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PROFESSOR: OK.

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00:00:22,150 --> 00:00:22,830
Let's start.

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00:00:22,830 --> 00:00:29,690
So last lecture we started
on the topic of doing

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00:00:29,690 --> 00:00:32,304
renormalization
in position space.

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00:00:35,580 --> 00:00:40,550
And the idea, let's say,
was to look at something

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00:00:40,550 --> 00:00:49,110
like the Ising model, whose
partition function is obtained,

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00:00:49,110 --> 00:00:53,560
if you have insides, by
summing over all the 2 to the n

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00:00:53,560 --> 00:01:02,470
configuration of a weight
that tends to align variables,

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00:01:02,470 --> 00:01:05,780
binary variables that
are next to each other.

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00:01:05,780 --> 00:01:08,130
And this next to each
other is indicated

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00:01:08,130 --> 00:01:13,500
by this nearest neighbor
symbol, sigma i, sigma j.

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00:01:13,500 --> 00:01:16,820
Potentially, we may want
to add the magnetic field

20
00:01:16,820 --> 00:01:18,590
like term that [INAUDIBLE].

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00:01:26,340 --> 00:01:31,870
The idea of the
renormalization group

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00:01:31,870 --> 00:01:34,960
is to obtain a similar
Hamiltonian that

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00:01:34,960 --> 00:01:41,010
describes interactions among
spins that are further apart.

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00:01:41,010 --> 00:01:43,340
We saw that we
could do this easily

25
00:01:43,340 --> 00:01:46,200
in the case of the
one-dimensional system.

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00:01:46,200 --> 00:01:54,290
Well, let's say you have a line,
and you have sites on the line.

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00:01:54,290 --> 00:01:57,280
Each one of them wants to
make their neighbor parallel

28
00:01:57,280 --> 00:01:58,930
to itself.

29
00:01:58,930 --> 00:02:02,430
And what we saw was
that I could easily

30
00:02:02,430 --> 00:02:12,840
get rid of every other spin
and keep one set of spins.

31
00:02:12,840 --> 00:02:17,410
If I do it that, I get
a partition function

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00:02:17,410 --> 00:02:20,720
that operates between
the remaining spins.

33
00:02:20,720 --> 00:02:26,350
Was very easy to sum over the
two variables that spin in

34
00:02:26,350 --> 00:02:29,550
between these two
could have and conclude

35
00:02:29,550 --> 00:02:34,340
that after this step, which
corresponds to removing half

36
00:02:34,340 --> 00:02:39,310
of the degrees of freedom, I
get a new interaction, k prime,

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00:02:39,310 --> 00:02:47,230
which was 1/2 log hyperbolic
cosine of 2K, if h was zero.

38
00:02:52,690 --> 00:02:57,190
And we saw that
that, which is also

39
00:02:57,190 --> 00:03:01,170
a prototype of other
systems in one dimension,

40
00:03:01,170 --> 00:03:06,940
basically is incapable of giving
you long-range order or phase

41
00:03:06,940 --> 00:03:12,570
transition at finite temperature
that corresponds to finite k.

42
00:03:12,570 --> 00:03:18,350
So the only place where you
could have potentially ordering

43
00:03:18,350 --> 00:03:21,420
over large landscape
is when k becomes

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00:03:21,420 --> 00:03:23,910
very large at zero temperature.

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00:03:23,910 --> 00:03:27,110
We saw how the correlation
length behaves and diverges

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00:03:27,110 --> 00:03:29,620
as you approach zero temperature
in this type of model.

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00:03:33,040 --> 00:03:35,170
Now, the next step
would be to look

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00:03:35,170 --> 00:03:37,780
at something that
is two-dimensional.

49
00:03:37,780 --> 00:03:45,070
And in this context,
I describe how

50
00:03:45,070 --> 00:03:49,460
it would be ideal if, let's say,
we start with a square lattice.

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00:03:49,460 --> 00:03:53,370
We have interactions
k between neighbors.

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00:03:53,370 --> 00:03:56,770
And we could potentially
do the same thing.

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00:03:56,770 --> 00:04:03,980
Let's say remove one
sublattice of spins,

54
00:04:03,980 --> 00:04:09,110
getting interactions among the
other sublattice of spins that

55
00:04:09,110 --> 00:04:12,040
would again
correspond to removing

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00:04:12,040 --> 00:04:14,640
half of the spins in the system.

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00:04:14,640 --> 00:04:16,599
But in terms of
length scale change,

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00:04:16,599 --> 00:04:19,700
it corresponds to
square root of 2.

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00:04:19,700 --> 00:04:22,580
This length compared
to the old length

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00:04:22,580 --> 00:04:27,560
is different by a factor
of square root of 2.

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00:04:27,560 --> 00:04:30,500
But the thing that
I also indicated

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00:04:30,500 --> 00:04:34,490
was that this spin is now
coupled to all four of them.

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00:04:34,490 --> 00:04:38,850
And once I remove the spin, I
can generate new interactions

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00:04:38,850 --> 00:04:42,030
operating between these spins.

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00:04:42,030 --> 00:04:45,630
In fact, you will generate
also a four-spin interaction,

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00:04:45,630 --> 00:04:48,380
so your space of
parameters is not

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00:04:48,380 --> 00:04:51,660
closed under this procedure.

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00:04:51,660 --> 00:04:57,810
The same applies to all
higher dimensional systems,

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00:04:57,810 --> 00:05:01,680
and so they are not really
solvable by this approach

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00:05:01,680 --> 00:05:05,470
unless you start making
some approximations.

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00:05:05,470 --> 00:05:09,010
So the particular approximation
that I introduced here

72
00:05:09,010 --> 00:05:13,750
was done and applied shortly
after Cavenaugh brought forth

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00:05:13,750 --> 00:05:18,210
this idea of removing degrees
of freedom and renormalization

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00:05:18,210 --> 00:05:22,830
by-- I'll write this
once, and hopefully I

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00:05:22,830 --> 00:05:24,350
will not make a mistake.

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00:05:28,600 --> 00:05:37,440
Niemeijer and van
Leeuwen And it's

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00:05:37,440 --> 00:05:44,750
a kind of cumulant expansion,
as I will describe shortly.

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00:05:44,750 --> 00:05:45,756
It's an approximation.

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00:05:48,540 --> 00:05:52,185
And rather than doing
the square lattice,

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00:05:52,185 --> 00:05:55,960
it is applied to the
triangular lattice.

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00:05:55,960 --> 00:05:57,550
And that's going
to be the hardest

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00:05:57,550 --> 00:06:00,812
part of this class for me,
to draw a triangular lattice.

83
00:06:19,895 --> 00:06:21,170
Not doing a good job.

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00:06:39,750 --> 00:06:47,530
So basically, we put our
spins, sigma i plus minus 1,

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00:06:47,530 --> 00:06:50,020
on the sides of this.

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00:06:50,020 --> 00:06:55,080
And we put an interaction k
that operates between neighbors.

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00:06:55,080 --> 00:06:59,330
And we want to do a
renormalization in which we

88
00:06:59,330 --> 00:07:02,920
reduce the number of
degrees of freedom.

89
00:07:02,920 --> 00:07:08,220
What Niemeijer and van Leeuwen
suggested was the following.

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00:07:08,220 --> 00:07:15,020
You can group the
sublattices of the triangular

91
00:07:15,020 --> 00:07:17,640
lattice into three.

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00:07:17,640 --> 00:07:21,590
I have indicated
them by 1, 2, 3.

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00:07:21,590 --> 00:07:32,250
Basically, there is going
to be some selection

94
00:07:32,250 --> 00:07:37,340
of sublattice sites
on this lattice.

95
00:07:37,340 --> 00:07:42,490
What they suggested was to
basically define cells--

96
00:07:42,490 --> 00:07:45,750
so this would be one cell.

97
00:07:45,750 --> 00:07:48,635
This would be another cell.

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00:07:51,500 --> 00:07:53,185
This would be another cell.

99
00:07:56,030 --> 00:07:59,710
This would be another
cell over here-- such

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00:07:59,710 --> 00:08:05,260
that every side of the
original lattice belongs to one

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00:08:05,260 --> 00:08:18,840
and only one site
of these cells.

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00:08:18,840 --> 00:08:19,340
OK.

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00:08:19,340 --> 00:08:25,030
So basically, I guess the
next one would come over here.

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00:08:25,030 --> 00:08:27,250
All right.

105
00:08:27,250 --> 00:08:36,559
So let's call the
site by label i.

106
00:08:36,559 --> 00:08:45,270
And let's give the cells label
that I will indicate by Greek.

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00:08:45,270 --> 00:08:48,480
So sites, I will
indicate i, j, cells,

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00:08:48,480 --> 00:08:52,980
by Greek letters,
alphabet, et cetera.

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00:08:52,980 --> 00:08:55,910
So that, for example,
we can regard

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00:08:55,910 --> 00:09:02,410
this as cell alpha,
this one, this triangle,

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00:09:02,410 --> 00:09:03,690
as forming cell beta.

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00:09:15,200 --> 00:09:18,750
Now, the idea of
Niemeijer and van Leeuwen

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00:09:18,750 --> 00:09:23,520
was that to each
cell, we are going

114
00:09:23,520 --> 00:09:29,850
to assign a new spin that is
reflective of the configuration

115
00:09:29,850 --> 00:09:32,860
of the site spins.

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00:09:32,860 --> 00:09:36,690
So for this, they propose
the majority rule.

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00:09:39,610 --> 00:09:45,250
Basically, they said that
we call the spin for site

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00:09:45,250 --> 00:09:51,920
alpha to be the majority
of the site spins.

119
00:10:04,550 --> 00:10:09,580
So basically, if all three
spins are plus or all three

120
00:10:09,580 --> 00:10:14,160
of them are minus, you
basically go to plus or minus.

121
00:10:14,160 --> 00:10:17,030
If two of them are plus
and one of them is minus,

122
00:10:17,030 --> 00:10:19,910
you would choose the
one that is a majority.

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00:10:19,910 --> 00:10:24,440
So you can see that this also
has only two possibilities,

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00:10:24,440 --> 00:10:28,210
plus 1 and minus 1, which
would not have taken place

125
00:10:28,210 --> 00:10:30,645
if I had tried to do a
majority of two sites.

126
00:10:30,645 --> 00:10:32,350
It would have worked
if I had chosen

127
00:10:32,350 --> 00:10:35,210
a majority of three sites
on a one-dimensional lattice

128
00:10:35,210 --> 00:10:36,890
clearly.

129
00:10:36,890 --> 00:10:39,540
So that's the rule.

130
00:10:39,540 --> 00:10:43,110
So you can see that for every
configuration that I have

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00:10:43,110 --> 00:10:46,670
originally, I can do
these kinds of averaging

132
00:10:46,670 --> 00:10:51,790
and define a configuration
that exists for the cells.

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00:10:51,790 --> 00:10:57,170
And the idea is that if I weigh
the initial configurations

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00:10:57,170 --> 00:10:59,320
according to the weight
where the nearest neighbor

135
00:10:59,320 --> 00:11:03,030
coupling is k, what is
the weight that governs

136
00:11:03,030 --> 00:11:09,350
these new configurations for
the averaged or majority cell

137
00:11:09,350 --> 00:11:09,850
spins?

138
00:11:12,430 --> 00:11:16,240
Now, to do this
problem exactly is

139
00:11:16,240 --> 00:11:19,900
subject to the same difficulty
that I mentioned before.

140
00:11:19,900 --> 00:11:22,810
That is, if I somehow
do an averaging

141
00:11:22,810 --> 00:11:26,390
over the spins in here
to get the majority,

142
00:11:26,390 --> 00:11:28,780
then I will generate
interactions

143
00:11:28,780 --> 00:11:31,820
that run over further
neighboring, as we

144
00:11:31,820 --> 00:11:34,060
will see shortly.

145
00:11:34,060 --> 00:11:38,850
So to remove that,
they introduced a kind

146
00:11:38,850 --> 00:11:44,410
of uncontrolled break-up
of the Hamiltonian

147
00:11:44,410 --> 00:11:45,670
that governed the system.

148
00:11:45,670 --> 00:11:49,220
That is, they wrote
beta H minus beta

149
00:11:49,220 --> 00:11:58,020
H, which is the sum over
all neighboring site spins,

150
00:11:58,020 --> 00:12:04,940
as the sum of a part that
then cancels perturbatively

151
00:12:04,940 --> 00:12:10,240
and a part of that will treat
as a correction to the part

152
00:12:10,240 --> 00:12:13,100
that we can solve exactly.

153
00:12:13,100 --> 00:12:28,300
The beta H zero is this
sum over all cells alpha.

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00:12:28,300 --> 00:12:30,460
What you do is
basically you just

155
00:12:30,460 --> 00:12:33,590
include the interactions
among the cell spins.

156
00:12:33,590 --> 00:12:42,620
So I have K sigma alpha 1
sigma alpha 2 sigma alpha

157
00:12:42,620 --> 00:12:49,020
2 sigma alpha 3 sigma
alpha 3 sigma alpha 1.

158
00:12:49,020 --> 00:12:53,700
So basically, these are the
interactions within a cell.

159
00:12:53,700 --> 00:12:54,960
What have I left out?

160
00:12:54,960 --> 00:12:59,245
I have left out the interactions
that operate between cells,

161
00:12:59,245 --> 00:13:02,260
so all of these things,
which, of course, are

162
00:13:02,260 --> 00:13:08,680
the same strength but, for
lack of better things to do,

163
00:13:08,680 --> 00:13:09,720
they said OK.

164
00:13:09,720 --> 00:13:14,060
We are going to sum over
all, you can see now,

165
00:13:14,060 --> 00:13:16,020
neighboring cells.

166
00:13:16,020 --> 00:13:18,215
So the things that
I left out, let's

167
00:13:18,215 --> 00:13:22,720
say, interactions between
this cell alpha and beta,

168
00:13:22,720 --> 00:13:27,180
evolve-- let's say
the spin number

169
00:13:27,180 --> 00:13:35,920
1 in this labeling of beta
times spin number 2 of alpha

170
00:13:35,920 --> 00:13:39,180
and spin number 3 of alpha.

171
00:13:39,180 --> 00:13:42,550
Now, of course, what
I call 1, 2, or 3

172
00:13:42,550 --> 00:13:47,190
will depend on the relative
orientation of the neighboring

173
00:13:47,190 --> 00:13:47,690
cells.

174
00:13:47,690 --> 00:13:49,640
But the idea is the
same, that basically,

175
00:13:49,640 --> 00:13:51,850
for each pair of
neighboring cells,

176
00:13:51,850 --> 00:13:55,870
there will be two of
these interactions.

177
00:13:55,870 --> 00:14:00,030
So again, these no a
priori reason to regard

178
00:14:00,030 --> 00:14:02,380
this as a perturbation.

179
00:14:02,380 --> 00:14:06,010
Both of them clearly carry the
same strength for the bond,

180
00:14:06,010 --> 00:14:09,840
this parameter K. OK?

181
00:14:09,840 --> 00:14:14,620
The justification
is only solvability.

182
00:14:14,620 --> 00:14:19,690
So the partition function
that I have to calculate,

183
00:14:19,690 --> 00:14:26,440
which is the sum of
all spin configuration

184
00:14:26,440 --> 00:14:34,270
with the original weight, I can
write as a sum over all spin

185
00:14:34,270 --> 00:14:42,810
configurations of e to the minus
beta H 0, and then minus u,

186
00:14:42,810 --> 00:14:44,710
and the idea of
perturbation is, of course,

187
00:14:44,710 --> 00:14:48,490
to write the part that depends
on u as a perturbation.

188
00:14:52,560 --> 00:14:54,220
It's expanding the exponential.

189
00:14:59,040 --> 00:15:02,650
Now, solvability
relies on the fact

190
00:15:02,650 --> 00:15:06,750
that the term that
just multiplies 1

191
00:15:06,750 --> 00:15:10,260
is the sum of
triplets that are only

192
00:15:10,260 --> 00:15:12,690
interacting between themselves.

193
00:15:12,690 --> 00:15:13,900
They don't see anybody else.

194
00:15:13,900 --> 00:15:16,690
So that's clearly
very easily solvable,

195
00:15:16,690 --> 00:15:19,880
and we can calculate
the partition functions

196
00:15:19,880 --> 00:15:22,636
0 that describes that.

197
00:15:26,270 --> 00:15:29,840
And then we can
start to evaluate

198
00:15:29,840 --> 00:15:33,870
all of those other terms,
once I have pulled out

199
00:15:33,870 --> 00:15:38,590
e to the minus beta H zero sum
over all configurations in Z 0.

200
00:15:38,590 --> 00:15:41,050
The series that I
will generate are

201
00:15:41,050 --> 00:15:44,750
averages of this
interaction calculated

202
00:15:44,750 --> 00:15:47,129
with the 0 [INAUDIBLE]
Hamiltonian.

203
00:15:52,120 --> 00:16:06,140
So my log of Z-- OK.

204
00:16:06,140 --> 00:16:11,090
That's how if I was to calculate
the problem perturbatively.

205
00:16:11,090 --> 00:16:15,990
Now I'm going to do something
that is slightly different.

206
00:16:15,990 --> 00:16:19,930
So what I will do
is, rather than

207
00:16:19,930 --> 00:16:23,180
do the sum that I
have indicated above,

208
00:16:23,180 --> 00:16:24,755
I will do a slight variation.

209
00:16:29,350 --> 00:16:35,990
I will sum only
over configurations.

210
00:16:35,990 --> 00:16:38,060
Maybe I should write
it in this fashion.

211
00:16:38,060 --> 00:16:41,660
I will sum only
over configurations

212
00:16:41,660 --> 00:16:48,940
that under averaging give me
a particular configuration

213
00:16:48,940 --> 00:16:52,630
of site spins.

214
00:16:52,630 --> 00:16:57,670
So, basically, let's say I pick
a configuration in which this

215
00:16:57,670 --> 00:17:01,030
is-- the cell spin is plus,
the cell spin is minus,

216
00:17:01,030 --> 00:17:05,349
whatever, some
configuration of cell spins.

217
00:17:05,349 --> 00:17:07,950
Now, depending on this
cell spin being plus,

218
00:17:07,950 --> 00:17:10,160
there are many
configurations-- not many--

219
00:17:10,160 --> 00:17:13,290
but there are four
configurations of the site

220
00:17:13,290 --> 00:17:16,250
spins that would correspond
to this being plus.

221
00:17:16,250 --> 00:17:19,480
There are four configurations
that would correspond to this.

222
00:17:19,480 --> 00:17:27,260
So basically, I specify
what configuration of cell

223
00:17:27,260 --> 00:17:30,710
spins I want, and I do this sum.

224
00:17:30,710 --> 00:17:34,930
So the answer there is
some kind of a weight that

225
00:17:34,930 --> 00:17:37,380
depends on my choice
of [INAUDIBLE] sigma

226
00:17:37,380 --> 00:17:39,680
alpha [INAUDIBLE].

227
00:17:39,680 --> 00:17:42,000
OK.

228
00:17:42,000 --> 00:17:44,240
So then I have the
same thing over here.

229
00:17:51,740 --> 00:17:56,510
And then, in principle,
all of these quantities

230
00:17:56,510 --> 00:18:02,000
will become a function of the
choice of my configuration.

231
00:18:08,190 --> 00:18:10,980
OK.

232
00:18:10,980 --> 00:18:17,180
So this is a weight
for cell configurations

233
00:18:17,180 --> 00:18:21,030
once I average out over all
site configurations that

234
00:18:21,030 --> 00:18:23,350
are compatible with that.

235
00:18:23,350 --> 00:18:30,520
So if I take the
log of this, I can

236
00:18:30,520 --> 00:18:39,190
think of that as an
effective Hamiltonian that

237
00:18:39,190 --> 00:18:43,880
operates on these variables.

238
00:18:43,880 --> 00:18:46,930
And this is what we
have usually been

239
00:18:46,930 --> 00:18:49,715
indicating by the
prime interactions.

240
00:18:55,300 --> 00:18:59,605
And so, if I take the
log of that expression,

241
00:18:59,605 --> 00:19:03,730
I will get log of z 0 that
is compatible with the choice

242
00:19:03,730 --> 00:19:09,770
of cell spins.

243
00:19:09,770 --> 00:19:12,480
And then the log
of this series--

244
00:19:12,480 --> 00:19:17,700
we've seen this many times--
it starts with minus U 0

245
00:19:17,700 --> 00:19:20,800
as a function of the
specified interactions.

246
00:19:20,800 --> 00:19:28,870
And then I have
the variance, again

247
00:19:28,870 --> 00:19:30,893
compatible with
the interactions.

248
00:19:35,794 --> 00:19:36,294
OK?

249
00:19:44,670 --> 00:19:48,880
So now it comes to basically
solving this problem.

250
00:19:48,880 --> 00:19:53,670
And I pick some particular cell.

251
00:19:53,670 --> 00:20:00,660
And I look at what
configurations

252
00:20:00,660 --> 00:20:04,620
I could have that
are compactible

253
00:20:04,620 --> 00:20:10,310
for a particular sign
of the cell spin.

254
00:20:10,310 --> 00:20:13,344
As far as the site
spins are concerned.

255
00:20:17,600 --> 00:20:21,875
And I will also
indicate what the weight

256
00:20:21,875 --> 00:20:29,130
is that I have to put for that
cell coming from minus beta H0.

257
00:20:32,820 --> 00:20:35,870
one thing that I
can certainly do

258
00:20:35,870 --> 00:20:40,070
is to have the cell spin plus.

259
00:20:40,070 --> 00:20:43,360
And I indicated that
that I can get either

260
00:20:43,360 --> 00:20:47,390
by having all three
spins to be plus or just

261
00:20:47,390 --> 00:20:50,370
the majority, which means
that one of the three

262
00:20:50,370 --> 00:20:51,525
can become minus.

263
00:20:54,410 --> 00:20:56,640
So there are these
configurations

264
00:20:56,640 --> 00:21:00,095
that are consistent, all
of them, with the sigma

265
00:21:00,095 --> 00:21:05,050
alpha prime, the
majority that is plus.

266
00:21:05,050 --> 00:21:09,180
And there are four
configurations

267
00:21:09,180 --> 00:21:12,837
that correspond to minuses we
shall obtain by essentially

268
00:21:12,837 --> 00:21:13,670
flipping everything.

269
00:21:22,530 --> 00:21:24,510
And the weights are
easy to figure out.

270
00:21:24,510 --> 00:21:28,660
Basically I have
a triplet of spins

271
00:21:28,660 --> 00:21:32,570
that are coupled by
interaction K. In this case,

272
00:21:32,570 --> 00:21:34,180
all three are positive.

273
00:21:34,180 --> 00:21:37,920
So I have three
into the K factor,

274
00:21:37,920 --> 00:21:41,680
so I will get e to the 3K.

275
00:21:41,680 --> 00:21:45,180
Whereas if one of them becomes
minus and two remain plus,

276
00:21:45,180 --> 00:21:48,470
you can see that
there are two unhappy

277
00:21:48,470 --> 00:21:50,100
misaligned configurations.

278
00:21:50,100 --> 00:21:53,800
So I will get minus K, minus
K, plus K. I will get e

279
00:21:53,800 --> 00:21:57,315
to the minus K. It doesn't
matter which one of these three

280
00:21:57,315 --> 00:21:57,815
it is.

281
00:22:03,270 --> 00:22:10,080
If all three are minuses, then
again, the sites are aligned.

282
00:22:10,080 --> 00:22:12,560
So I will get e to the 3K.

283
00:22:12,560 --> 00:22:16,060
If two minuses and 1
plus, 2 bonds are unhappy.

284
00:22:16,060 --> 00:22:17,275
One is happy.

285
00:22:17,275 --> 00:22:20,060
I will get e to the minus
K, e to the minus K,

286
00:22:20,060 --> 00:22:24,790
e to the minus K.

287
00:22:24,790 --> 00:22:31,930
So once I have specified
what my cell spin is,

288
00:22:31,930 --> 00:22:36,900
the contribution to
the partition function

289
00:22:36,900 --> 00:22:40,870
is obtained by summing over
contributions things that

290
00:22:40,870 --> 00:22:43,430
are compatible with that.

291
00:22:43,430 --> 00:22:47,480
So what is it if I specify
that my cell spin is plus?

292
00:22:47,480 --> 00:22:49,390
The contribution to
the partition function

293
00:22:49,390 --> 00:22:53,640
is e to the 3K plus e
to the 3 to the minus K.

294
00:22:53,640 --> 00:22:56,430
It's actually exactly
the same thing

295
00:22:56,430 --> 00:22:59,500
if I had specified
that it is minus.

296
00:22:59,500 --> 00:23:04,930
So we see that this
factor, irrespective

297
00:23:04,930 --> 00:23:08,370
of whether the choice here
is that the cell spin is

298
00:23:08,370 --> 00:23:16,070
plus or minus, is log of e to
the 3K plus 3e to the minus

299
00:23:16,070 --> 00:23:22,670
K. Pair any one of the cells
and how many cells I have,

300
00:23:22,670 --> 00:23:26,130
1/3 of the number of sites
with the number of sites

301
00:23:26,130 --> 00:23:31,630
I had indicated by N,
this would be N over 3.

302
00:23:31,630 --> 00:23:32,130
OK.

303
00:23:35,230 --> 00:23:38,180
Now, let's see what
this U average is.

304
00:23:42,680 --> 00:23:53,480
So U-- I made one sign error.

305
00:23:53,480 --> 00:23:56,610
I put both of them
as minuses, which

306
00:23:56,610 --> 00:24:02,830
means that in the
notation I had-- no.

307
00:24:02,830 --> 00:24:04,630
That's fine.

308
00:24:04,630 --> 00:24:05,530
Minus U0.

309
00:24:08,570 --> 00:24:18,820
So I put the minus sign here
-- is plus K sum over all pairs

310
00:24:18,820 --> 00:24:22,080
that are neighboring
each other, for example,

311
00:24:22,080 --> 00:24:24,530
like the pair alpha
beta I have indicated,

312
00:24:24,530 --> 00:24:27,960
but any other pair
of neighboring cells.

313
00:24:27,960 --> 00:24:31,450
I have to write an
expression such as this.

314
00:24:31,450 --> 00:24:34,570
So I have the K. I have sigma.

315
00:24:34,570 --> 00:24:47,480
I have beta 1 sigma alpha 2
plus sigma beta 1 sigma alpha 3.

316
00:24:50,290 --> 00:24:54,050
So this is the expression
that I have for you.

317
00:24:54,050 --> 00:24:57,960
I have to take the average
of this quantity, basically

318
00:24:57,960 --> 00:24:59,695
the average of the sum.

319
00:24:59,695 --> 00:25:01,375
I will have two
of these averages.

320
00:25:04,462 --> 00:25:08,170
Now in my zero to
order weight, there

321
00:25:08,170 --> 00:25:13,150
is no coupling between this
cell and any other cell.

322
00:25:13,150 --> 00:25:17,540
So what the spin on
each cell is on average

323
00:25:17,540 --> 00:25:19,560
cares nothing about
what the spin is

324
00:25:19,560 --> 00:25:24,850
on any other cell, which
means that these averages are

325
00:25:24,850 --> 00:25:26,230
independent of each other.

326
00:25:26,230 --> 00:25:28,044
I can write it in this fashion.

327
00:25:30,820 --> 00:25:35,180
So all I need to
do is to calculate

328
00:25:35,180 --> 00:25:40,050
the average of one
of these columns,

329
00:25:40,050 --> 00:25:44,600
given that I have
specified what the cell is.

330
00:25:44,600 --> 00:25:51,585
So let's pick, let's say sigma
alpha 1 average in this zero

331
00:25:51,585 --> 00:25:54,540
to order weight.

332
00:25:54,540 --> 00:25:56,150
Now I can see
immediately that I will

333
00:25:56,150 --> 00:26:00,980
have two possibilities, the
top four or the bottom four.

334
00:26:00,980 --> 00:26:07,480
The top four corresponds
to sigma cell being plus.

335
00:26:07,480 --> 00:26:10,705
The bottom four correspond to
the sigma cell being minus.

336
00:26:13,620 --> 00:26:15,740
So the top four is
essentially I have

337
00:26:15,740 --> 00:26:18,950
to look at the average
on this column.

338
00:26:18,950 --> 00:26:21,910
I have-- it is either
plus, and then I

339
00:26:21,910 --> 00:26:23,475
get a weight e to the 3K.

340
00:26:26,550 --> 00:26:28,050
Or it is minus.

341
00:26:28,050 --> 00:26:32,570
I get weight e to the minus
K plus e to the minus k plus

342
00:26:32,570 --> 00:26:33,960
e to the minus k.

343
00:26:33,960 --> 00:26:38,470
So once I add 2 plus
e to the minus K

344
00:26:38,470 --> 00:26:43,710
and subtract 1e to the
minus K, I really get this.

345
00:26:43,710 --> 00:26:47,100
Now, of course, I have to
normalize by the weights

346
00:26:47,100 --> 00:26:48,630
that I have per cell.

347
00:26:48,630 --> 00:26:51,040
And the weights are
really these factors

348
00:26:51,040 --> 00:27:01,350
but divided by e to the
3K plus 3 to the minus K.

349
00:27:01,350 --> 00:27:05,810
Whereas if I had specified
that the cell spin is minus,

350
00:27:05,810 --> 00:27:08,510
and I wanted to calculate
the average here,

351
00:27:08,510 --> 00:27:11,380
I would be dealing
with these numbers.

352
00:27:11,380 --> 00:27:18,710
You can see that I will
have a minus e to the 3K.

353
00:27:18,710 --> 00:27:22,880
I will have one plus and two
minuses e to the minus K,

354
00:27:22,880 --> 00:27:26,580
so I will get minus
e to the minus K e

355
00:27:26,580 --> 00:27:31,200
to the 3K plus 3e to the minus
K, which is the normalizing

356
00:27:31,200 --> 00:27:34,120
weight.

357
00:27:34,120 --> 00:27:36,490
So it's just minus
the other one.

358
00:27:36,490 --> 00:27:40,390
And I can put these two
together and write it as e

359
00:27:40,390 --> 00:27:44,100
to the 3K plus e
to the minus K e

360
00:27:44,100 --> 00:27:51,160
to the 3K plus 3e to the
minus K sigma alpha prime.

361
00:27:51,160 --> 00:27:55,510
So the average of any one
of these three site spins

362
00:27:55,510 --> 00:28:01,580
is simply proportional to what
you said was the cell spin.

363
00:28:01,580 --> 00:28:05,754
The constant of proportionality
depends on K according to this.

364
00:28:08,820 --> 00:28:11,850
So now if I substitute this
over here, what do I find?

365
00:28:11,850 --> 00:28:15,420
I will find that
minus U at the lowest

366
00:28:15,420 --> 00:28:22,330
level, each one of these factors
will give me the same thing.

367
00:28:22,330 --> 00:28:23,830
So this K becomes 2K.

368
00:28:26,460 --> 00:28:33,250
I have a sum over alpha and
betas that are neighboring.

369
00:28:33,250 --> 00:28:35,850
Each one of these
sigmas I will replace

370
00:28:35,850 --> 00:28:40,570
by the corresponding
average here.

371
00:28:40,570 --> 00:28:45,520
Add the cost of multiplying
by one of these factors.

372
00:28:45,520 --> 00:28:49,450
And there are two such factors.

373
00:28:49,450 --> 00:28:50,990
So I basically get this.

374
00:28:56,010 --> 00:29:00,640
So add this order in the
series that I have written,

375
00:29:00,640 --> 00:29:05,650
if I forget about all of those
things, what has happened?

376
00:29:05,650 --> 00:29:10,570
I see that the weight that
governs the cell spins

377
00:29:10,570 --> 00:29:14,870
is again something that only
couples the nearest neighbor

378
00:29:14,870 --> 00:29:21,750
cells with a new interaction
that I can call K prime.

379
00:29:21,750 --> 00:29:26,186
And this new
interaction, K prime,

380
00:29:26,186 --> 00:29:32,950
is simply 2K into the
3K plus e to the minus K

381
00:29:32,950 --> 00:29:37,325
e to the 3K plus 3 e
to the minus K squared.

382
00:29:46,980 --> 00:29:53,250
So presumably, again, if I think
of the axes' possible values

383
00:29:53,250 --> 00:29:58,050
of K running all the way
from no coupling at 0

384
00:29:58,050 --> 00:30:01,190
to very strongly
coupling at infinity,

385
00:30:01,190 --> 00:30:06,170
this tells me under rescaling
where the parameters go.

386
00:30:06,170 --> 00:30:10,360
If I start here, where
do I go back and forth?

387
00:30:10,360 --> 00:30:15,220
So let's follow the path that
we followed in one dimension.

388
00:30:15,220 --> 00:30:17,380
We expect something
to correspond

389
00:30:17,380 --> 00:30:21,680
to essentially no
coupling at all.

390
00:30:21,680 --> 00:30:26,615
So we look at the limit
where K goes to 0.

391
00:30:29,620 --> 00:30:31,900
Then you can see that
what is happening here

392
00:30:31,900 --> 00:30:37,750
is that K prime is 2K.

393
00:30:37,750 --> 00:30:43,170
And then from here, when K
goes to 0, all of these factors

394
00:30:43,170 --> 00:30:45,060
become 1.

395
00:30:45,060 --> 00:30:48,980
So numerator is 2,
denominator is 4.

396
00:30:48,980 --> 00:30:51,840
The whole thing is squared.

397
00:30:51,840 --> 00:30:58,170
So basically in that limit,
the interaction gets halved.

398
00:30:58,170 --> 00:31:04,240
So if I have a very
[INAUDIBLE] coupling of 1/8,

399
00:31:04,240 --> 00:31:09,320
then it becomes 1/16 and
then it becomes 1 over 32.

400
00:31:09,320 --> 00:31:12,560
I get pulled towards this.

401
00:31:12,560 --> 00:31:15,710
So presumably
anything that is here

402
00:31:15,710 --> 00:31:18,830
will at long distance
look disordered, just

403
00:31:18,830 --> 00:31:21,450
like one dimension.

404
00:31:21,450 --> 00:31:23,810
But now let's look
at the other limit.

405
00:31:23,810 --> 00:31:29,570
What happens when K is very
large, K goes to infinity?

406
00:31:29,570 --> 00:31:36,540
Then K prime is-- well,
there's the 2K out front,

407
00:31:36,540 --> 00:31:37,530
but that's it.

408
00:31:37,530 --> 00:31:42,910
Because e to the 3K is going to
dominate over e to the minus K

409
00:31:42,910 --> 00:31:47,790
when K is large, and
this ratio goes to 1.

410
00:31:47,790 --> 00:31:51,600
So we see that if I
start with a K of 1,000,

411
00:31:51,600 --> 00:31:55,110
then I go to 2,000 to
4,000, and basically I

412
00:31:55,110 --> 00:31:58,440
get pulled towards a
behavior of infinity.

413
00:31:58,440 --> 00:32:01,270
So this is different
from one dimension.

414
00:32:01,270 --> 00:32:04,640
In one dimension, you
were always going to 0.

415
00:32:04,640 --> 00:32:07,620
Now we can see that in
this two-dimensional model,

416
00:32:07,620 --> 00:32:11,510
recoupling disappears,
goes to no coupling.

417
00:32:11,510 --> 00:32:15,600
Strong enough coupling goes to
everybody's following in line

418
00:32:15,600 --> 00:32:18,700
and doing the same
thing at large scale.

419
00:32:18,700 --> 00:32:21,460
So we can very well
guess that there

420
00:32:21,460 --> 00:32:25,560
should be some point
in between that

421
00:32:25,560 --> 00:32:28,950
separates these
two types of flows.

422
00:32:28,950 --> 00:32:33,490
And that is going to
be the point where

423
00:32:33,490 --> 00:32:36,370
I would have KC,
or let's call it--

424
00:32:36,370 --> 00:32:39,550
I guess I call it K
star in the notes.

425
00:32:39,550 --> 00:32:41,270
So let's call it K star.

426
00:32:41,270 --> 00:32:50,810
So K star is 2K star e to the 3K
star plus e to the minus K star

427
00:32:50,810 --> 00:32:56,990
e to the 3K star plus e to the
minus-- 3e to the minus K star

428
00:32:56,990 --> 00:32:59,850
squared.

429
00:32:59,850 --> 00:33:02,010
So we can drop out the K star.

430
00:33:02,010 --> 00:33:04,360
You can see that what
you have to solve

431
00:33:04,360 --> 00:33:09,320
is e to the 3K star plus
e to the minus K star

432
00:33:09,320 --> 00:33:16,210
divided by e to the 3K star
plus 3e to the minus K star.

433
00:33:16,210 --> 00:33:19,910
This ratio is 1 over
square root of 2.

434
00:33:23,540 --> 00:33:28,040
I can multiply everything
by e to the plus K star

435
00:33:28,040 --> 00:33:30,140
so that this becomes 1.

436
00:33:30,140 --> 00:33:35,080
And I have an algebraic equation
to solve for e to the 4K star.

437
00:33:35,080 --> 00:33:39,120
So I will get root
2e to the 4K star

438
00:33:39,120 --> 00:33:47,220
plus root 2 is e to
the 4K star plus 1.

439
00:33:47,220 --> 00:33:57,080
And I get the value of K star,
which is 1/4 log of-- whoops,

440
00:33:57,080 --> 00:34:09,306
this was a 3-- 3 minus root
2 divided by root 2 minus 1.

441
00:34:12,708 --> 00:34:15,060
You put it in the
calculator, and it

442
00:34:15,060 --> 00:34:24,790
becomes something that
is of the order of 0.233.

443
00:34:24,790 --> 00:34:25,290
No, 27.

444
00:34:32,530 --> 00:34:35,679
So yes?

445
00:34:35,679 --> 00:34:39,050
AUDIENCE: So, first
thing, do we want

446
00:34:39,050 --> 00:34:43,889
to name what is the length
factor by which we change

447
00:34:43,889 --> 00:34:45,064
the characteristic length?

448
00:34:45,064 --> 00:34:45,980
PROFESSOR: Absolutely.

449
00:34:45,980 --> 00:34:46,860
Yes.

450
00:34:46,860 --> 00:34:48,130
So the next-- yeah.

451
00:34:48,130 --> 00:34:51,250
AUDIENCE: But we never kind
of bothered to do it so far.

452
00:34:51,250 --> 00:34:53,429
PROFESSOR: We will
need to do immediately.

453
00:34:53,429 --> 00:34:55,130
So just hold on a second.

454
00:34:55,130 --> 00:34:58,170
The next thing, I
need this B factor.

455
00:34:58,170 --> 00:35:01,230
But it's obvious, I have
reduced the number of degrees

456
00:35:01,230 --> 00:35:04,370
of freedom by 3,
so the length scale

457
00:35:04,370 --> 00:35:08,890
must have in two dimensions
increased by square root of 3.

458
00:35:08,890 --> 00:35:14,250
And you can do also the
algebra analogous to this

459
00:35:14,250 --> 00:35:16,630
to convince you that
the distance, let's say,

460
00:35:16,630 --> 00:35:19,960
from the center of this triangle
to the center of that triangle

461
00:35:19,960 --> 00:35:23,250
is exactly [INAUDIBLE].

462
00:35:23,250 --> 00:35:25,122
AUDIENCE: Also,
when you're writing

463
00:35:25,122 --> 00:35:26,416
the cumulant expansion--

464
00:35:26,416 --> 00:35:27,040
PROFESSOR: Yes.

465
00:35:27,040 --> 00:35:31,890
AUDIENCE: In all of
our previous occasions

466
00:35:31,890 --> 00:35:35,090
when we did perturbations,
the convergence of the series

467
00:35:35,090 --> 00:35:38,410
was kind of reassured because
every perturbation was

468
00:35:38,410 --> 00:35:40,745
proportional to
some scalar number

469
00:35:40,745 --> 00:35:43,330
that we claimed to be
small, and thus series

470
00:35:43,330 --> 00:35:44,590
would hopefully converge.

471
00:35:44,590 --> 00:35:44,850
PROFESSOR: Right.

472
00:35:44,850 --> 00:35:46,475
AUDIENCE: But In this
case, how can you

473
00:35:46,475 --> 00:35:50,200
be sure that for
modified interaction

474
00:35:50,200 --> 00:35:53,016
and renormalized version,
you don't need [INAUDIBLE]?

475
00:35:53,016 --> 00:35:56,300
PROFESSOR: Well, let me
first slightly correct

476
00:35:56,300 --> 00:35:59,540
what you said before I
think you meant correctly,

477
00:35:59,540 --> 00:36:02,750
which is that previously
we had parameters

478
00:36:02,750 --> 00:36:05,870
that we were
ensuring were small.

479
00:36:05,870 --> 00:36:08,070
That did not guarantee
the convergence

480
00:36:08,070 --> 00:36:11,050
of the series or the lattice.

481
00:36:11,050 --> 00:36:14,200
In this case, we don't
have even a parameter

482
00:36:14,200 --> 00:36:16,370
that we can make small.

483
00:36:16,370 --> 00:36:18,240
So the only thing
that we can do,

484
00:36:18,240 --> 00:36:20,970
and I will briefly
mention that, is

485
00:36:20,970 --> 00:36:24,310
to basically see what
happens if we include

486
00:36:24,310 --> 00:36:28,530
more and more terms in that
series and we compare results

487
00:36:28,530 --> 00:36:32,690
and whether there is
some convergence or not.

488
00:36:32,690 --> 00:36:33,190
Yes?

489
00:36:33,190 --> 00:36:34,648
AUDIENCE: Can you
explain again how

490
00:36:34,648 --> 00:36:36,800
we got the K prime equation?

491
00:36:36,800 --> 00:36:37,920
PROFESSOR: OK.

492
00:36:37,920 --> 00:36:44,790
So I said that I have
some configuration

493
00:36:44,790 --> 00:36:45,990
of the cell spins.

494
00:36:45,990 --> 00:36:52,080
Let's say the configuration
is plus plus minus plus.

495
00:36:52,080 --> 00:36:54,085
Whatever, some configuration.

496
00:36:54,085 --> 00:36:57,300
Now there are many
configurations of site spins

497
00:36:57,300 --> 00:36:59,360
that correspond to that.

498
00:36:59,360 --> 00:37:02,610
So the weight of
this configuration

499
00:37:02,610 --> 00:37:06,430
is obtained by summing over the
weights of all configurations

500
00:37:06,430 --> 00:37:08,950
of site spins that are
compatible with that.

501
00:37:08,950 --> 00:37:12,580
And that was a series
that we had over here.

502
00:37:12,580 --> 00:37:16,690
And K prime, or the
interaction, typically we

503
00:37:16,690 --> 00:37:20,940
put in the exponent, so I
have to take a log of this

504
00:37:20,940 --> 00:37:23,100
to see what the
interactions are.

505
00:37:23,100 --> 00:37:25,720
The log has this
series that starts

506
00:37:25,720 --> 00:37:28,179
with the average of
this interaction.

507
00:37:28,179 --> 00:37:29,540
OK?

508
00:37:29,540 --> 00:37:35,810
So this was the formula
for U. It's over here.

509
00:37:35,810 --> 00:37:40,100
And then here, it says I have
to take an average of it.

510
00:37:40,100 --> 00:37:42,910
Average, given that I have
specified what the cell spins

511
00:37:42,910 --> 00:37:44,430
are.

512
00:37:44,430 --> 00:37:46,670
And I see that that
average is really

513
00:37:46,670 --> 00:37:50,580
product of averages
of site spins.

514
00:37:50,580 --> 00:37:54,890
And I was able to evaluate
the average of a site spin,

515
00:37:54,890 --> 00:37:58,270
and I found that up to some
proportionality constant,

516
00:37:58,270 --> 00:38:00,360
it was the cell spin.

517
00:38:00,360 --> 00:38:04,460
So if the cell spin is
specified to be plus,

518
00:38:04,460 --> 00:38:07,610
the average of each one of the
site spins tends to be plus.

519
00:38:07,610 --> 00:38:11,160
If the cell spin is
specified to be minus,

520
00:38:11,160 --> 00:38:13,800
since I'm looking at this
subset of configuration,

521
00:38:13,800 --> 00:38:16,840
the average is
likely to be minus.

522
00:38:16,840 --> 00:38:19,600
And that proportionality
factor is here.

523
00:38:19,600 --> 00:38:22,250
I put that proportionality
factor here,

524
00:38:22,250 --> 00:38:26,580
and I see that this average is
a product of neighboring cell

525
00:38:26,580 --> 00:38:31,100
spins, which are weighted
by this factor, which

526
00:38:31,100 --> 00:38:35,070
is like the original weights
that you write, except

527
00:38:35,070 --> 00:38:38,732
with a new K. Yes?

528
00:38:38,732 --> 00:38:42,071
AUDIENCE: So after
renormalization,

529
00:38:42,071 --> 00:38:47,142
we get some new kind of
lattice, which is not random.

530
00:38:47,142 --> 00:38:48,534
It's completely new.

531
00:38:48,534 --> 00:38:51,734
Because what you did here is
you take out certain cells--

532
00:38:51,734 --> 00:38:52,400
PROFESSOR: Yeah.

533
00:38:52,400 --> 00:38:53,650
AUDIENCE: And call
them [INAUDIBLE].

534
00:38:53,650 --> 00:38:54,358
PROFESSOR: Right.

535
00:38:54,358 --> 00:38:56,805
But what is this new lattice?

536
00:38:56,805 --> 00:38:59,330
This new lattice is
a triangular lattice

537
00:38:59,330 --> 00:39:03,640
that this is rotated with
respect to the original one.

538
00:39:03,640 --> 00:39:05,790
So it's exactly the
same lattice as before.

539
00:39:05,790 --> 00:39:06,900
It's not a random lattice.

540
00:39:06,900 --> 00:39:07,580
AUDIENCE: Yes.

541
00:39:07,580 --> 00:39:11,530
But on the initial
lattice, you specified

542
00:39:11,530 --> 00:39:13,640
that these cells
would contribute to--

543
00:39:13,640 --> 00:39:14,520
PROFESSOR: Yes.

544
00:39:14,520 --> 00:39:16,340
I separated K and K prime.

545
00:39:16,340 --> 00:39:16,870
Yes.

546
00:39:16,870 --> 00:39:18,529
K and U, yes.

547
00:39:18,529 --> 00:39:19,070
AUDIENCE: OK.

548
00:39:19,070 --> 00:39:22,430
So if you want to do a
renormalization group again,

549
00:39:22,430 --> 00:39:23,390
we'll need to--

550
00:39:23,390 --> 00:39:24,100
PROFESSOR: Yeah.

551
00:39:24,100 --> 00:39:24,847
Do this.

552
00:39:24,847 --> 00:39:26,040
AUDIENCE: Again [INAUDIBLE].

553
00:39:26,040 --> 00:39:26,831
PROFESSOR: Exactly.

554
00:39:26,831 --> 00:39:27,650
Yeah.

555
00:39:27,650 --> 00:39:30,560
But we do it once, and we
have the recursion relation.

556
00:39:30,560 --> 00:39:32,210
And then we stop.

557
00:39:32,210 --> 00:39:33,160
AUDIENCE: Yeah.

558
00:39:33,160 --> 00:39:34,700
PROFESSOR: OK.

559
00:39:34,700 --> 00:39:35,200
Yes?

560
00:39:35,200 --> 00:39:40,300
AUDIENCE: Is this possible
for other odd number lattices?

561
00:39:40,300 --> 00:39:42,680
Will you still
preserve the parameter?

562
00:39:42,680 --> 00:39:43,570
PROFESSOR: Yes.

563
00:39:43,570 --> 00:39:46,140
It's even possible
for square lattices

564
00:39:46,140 --> 00:39:47,559
with some modification,
and that's

565
00:39:47,559 --> 00:39:49,225
what you'll have in
one of the problems.

566
00:39:55,141 --> 00:39:55,640
OK?

567
00:40:00,980 --> 00:40:01,840
Fine.

568
00:40:01,840 --> 00:40:03,100
But the point is-- OK.

569
00:40:03,100 --> 00:40:04,210
So I stopped here.

570
00:40:04,210 --> 00:40:10,760
So K star was 0.27, which is
the coupling that separates

571
00:40:10,760 --> 00:40:13,420
places where you go
to uncorrelated spins,

572
00:40:13,420 --> 00:40:17,010
places you go to everything
ordered together.

573
00:40:17,010 --> 00:40:20,200
It turns out that the
triangular lattice is something

574
00:40:20,200 --> 00:40:21,850
that one can solve exactly.

575
00:40:21,850 --> 00:40:23,400
It's one of the few things.

576
00:40:23,400 --> 00:40:25,570
And you'll have the pleasure
of serving that also

577
00:40:25,570 --> 00:40:27,260
in a problem set.

578
00:40:27,260 --> 00:40:32,620
And you will show that KC, the
correct value of the coupling,

579
00:40:32,620 --> 00:40:36,630
is something like 0.33.

580
00:40:36,630 --> 00:40:41,865
So that gives you an idea of how
good or bad this approximation

581
00:40:41,865 --> 00:40:43,510
is.

582
00:40:43,510 --> 00:40:46,260
But the point in any
case is that the location

583
00:40:46,260 --> 00:40:48,050
of the coupling is
not that important.

584
00:40:48,050 --> 00:40:51,304
We have discussed that
it is non-universal.

585
00:40:51,304 --> 00:40:53,345
The thing that maybe we
should be more interested

586
00:40:53,345 --> 00:40:59,320
in is what happens if I'm
in the vicinity of this,

587
00:40:59,320 --> 00:41:01,920
how rapidly do I move away?

588
00:41:01,920 --> 00:41:04,910
And actually I have to show
that we are moving away.

589
00:41:04,910 --> 00:41:07,610
But because of topology,
it's more or less obvious

590
00:41:07,610 --> 00:41:09,590
that it should be that way.

591
00:41:09,590 --> 00:41:13,220
So what I need to do is
evaluate this at K star.

592
00:41:13,220 --> 00:41:14,340
OK.

593
00:41:14,340 --> 00:41:18,100
Now you can see that K
prime is a function of K.

594
00:41:18,100 --> 00:41:20,680
So what you need to do
is to take derivatives.

595
00:41:20,680 --> 00:41:24,560
So thers' some
algebra involved here.

596
00:41:24,560 --> 00:41:27,810
And then, once you have
taken the derivative,

597
00:41:27,810 --> 00:41:30,920
you have to put the
value of K star.

598
00:41:30,920 --> 00:41:33,840
And here, some
calculator is necessary.

599
00:41:33,840 --> 00:41:36,870
And at the end of the day, the
number that you get, I believe,

600
00:41:36,870 --> 00:41:39,460
is something like 1.62.

601
00:41:39,460 --> 00:41:41,280
Yes.

602
00:41:41,280 --> 00:41:44,180
And so that says
since being it's

603
00:41:44,180 --> 00:41:48,240
larger than 1, that you
will be pushed away.

604
00:41:48,240 --> 00:41:52,040
But these things have
been important to us

605
00:41:52,040 --> 00:41:56,566
as indicators of
these exponents.

606
00:41:56,566 --> 00:41:59,640
In particular, I'm on the
subspace that has symmetry,

607
00:41:59,640 --> 00:42:03,470
so I should be
calculating yt here.

608
00:42:03,470 --> 00:42:07,260
As was pointed out,
important to this step

609
00:42:07,260 --> 00:42:10,440
is knowing what the value
of B is, which we can either

610
00:42:10,440 --> 00:42:12,910
look at by the ratio of
the lattice constants

611
00:42:12,910 --> 00:42:16,630
or by the fact that I have
removed one third of the spins.

612
00:42:16,630 --> 00:42:19,970
It has to be root
3 to the power yt.

613
00:42:19,970 --> 00:42:27,005
So my yt is log of 1.62
divided by log of root 3.

614
00:42:27,005 --> 00:42:29,890
So again you go and
look at your calculator,

615
00:42:29,890 --> 00:42:37,360
and the answer comes
out to be 0.88.

616
00:42:37,360 --> 00:42:42,300
Now the exact value of yt for
all two-dimensionalizing models

617
00:42:42,300 --> 00:42:43,940
is 1.

618
00:42:43,940 --> 00:42:48,070
So again, this is an
indicator of how good or bad

619
00:42:48,070 --> 00:42:51,791
you have done at this
ordering perturbation tier.

620
00:42:51,791 --> 00:42:53,540
OK.

621
00:42:53,540 --> 00:42:57,880
Now, answering the question
that you had before,

622
00:42:57,880 --> 00:43:01,380
suppose I were to go
to order of U squared?

623
00:43:04,105 --> 00:43:08,260
Now, order of U
squared, I have to take

624
00:43:08,260 --> 00:43:12,080
this kind of interaction,
which is bilinear.

625
00:43:12,080 --> 00:43:16,080
Let's say pair of spins
here, multiply two of them,

626
00:43:16,080 --> 00:43:19,900
so I will get a pair of spins
here and pair of spins there.

627
00:43:19,900 --> 00:43:23,150
As long as they are
distinct locations when

628
00:43:23,150 --> 00:43:28,440
I subtract the average squared,
they will subtract out.

629
00:43:28,440 --> 00:43:31,650
So the only place where I
will get something non-trivial

630
00:43:31,650 --> 00:43:35,190
is if I pick one
here and one here.

631
00:43:35,190 --> 00:43:38,240
And by that kind
of reasoning, you

632
00:43:38,240 --> 00:43:42,920
can convince yourself that
what happens at next order

633
00:43:42,920 --> 00:43:46,590
is that in addition to
interactions between neighbors,

634
00:43:46,590 --> 00:43:50,550
you will degenerate interactions
between things that are two

635
00:43:50,550 --> 00:43:56,390
apart and things that
are, well, three apart,

636
00:43:56,390 --> 00:43:58,890
so basically next
nearest neighbors

637
00:43:58,890 --> 00:44:01,380
and next next nearest neighbors.

638
00:44:01,380 --> 00:44:03,590
So what you will
have, even if you

639
00:44:03,590 --> 00:44:07,770
start with a form
such as this, you

640
00:44:07,770 --> 00:44:16,649
will generate next nearest
neighbor and next next nearest

641
00:44:16,649 --> 00:44:17,565
neighbor interactions.

642
00:44:21,610 --> 00:44:25,250
Let's call them K, L, M.

643
00:44:25,250 --> 00:44:30,490
So to be consistent, you have
to go back to the original model

644
00:44:30,490 --> 00:44:35,270
and put the three interactions
and construct recursion

645
00:44:35,270 --> 00:44:37,260
relations from the
three parameters,

646
00:44:37,260 --> 00:44:41,090
K, L, M, to the new
three parameters.

647
00:44:41,090 --> 00:44:43,170
More or less following
this procedure,

648
00:44:43,170 --> 00:44:45,580
it's several pages of algebra.

649
00:44:45,580 --> 00:44:47,370
So I won't do it.

650
00:44:47,370 --> 00:44:49,500
Niemeijer and van
Leeuwen did it,

651
00:44:49,500 --> 00:44:53,340
and they calculated
the yt at next order

652
00:44:53,340 --> 00:44:56,530
by finding the fixed point in
this three-dimensional space.

653
00:44:56,530 --> 00:45:00,510
It has one relevant direction,
and that one relevant direction

654
00:45:00,510 --> 00:45:06,840
gave them an eigenvalue that
was extremely close to 1.

655
00:45:06,840 --> 00:45:12,980
So I don't believe anybody
has taken this to next order.

656
00:45:12,980 --> 00:45:17,290
You've got good enough,
might as well stop.

657
00:45:17,290 --> 00:45:20,440
I think it's not going
to improve and get better

658
00:45:20,440 --> 00:45:23,560
because this is an
uncontrolled approximation.

659
00:45:23,560 --> 00:45:25,895
So it's likely to be
one of those cases,

660
00:45:25,895 --> 00:45:28,385
that you asymptotically
approach the good result

661
00:45:28,385 --> 00:45:29,830
and then move away.

662
00:45:33,520 --> 00:45:37,030
Now once I have
yt, I can naturally

663
00:45:37,030 --> 00:45:40,520
calculate exponents
such as alpha.

664
00:45:40,520 --> 00:45:44,760
First of all U,
which is 1 over yt.

665
00:45:44,760 --> 00:45:49,210
1 over 0.88 is
something like 1.13.

666
00:45:49,210 --> 00:45:53,330
And the exact result would be
the inverse of 1 which is 1.

667
00:45:53,330 --> 00:46:00,746
And I can calculate alpha, which
is 2 minus d, which is 2 mu.

668
00:46:00,746 --> 00:46:07,150
With that value of U,
I will get minus 0.26.

669
00:46:07,150 --> 00:46:09,630
Again, the correct result
would be corresponding

670
00:46:09,630 --> 00:46:11,910
to a logarithmic divergence.

671
00:46:11,910 --> 00:46:16,290
So this zeroed order, OK.

672
00:46:16,290 --> 00:46:23,560
Those things, let's say, for
the exponents to 10%, 20%.

673
00:46:23,560 --> 00:46:26,430
You would say that, OK,
what about other exponents,

674
00:46:26,430 --> 00:46:30,100
such as beta,
gamma, and so forth.

675
00:46:30,100 --> 00:46:36,305
Clearly, to get those exponents,
I also need to have yh.

676
00:46:36,305 --> 00:46:38,250
OK.

677
00:46:38,250 --> 00:46:54,460
So to get yh, I will add, as
an additional perturbation,

678
00:46:54,460 --> 00:47:00,120
a term which is h sum
over i sigma i, which

679
00:47:00,120 --> 00:47:04,730
is, of course, the same thing
as sum over alpha, sigma alpha

680
00:47:04,730 --> 00:47:10,640
of 1 plus sigma alpha
2 plus sigma alpha 3.

681
00:47:15,440 --> 00:47:17,740
And if I regard this
as a perturbation,

682
00:47:17,740 --> 00:47:21,080
you can see that in the
perturbative scheme,

683
00:47:21,080 --> 00:47:25,200
this would go under
the transformation

684
00:47:25,200 --> 00:47:31,720
to the average of this
quantity, and that the average

685
00:47:31,720 --> 00:47:35,890
of this quantity will
give me 3 for each cell.

686
00:47:35,890 --> 00:47:39,370
So I will get 3h.

687
00:47:39,370 --> 00:47:41,900
And for each cell, I
will get the average

688
00:47:41,900 --> 00:47:44,810
of a site spin, which is
related to the cell spin

689
00:47:44,810 --> 00:47:46,940
through this factor
that we calculated,

690
00:47:46,940 --> 00:47:53,250
e to the 3K e to the minus k e
to the 3k plus 3e to the minus

691
00:47:53,250 --> 00:47:56,440
k sigma alpha prime.

692
00:47:56,440 --> 00:48:00,490
So we can see that by
generating h prime,

693
00:48:00,490 --> 00:48:04,710
which is 3h times
e to the 3K plus

694
00:48:04,710 --> 00:48:13,450
e to the minus K e to the
3K plus 3e to the minus K.

695
00:48:13,450 --> 00:48:19,800
And I can evaluate d
to the yh as dh prime

696
00:48:19,800 --> 00:48:24,090
by dh evaluated at
the fixed point.

697
00:48:24,090 --> 00:48:30,000
So I will get essentially
3 times this factor

698
00:48:30,000 --> 00:48:32,730
evaluated at the fixed point.

699
00:48:32,730 --> 00:48:35,070
But we can see that
at the fixed point,

700
00:48:35,070 --> 00:48:38,070
this factor is 1 over root 2.

701
00:48:38,070 --> 00:48:41,160
So the answer is 3 over root 2.

702
00:48:41,160 --> 00:48:48,170
And my yh would be the log of 3
over root 2 divided by log of b

703
00:48:48,170 --> 00:48:52,670
that we said is
square root of 3.

704
00:48:52,670 --> 00:48:56,440
Put it in the calculator,
you get a number

705
00:48:56,440 --> 00:48:59,670
that is of the order of 1.4.

706
00:48:59,670 --> 00:49:07,310
And exact yh is 1.875.

707
00:49:07,310 --> 00:49:10,990
So again, once you have yh,
you can go and calculate

708
00:49:10,990 --> 00:49:13,350
through the exponent
scaling relations all

709
00:49:13,350 --> 00:49:15,968
the other exponents
that you have like beta.

710
00:49:19,420 --> 00:49:23,220
So not bad,
considering that if you

711
00:49:23,220 --> 00:49:26,250
wanted to go through epsilon
expansion, how much difficulty

712
00:49:26,250 --> 00:49:27,650
you would have.

713
00:49:27,650 --> 00:49:30,550
And in any case, we
are at two dimensions,

714
00:49:30,550 --> 00:49:33,190
which is far away from four.

715
00:49:33,190 --> 00:49:35,990
And getting results
at 2 is worse than

716
00:49:35,990 --> 00:49:38,070
trying to get results
at three dimensions.

717
00:49:47,990 --> 00:49:51,620
Now we want to do the procedure
as an approximation that

718
00:49:51,620 --> 00:49:54,240
is even simpler than this.

719
00:49:54,240 --> 00:49:58,420
And for that-- so that was
the Niemeijer-van Leeuwen

720
00:49:58,420 --> 00:49:59,790
procedure.

721
00:49:59,790 --> 00:50:03,140
The next one is due to
Kadanoff again and Migdal.

722
00:50:10,230 --> 00:50:11,650
And it's called bond-moving.

723
00:50:16,360 --> 00:50:19,550
And again, we have to
do an approximation.

724
00:50:19,550 --> 00:50:23,326
You can't do things exact.

725
00:50:23,326 --> 00:50:28,800
So let's demonstrate that
by a square lattice, which

726
00:50:28,800 --> 00:50:31,987
is much easier to draw than
the triangular lattice.

727
00:50:45,930 --> 00:50:50,650
And let's kind of
follow the procedure

728
00:50:50,650 --> 00:50:54,520
that we had for the
one-dimensional case.

729
00:50:54,520 --> 00:51:00,230
Let's say we want to do
rescaling by a factor of 2.

730
00:51:00,230 --> 00:51:04,360
And I want to keep this
spin, this spin, this spin,

731
00:51:04,360 --> 00:51:12,550
this spin, this spin and get
rid of all of the other spins

732
00:51:12,550 --> 00:51:13,910
that I have.

733
00:51:13,910 --> 00:51:16,930
Not the circular
round, much as I

734
00:51:16,930 --> 00:51:20,740
did for the
one-dimensional case.

735
00:51:20,740 --> 00:51:25,390
And the problem is
that if I'm summing

736
00:51:25,390 --> 00:51:28,800
over this spin over
here, there are

737
00:51:28,800 --> 00:51:34,890
paths that connect that
spin to other spins.

738
00:51:34,890 --> 00:51:38,430
So by necessity, once I sum
over all of these spins,

739
00:51:38,430 --> 00:51:41,416
I will generate all
kinds of interactions.

740
00:51:44,410 --> 00:51:49,860
So the problem is all of these
paths that connect things.

741
00:51:49,860 --> 00:51:53,410
So maybe-- and this is
called bond-moving--

742
00:51:53,410 --> 00:51:58,160
maybe I can remove all of these
things that are going to cause

743
00:51:58,160 --> 00:51:59,081
problem for--

744
00:52:04,490 --> 00:52:07,160
So if I do that, then
the only connection

745
00:52:07,160 --> 00:52:11,080
between this spin and this
spin comes from that side,

746
00:52:11,080 --> 00:52:14,480
between this spin and that
spin comes from that side.

747
00:52:14,480 --> 00:52:19,070
And if the original interaction
was K and I sum over this,

748
00:52:19,070 --> 00:52:25,580
I will get K prime, which is
what I have over there, 1/2 log

749
00:52:25,580 --> 00:52:30,230
cos 2K, because the
only thing that I did

750
00:52:30,230 --> 00:52:34,300
was to connect this
site to two neighbors,

751
00:52:34,300 --> 00:52:36,220
and then effectively,
it's the same thing

752
00:52:36,220 --> 00:52:39,010
as I was doing
for one dimension.

753
00:52:39,010 --> 00:52:41,960
So clearly this is a
very bad approximation,

754
00:52:41,960 --> 00:52:44,340
because I have reproduced
the same result

755
00:52:44,340 --> 00:52:47,410
as one dimension for the
two-dimensional case.

756
00:52:47,410 --> 00:52:50,380
And the reason is that
I weakened the lattice

757
00:52:50,380 --> 00:52:53,690
so drastically, I removed
most of the bonds.

758
00:52:53,690 --> 00:53:00,830
So there isn't that much weight
for the lattice to order.

759
00:53:00,830 --> 00:53:04,940
Kadanoff and Migdal
suggested was, OK.

760
00:53:04,940 --> 00:53:07,770
Let's not to remove these bonds.

761
00:53:07,770 --> 00:53:12,070
Just move them to some place
that they don't cause any harm.

762
00:53:12,070 --> 00:53:15,510
So I take this bond and
I strengthen this bond.

763
00:53:15,510 --> 00:53:18,290
I take this bond,
strengthen this one.

764
00:53:18,290 --> 00:53:19,890
This one goes to this one.

765
00:53:19,890 --> 00:53:22,320
Essentially what
happens is you can

766
00:53:22,320 --> 00:53:28,810
see that each one of the bonds
has been strengthened by 2.

767
00:53:28,810 --> 00:53:31,410
So I have this because
of the strength.

768
00:53:35,340 --> 00:53:40,200
So as simple as you can get to
construct a potential recursion

769
00:53:40,200 --> 00:53:43,070
relation for this
square lattice.

770
00:53:43,070 --> 00:53:46,890
So this is a way
that the parameter K

771
00:53:46,890 --> 00:53:51,520
changes, going
from 0 to infinity.

772
00:53:51,520 --> 00:53:54,260
And we can do the same thing
that we did over there.

773
00:53:54,260 --> 00:54:02,740
So we can check that for K going
zero, if I look at K prime,

774
00:54:02,740 --> 00:54:08,660
it is approximately 1/2 log of
hyperbolic cosine of something

775
00:54:08,660 --> 00:54:10,720
that is close to 0.

776
00:54:10,720 --> 00:54:15,230
So that becomes 1 plus the
square of this quantity, which

777
00:54:15,230 --> 00:54:19,110
is 4K squared over 2.

778
00:54:19,110 --> 00:54:26,856
And taking the log of that,
it becomes 4K squared.

779
00:54:26,856 --> 00:54:30,170
The factor of 4 does
not really matter.

780
00:54:30,170 --> 00:54:33,490
If K is very small,
like 1 over 100,

781
00:54:33,490 --> 00:54:35,820
K squared would be
10 to the minus 4.

782
00:54:35,820 --> 00:54:41,750
So basically, you certainly
have the expected behavior

783
00:54:41,750 --> 00:54:46,340
of becoming disordered if you
have a [INAUDIBLE] interaction.

784
00:54:46,340 --> 00:54:48,790
If you have a strong
enough coupling,

785
00:54:48,790 --> 00:54:50,860
however, are we
different from what

786
00:54:50,860 --> 00:54:53,110
we did for the
one-dimensional case?

787
00:54:53,110 --> 00:54:55,560
Well, the answer is
that in this case,

788
00:54:55,560 --> 00:55:04,260
K prime is 1/2 log
hyperbolic cosine of 4K,

789
00:55:04,260 --> 00:55:09,320
starts as e to the 4K plus e
to the minus 4K divided by 2.

790
00:55:09,320 --> 00:55:12,360
e to the minus 4K I can ignore.

791
00:55:12,360 --> 00:55:16,680
So you can see that in
this case, I will have 2K.

792
00:55:16,680 --> 00:55:20,440
I can even ignore the minus
log 2, which previously

793
00:55:20,440 --> 00:55:23,950
was so important because
previously we had 1 here

794
00:55:23,950 --> 00:55:27,710
and now it became 2, which means
that if I start to be 10,000,

795
00:55:27,710 --> 00:55:32,010
it will become 20,000,
40,000 and now you're

796
00:55:32,010 --> 00:55:34,220
going this direction.

797
00:55:34,220 --> 00:55:37,350
So again, by necessity
almost, I must

798
00:55:37,350 --> 00:55:42,590
have a fixed point at
some value in between.

799
00:55:42,590 --> 00:55:48,290
So I essentially have
to solve for K star

800
00:55:48,290 --> 00:55:57,510
as K star is 1/2
log cos of 4K star.

801
00:55:57,510 --> 00:56:01,710
You can recast this as some
algebraic equation in terms

802
00:56:01,710 --> 00:56:06,920
of e to the 4k e to the
K star and manipulate it.

803
00:56:06,920 --> 00:56:09,920
And after you do
your algebra, you

804
00:56:09,920 --> 00:56:14,530
will eventually come up
with a value of K star,

805
00:56:14,530 --> 00:56:24,280
which I believe is 0.3.

806
00:56:24,280 --> 00:56:29,370
You can ask, well, the square
lattice we will solve in class.

807
00:56:29,370 --> 00:56:33,000
I said the triangular lattice
I will leave for you to solve.

808
00:56:33,000 --> 00:56:38,140
KC for the square lattice
is something like 0.44.

809
00:56:38,140 --> 00:56:44,150
So you are off by about 25%.

810
00:56:44,150 --> 00:56:46,890
Of course, again, the quantity
that you're interested in

811
00:56:46,890 --> 00:56:49,550
is b to t yt.

812
00:56:49,550 --> 00:56:51,550
b is 2 in this case.

813
00:56:51,550 --> 00:56:54,370
The length scale has
changed by a factor of 2,

814
00:56:54,370 --> 00:57:00,530
which is dk prime by
dk evaluated as K star,

815
00:57:00,530 --> 00:57:04,140
again, a combination of doing
the algebra of derivatives,

816
00:57:04,140 --> 00:57:09,270
evaluating at K star, and
then ultimately taking the log

817
00:57:09,270 --> 00:57:11,660
to convert it to a yt.

818
00:57:11,660 --> 00:57:16,460
And you come up with a value
of yt that is around 0.75.

819
00:57:16,460 --> 00:57:20,360
And, as I said, the exact
yt, which doesn't depend on

820
00:57:20,360 --> 00:57:22,320
whether you are dealing
with a square lattice

821
00:57:22,320 --> 00:57:24,510
or a triangular
lattice-- it's only

822
00:57:24,510 --> 00:57:28,100
a function of symmetry
and dimensionality-- is 1.

823
00:57:32,710 --> 00:57:38,570
So you can see that gradually we
are simplifying the complexity.

824
00:57:38,570 --> 00:57:42,110
Now we could, within
this approximation,

825
00:57:42,110 --> 00:57:45,980
solve everything
within one panel.

826
00:57:45,980 --> 00:57:48,410
Now this kind of
approximation, again,

827
00:57:48,410 --> 00:57:50,800
is not particularly very good.

828
00:57:50,800 --> 00:57:55,760
But it's a quick and dirty
way of getting results.

829
00:57:55,760 --> 00:57:59,900
And the advantage of it
is that you can do this

830
00:57:59,900 --> 00:58:04,459
not only in two dimensions, but
in higher dimensions as well.

831
00:58:21,450 --> 00:58:28,340
So let's say that you
had a cubic lattice

832
00:58:28,340 --> 00:58:32,520
and you were doing rescaling
by a factor of 2, which

833
00:58:32,520 --> 00:58:38,580
means that originally
you had spins along

834
00:58:38,580 --> 00:58:42,440
the various diagonals
and so forth--

835
00:58:42,440 --> 00:58:50,300
around the various partitions
of a square of size 2 x 2,

836
00:58:50,300 --> 00:58:55,490
and you want to keep
interactions among these

837
00:58:55,490 --> 00:59:00,120
and get rid of the interactions
among all of the places

838
00:59:00,120 --> 00:59:01,430
that you are not interested in.

839
00:59:04,070 --> 00:59:08,370
And the way that you do
that is precisely as before.

840
00:59:08,370 --> 00:59:12,071
You move these interactions
and strengthen the things

841
00:59:12,071 --> 00:59:13,070
that you have over here.

842
00:59:16,190 --> 00:59:18,510
Now whereas the number
of bonds that you

843
00:59:18,510 --> 00:59:23,390
had to move for the
square lattice was 2--

844
00:59:23,390 --> 00:59:26,860
the enhancement factor
was 2-- turns out

845
00:59:26,860 --> 00:59:31,880
that the enhancement factor in
three dimensions would be 4.

846
00:59:31,880 --> 00:59:35,690
You essentially have to take
one from here, one from there,

847
00:59:35,690 --> 00:59:36,970
one from there.

848
00:59:36,970 --> 00:59:40,460
So 1 plus 3 becomes 4.

849
00:59:40,460 --> 00:59:42,500
And you can convince
yourself that if I

850
00:59:42,500 --> 00:59:55,790
had done this in d-dimensional
hypercubic lattice, what

851
00:59:55,790 --> 01:00:00,860
I would have gotten is again
the one-dimensional recursion

852
01:00:00,860 --> 01:00:11,770
relation, except for this
enhancement factor, which

853
01:00:11,770 --> 01:00:19,490
is 2 to the power of d
minus 1 in d dimensions.

854
01:00:19,490 --> 01:00:25,250
Actually, I could even do
that for rescaling rather

855
01:00:25,250 --> 01:00:30,380
by a factor 2, by a factor of b.

856
01:00:30,380 --> 01:00:33,850
And hopefully, you
can convince yourself

857
01:00:33,850 --> 01:00:38,621
that it will become b to
the d minus 1 times 2K.

858
01:00:41,390 --> 01:00:46,620
And essentially that
factor is a cross section

859
01:00:46,620 --> 01:00:48,910
that you have to remove.

860
01:00:48,910 --> 01:00:56,020
So the cross-sectional area that
you encounter grows as the size

861
01:00:56,020 --> 01:00:58,560
to the power of d minus 1.

862
01:00:58,560 --> 01:01:02,540
And a kind of obvious
consequence of that

863
01:01:02,540 --> 01:01:06,470
is that if I go to the limit
of K going to infinity,

864
01:01:06,470 --> 01:01:13,020
you can see that K prime b would
go like b to the d minus 1K.

865
01:01:13,020 --> 01:01:18,200
Essentially, if you were to
have some kind of a system

866
01:01:18,200 --> 01:01:21,620
and you make pluses on one
side, minuses on the one side,

867
01:01:21,620 --> 01:01:24,880
to break it, then,
the number of bonds

868
01:01:24,880 --> 01:01:26,880
that you would
have to break would

869
01:01:26,880 --> 01:01:29,430
grow like the
cross-sectional area.

870
01:01:29,430 --> 01:01:33,630
So that's where that comes from.

871
01:01:33,630 --> 01:01:39,380
It turns out that, again,
this approach is exact

872
01:01:39,380 --> 01:01:42,710
as we've seen for one dimension.

873
01:01:42,710 --> 01:01:48,860
As we go to higher dimensions,
it becomes worse and worse.

874
01:01:48,860 --> 01:01:52,170
So I showed you how bad
it was in two dimensions.

875
01:01:52,170 --> 01:01:55,130
If I calculate the fixed point
and the exponents in three

876
01:01:55,130 --> 01:01:59,920
dimensions compared to our best
numerical result, it is off by,

877
01:01:59,920 --> 01:02:04,560
I don't know, 40%, 50%,
whereas 25% over there.

878
01:02:04,560 --> 01:02:07,350
So it gradually gets
worse and worse.

879
01:02:07,350 --> 01:02:10,985
And so one approach that
people have tried to do,

880
01:02:10,985 --> 01:02:13,700
which, again, doesn't
seem to be very rigorous

881
01:02:13,700 --> 01:02:18,085
is to convert this into an
expansion on dimensionality

882
01:02:18,085 --> 01:02:20,210
of 1.

883
01:02:20,210 --> 01:02:22,980
So it's roughly
correct that it's

884
01:02:22,980 --> 01:02:27,630
going to be close to 1--
correct close to one dimension.

885
01:02:27,630 --> 01:02:30,400
But as opposed to the
previous epsilon expansion,

886
01:02:30,400 --> 01:02:33,690
there doesn't seem to be a
controlled way to do this.

887
01:02:39,670 --> 01:02:43,850
I showed you how to do
this for Ising models.

888
01:02:43,850 --> 01:02:53,360
Actually, you can do
this for any spin model.

889
01:02:59,910 --> 01:03:04,530
So let's imagine that we have
some kind of a model that's

890
01:03:04,530 --> 01:03:07,290
in one dimension.

891
01:03:07,290 --> 01:03:11,515
At each site, we
have some variable

892
01:03:11,515 --> 01:03:15,130
that as i that I will
not specify what it is,

893
01:03:15,130 --> 01:03:18,480
how many variables-- how
many values it takes.

894
01:03:18,480 --> 01:03:23,370
But it interacts only with
its neighboring sites.

895
01:03:23,370 --> 01:03:29,940
And so presumably, there
is some interaction

896
01:03:29,940 --> 01:03:32,770
that depends on the two sites.

897
01:03:32,770 --> 01:03:36,480
There may be multiple
couplings implicit in this

898
01:03:36,480 --> 01:03:42,170
if I try to write this in terms
of the dot product of spins

899
01:03:42,170 --> 01:03:45,690
or things like this.

900
01:03:45,690 --> 01:03:51,070
So if I were to calculate
the partition function in one

901
01:03:51,070 --> 01:03:55,320
dimension-- I already
mentioned this last time--

902
01:03:55,320 --> 01:04:00,320
I have to do a sum
over what this spin is

903
01:04:00,320 --> 01:04:10,310
of e to the K of si si
plus 1 and a product

904
01:04:10,310 --> 01:04:13,810
over subsequent sites.

905
01:04:13,810 --> 01:04:20,910
And if I regard this as a
matrix, which is generally

906
01:04:20,910 --> 01:04:24,430
called a transfer
matrix, you can

907
01:04:24,430 --> 01:04:28,360
see that this
multiplication involving

908
01:04:28,360 --> 01:04:32,270
the sum over all of the
spins is equivalent to matrix

909
01:04:32,270 --> 01:04:34,020
multiplication.

910
01:04:34,020 --> 01:04:38,640
And, in particular, if I have
periodic boundary conditions

911
01:04:38,640 --> 01:04:42,890
in which the last spin
couples to the first spin,

912
01:04:42,890 --> 01:04:49,070
I would have trace of
T to the power of N,

913
01:04:49,070 --> 01:04:56,434
where t is essentially this,
e to the K of si si plus 1.

914
01:05:00,900 --> 01:05:09,570
Now, clearly I can
write this as trace of T

915
01:05:09,570 --> 01:05:14,070
squared to the
power of N over 2.

916
01:05:14,070 --> 01:05:15,960
Right.

917
01:05:15,960 --> 01:05:21,010
And this I can regard as
the partition function

918
01:05:21,010 --> 01:05:25,990
of a system that has
half as many spins.

919
01:05:25,990 --> 01:05:28,660
So I have performed the
renormalization group

920
01:05:28,660 --> 01:05:31,310
like what we are doing
in one dimension.

921
01:05:31,310 --> 01:05:34,550
I have T prime is T squared.

922
01:05:34,550 --> 01:05:37,880
And in general, you can see
that I can write this as N

923
01:05:37,880 --> 01:05:41,540
to the b N over b.

924
01:05:41,540 --> 01:05:45,610
So the result of
renormalization by a factor of b

925
01:05:45,610 --> 01:05:49,860
is simply one dimension to
take the matrix that you have

926
01:05:49,860 --> 01:05:53,690
and raise it to b power.

927
01:05:53,690 --> 01:05:54,396
OK.

928
01:05:54,396 --> 01:06:06,060
And so I could parameterize my
T by a set of interactions K,

929
01:06:06,060 --> 01:06:08,570
like we do for the Ising model.

930
01:06:08,570 --> 01:06:14,340
Raise it to the power of b,
and I would generate the matrix

931
01:06:14,340 --> 01:06:17,860
that I could then
parametrize by K prime,

932
01:06:17,860 --> 01:06:20,540
and I would have the
relationship between K prime

933
01:06:20,540 --> 01:06:25,160
and K in one dimension.

934
01:06:25,160 --> 01:06:29,560
So this is d equals 1.

935
01:06:29,560 --> 01:06:40,875
And the way to generalize
this Migdal Kadanoff, RG,

936
01:06:40,875 --> 01:06:45,510
to a very general
system is simply

937
01:06:45,510 --> 01:06:47,700
to enhance the couplings.

938
01:06:47,700 --> 01:06:50,570
So basically, what
I would write down

939
01:06:50,570 --> 01:06:56,140
is T prime, which, after
rescaling by a factor of b,

940
01:06:56,140 --> 01:07:01,000
is a function of a set of
parameters that I will K prime,

941
01:07:01,000 --> 01:07:05,540
is obtained by taking the
matrix that I have for one set

942
01:07:05,540 --> 01:07:11,800
of couplings and raise
it to the power of b.

943
01:07:11,800 --> 01:07:14,950
This is the exact
one-dimensional result.

944
01:07:14,950 --> 01:07:19,125
And if I want to construct this
approximation in d dimensions,

945
01:07:19,125 --> 01:07:20,520
I will just do this.

946
01:07:24,890 --> 01:07:32,620
So for a while, before people
had sufficiently powerful

947
01:07:32,620 --> 01:07:37,160
computers to maybe
simulate things as easily,

948
01:07:37,160 --> 01:07:42,290
this was a good way to estimate
locations and exponents

949
01:07:42,290 --> 01:07:46,370
of phase diagrams,
critical exponents,

950
01:07:46,370 --> 01:07:49,850
et cetera for essentially
complicated problems

951
01:07:49,850 --> 01:07:53,760
that could have a set
of parameters even here.

952
01:07:53,760 --> 01:07:56,350
Nowadays, as I
said, you probably

953
01:07:56,350 --> 01:07:59,734
can do things much more easily
by computer simulations.

954
01:08:12,900 --> 01:08:17,569
So I guess I still have
another 10 minutes.

955
01:08:17,569 --> 01:08:21,949
I probably don't want
to start on the topic

956
01:08:21,949 --> 01:08:24,310
of the next lecture.

957
01:08:24,310 --> 01:08:27,439
But maybe what I'll do is
I'll expand a little bit

958
01:08:27,439 --> 01:08:32,370
on something that I mentioned
very rapidly last lecture,

959
01:08:32,370 --> 01:08:40,600
which is that in this
one-dimensional model where

960
01:08:40,600 --> 01:08:54,520
I solve the problem by
transfer matrix, what I have is

961
01:08:54,520 --> 01:08:59,939
that the partition function
is trace of some matrix raised

962
01:08:59,939 --> 01:09:03,229
to the N power.

963
01:09:03,229 --> 01:09:07,410
And if I diagonalize the
matrix, what I will get

964
01:09:07,410 --> 01:09:12,450
is the sum over all eigenvalues
raised to the N power.

965
01:09:16,180 --> 01:09:20,880
Now note that we expect
phase transitions to occur,

966
01:09:20,880 --> 01:09:27,250
not for any finite system,
but only in the limit where

967
01:09:27,250 --> 01:09:28,899
there are many
degrees of freedom.

968
01:09:31,450 --> 01:09:34,600
And, actually, if
I have such a sum

969
01:09:34,600 --> 01:09:38,280
as this in the limit of
very large number of degrees

970
01:09:38,280 --> 01:09:46,270
of freedom, this becomes
lambda max to the power of N.

971
01:09:46,270 --> 01:09:50,310
Now in order to get any one of
these series of eigenvalues,

972
01:09:50,310 --> 01:09:52,109
what should I do?

973
01:09:52,109 --> 01:09:56,370
I should take a matrix,
which is this e to the T--

974
01:09:56,370 --> 01:09:59,110
e to the strength
of the interactions.

975
01:09:59,110 --> 01:10:04,040
What did I write? e to
the K of S and S prime.

976
01:10:04,040 --> 01:10:07,020
So there is a matrix.

977
01:10:07,020 --> 01:10:10,392
All of its elements
are Boltzmann weights.

978
01:10:10,392 --> 01:10:11,350
There are all positive.

979
01:10:17,860 --> 01:10:21,370
And find the eigenvalues
of this matrix.

980
01:10:21,370 --> 01:10:29,030
Now for the case of the Ising
model without a magnetic field,

981
01:10:29,030 --> 01:10:30,470
the matrix is 2 x 2.

982
01:10:30,470 --> 01:10:34,290
It's e to the K, corresponding
to the diagonal terms

983
01:10:34,290 --> 01:10:37,620
where the spins are
parallel, e to the minus K

984
01:10:37,620 --> 01:10:41,140
when the spins are antiparallel.

985
01:10:41,140 --> 01:10:45,350
And clearly you can see that the
eigenvalues corresponding to 1,

986
01:10:45,350 --> 01:10:53,050
1 or 1, minus 1 as eigenvectors,
are hyperbolic cosine of K--

987
01:10:53,050 --> 01:10:54,940
strike this.

988
01:10:54,940 --> 01:10:59,765
e to the K plus e to the
minus K e to the K minus

989
01:10:59,765 --> 01:11:06,450
e to the minus K. You
can see that to get this,

990
01:11:06,450 --> 01:11:10,490
all I had to do was to
diagonalize a matrix that

991
01:11:10,490 --> 01:11:16,000
corresponded to one
bond, if you like.

992
01:11:16,000 --> 01:11:19,650
And just as in here,
there is no reason

993
01:11:19,650 --> 01:11:23,010
to expect that these
eigenvalues, which

994
01:11:23,010 --> 01:11:25,420
depend on this
set of parameters,

995
01:11:25,420 --> 01:11:29,550
should non-analytical functions.

996
01:11:29,550 --> 01:11:32,070
There is no reason
for non-analyticity

997
01:11:32,070 --> 01:11:34,705
as long as you are dealing
with a single bond.

998
01:11:34,705 --> 01:11:40,190
We expect non-analyticities
at their limit of large N.

999
01:11:40,190 --> 01:11:46,810
So if each one of these
is analytic function of K,

1000
01:11:46,810 --> 01:11:50,505
the only way spanning the K
axis that I could encounter

1001
01:11:50,505 --> 01:11:55,860
in non-analyticity is if two
of these eigenvalues cross,

1002
01:11:55,860 --> 01:11:59,730
because we have seen a
potential mechanism for phase

1003
01:11:59,730 --> 01:12:00,470
transitions.

1004
01:12:00,470 --> 01:12:02,816
We discussed this in A333.

1005
01:12:02,816 --> 01:12:05,310
That if we have sum
of contributions,

1006
01:12:05,310 --> 01:12:08,302
that each one of them is
exponentially large in N,

1007
01:12:08,302 --> 01:12:10,820
and two of these
contributions cross,

1008
01:12:10,820 --> 01:12:14,020
then your partition function
will jump from one hill

1009
01:12:14,020 --> 01:12:14,870
to another hill.

1010
01:12:14,870 --> 01:12:16,970
And you will have a
discontinuity, let's say,

1011
01:12:16,970 --> 01:12:19,260
in derivatives or whatever.

1012
01:12:19,260 --> 01:12:22,690
So the potential
mechanism that I can have

1013
01:12:22,690 --> 01:12:26,040
is that if I have as
a function of changing

1014
01:12:26,040 --> 01:12:29,460
one of my parameters or a
bunch of these parameters,

1015
01:12:29,460 --> 01:12:31,760
the ordering of
these eigenvalues--

1016
01:12:31,760 --> 01:12:34,610
let's say lambda 0 is
the largest one, lambda

1017
01:12:34,610 --> 01:12:37,070
1 is the next one,
lambda 2, et cetera.

1018
01:12:37,070 --> 01:12:41,180
So each one of them
is going its own way.

1019
01:12:41,180 --> 01:12:47,500
If the largest one suddenly
gets crossed by something else,

1020
01:12:47,500 --> 01:12:52,010
then you willl basically abandon
one eigenvalue for another,

1021
01:12:52,010 --> 01:12:55,490
and you will have a mechanism
for a phase transition.

1022
01:12:55,490 --> 01:12:59,680
So what I told you was that
there is a theorem that

1023
01:12:59,680 --> 01:13:03,320
for a matrix where all of
the eigenvalues are positive,

1024
01:13:03,320 --> 01:13:05,650
this will never happen.

1025
01:13:05,650 --> 01:13:11,450
The largest eigenvalue
will remain non-degenerate.

1026
01:13:11,450 --> 01:13:13,720
And there is some
analog of this,

1027
01:13:13,720 --> 01:13:16,830
probably you've seen
in quantum mechanics,

1028
01:13:16,830 --> 01:13:19,090
that if you have a
potential, the ground

1029
01:13:19,090 --> 01:13:20,640
state is non-degenerate.

1030
01:13:20,640 --> 01:13:23,200
The ground state is
always a function

1031
01:13:23,200 --> 01:13:26,390
that is positive everywhere.

1032
01:13:26,390 --> 01:13:28,210
And the next
excitation would have

1033
01:13:28,210 --> 01:13:30,860
to have a node go
from plus to minus.

1034
01:13:30,860 --> 01:13:34,490
And somehow you cannot
have the eigenvalues cross.

1035
01:13:34,490 --> 01:13:39,060
So in a similar sense, it turns
out the largest eigenvalue,

1036
01:13:39,060 --> 01:13:41,830
the largest eigenvector
for these matrices,

1037
01:13:41,830 --> 01:13:44,590
will have all of the
elements positive,

1038
01:13:44,590 --> 01:13:46,800
and it cannot become degenerate.

1039
01:13:46,800 --> 01:13:51,390
And so you are guaranteed
that this will not happen.

1040
01:13:51,390 --> 01:13:53,890
Now the second part of
this story that I briefly

1041
01:13:53,890 --> 01:13:58,530
mentioned was you can repeat
this for two dimensions,

1042
01:13:58,530 --> 01:14:01,860
for three dimensions,
higher dimensional things.

1043
01:14:01,860 --> 01:14:04,640
So one thing that
you could do is,

1044
01:14:04,640 --> 01:14:08,310
rather than sorting the
Ising model on a line,

1045
01:14:08,310 --> 01:14:13,010
you can solve it on a ladder,
or a ladder that has two rungs.

1046
01:14:13,010 --> 01:14:17,910
So solving the Ising
model on this structure

1047
01:14:17,910 --> 01:14:19,840
is not very
difficult because you

1048
01:14:19,840 --> 01:14:25,330
can say that there are eight
possible values that this

1049
01:14:25,330 --> 01:14:26,910
can take.

1050
01:14:26,910 --> 01:14:29,095
And so I can construct
a matrix that

1051
01:14:29,095 --> 01:14:33,810
is 8 x 8 that tells me how I
go from the choice of eight

1052
01:14:33,810 --> 01:14:38,370
possibilities here to the
eight possibilities there.

1053
01:14:38,370 --> 01:14:41,820
And I will have an 8 x 8 matrix
that has these properties

1054
01:14:41,820 --> 01:14:44,570
and will satisfy this.

1055
01:14:44,570 --> 01:14:50,090
It will be true if
I go to a 4 strip.

1056
01:14:50,090 --> 01:14:52,330
It will be a 16 x 16.

1057
01:14:52,330 --> 01:14:53,230
No problem.

1058
01:14:53,230 --> 01:14:55,180
I can keep going.

1059
01:14:55,180 --> 01:14:58,520
And it would say, well,
the two-dimensional--

1060
01:14:58,520 --> 01:15:01,160
and also you could do
three-dimensional or higher

1061
01:15:01,160 --> 01:15:05,226
dimensional models-- should
not have a phase transition.

1062
01:15:05,226 --> 01:15:08,180
Well, it turns out
that all of this

1063
01:15:08,180 --> 01:15:11,930
relies on having
a finite matrix.

1064
01:15:11,930 --> 01:15:17,220
And what Onsager showed
was that, indeed,

1065
01:15:17,220 --> 01:15:20,180
for any finite strip, you
would have a situation

1066
01:15:20,180 --> 01:15:22,375
such as this-- actually
more accurately

1067
01:15:22,375 --> 01:15:28,720
if I were to draw a
situation such as this, where

1068
01:15:28,720 --> 01:15:32,860
two eigenvalues would
approach but will never cross.

1069
01:15:32,860 --> 01:15:35,470
And one can show that
the gap between them

1070
01:15:35,470 --> 01:15:40,930
will scale as something
like 1 over this length.

1071
01:15:40,930 --> 01:15:45,680
And so in the limit where you
go to a large enough system,

1072
01:15:45,680 --> 01:15:48,875
you have the
possibility of ascension

1073
01:15:48,875 --> 01:15:54,390
to some singularity when two
eigenvalues touch each other.

1074
01:15:54,390 --> 01:15:57,780
So this scenario is very
well known and studied

1075
01:15:57,780 --> 01:15:58,830
in two dimensions.

1076
01:15:58,830 --> 01:16:00,270
In higher dimensions,
we actually

1077
01:16:00,270 --> 01:16:04,550
don't really know what happens.

1078
01:16:04,550 --> 01:16:05,050
OK.

1079
01:16:05,050 --> 01:16:06,687
Any questions?

1080
01:16:06,687 --> 01:16:08,978
AUDIENCE: So it appears that
there are or there are not

1081
01:16:08,978 --> 01:16:11,192
phase transitions [INAUDIBLE]?

1082
01:16:11,192 --> 01:16:13,630
PROFESSOR: Well, we
showed-- we were discussing

1083
01:16:13,630 --> 01:16:16,670
phase transitions for
the triangular lattice,

1084
01:16:16,670 --> 01:16:18,320
for the square lattice.

1085
01:16:18,320 --> 01:16:20,857
I even told you what the
critical coupling is.

1086
01:16:20,857 --> 01:16:23,190
AUDIENCE: But it seems to me
that the conclusion of what

1087
01:16:23,190 --> 01:16:26,230
you're-- of this part
is that there aren't.

1088
01:16:26,230 --> 01:16:29,356
PROFESSOR: As long as you
have a finite strip, no.

1089
01:16:29,356 --> 01:16:33,960
But if you have a 2-- an
infinite strip, you do.

1090
01:16:33,960 --> 01:16:36,890
So what I've shown you
here is the following.

1091
01:16:36,890 --> 01:16:43,260
If I have an L x N system
in which you keep L finite

1092
01:16:43,260 --> 01:16:47,650
and set N going to infinity,
you won't see a singularity.

1093
01:16:47,650 --> 01:16:53,130
But if I have an N x N system,
and I said N goes to infinity,

1094
01:16:53,130 --> 01:16:56,240
I will encounter a
singularity in the limit

1095
01:16:56,240 --> 01:16:57,616
of N going to infinity.

1096
01:17:01,360 --> 01:17:03,370
Again, very roughly,
one can also

1097
01:17:03,370 --> 01:17:07,610
should develop a physical
picture of what's going on.

1098
01:17:07,610 --> 01:17:13,940
So let's imagine that
you have a system that

1099
01:17:13,940 --> 01:17:20,040
is a very large number but
finite in one direction.

1100
01:17:20,040 --> 01:17:26,070
And this other direction can
be two, can be three, whatever.

1101
01:17:26,070 --> 01:17:28,960
You have a finite size.

1102
01:17:28,960 --> 01:17:33,620
But in this other
direction, you basically

1103
01:17:33,620 --> 01:17:37,150
can go as large as you like.

1104
01:17:37,150 --> 01:17:42,640
Now presumably this
two-dimensional model

1105
01:17:42,640 --> 01:17:46,750
has a phase transition if
it was infinite x infinite.

1106
01:17:46,750 --> 01:17:49,150
And on approaching
that phase transition,

1107
01:17:49,150 --> 01:17:50,950
there would be a
correlation length

1108
01:17:50,950 --> 01:17:54,460
that would diverge
with this exponent nu.

1109
01:17:54,460 --> 01:17:57,280
So let's say I am sufficiently
far away from the phase

1110
01:17:57,280 --> 01:18:01,680
transition that the correlation
length is something like this.

1111
01:18:01,680 --> 01:18:05,270
So this patch of spins
knows about each other.

1112
01:18:05,270 --> 01:18:07,580
If I go closer to
the transition,

1113
01:18:07,580 --> 01:18:09,395
it will grow bigger and bigger.

1114
01:18:09,395 --> 01:18:12,840
At some point, it will fit
the size of the system,

1115
01:18:12,840 --> 01:18:15,950
and then it cannot
grow any further.

1116
01:18:15,950 --> 01:18:18,520
So beyond that,
what you will see

1117
01:18:18,520 --> 01:18:20,420
is essentially
there is one patch

1118
01:18:20,420 --> 01:18:23,780
here, one patch
here, one patch here.

1119
01:18:23,780 --> 01:18:27,000
And you are back to a
one-dimensional system.

1120
01:18:27,000 --> 01:18:29,630
So what happens is that
your correlation length

1121
01:18:29,630 --> 01:18:33,560
starts to grow as if you were
in two dimensions or three

1122
01:18:33,560 --> 01:18:34,650
dimensions.

1123
01:18:34,650 --> 01:18:37,480
Once it hits the
size of the system,

1124
01:18:37,480 --> 01:18:38,760
then it has to saturate.

1125
01:18:38,760 --> 01:18:40,610
It cannot grow any bigger.

1126
01:18:40,610 --> 01:18:45,930
And then this block becomes
independent of the next block.

1127
01:18:45,930 --> 01:18:50,080
So essentially, you would say
that you would have effectively

1128
01:18:50,080 --> 01:18:54,260
a one-dimensional system where
the number of blocks that you

1129
01:18:54,260 --> 01:19:06,980
have is of the
order of N over L.

1130
01:19:06,980 --> 01:19:11,380
So what we are going to do
starting from next lecture

1131
01:19:11,380 --> 01:19:15,560
is to develop again a more
systematic approach, which

1132
01:19:15,560 --> 01:19:18,920
is a series expansion about
either low temperatures

1133
01:19:18,920 --> 01:19:22,870
or more usefully about
high temperature.

1134
01:19:22,870 --> 01:19:26,160
And then we will take that
high temperature expansion

1135
01:19:26,160 --> 01:19:28,530
and gradually go
in the direction

1136
01:19:28,530 --> 01:19:31,240
to solve these
two-dimensionalizing models

1137
01:19:31,240 --> 01:19:31,980
exactly.

1138
01:19:31,980 --> 01:19:36,000
And so we will
see why I told you

1139
01:19:36,000 --> 01:19:39,300
some of these results
about exact value of KC,

1140
01:19:39,300 --> 01:19:40,630
exact value yT.

1141
01:19:40,630 --> 01:19:43,080
Where do they come from?