,

891
01:05:23,850 --> 01:05:28,790
if I evaluate it at
h equals to zero.

892
01:05:28,790 --> 01:05:35,660
So the statement is that
if I'm at zero field,

893
01:05:35,660 --> 01:05:41,640
and I put on a small
infinitesimal magnetic field,

894
01:05:41,640 --> 01:05:44,080
it tends to overturn
all of the spins

895
01:05:44,080 --> 01:05:46,380
in the direction of the field.

896
01:05:46,380 --> 01:05:50,010
And the further you are
close to k to infinity,

897
01:05:50,010 --> 01:05:53,270
there are larger responses
that you would see.

898
01:05:53,270 --> 01:05:55,130
The susceptibility
of the response

899
01:05:55,130 --> 01:05:58,070
diverges as k goes to infinity.

900
01:05:58,070 --> 01:06:02,780
So in some sense, this model
does not have phase transition,

901
01:06:02,780 --> 01:06:05,970
but it demonstrates some
signatures of the phase

902
01:06:05,970 --> 01:06:08,910
transition, such as
diverging correlation length

903
01:06:08,910 --> 01:06:11,530
and diverging
susceptibility if you

904
01:06:11,530 --> 01:06:14,340
go to very, very strong
nearest neighbor coupling.

905
01:06:19,840 --> 01:06:23,170
There is one other
relationship that we had.

906
01:06:23,170 --> 01:06:28,410
That is, the susceptibility
is an integral

907
01:06:28,410 --> 01:06:37,120
in d dimension over
the correlation length,

908
01:06:37,120 --> 01:06:40,120
which we could say
in one dimension--

909
01:06:40,120 --> 01:06:42,710
well, actually, let's
write it in d dimension.

910
01:06:42,710 --> 01:06:47,550
e to the minus x
over c divided by x.

911
01:06:47,550 --> 01:06:52,940
We introduce an exponent,
eta, to describe

912
01:06:52,940 --> 01:06:55,990
the decay of correlations.

913
01:06:55,990 --> 01:07:00,050
So this would generally behave
like c to the 2 minus eta.

914
01:07:05,680 --> 01:07:11,590
Now, in this case, we see that
both the correlation length

915
01:07:11,590 --> 01:07:17,840
and susceptibility diverge with
the same behavior, e to the 2k.

916
01:07:17,840 --> 01:07:20,050
They're proportional
to each other.

917
01:07:20,050 --> 01:07:23,250
So that immediately
tells me that for the one

918
01:07:23,250 --> 01:07:26,070
dimensional system
that we are looking at,

919
01:07:26,070 --> 01:07:29,700
eta is equal to one.

920
01:07:29,700 --> 01:07:34,170
And if I substitute back
here, so eta is one,

921
01:07:34,170 --> 01:07:37,180
the dimension is one,
and the two cancels.

922
01:07:37,180 --> 01:07:39,890
Essentially, it says that
the correlation length

923
01:07:39,890 --> 01:07:45,490
in one dimension have a
pure exponential decay.

924
01:07:45,490 --> 01:07:48,010
They don't have this
sub leading power load

925
01:07:48,010 --> 01:07:51,310
that you would have
in higher dimensions.

926
01:07:51,310 --> 01:07:53,640
So when you do
things exactly, you

927
01:07:53,640 --> 01:07:57,510
will also be able
to verify that.

928
01:07:57,510 --> 01:08:03,230
So all of the predictions
of this position space

929
01:08:03,230 --> 01:08:09,390
RG method that we can carry out
in this one dimensional example

930
01:08:09,390 --> 01:08:14,920
very easily, you can
also calculate and obtain

931
01:08:14,920 --> 01:08:18,229
through the method that
is called transfer matrix,

932
01:08:18,229 --> 01:08:20,319
and is the subject
of the problem set

933
01:08:20,319 --> 01:08:21,487
that was posted yesterday.

934
01:08:27,220 --> 01:08:32,090
Also, you can see that
this approach will

935
01:08:32,090 --> 01:08:35,890
work for any system
in one dimension.

936
01:08:35,890 --> 01:08:44,040
All I really need to ensure is
that the b that I write down

937
01:08:44,040 --> 01:08:47,229
is sufficiently
general to include

938
01:08:47,229 --> 01:08:51,720
all possible interactions that
I can write between two spins.

939
01:08:51,720 --> 01:08:55,200
Because if I have some
subset of those interactions,

940
01:08:55,200 --> 01:08:58,350
and then I do this procedure
that I have over here

941
01:08:58,350 --> 01:09:02,189
and then take the
log, there's no reason

942
01:09:02,189 --> 01:09:05,160
why that would not
generate all things that

943
01:09:05,160 --> 01:09:07,130
are consistent with symmetry.

944
01:09:07,130 --> 01:09:10,300
So you really have to
put all possible terms,

945
01:09:10,300 --> 01:09:13,300
and then you will get
all possible terms here,

946
01:09:13,300 --> 01:09:16,960
and there would be a recursion
relation that would relate.

947
01:09:16,960 --> 01:09:20,720
You can do this very easily, for
example, for the Potts model.

948
01:09:20,720 --> 01:09:23,759
For the continuous spin systems,
it becomes more difficult.

949
01:09:33,839 --> 01:09:42,380
Now let's say briefly as to why
we can solve things exactly,

950
01:09:42,380 --> 01:09:45,210
let's say, for the one
dimensionalizing model

951
01:09:45,210 --> 01:09:46,859
by this procedure.

952
01:09:46,859 --> 01:09:51,540
But this procedure we cannot
do in higher dimensions.

953
01:09:51,540 --> 01:09:56,640
So let's, for example, think
that we have a square lattice.

954
01:09:56,640 --> 01:09:59,045
Just generalize what we
had to two dimensions.

955
01:10:03,170 --> 01:10:10,940
And let's say that we want
to eliminate this spin

956
01:10:10,940 --> 01:10:14,180
and-- well, let's see,
what's the best way?

957
01:10:14,180 --> 01:10:18,730
Yeah, we want to eliminate
a checkerboard of spins.

958
01:10:18,730 --> 01:10:22,750
So we want to eliminate
half of the spins.

959
01:10:22,750 --> 01:10:26,255
Let's say the white
squares on a checkerboard.

960
01:10:29,750 --> 01:10:34,090
And if I were to eliminate these
spins, like I did over here,

961
01:10:34,090 --> 01:10:37,520
I should be able to
generate interactions

962
01:10:37,520 --> 01:10:41,640
among spins that are left over.

963
01:10:41,640 --> 01:10:42,870
So you say, fine.

964
01:10:42,870 --> 01:10:45,500
Let's pick these four spins.

965
01:10:45,500 --> 01:10:52,030
Sigma 1, sigma 2,
sigma 3, sigma 4,

966
01:10:52,030 --> 01:10:55,070
that are connected
to this spin, s,

967
01:10:55,070 --> 01:10:59,670
that is sitting in the middle,
that I have to eliminate.

968
01:10:59,670 --> 01:11:03,930
So let's also stay
in the space where

969
01:11:03,930 --> 01:11:07,900
h equals to zero,
just for simplicity.

970
01:11:07,900 --> 01:11:10,760
So the result of
eliminating that is

971
01:11:10,760 --> 01:11:14,080
I have to do a sum over s.

972
01:11:14,080 --> 01:11:19,500
I have e to the-- let's say
the original interaction is k,

973
01:11:19,500 --> 01:11:27,611
and s is coupled by these bonds
to sigma 1, sigma 2, sigma 3,

974
01:11:27,611 --> 01:11:28,110
sigma 4.

975
01:11:33,070 --> 01:11:35,940
Now, summing over the
two possible values of s

976
01:11:35,940 --> 01:11:36,920
is very simple.

977
01:11:36,920 --> 01:11:40,950
It just gives me e to
the k times the sum plus

978
01:11:40,950 --> 01:11:42,850
e to the minus k, that sum.

979
01:11:42,850 --> 01:11:48,110
So it's the same thing as 2
hyperbolic cosine of k, sigma

980
01:11:48,110 --> 01:11:51,220
1 plus sigma 2 plus
sigma 3 plus sigma 4.

981
01:11:55,960 --> 01:11:57,660
We say good.

982
01:11:57,660 --> 01:12:00,370
I had something like
that, and I took a log

983
01:12:00,370 --> 01:12:02,530
so that I got the k prime.

984
01:12:02,530 --> 01:12:07,030
So maybe I'll do something
like a recasting of this,

985
01:12:07,030 --> 01:12:10,760
and maybe a recasting of this
will give me some constant,

986
01:12:10,760 --> 01:12:13,450
will give me some
kind of a k prime,

987
01:12:13,450 --> 01:12:19,270
and then I will have sigma
1, sigma 2, sigma 2, sigma 3,

988
01:12:19,270 --> 01:12:22,890
sigma 3, sigma 4,
sigma 4, sigma 1.

989
01:12:26,240 --> 01:12:28,870
But you immediately see that
that cannot be the entire

990
01:12:28,870 --> 01:12:30,590
story.

991
01:12:30,590 --> 01:12:35,220
Because there is really
no ordering among sigma 1,

992
01:12:35,220 --> 01:12:38,140
sigma 2, sigma 3, sigma 4.

993
01:12:38,140 --> 01:12:40,760
So clearly, because of
the symmetries of this,

994
01:12:40,760 --> 01:12:45,530
you will generate also sigma 1
sigma 3 plus sigma 2 sigma 4.

995
01:12:51,260 --> 01:12:56,610
That is, eliminating this
spin will generate for you

996
01:12:56,610 --> 01:12:59,800
interactions among
these, but also

997
01:12:59,800 --> 01:13:03,640
interactions that go like this.

998
01:13:03,640 --> 01:13:06,960
And in fact, if you
do it carefully,

999
01:13:06,960 --> 01:13:10,780
you'll find that you will also
generate an interaction that

1000
01:13:10,780 --> 01:13:14,465
is product of all four spins.

1001
01:13:14,465 --> 01:13:18,350
You will generate something
that involves all of the force.

1002
01:13:18,350 --> 01:13:21,550
So basically, because
of the way of geometry,

1003
01:13:21,550 --> 01:13:25,110
et cetera, that you have
in higher dimensions,

1004
01:13:25,110 --> 01:13:28,340
there is no trick that
is analogous to what

1005
01:13:28,340 --> 01:13:31,980
we could do in one dimension
where you would eliminate

1006
01:13:31,980 --> 01:13:37,140
some subset of spins and not
generate longer and longer

1007
01:13:37,140 --> 01:13:40,920
range interactions, interactions
that you did not have.

1008
01:13:40,920 --> 01:13:43,840
You could say, OK, I
will start including

1009
01:13:43,840 --> 01:13:46,580
all of these
interactions and then

1010
01:13:46,580 --> 01:13:50,020
have a higher, larger
parameter space.

1011
01:13:50,020 --> 01:13:51,880
But then you do
something else, you'll

1012
01:13:51,880 --> 01:13:54,790
find that you need to
include further and further

1013
01:13:54,790 --> 01:13:57,360
neighboring interactions.

1014
01:13:57,360 --> 01:14:04,000
So unless you do some kind of
termination or approximation,

1015
01:14:04,000 --> 01:14:07,140
which we will do
next time, then there

1016
01:14:07,140 --> 01:14:13,008
is no way to do this exactly
in higher dimensions.

1017
01:14:13,008 --> 01:14:14,111
AUDIENCE: Question.

1018
01:14:14,111 --> 01:14:14,735
PROFESSOR: Yes.

1019
01:14:17,450 --> 01:14:20,730
AUDIENCE: I mean, are you
putting any sort of weight

1020
01:14:20,730 --> 01:14:26,610
on the fact that, for
example, sigma 1 and sigma 3

1021
01:14:26,610 --> 01:14:29,010
are farther apart than
sigma 1 and sigma 2,

1022
01:14:29,010 --> 01:14:34,680
or are we using taxi cab
distances on this lattice?

1023
01:14:34,680 --> 01:14:40,220
PROFESSOR: Well, we are
thinking of an original model

1024
01:14:40,220 --> 01:14:42,660
that we would like
to solve, in which

1025
01:14:42,660 --> 01:14:45,670
I have specified that
things are coupled only

1026
01:14:45,670 --> 01:14:47,900
to nearest neighbors.

1027
01:14:47,900 --> 01:14:50,910
So the ones that correspond
to sigma 1, sigma 3,

1028
01:14:50,910 --> 01:14:52,520
are next nearest neighbors.

1029
01:14:52,520 --> 01:14:55,720
They're certainly farther
apart on the lattice.

1030
01:14:55,720 --> 01:14:59,430
You could say, well,
there's some justification,

1031
01:14:59,430 --> 01:15:03,180
if these are ions
and they have spins,

1032
01:15:03,180 --> 01:15:06,260
to have some weaker interaction
that goes across here.

1033
01:15:06,260 --> 01:15:09,350
There has to be some
notion of space.

1034
01:15:09,350 --> 01:15:13,360
I don't want to couple everybody
to everybody else equivalently.

1035
01:15:13,360 --> 01:15:17,520
But if I include this,
then I have further more

1036
01:15:17,520 --> 01:15:19,670
complicated Hamiltonian.

1037
01:15:19,670 --> 01:15:22,180
And when I do RG,
I will generate

1038
01:15:22,180 --> 01:15:23,922
an even more
complicated Hamiltonian.

1039
01:15:27,300 --> 01:15:28,164
Yes.

1040
01:15:28,164 --> 01:15:29,080
AUDIENCE: [INAUDIBLE].

1041
01:15:35,330 --> 01:15:39,039
PROFESSOR: Question is, suppose
I have a square lattice.

1042
01:15:39,039 --> 01:15:39,957
Let's go here.

1043
01:15:43,170 --> 01:15:46,640
And the suggestion
is, why don't I

1044
01:15:46,640 --> 01:15:50,730
eliminate all of
the spins over here,

1045
01:15:50,730 --> 01:15:53,740
maybe all of the
spins over here?

1046
01:15:53,740 --> 01:15:56,620
So the problem is that I
will generate interactions

1047
01:15:56,620 --> 01:16:00,212
that not only go like this, but
interactions that go like this.

1048
01:16:03,030 --> 01:16:09,040
So the idea of what happens is
that imagine that there's these

1049
01:16:09,040 --> 01:16:11,240
spins that you're eliminating.

1050
01:16:11,240 --> 01:16:14,790
As long as there's paths that
connect the spins that you're

1051
01:16:14,790 --> 01:16:18,620
eliminating to any
other spin, you

1052
01:16:18,620 --> 01:16:20,220
will generate that
kind of couple.

1053
01:16:28,650 --> 01:16:31,860
Again, the reason that the
one dimensional model works

1054
01:16:31,860 --> 01:16:35,920
is also related to
its exact solvability

1055
01:16:35,920 --> 01:16:38,700
by this transfer matrix method.

1056
01:16:38,700 --> 01:16:43,070
So I will briefly mention
that in the last five minutes.

1057
01:16:47,780 --> 01:16:53,300
So for one dimensional
models, the partition function

1058
01:16:53,300 --> 01:16:57,730
is a sum over whatever
degree of freedom you have.

1059
01:16:57,730 --> 01:17:01,410
Could be Ising, Potts,
xy, doesn't matter.

1060
01:17:01,410 --> 01:17:04,480
But the point is
that the interaction

1061
01:17:04,480 --> 01:17:10,820
is a sum of bonds that connect
one site to the next site.

1062
01:17:10,820 --> 01:17:18,566
I can write this as a product of
e to the b of SI and SI plus 1.

1063
01:17:23,920 --> 01:17:27,320
Now, I can regard
this entity-- let's

1064
01:17:27,320 --> 01:17:31,710
say I have the Potts
model q values.

1065
01:17:31,710 --> 01:17:33,490
This is q possible values.

1066
01:17:33,490 --> 01:17:34,940
This is q possible values.

1067
01:17:34,940 --> 01:17:38,530
So there are q squared possible
values of the interaction.

1068
01:17:38,530 --> 01:17:40,790
And there is q squared
possible values

1069
01:17:40,790 --> 01:17:42,930
of this Boltzmann weight.

1070
01:17:42,930 --> 01:17:49,170
I can regard that
as a matrix, but I

1071
01:17:49,170 --> 01:17:50,520
can write in this fashion.

1072
01:17:53,690 --> 01:17:56,020
And so what you
have over there, you

1073
01:17:56,020 --> 01:18:07,240
can see is effectively you have
s1, ts2, s2, ts3, and so forth.

1074
01:18:07,240 --> 01:18:10,090
And if I use periodic boundary
condition like the one

1075
01:18:10,090 --> 01:18:14,070
that I indicated there so
that the last one is connected

1076
01:18:14,070 --> 01:18:22,190
to the first one, and then I
do a sum over all of these s's,

1077
01:18:22,190 --> 01:18:28,130
this is just the
product of two matrices.

1078
01:18:28,130 --> 01:18:33,800
So this is going to become, when
I sum over s2, s1 t squared s3

1079
01:18:33,800 --> 01:18:35,055
and so forth.

1080
01:18:35,055 --> 01:18:37,410
You can see that
the end result is

1081
01:18:37,410 --> 01:18:41,840
trace of the matrix t
raised to the power of n.

1082
01:18:45,820 --> 01:18:50,220
Now, the trace you can
calculate in any representation

1083
01:18:50,220 --> 01:18:51,980
of the matrix.

1084
01:18:51,980 --> 01:18:54,350
If you manage to find
the representation where

1085
01:18:54,350 --> 01:18:57,700
t is diagonal, then
the trace would

1086
01:18:57,700 --> 01:19:02,280
be sum over alpha
lambda alpha to the n,

1087
01:19:02,280 --> 01:19:07,240
where these lambdas are the
eigenvalues of this matrix.

1088
01:19:07,240 --> 01:19:09,690
And if n is very large,
the thermodynamic

1089
01:19:09,690 --> 01:19:12,630
limit that we are
interested, it will

1090
01:19:12,630 --> 01:19:14,460
be dominated by the
largest eigenvalue.

1091
01:19:21,460 --> 01:19:25,010
Now, if I write this for
something like Potts model

1092
01:19:25,010 --> 01:19:29,180
or any of the spin models that
I had indicated over there,

1093
01:19:29,180 --> 01:19:32,070
you can see that all
elements of this matrix

1094
01:19:32,070 --> 01:19:37,440
being these Boltzmann
weights are plus, positive.

1095
01:19:37,440 --> 01:19:47,550
Now, there's a theorem called
Frobenius's theorem, which

1096
01:19:47,550 --> 01:19:53,120
states that if you have a
matrix, all of its eigenvalues

1097
01:19:53,120 --> 01:20:03,880
are positive, then the largest
eigenvalues is non-degenerate.

1098
01:20:06,810 --> 01:20:12,170
So what that means is that if
this matrix is characterized

1099
01:20:12,170 --> 01:20:17,630
by a set of parameters, like
our k's, et cetera, and I

1100
01:20:17,630 --> 01:20:24,120
change that parameter,
k, well the eigenvalue

1101
01:20:24,120 --> 01:20:26,490
is obtained by looking
at a single matrix.

1102
01:20:26,490 --> 01:20:29,330
It doesn't know
anything about that.

1103
01:20:29,330 --> 01:20:32,520
The only way that the
eigenvalue can become singular

1104
01:20:32,520 --> 01:20:36,330
is if two eigenvalues
cross each other.

1105
01:20:36,330 --> 01:20:41,070
And since Frobenius's
theorem does not allow that,

1106
01:20:41,070 --> 01:20:45,290
you conclude that this
largest eigenvalue

1107
01:20:45,290 --> 01:20:49,330
has to be a perfectly nice
analytical function of all

1108
01:20:49,330 --> 01:20:53,790
of the parameters that go into
constructing this Hamiltonian.

1109
01:20:53,790 --> 01:20:58,190
And that's a mathematical way
of saying that there is no phase

1110
01:20:58,190 --> 01:21:05,850
transition for one dimensional
model because you cannot have

1111
01:21:05,850 --> 01:21:10,760
a crossing of eigenvalues, and
there is no singularity that

1112
01:21:10,760 --> 01:21:11,600
can take place.

1113
01:21:15,630 --> 01:21:19,150
Now, an interesting then
question or caveat to that

1114
01:21:19,150 --> 01:21:24,690
comes from the very question
that was asked over here.

1115
01:21:24,690 --> 01:21:31,380
What if I have, let's say,
a two dimensional model,

1116
01:21:31,380 --> 01:21:35,470
and I regard it essentially
as a complicated one

1117
01:21:35,470 --> 01:21:41,160
dimensional model in
which I have a complicated

1118
01:21:41,160 --> 01:21:46,240
multi-variable
thing over one side,

1119
01:21:46,240 --> 01:21:49,650
and then I can go through this
exact same procedure over here

1120
01:21:49,650 --> 01:21:52,090
also?

1121
01:21:52,090 --> 01:21:56,860
And then I would have to
diagonalize this huge matrix.

1122
01:21:56,860 --> 01:22:01,596
So if this is l, it would be a
2 to the l by 2 to the l matrix.

1123
01:22:04,270 --> 01:22:08,340
And you may naively
think that, again,

1124
01:22:08,340 --> 01:22:09,910
according to
Frobenius's theorem,

1125
01:22:09,910 --> 01:22:13,020
there should be no
phase transition.

1126
01:22:13,020 --> 01:22:17,480
Now, this is exactly what
Lars Onsager did in order

1127
01:22:17,480 --> 01:22:19,390
to solve the two
dimensionalizing model.

1128
01:22:19,390 --> 01:22:22,320
He constructed this matrix
and was clever enough

1129
01:22:22,320 --> 01:22:27,060
to diagonalize it and show
that in the limit of l going

1130
01:22:27,060 --> 01:22:30,010
to infinity, then the
Frobenius's theorem

1131
01:22:30,010 --> 01:22:33,120
can and will be violated.

1132
01:22:33,120 --> 01:22:37,070
And so that's something
that we will also

1133
01:22:37,070 --> 01:22:41,570
discuss in some of
our future lectures.

1134
01:22:41,570 --> 01:22:43,265
Yes.

1135
01:22:43,265 --> 01:22:45,140
AUDIENCE: But it won't
be violated in the one

1136
01:22:45,140 --> 01:22:48,450
dimensional case, even
if n goes to infinity?

1137
01:22:48,450 --> 01:22:51,850
PROFESSOR: Yeah, because
n only appears over here.

1138
01:22:51,850 --> 01:22:55,770
Lambda max is a perfectly
analytic function.