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PROFESSOR: So last
time, we started

9
00:00:27,970 --> 00:00:32,032
looking at the system
of [? spins. ?] So there

10
00:00:32,032 --> 00:00:36,640
was a field S of
x on the lattice.

11
00:00:36,640 --> 00:00:41,840
And the energy cost was
proportional to differences

12
00:00:41,840 --> 00:00:45,280
of space on two
neighboring sites,

13
00:00:45,280 --> 00:00:50,290
which if we go to the continuum,
became something like gradient

14
00:00:50,290 --> 00:00:55,110
of the vector S. You have to
integrate this, of course,

15
00:00:55,110 --> 00:00:58,372
over all space.

16
00:00:58,372 --> 00:01:01,580
We gave this a rate of K over 2.

17
00:01:01,580 --> 00:01:08,200
There was some energy costs
[INAUDIBLE] of this [? form ?]

18
00:01:08,200 --> 00:01:10,760
so that the particular
configuration

19
00:01:10,760 --> 00:01:13,630
was weighted by this factor.

20
00:01:13,630 --> 00:01:16,830
And to calculate the
partition function,

21
00:01:16,830 --> 00:01:21,940
we had to integrate over
all configurations of this

22
00:01:21,940 --> 00:01:26,980
over this field S.

23
00:01:26,980 --> 00:01:31,670
And the constraint that we had
was that this was a unit vector

24
00:01:31,670 --> 00:01:39,522
so that this was an n component
field whose magnitude was 1,

25
00:01:39,522 --> 00:01:40,380
OK?

26
00:01:40,380 --> 00:01:43,460
So this is what we
want to calculate.

27
00:01:43,460 --> 00:01:48,200
Again, whenever we are writing
an expression such as this,

28
00:01:48,200 --> 00:01:52,760
thinking that we started with
some average system [INAUDIBLE]

29
00:01:52,760 --> 00:01:55,780
some kind of a coarse graining.

30
00:01:55,780 --> 00:01:58,400
There is a short distance
[INAUDIBLE] [? replacing ?]

31
00:01:58,400 --> 00:02:01,570
all of these tiers.

32
00:02:01,570 --> 00:02:07,360
Now what we can do is
imagine that this vector

33
00:02:07,360 --> 00:02:10,300
S in its ground
state, let's say,

34
00:02:10,300 --> 00:02:15,490
is pointing in some particular
direction throughout the system

35
00:02:15,490 --> 00:02:18,030
and that fluctuations
around this ground

36
00:02:18,030 --> 00:02:21,950
state in the
transverse direction

37
00:02:21,950 --> 00:02:24,670
are characterized by
some vector of pi that

38
00:02:24,670 --> 00:02:27,690
is n minus 1 dimensional.

39
00:02:27,690 --> 00:02:32,660
And so this partition function
can be written entirely

40
00:02:32,660 --> 00:02:37,360
[? rather ?] than fluctuations
of the unit vector S in terms

41
00:02:37,360 --> 00:02:40,950
of the fluctuations of these
transverse [? coordinates ?]

42
00:02:40,950 --> 00:02:43,110
[INAUDIBLE] pi.

43
00:02:43,110 --> 00:02:46,310
And we saw that the
appropriate weight

44
00:02:46,310 --> 00:02:51,410
for this n minus 1
component vector of pi

45
00:02:51,410 --> 00:02:54,950
has within it a factor of
something like square root of 1

46
00:02:54,950 --> 00:02:56,140
minus pi squared.

47
00:02:56,140 --> 00:03:00,520
There's an overall factor
of 2, but it doesn't matter.

48
00:03:00,520 --> 00:03:05,500
And essentially, this says that
because you have a unit vector,

49
00:03:05,500 --> 00:03:09,316
this pi cannot get too big.

50
00:03:09,316 --> 00:03:11,700
You have to pay
[? a cost ?] here,

51
00:03:11,700 --> 00:03:14,760
certainly not larger than 1.

52
00:03:14,760 --> 00:03:20,960
And then the expression
for the energy costs

53
00:03:20,960 --> 00:03:24,680
can be written in terms of
two parts [? where it ?]

54
00:03:24,680 --> 00:03:31,730
is the gradient
of this vector pi.

55
00:03:31,730 --> 00:03:33,830
But then there's
also the gradient

56
00:03:33,830 --> 00:03:42,140
in the other direction
which has magnitude

57
00:03:42,140 --> 00:03:45,030
square root of 1
minus pi squared.

58
00:03:45,030 --> 00:03:48,830
So we have this
gradient squared.

59
00:03:48,830 --> 00:03:52,400
And so basically,
these are two ways

60
00:03:52,400 --> 00:03:56,115
of writing the same thing, OK?

61
00:03:59,680 --> 00:04:05,340
So we looked at this
and we said that once

62
00:04:05,340 --> 00:04:09,770
we include all of these
terms, what we have here is

63
00:04:09,770 --> 00:04:14,410
a non-linear [? activity ?]
that includes,

64
00:04:14,410 --> 00:04:19,940
for example, interactions
among the various modes.

65
00:04:19,940 --> 00:04:23,915
And one particular
leading order term

66
00:04:23,915 --> 00:04:29,000
is if we expand this square
root of 1 minus pi squared,

67
00:04:29,000 --> 00:04:32,880
if it would be something
like pi square root of pi,

68
00:04:32,880 --> 00:04:37,260
so a particular term in this
expansion as the from pi

69
00:04:37,260 --> 00:04:42,938
[? grad ?] pi multiplied
with pi [? grad ?] pi.

70
00:04:42,938 --> 00:04:46,802
[INAUDIBLE], OK?

71
00:04:49,700 --> 00:04:57,626
So a particular way of dealing
these kinds of theories

72
00:04:57,626 --> 00:05:03,652
is to regard all of these
things as interactions

73
00:05:03,652 --> 00:05:08,500
and perturbations with respect
to a Gaussian weight which

74
00:05:08,500 --> 00:05:12,970
we can compute easily.

75
00:05:12,970 --> 00:05:18,860
And then you can either do that
perturbation straightforwardly

76
00:05:18,860 --> 00:05:22,280
or from the beginning to a
perturbative [? origin, ?]

77
00:05:22,280 --> 00:05:24,080
which is the route
that we chose.

78
00:05:30,440 --> 00:05:37,270
And this amount to
changing the short distance

79
00:05:37,270 --> 00:05:40,220
cut off that we
have here that is

80
00:05:40,220 --> 00:05:46,220
a to be b times a and
averaging over all [? nodes ?]

81
00:05:46,220 --> 00:05:53,330
within that distance short
wavelength between a and ba.

82
00:05:53,330 --> 00:06:01,680
And once we do that, we
arrive at a new interaction.

83
00:06:01,680 --> 00:06:08,214
So the first step is
to do a coarse graining

84
00:06:08,214 --> 00:06:10,504
between the range a and ba.

85
00:06:13,250 --> 00:06:18,750
But then steps two
and three amount

86
00:06:18,750 --> 00:06:27,170
to a rescaling in position space
so that the cut-off comes back

87
00:06:27,170 --> 00:06:38,480
to ba and the corresponding
thing in the spin space

88
00:06:38,480 --> 00:06:42,600
so that we start with a
partition function that

89
00:06:42,600 --> 00:06:44,610
describes unit vectors.

90
00:06:44,610 --> 00:06:47,660
And after this
transformation, we

91
00:06:47,660 --> 00:06:50,200
end up with a new
partition function

92
00:06:50,200 --> 00:06:53,360
that also describes
unit vectors so

93
00:06:53,360 --> 00:06:57,120
that after all of
these three procedures,

94
00:06:57,120 --> 00:07:00,290
we hope that we are
back to exactly the form

95
00:07:00,290 --> 00:07:05,230
that we had at the beginning
with the same cut-off,

96
00:07:05,230 --> 00:07:09,510
with the same unit
vector constraint,

97
00:07:09,510 --> 00:07:15,660
but potentially with a new
interaction parameter K

98
00:07:15,660 --> 00:07:19,240
And calculating
what this new K is

99
00:07:19,240 --> 00:07:24,550
after we've scaled by a factor
of b, the parts that correspond

100
00:07:24,550 --> 00:07:29,010
to 2 and 3 are
immediately obvious.

101
00:07:29,010 --> 00:07:35,960
Because whenever I see x, I
have to replace it wit bx prime.

102
00:07:35,960 --> 00:07:38,250
And so from
integration, I then get

103
00:07:38,250 --> 00:07:41,740
a factor of b to the d
from the two gradients,

104
00:07:41,740 --> 00:07:44,830
I would get a factor of minus 2.

105
00:07:44,830 --> 00:07:52,410
So the step that corresponds
to this is trivial.

106
00:07:52,410 --> 00:07:57,130
The step that corresponds to
replacing S with zeta S prime

107
00:07:57,130 --> 00:07:58,020
is also trivial.

108
00:07:58,020 --> 00:08:00,680
And it will give
you zeta squared.

109
00:08:00,680 --> 00:08:03,680
Or do I have yet to
tell you what is?

110
00:08:03,680 --> 00:08:07,000
We'll do that shortly.

111
00:08:07,000 --> 00:08:11,840
And finally, the first step,
which was the coarse graining,

112
00:08:11,840 --> 00:08:17,710
we found that what it did
was that it replaced K

113
00:08:17,710 --> 00:08:21,055
by a strongly-- sorry.

114
00:08:21,055 --> 00:08:22,540
I didn't expect this.

115
00:08:28,432 --> 00:08:41,200
All right-- the factor K, which
is larger by a certain amount.

116
00:08:41,200 --> 00:08:45,660
And the mathematical
justification

117
00:08:45,660 --> 00:08:53,070
that I gave for this is we
look at this expression,

118
00:08:53,070 --> 00:08:55,410
and we see that in
this expression,

119
00:08:55,410 --> 00:08:59,690
each one of these pi's
can be a long wavelength

120
00:08:59,690 --> 00:09:03,670
fluctuation or a short
wavelength fluctuation.

121
00:09:03,670 --> 00:09:08,960
Among the many possibilities
is when these 2 pi's

122
00:09:08,960 --> 00:09:12,480
that are sitting
out front correspond

123
00:09:12,480 --> 00:09:15,800
to the short wavelength
fluctuations.

124
00:09:15,800 --> 00:09:19,040
These correspond to the long
wavelength fluctuations.

125
00:09:24,450 --> 00:09:29,710
And you can see that
averaging over these two

126
00:09:29,710 --> 00:09:33,390
will generate an
interaction that

127
00:09:33,390 --> 00:09:36,740
looks like gradient
of pi lesser squared.

128
00:09:36,740 --> 00:09:42,380
And that will change the
coefficient over here

129
00:09:42,380 --> 00:09:46,620
by an amount that is clearly
proportional to the average

130
00:09:46,620 --> 00:09:48,540
of pi greater squared.

131
00:09:54,070 --> 00:09:56,440
And that we can see
in Fourier space

132
00:09:56,440 --> 00:10:00,350
is simply 1 over
Kq squared over KK

133
00:10:00,350 --> 00:10:06,740
squared for modes that
have wave number K.

134
00:10:06,740 --> 00:10:09,790
So if I, rather than
write this in real space,

135
00:10:09,790 --> 00:10:11,620
I write it in
Fourier space, this

136
00:10:11,620 --> 00:10:14,030
is what I would get
for the average.

137
00:10:14,030 --> 00:10:18,790
And in real space, I have
to integrate over this K

138
00:10:18,790 --> 00:10:23,050
appropriately [INAUDIBLE] within
the wave numbers lambda over

139
00:10:23,050 --> 00:10:33,320
[? real ?] [? light. ?] And this
is clearly something that is

140
00:10:33,320 --> 00:10:40,960
inversely proportional to K. And
the result of this integration

141
00:10:40,960 --> 00:10:44,570
of 1 over K squared-- we
simply gave a [? 9 ?],

142
00:10:44,570 --> 00:10:49,530
which was i sub d of b.

143
00:10:49,530 --> 00:10:51,415
Because it depends
on the dimension.

144
00:10:51,415 --> 00:10:53,586
It depends on
[? et ?] [? cetera. ?]

145
00:10:53,586 --> 00:10:54,265
AUDIENCE: Sir?

146
00:10:54,265 --> 00:10:54,890
PROFESSOR: Yes?

147
00:10:54,890 --> 00:10:55,698
AUDIENCE: Shouldn't
there be an exponential

148
00:10:55,698 --> 00:10:56,674
inside the integral?

149
00:10:59,040 --> 00:11:00,915
PROFESSOR: Why should
there be an exponential

150
00:11:00,915 --> 00:11:01,740
inside the integral?

151
00:11:01,740 --> 00:11:03,948
AUDIENCE: Oh, I thought we
were Fourier transforming.

152
00:11:03,948 --> 00:11:06,640
PROFESSOR: OK, it is true,
when we Fourier transform

153
00:11:06,640 --> 00:11:10,630
for each pi, we will have
a factor of e to the IK.

154
00:11:10,630 --> 00:11:16,190
If we have 2 pi's, I will get
e to the IK e to the IK prime.

155
00:11:16,190 --> 00:11:18,420
But the averaging
[INAUDIBLE] set

156
00:11:18,420 --> 00:11:20,940
K and K prime can be
opposite each other.

157
00:11:20,940 --> 00:11:22,710
So the exponentials disappear.

158
00:11:22,710 --> 00:11:28,190
So always remember the
integral of any field squared

159
00:11:28,190 --> 00:11:32,300
in real space is the same
thing as the integral

160
00:11:32,300 --> 00:11:34,700
of that field squared
in Fourier space.

161
00:11:34,700 --> 00:11:36,710
This is one of
the first theorems

162
00:11:36,710 --> 00:11:38,030
of Fourier transformation.

163
00:11:41,190 --> 00:11:46,430
OK, so this is a correction
that goes like [INAUDIBLE].

164
00:11:49,890 --> 00:11:57,210
And last time, to give you a
kind of visual demonstration

165
00:11:57,210 --> 00:12:01,960
of what this factor is, I
said that it is similar,

166
00:12:01,960 --> 00:12:06,250
but by no means
identical to something

167
00:12:06,250 --> 00:12:16,940
like this, which is that a mode
by itself has very low energy.

168
00:12:16,940 --> 00:12:21,170
But because we have coupling
among different modes,

169
00:12:21,170 --> 00:12:24,220
here for the Goldstone
modes of the surface,

170
00:12:24,220 --> 00:12:27,740
but here for the Goldstone
modes of the spin,

171
00:12:27,740 --> 00:12:33,980
the presence of a certain amount
of short range fluctuations

172
00:12:33,980 --> 00:12:40,390
will stiffen the modes that you
have for longer wavelengths.

173
00:12:40,390 --> 00:12:43,740
Now, I'm not saying that these
two problems are mathematically

174
00:12:43,740 --> 00:12:45,040
identical.

175
00:12:45,040 --> 00:12:49,390
All I'm showing you
is that the coupling

176
00:12:49,390 --> 00:12:52,390
between the short and
long wavelength modes

177
00:12:52,390 --> 00:12:59,000
can lead to a stiffening of
the modes over long distances

178
00:12:59,000 --> 00:13:03,390
because they have to fight
off the [? rails ?] that

179
00:13:03,390 --> 00:13:06,440
have been established by
shorter wavelength modes.

180
00:13:06,440 --> 00:13:08,410
You have to try to undo them.

181
00:13:08,410 --> 00:13:12,870
And that's an
additional cost, OK?

182
00:13:12,870 --> 00:13:21,570
Now, that stiffening over here
is opposed by a factor of zeta

183
00:13:21,570 --> 00:13:23,180
over here.

184
00:13:23,180 --> 00:13:24,860
Essentially, we
said that we have

185
00:13:24,860 --> 00:13:30,240
to ensure that what we
are seeing after the three

186
00:13:30,240 --> 00:13:34,500
steps of RG is a
description of a [? TOD ?]

187
00:13:34,500 --> 00:13:39,640
that has the same short distance
cut-off and the same length

188
00:13:39,640 --> 00:13:42,860
so that the two partition
functions can map on

189
00:13:42,860 --> 00:13:44,390
to each other.

190
00:13:44,390 --> 00:13:48,820
And again, another
visual demonstration

191
00:13:48,820 --> 00:13:54,556
is that you can decompose
the spin over here

192
00:13:54,556 --> 00:13:59,510
to a superposition of short
and long wavelength modes.

193
00:13:59,510 --> 00:14:03,085
And we are averaging over
these short wavelength modes.

194
00:14:03,085 --> 00:14:07,090
And because of that, we will
see that the effective length

195
00:14:07,090 --> 00:14:12,210
once that averaging has been
performed has been reduced.

196
00:14:12,210 --> 00:14:17,600
It has been reduced because I
will write this as 1 minus pi

197
00:14:17,600 --> 00:14:20,920
squared over 2 to
the lowest order.

198
00:14:20,920 --> 00:14:25,443
And pi squared has n
minus 1 components.

199
00:14:25,443 --> 00:14:28,670
So this is n minus 1 over 2.

200
00:14:28,670 --> 00:14:31,790
And then I have
to integrate over

201
00:14:31,790 --> 00:14:38,720
all of the modes pi
alpha of K in this range.

202
00:14:38,720 --> 00:14:43,330
So I'm performing exactly
the same integral as above.

203
00:14:43,330 --> 00:14:53,410
So the reduction is precisely
the same integral as above, OK?

204
00:14:53,410 --> 00:15:00,612
So the three steps of RG
performed for this model

205
00:15:00,612 --> 00:15:04,940
to lowest order
in this inverse K,

206
00:15:04,940 --> 00:15:09,310
our [? temperature-like ?]
variable is given by this one

207
00:15:09,310 --> 00:15:13,590
[INAUDIBLE] once I substitute
the value of zeta over there.

208
00:15:13,590 --> 00:15:18,070
So you can see that
the answer K prime of b

209
00:15:18,070 --> 00:15:21,590
is going to b to
the b minus 2 zeta

210
00:15:21,590 --> 00:15:24,470
squared-- ah, that's right.

211
00:15:24,470 --> 00:15:26,322
For zeta squared,
essentially the square

212
00:15:26,322 --> 00:15:31,180
of that, 1 minus
n minus 1 over K,

213
00:15:31,180 --> 00:15:33,797
the 2 disappears
once I squared it.

214
00:15:33,797 --> 00:15:39,120
Id of b divided by K-- that
comes from zeta squared.

215
00:15:39,120 --> 00:15:44,990
And from here I can get
the plus Id of be over K.

216
00:15:44,990 --> 00:15:48,960
And the whole thing gets
multiplied by this K.

217
00:15:48,960 --> 00:15:52,955
And there is still terms at
the order of temperature of 1

218
00:15:52,955 --> 00:15:59,020
over K squared, OK?

219
00:15:59,020 --> 00:16:04,960
And finally, we are going
to do the same choice

220
00:16:04,960 --> 00:16:09,090
that we were doing for
our epsilon expansion.

221
00:16:09,090 --> 00:16:12,800
That is, choose a
rescaling factor that

222
00:16:12,800 --> 00:16:15,631
is just slightly larger than 1.

223
00:16:15,631 --> 00:16:16,130
Yes?

224
00:16:16,130 --> 00:16:20,580
AUDIENCE: Sir, you're at n
minus 1 over K times Id over K.

225
00:16:20,580 --> 00:16:24,490
PROFESSOR: Yeah, I
gave this too much.

226
00:16:24,490 --> 00:16:25,424
Thank you.

227
00:16:28,230 --> 00:16:31,100
OK, thank You.

228
00:16:31,100 --> 00:16:38,120
And we will write K prime at Kb
to be K plus delta l dK by dl.

229
00:16:41,570 --> 00:16:46,980
And we note that for
calculating this Id of b, when

230
00:16:46,980 --> 00:16:52,010
b goes to 1 plus delta
l, all I need to do

231
00:16:52,010 --> 00:16:58,240
is to evaluate this integrand
essentially on the shell.

232
00:16:58,240 --> 00:17:04,310
So what I will get is lambda
to the power of d minus 2.

233
00:17:04,310 --> 00:17:10,270
The surface area of a unit
sphere divided by 2 pi

234
00:17:10,270 --> 00:17:12,329
to the d, which is the
combination that we

235
00:17:12,329 --> 00:17:16,700
have been calling K sub d, OK?

236
00:17:19,849 --> 00:17:27,400
So once I do that, I will
get that the dK by dl, OK?

237
00:17:27,400 --> 00:17:29,050
What do I get?

238
00:17:29,050 --> 00:17:36,980
I will get a d minus
2 here times K.

239
00:17:36,980 --> 00:17:39,700
And then I will get
these two factors.

240
00:17:39,700 --> 00:17:41,560
There's n minus 1.

241
00:17:41,560 --> 00:17:42,930
And then there's 1 here.

242
00:17:42,930 --> 00:17:44,810
So that becomes n minus 2.

243
00:17:47,678 --> 00:17:56,170
I have a Idb, which is Kd
lambda to the d minus 2.

244
00:17:56,170 --> 00:17:59,120
And then the 1 over
K and K disappear.

245
00:17:59,120 --> 00:18:02,080
And so that's the
expression that we have.

246
00:18:07,730 --> 00:18:08,380
Yes?

247
00:18:08,380 --> 00:18:13,170
AUDIENCE: Sorry, is the Kd
[? some ?] angle factor again?

248
00:18:13,170 --> 00:18:18,820
PROFESSOR: OK, so you have
to do this integration, which

249
00:18:18,820 --> 00:18:24,140
is written as the surface
area inside an angle,

250
00:18:24,140 --> 00:18:29,590
K to the d minus 1 dK
divided by 2 pi to the d.

251
00:18:29,590 --> 00:18:33,009
This is the combination that
we have always called K sub d.

252
00:18:33,009 --> 00:18:33,550
AUDIENCE: OK.

253
00:18:40,410 --> 00:18:43,070
PROFESSOR: OK?

254
00:18:43,070 --> 00:18:46,343
Now, it actually
makes more sense

255
00:18:46,343 --> 00:18:50,490
since we are making a
low temperature expansion

256
00:18:50,490 --> 00:18:57,734
to define a T that is
simply 1 over K. Its

257
00:18:57,734 --> 00:19:00,120
again, dimensionless.

258
00:19:00,120 --> 00:19:04,330
And then clearly
dT by dl is going

259
00:19:04,330 --> 00:19:12,759
to be minus K squared dK by
dl minus 1 over K squared.

260
00:19:17,450 --> 00:19:24,840
Minus 1 over K squared becomes
minus T squared dK by dl.

261
00:19:24,840 --> 00:19:27,730
So I just have to
multiply the expression

262
00:19:27,730 --> 00:19:32,790
that I have up here
with minus T squared,

263
00:19:32,790 --> 00:19:35,253
recognizing that TK is 1.

264
00:19:35,253 --> 00:19:39,210
So I end up with the
[? recursion ?] convention

265
00:19:39,210 --> 00:19:45,640
for T, which is
minus d minus 2T.

266
00:19:45,640 --> 00:19:53,800
And then it becomes plus n minus
2 Kd lambda to the d minus 2 T

267
00:19:53,800 --> 00:19:54,550
squared.

268
00:19:54,550 --> 00:19:57,390
And presumably, there
are high order terms

269
00:19:57,390 --> 00:20:01,800
that we have not
bothered to calculate.

270
00:20:01,800 --> 00:20:05,980
So this is the
[INAUDIBLE] we focused on.

271
00:20:11,646 --> 00:20:12,146
OK?

272
00:20:15,130 --> 00:20:20,640
So let's see whether this
expression makes sense.

273
00:20:20,640 --> 00:20:27,930
So if I'm looking at dimensions
that are less than 2,

274
00:20:27,930 --> 00:20:32,760
then the linear term
in the expression

275
00:20:32,760 --> 00:20:41,920
is positive, which means that if
I'm looking at the temperature

276
00:20:41,920 --> 00:20:48,560
axis and this is 0, and
I start with a value that

277
00:20:48,560 --> 00:20:55,770
is slightly positive,
because of this term,

278
00:20:55,770 --> 00:21:00,700
it will be pushed larger
and larger values.

279
00:21:00,700 --> 00:21:03,890
So you think that
you have a system

280
00:21:03,890 --> 00:21:05,965
at very low temperatures.

281
00:21:05,965 --> 00:21:08,600
You look at it at larger
and larger scales,

282
00:21:08,600 --> 00:21:10,930
and you find that it
becomes effectively

283
00:21:10,930 --> 00:21:13,380
something that has
higher temperature

284
00:21:13,380 --> 00:21:16,260
and becomes more
and more disordered.

285
00:21:16,260 --> 00:21:23,190
So basically, this is a
manifestation of something

286
00:21:23,190 --> 00:21:31,010
that we had said before,
Mermin-Wagner theorem, which

287
00:21:31,010 --> 00:21:42,440
is no [? long ?] range
order in d less than 2, OK?

288
00:21:42,440 --> 00:21:51,470
Now, if I go to the other
limit, d greater than 2,

289
00:21:51,470 --> 00:21:57,660
then something
interesting happens,

290
00:21:57,660 --> 00:22:04,700
in that the linear
term is negative.

291
00:22:04,700 --> 00:22:09,700
So if I start with a
sufficiently small temperature

292
00:22:09,700 --> 00:22:13,360
or a large enough coupling,
it will get stronger

293
00:22:13,360 --> 00:22:17,450
as we go towards
an ordered phase,

294
00:22:17,450 --> 00:22:21,540
whereas the quadratic
term for n greater than 2

295
00:22:21,540 --> 00:22:25,545
has the opposite sign--
this is n greater than 2--

296
00:22:25,545 --> 00:22:32,010
and pushes me towards disorder,
which means that there should

297
00:22:32,010 --> 00:22:39,570
be a fixed point that
separates the two behaviors.

298
00:22:39,570 --> 00:22:45,090
Any temperature lower than this
will give me an ordered phase.

299
00:22:45,090 --> 00:22:47,800
Any temperature higher
than this will give me

300
00:22:47,800 --> 00:22:50,250
a disordered phase.

301
00:22:50,250 --> 00:22:55,150
And suddenly, we see that
we have potentially a way

302
00:22:55,150 --> 00:23:00,430
of figuring out what the phase
transition is because this T

303
00:23:00,430 --> 00:23:06,630
star is a location that we
can perturbatively access.

304
00:23:06,630 --> 00:23:10,280
Because we set this
to 0, and we find

305
00:23:10,280 --> 00:23:16,250
that T star is
equal to d minus 2

306
00:23:16,250 --> 00:23:26,430
divided by n minus 2 Kd
lambda to the d minus 2.

307
00:23:26,430 --> 00:23:33,830
So now in order to have
a theory that makes sense

308
00:23:33,830 --> 00:23:38,130
in the sense of the perturbation
that we have carried out,

309
00:23:38,130 --> 00:23:41,570
we have to make sure
that this is small.

310
00:23:41,570 --> 00:23:47,706
So we can do that by assuming
that this quantity d minus 2

311
00:23:47,706 --> 00:23:52,853
is a small quantity in making
an expansion in d minus 2, OK?

312
00:23:56,280 --> 00:24:00,150
So in particular,
T star itself we

313
00:24:00,150 --> 00:24:05,590
expect to be related to
transition temperature, not

314
00:24:05,590 --> 00:24:07,490
something that is universal.

315
00:24:07,490 --> 00:24:10,700
But exponents are universal.

316
00:24:10,700 --> 00:24:14,950
So what we do is we
look at d by dl of delta

317
00:24:14,950 --> 00:24:18,390
T. Delta T is,
let's say, T minus T

318
00:24:18,390 --> 00:24:21,060
star in one direction
or the other.

319
00:24:21,060 --> 00:24:22,860
[INAUDIBLE]

320
00:24:22,860 --> 00:24:26,490
And for that, what
I need to do it

321
00:24:26,490 --> 00:24:29,240
to linearize this expression.

322
00:24:29,240 --> 00:24:33,046
So I will get a minus
epsilon from her.

323
00:24:33,046 --> 00:24:39,334
And from here, I will
get 2 n minus 2 Kd lambda

324
00:24:39,334 --> 00:24:49,570
to the d minus 2 T
star times delta T.

325
00:24:49,570 --> 00:24:53,315
I just took the derivative,
evaluated the T star.

326
00:24:53,315 --> 00:24:58,530
And we can see that this
combination is precisely

327
00:24:58,530 --> 00:25:02,510
the combination that I
have to solve for T star.

328
00:25:02,510 --> 00:25:05,780
So this really becomes
another factor of epsilon.

329
00:25:05,780 --> 00:25:09,370
I have minus epsilon
plus [? 2 ?] epsilon.

330
00:25:09,370 --> 00:25:13,960
So this is epsilon delta
T. So that tells me

331
00:25:13,960 --> 00:25:25,520
that my thermal
eigenvalue is epsilon,

332
00:25:25,520 --> 00:25:29,300
a disorder clearly
independent of n.

333
00:25:32,480 --> 00:25:36,170
Now, we've seen that in
order to fully characterize

334
00:25:36,170 --> 00:25:42,440
the exponent, including things
like magnetization, et cetera,

335
00:25:42,440 --> 00:25:47,690
it makes sense to also put
a magnetic field direction

336
00:25:47,690 --> 00:25:52,900
and figure out how rapidly you
go along the magnetic field

337
00:25:52,900 --> 00:25:55,060
direction.

338
00:25:55,060 --> 00:25:59,530
So for that, one
way of doing this

339
00:25:59,530 --> 00:26:06,698
is to go and add to this term,
which is h integral S of x.

340
00:26:10,040 --> 00:26:12,950
And you can see very easily
that under these steps

341
00:26:12,950 --> 00:26:15,580
of the transformation,
essentially the only thing that

342
00:26:15,580 --> 00:26:20,240
happens is that I will
get h prime at scale

343
00:26:20,240 --> 00:26:23,380
d is h from the integration.

344
00:26:23,380 --> 00:26:26,490
I will get a factor
of b to the d.

345
00:26:26,490 --> 00:26:29,050
From the replacement
of s with s prime,

346
00:26:29,050 --> 00:26:30,560
I will get a factor of zeta.

347
00:26:34,230 --> 00:26:39,290
So this combination
is simply my yh.

348
00:26:39,290 --> 00:26:42,666
And just bringing a
little bit of manipulation

349
00:26:42,666 --> 00:26:47,730
will tell you that yh
is d minus the part that

350
00:26:47,730 --> 00:26:53,280
comes from zeta,
which is n minus 1

351
00:26:53,280 --> 00:27:00,980
over 2 Id of b, which is
lambda to the d minus 2 Kd.

352
00:27:00,980 --> 00:27:04,280
And then we have T star.

353
00:27:07,380 --> 00:27:11,200
And again, you
substitute for lambda

354
00:27:11,200 --> 00:27:16,520
to the d minus 2 Kd T star
on what we have over here.

355
00:27:16,520 --> 00:27:22,750
And you get this to
be d minus n minus 1

356
00:27:22,750 --> 00:27:25,360
over 2n minus 2 epsilon.

357
00:27:28,040 --> 00:27:31,290
And again, to be consistent
to order of epsilon,

358
00:27:31,290 --> 00:27:36,650
this d you will have to
replace with 2 plus epsilon.

359
00:27:36,650 --> 00:27:39,480
And a little bit of
manipulation will give you

360
00:27:39,480 --> 00:27:50,240
yh, which is 2 minus n minus 3
divided by 2n minus 2 epsilon.

361
00:27:59,930 --> 00:28:07,270
I did this calculation of the
two exponents rather rapidly.

362
00:28:07,270 --> 00:28:12,036
The reason for that is they
are not particularly useful.

363
00:28:12,036 --> 00:28:15,730
That is, whereas
we saw that coming

364
00:28:15,730 --> 00:28:20,080
from four dimensions the
epsilon expansion was very

365
00:28:20,080 --> 00:28:23,600
useful to give us
corrections to the [? mean ?]

366
00:28:23,600 --> 00:28:29,390
field values of 1/2, for
example, for mu to order of 10%

367
00:28:29,390 --> 00:28:32,780
or so already by setting
epsilon equals to 1.

368
00:28:32,780 --> 00:28:36,670
If I, for example, put
here epsilon equal to 1

369
00:28:36,670 --> 00:28:39,610
to [? access ?] 3
dimensions, I will

370
00:28:39,610 --> 00:28:42,310
conclude that mu, which
is the inverse of yT

371
00:28:42,310 --> 00:28:45,990
is 1 in 3 dimensions
independent of n.

372
00:28:45,990 --> 00:28:50,570
And let's say for super
fluid it is closer to 2/3.

373
00:28:50,570 --> 00:28:56,000
And so essentially, this
expansion in some sense

374
00:28:56,000 --> 00:29:01,030
is much further away from 3
dimensions than the 4 minus

375
00:29:01,030 --> 00:29:03,610
epsilon coming
from 4 dimensions,

376
00:29:03,610 --> 00:29:06,030
although numerically,
we would have

377
00:29:06,030 --> 00:29:09,730
said epsilon equaled
1 to both of them.

378
00:29:09,730 --> 00:29:20,485
So nobody really has taken
much advantage of this 2 plus

379
00:29:20,485 --> 00:29:23,810
epsilon expansion.

380
00:29:23,810 --> 00:29:26,440
So why is it useful?

381
00:29:26,440 --> 00:29:30,200
The reason that this is
useful is the third case

382
00:29:30,200 --> 00:29:32,276
that I have not
explained, yet, which

383
00:29:32,276 --> 00:29:38,150
is, what happens if you sit
exactly in 2 dimensions, OK?

384
00:29:38,150 --> 00:29:41,620
So if we sit exactly
in 2 dimensions,

385
00:29:41,620 --> 00:29:43,870
this first term disappears.

386
00:29:43,870 --> 00:29:46,940
And you can see
that the behavioral

387
00:29:46,940 --> 00:29:50,180
is determined by
the second order

388
00:29:50,180 --> 00:29:54,360
and [? then ?] depends
on the value of n.

389
00:29:54,360 --> 00:29:59,320
So if I look at n, let's
say, that is less than 2,

390
00:29:59,320 --> 00:30:04,210
then what I will see is
that along the temperature

391
00:30:04,210 --> 00:30:09,180
axis, the quadratic
term-- the linear term

392
00:30:09,180 --> 00:30:12,580
is absent-- the quadratic
term, let's say, for n

393
00:30:12,580 --> 00:30:16,080
equals 1 is negative.

394
00:30:16,080 --> 00:30:19,540
And you're being
pushed quadratically

395
00:30:19,540 --> 00:30:21,550
very slowly towards 0.

396
00:30:21,550 --> 00:30:25,165
The one example that
we know is indeed n

397
00:30:25,165 --> 00:30:27,130
equals 1, the Ising model.

398
00:30:27,130 --> 00:30:31,560
And we know that the Ising
model in 2 dimensions

399
00:30:31,560 --> 00:30:32,850
has an ordered phase.

400
00:30:32,850 --> 00:30:35,305
It shouldn't really even
be described by this

401
00:30:35,305 --> 00:30:38,990
because there are
no Goldstone modes.

402
00:30:38,990 --> 00:30:43,941
But n greater than 2, like
n equals 3-- the Heisenberg

403
00:30:43,941 --> 00:30:48,270
model, is interesting.

404
00:30:48,270 --> 00:30:54,435
And what we see is that here the
second order term is positive.

405
00:30:54,435 --> 00:30:59,690
And it is pushing you
towards high temperatures.

406
00:30:59,690 --> 00:31:02,928
So you can see a
disordered behavior.

407
00:31:02,928 --> 00:31:06,230
And what this calculation
tells you that is useful

408
00:31:06,230 --> 00:31:09,740
that you wouldn't
have known otherwise

409
00:31:09,740 --> 00:31:12,800
is, what is the
correlation [? length? ?]

410
00:31:12,800 --> 00:31:19,090
Because the recursion relation
is now dT by dl is, let's say,

411
00:31:19,090 --> 00:31:20,270
n minus 2.

412
00:31:23,150 --> 00:31:24,480
And my d equals to 2.

413
00:31:24,480 --> 00:31:27,280
The lambda to the d
minus 2, I can ignore.

414
00:31:27,280 --> 00:31:34,160
Kd is 2 pi from the [? solid ?]
angle divided by 2 pi squared.

415
00:31:34,160 --> 00:31:38,980
So that's 1 over 2 pi
times T squared, OK?

416
00:31:38,980 --> 00:31:43,010
So that's the recursion relation
that we are dealing with.

417
00:31:43,010 --> 00:31:46,576
I can divide by
1 over T squared.

418
00:31:46,576 --> 00:31:53,640
And then this becomes d
by dl of minus 1 over T

419
00:31:53,640 --> 00:31:57,670
equals n minus 2
divided by 2 pi.

420
00:32:00,600 --> 00:32:09,330
I can integrate this from, say,
some initial value minus 1 over

421
00:32:09,330 --> 00:32:16,254
some initial temperature
to some temperature where

422
00:32:16,254 --> 00:32:18,775
I'm at [? length ?]
[? scale ?] l.

423
00:32:18,775 --> 00:32:25,582
What I would have on the right
hand side would be n minus 2

424
00:32:25,582 --> 00:32:29,940
over [? 2yl ?], OK?

425
00:32:29,940 --> 00:32:35,800
So I start very, very close
to the origin T equals 0.

426
00:32:35,800 --> 00:32:40,700
So I have a very strong
coupling at the beginning.

427
00:32:40,700 --> 00:32:42,310
So this factor is huge.

428
00:32:42,310 --> 00:32:44,440
T is [? more than ?]
1 over T is huge.

429
00:32:44,440 --> 00:32:47,930
I have a huge coupling.

430
00:32:47,930 --> 00:32:54,390
And then I rescale to a point
where the coupling has become

431
00:32:54,390 --> 00:33:00,470
weak, let's say some number
of order of 1, order of 1

432
00:33:00,470 --> 00:33:01,970
or order of 0.

433
00:33:01,970 --> 00:33:04,840
In any case, it is
overwhelmingly smaller

434
00:33:04,840 --> 00:33:07,550
than this.

435
00:33:07,550 --> 00:33:09,470
How far did I have to go?

436
00:33:09,470 --> 00:33:12,294
I had to rescale
by a factor of l

437
00:33:12,294 --> 00:33:13,710
that is related
to the temperature

438
00:33:13,710 --> 00:33:18,150
that I started with
by this factor,

439
00:33:18,150 --> 00:33:20,443
except that I forgot
the minus that I

440
00:33:20,443 --> 00:33:23,470
had in front of the whole thing.

441
00:33:23,470 --> 00:33:31,430
So the resulting l will
be large and positive.

442
00:33:31,430 --> 00:33:37,100
And the correlation length--
the length scale at which

443
00:33:37,100 --> 00:33:39,020
we arrived at the
coupling, which

444
00:33:39,020 --> 00:33:45,070
is of the order of 1 or 0,
is whatever my initial length

445
00:33:45,070 --> 00:33:51,090
scale was times this factor
that I have rescaled by,

446
00:33:51,090 --> 00:33:54,290
b, which is e to the l.

447
00:33:54,290 --> 00:34:02,268
And so this is a exponential
of n minus 1 over 2 pi

448
00:34:02,268 --> 00:34:09,270
times 1 over T. The
statement is that if you're

449
00:34:09,270 --> 00:34:13,914
having 2 dimensions, a
system of, let's say,

450
00:34:13,914 --> 00:34:17,679
3 component spins-- and
that is something that

451
00:34:17,679 --> 00:34:21,690
has a lot of experimental
realizations--

452
00:34:21,690 --> 00:34:25,790
you find that as you go
towards low temperature,

453
00:34:25,790 --> 00:34:28,520
the size of domains
that are ordered

454
00:34:28,520 --> 00:34:34,580
diverges according to
this nice universal form.

455
00:34:34,580 --> 00:34:41,489
And let's say around
1995 or so, when

456
00:34:41,489 --> 00:34:44,530
people had these high
temperature superconductors

457
00:34:44,530 --> 00:34:49,280
which are effectively 2
dimensional layers of magnets--

458
00:34:49,280 --> 00:34:50,949
they're actually
antiferromagnets,

459
00:34:50,949 --> 00:34:53,730
but they are still described
by this [INAUDIBLE] with n

460
00:34:53,730 --> 00:34:57,450
equals 3-- there were
lots of x-ray studies

461
00:34:57,450 --> 00:35:01,930
of what happens to the ordering
of these antiferromagnetic

462
00:35:01,930 --> 00:35:04,730
copper oxide layers as you
go to low temperatures.

463
00:35:04,730 --> 00:35:10,566
And this form was very
much used and confirmed.

464
00:35:10,566 --> 00:35:16,610
OK, so that's really one thing
that one can get from this

465
00:35:16,610 --> 00:35:19,344
analysis that has been
explicitly [? confirmed ?]

466
00:35:19,344 --> 00:35:20,260
[? for ?] experiments.

467
00:35:23,290 --> 00:35:25,526
And finally, there's
one case in this

468
00:35:25,526 --> 00:35:32,060
that I have not mentioned
so far, which is n equals 2.

469
00:35:32,060 --> 00:35:37,780
And when I am at n
equals 2, what I have is

470
00:35:37,780 --> 00:35:42,350
that the first and second
order terms in this series

471
00:35:42,350 --> 00:35:46,400
are both vanishing.

472
00:35:46,400 --> 00:35:53,710
And I really at this stage don't
quite know what is happening.

473
00:35:53,710 --> 00:35:55,800
But we can think
about it a little bit.

474
00:35:55,800 --> 00:36:00,160
And you can see that if you are
n equals 2, then essentially,

475
00:36:00,160 --> 00:36:03,830
you have a 1 component angle.

476
00:36:03,830 --> 00:36:07,510
And if I write the theory
in terms of the angle theta,

477
00:36:07,510 --> 00:36:11,000
let's say, between
neighboring spins,

478
00:36:11,000 --> 00:36:15,960
then the expansions would simply
be gradient of theta squared.

479
00:36:15,960 --> 00:36:21,560
And there isn't any other
mode to couple with.

480
00:36:21,560 --> 00:36:24,430
You may worry a little bit
about gradient of theta

481
00:36:24,430 --> 00:36:26,290
to the 4th and such things.

482
00:36:26,290 --> 00:36:28,640
But a little bit of
thinking will convince you

483
00:36:28,640 --> 00:36:31,150
that all of those
terms are irrelevant.

484
00:36:31,150 --> 00:36:35,280
So as far as we can
show, there is reason

485
00:36:35,280 --> 00:36:38,820
that essentially
this series for n

486
00:36:38,820 --> 00:36:44,460
equals 2 is 0 at
all orders, which

487
00:36:44,460 --> 00:36:50,902
means that as far as this
analysis is concerned,

488
00:36:50,902 --> 00:36:55,550
there is a kind of a
line of fixed points.

489
00:36:55,550 --> 00:36:58,610
You start with any
temperature, and you

490
00:36:58,610 --> 00:37:04,100
will stay at that
temperature, OK?

491
00:37:04,100 --> 00:37:08,490
Still you would say that even if
you have this gradient of theta

492
00:37:08,490 --> 00:37:12,260
squared type of theory,
the fluctuations that you

493
00:37:12,260 --> 00:37:18,630
have are solutions
of 1 over q squared.

494
00:37:18,630 --> 00:37:22,690
And the integral of 1 over
2 squared in 2 dimensions

495
00:37:22,690 --> 00:37:25,480
is logarithmically divergent.

496
00:37:25,480 --> 00:37:30,760
So the more correct statement
of the Mermin-Wagner's theorem

497
00:37:30,760 --> 00:37:32,830
is that there should
be no long range

498
00:37:32,830 --> 00:37:37,310
order in d less
than or equal to 2.

499
00:37:37,310 --> 00:37:40,750
Because for d equals
2 also, you have

500
00:37:40,750 --> 00:37:44,790
these logarithmic
divergence of fluctuations.

501
00:37:44,790 --> 00:37:47,930
So you may have thought that you
are pointing along, say, the y

502
00:37:47,930 --> 00:37:49,140
direction.

503
00:37:49,140 --> 00:37:51,380
But you average more
and more, and you

504
00:37:51,380 --> 00:37:54,730
see that the extent of
the fluctuations in angle

505
00:37:54,730 --> 00:37:56,560
are growing logarithmically.

506
00:37:56,560 --> 00:38:00,470
You say that once that logarithm
becomes of the order of pi,

507
00:38:00,470 --> 00:38:02,620
I have no idea
where my angle is.

508
00:38:02,620 --> 00:38:05,200
There should be no
true long range order.

509
00:38:05,200 --> 00:38:10,110
And I'm not going to try
to interpret this too much.

510
00:38:10,110 --> 00:38:12,920
I just say that
Mermin-Wagner's theorem

511
00:38:12,920 --> 00:38:16,260
says that there should
be no true long range

512
00:38:16,260 --> 00:38:20,890
order in systems that have
continuous symmetry in 2

513
00:38:20,890 --> 00:38:24,850
dimensions and below, OK?

514
00:38:30,530 --> 00:38:36,810
And that statement is
correct, except that

515
00:38:36,810 --> 00:38:47,180
around that same time, Stanley
and Kaplan did low temperature

516
00:38:47,180 --> 00:38:55,780
series analysis-- actually,
no, I'm incorrect--

517
00:38:55,780 --> 00:39:06,370
did high temperatures
series of these spin models

518
00:39:06,370 --> 00:39:08,740
in 2 dimensions.

519
00:39:08,740 --> 00:39:12,625
And what they found was,
OK, let's [? re-plot ?]

520
00:39:12,625 --> 00:39:16,840
susceptibility as a
function of temperature.

521
00:39:16,840 --> 00:39:19,950
We calculate our best
estimate of susceptibility

522
00:39:19,950 --> 00:39:22,300
from the high
temperature series.

523
00:39:22,300 --> 00:39:25,440
And what they do is, let's
say, they look at the system

524
00:39:25,440 --> 00:39:29,170
that corresponds to n equals 3.

525
00:39:29,170 --> 00:39:35,420
And they see that the
susceptibility diverges only

526
00:39:35,420 --> 00:39:39,210
when you get in the
vicinity of 0 temperature,

527
00:39:39,210 --> 00:39:43,390
which is consistent with all
of these statements that first

528
00:39:43,390 --> 00:39:46,240
of all, this is a [? direct ?]
correlation [? when it ?] only

529
00:39:46,240 --> 00:39:47,760
diverges at 0 temperature.

530
00:39:47,760 --> 00:39:51,790
And divergence of susceptibility
has to be coupled through that.

531
00:39:51,790 --> 00:39:55,090
And therefore, really,
the only exciting thing

532
00:39:55,090 --> 00:39:58,315
is right at 0 temperature,
there is no region where

533
00:39:58,315 --> 00:40:03,020
this is long range
order, except that when

534
00:40:03,020 --> 00:40:09,050
they did the analysis
for n equals 2,

535
00:40:09,050 --> 00:40:14,590
they kept getting
signature that there

536
00:40:14,590 --> 00:40:18,390
is a phase transition at
a finite temperature in d

537
00:40:18,390 --> 00:40:23,750
equals 2 for this xy
model that described

538
00:40:23,750 --> 00:40:26,680
by just an [INAUDIBLE], OK?

539
00:40:26,680 --> 00:40:41,200
So there is lots of numerical
evidence of phase transition

540
00:40:41,200 --> 00:40:47,225
for n equals 2 in d
equals [? 2, ?] OK?

541
00:40:51,040 --> 00:40:56,350
So this is another
one of those puzzles

542
00:40:56,350 --> 00:41:02,120
which [INAUDIBLE]
if we interpret

543
00:41:02,120 --> 00:41:08,350
the existence of a diverging
susceptibility in the way

544
00:41:08,350 --> 00:41:10,770
that we are used to,
let's say the Ising model

545
00:41:10,770 --> 00:41:13,130
and all the models
that we have discussed

546
00:41:13,130 --> 00:41:18,050
so far, in all cases that
we have seen, essentially,

547
00:41:18,050 --> 00:41:25,170
the divergence of
the susceptibility

548
00:41:25,170 --> 00:41:28,970
was an indicator of the onset
of true long range order

549
00:41:28,970 --> 00:41:34,050
so that on the other side, you
had something like a magnet.

550
00:41:34,050 --> 00:41:38,570
But that is rigorously
ruled out by Mermin-Wagner.

551
00:41:38,570 --> 00:41:42,820
So the question is, can
we have a phase transition

552
00:41:42,820 --> 00:41:47,200
in the absence of
symmetry breaking?

553
00:41:47,200 --> 00:41:49,080
All right?

554
00:41:49,080 --> 00:41:55,980
And we already saw one example
of that a couple of lectures

555
00:41:55,980 --> 00:41:59,770
back when we were doing the
dual of the 3-dimensional Ising

556
00:41:59,770 --> 00:42:01,330
model.

557
00:42:01,330 --> 00:42:05,550
We saw that the 3-dimensional
Ising model, its dual

558
00:42:05,550 --> 00:42:10,110
had a phase transition but
was rigorously prevented

559
00:42:10,110 --> 00:42:13,090
from having true
long range order.

560
00:42:13,090 --> 00:42:17,090
So there, how did we distinguish
the different phases?

561
00:42:17,090 --> 00:42:20,580
We found some appropriate
correlation function.

562
00:42:20,580 --> 00:42:23,100
And we showed that that
correlation function

563
00:42:23,100 --> 00:42:26,730
had different behaviors at
high and low temperature.

564
00:42:26,730 --> 00:42:30,650
And these two different
behaviors could not be matched.

565
00:42:30,650 --> 00:42:34,350
And so the phase
transition was an indicator

566
00:42:34,350 --> 00:42:37,850
of the switch-over in the
behavior of the correlation

567
00:42:37,850 --> 00:42:40,190
functions.

568
00:42:40,190 --> 00:42:45,270
So here, let's examine
the correlation functions

569
00:42:45,270 --> 00:42:46,400
of our model.

570
00:42:52,320 --> 00:42:54,460
And the simplest
correlation that we

571
00:42:54,460 --> 00:42:59,830
can think of for a system that
is described by unit spins

572
00:42:59,830 --> 00:43:04,060
is to look at the spin at
some location and the spin

573
00:43:04,060 --> 00:43:07,230
at some far away
location and ask

574
00:43:07,230 --> 00:43:10,730
how correlated they
are to each other?

575
00:43:10,730 --> 00:43:15,080
And so basically, there is
some kind of, let's say,

576
00:43:15,080 --> 00:43:16,170
underlying lattice.

577
00:43:20,510 --> 00:43:27,060
And we pick 2 points
at 0 and at r.

578
00:43:29,980 --> 00:43:34,352
And we ask, what
is the dot product

579
00:43:34,352 --> 00:43:37,225
of the spins that we have
at these 2 locations?

580
00:43:40,760 --> 00:43:44,033
And clearly, this is invariant
under the [? global ?]

581
00:43:44,033 --> 00:43:45,870
rotation.

582
00:43:45,870 --> 00:43:49,040
What I can do is I can
pick some kind of axis

583
00:43:49,040 --> 00:43:53,040
and define angles with
respect to some axis.

584
00:43:53,040 --> 00:43:55,590
Let's say with respect
to the x direction,

585
00:43:55,590 --> 00:43:58,200
I define an angle theta.

586
00:43:58,200 --> 00:44:02,120
And then clearly, this
is the expectation value

587
00:44:02,120 --> 00:44:05,960
of cosine of theta
0 minus theta r.

588
00:44:10,720 --> 00:44:14,800
Now, this quantity
I can asymptotically

589
00:44:14,800 --> 00:44:16,950
calculate both at
high temperatures

590
00:44:16,950 --> 00:44:20,190
and low temperatures
and compare them.

591
00:44:20,190 --> 00:44:22,116
So let's do a high T expansion.

592
00:44:25,380 --> 00:44:28,820
For the high T expansion,
I sort of go back

593
00:44:28,820 --> 00:44:37,470
to the discrete model and
say that what I have here

594
00:44:37,470 --> 00:44:42,340
is a system that
is characterized

595
00:44:42,340 --> 00:44:52,020
by a bunch of angles that I
have to integrate in theta i.

596
00:44:52,020 --> 00:44:58,690
I have the cosine of
theta 0 minus theta r.

597
00:44:58,690 --> 00:45:03,910
And I have a weight that wants
to make near neighbours to be

598
00:45:03,910 --> 00:45:05,400
parallel.

599
00:45:05,400 --> 00:45:09,140
And so I will write it as
product over nearest neighbors,

600
00:45:09,140 --> 00:45:15,826
p to the K cosine of
theta i minus theta j.

601
00:45:15,826 --> 00:45:19,750
OK, so the dot
product of 2 spins

602
00:45:19,750 --> 00:45:23,160
I have written as the
cosine between [? nearest ?]

603
00:45:23,160 --> 00:45:24,410
neighbors.

604
00:45:24,410 --> 00:45:33,344
And of course, I have to
then divide by [INAUDIBLE].

605
00:45:41,790 --> 00:45:47,820
Now, if I'm doing the high
temperature expansion,

606
00:45:47,820 --> 00:45:50,450
that means that this
coupling constant K

607
00:45:50,450 --> 00:45:53,050
scaled by temperature is known.

608
00:45:53,050 --> 00:45:59,320
And I can expand this as 1 plus
K cosine of theta i minus theta

609
00:45:59,320 --> 00:46:08,160
j [? plus ?] [? higher ?] orders
in powers of K of course, OK?

610
00:46:08,160 --> 00:46:12,110
Now, this looks to
have the same structure

611
00:46:12,110 --> 00:46:15,470
as we had for the Ising model.

612
00:46:15,470 --> 00:46:18,920
In the Ising model, I had
something like sigma i sigma j.

613
00:46:18,920 --> 00:46:22,120
And if I had a sigma
by itself and I

614
00:46:22,120 --> 00:46:26,130
summed over the possible
values, I would get 0.

615
00:46:26,130 --> 00:46:31,630
Here, I have something
like a cosine of an angle.

616
00:46:31,630 --> 00:46:38,570
And if I integrate, let's say, d
theta 0 cosine of theta 0 minus

617
00:46:38,570 --> 00:46:44,570
something, just because
theta 0 can be both positive

618
00:46:44,570 --> 00:46:46,520
as [? it just ?] goes
over the entire angle,

619
00:46:46,520 --> 00:46:47,670
this will give me 0.

620
00:46:50,400 --> 00:46:56,480
So this cosine I
better get rid of.

621
00:46:56,480 --> 00:47:01,380
And the way that I
can do that is let's

622
00:47:01,380 --> 00:47:05,606
say I multiply cosine
of theta 0 minus theta r

623
00:47:05,606 --> 00:47:07,530
with one of the
terms that I would

624
00:47:07,530 --> 00:47:11,560
get in the expansion, such
as, let's say, a factor of K

625
00:47:11,560 --> 00:47:15,154
cosine of theta 0 minus theta 1.

626
00:47:15,154 --> 00:47:17,830
So if I call the
next one theta 1,

627
00:47:17,830 --> 00:47:20,510
I will have a term
in the expansion that

628
00:47:20,510 --> 00:47:28,710
is cosine of theta
0 minus theta 1, OK?

629
00:47:28,710 --> 00:47:34,140
Then this will be non-zero
because I can certainly

630
00:47:34,140 --> 00:47:35,320
change the origin.

631
00:47:35,320 --> 00:47:46,610
I can write this as integral
d theta 0 minus theta 1--

632
00:47:46,610 --> 00:47:49,270
I can call phi.

633
00:47:49,270 --> 00:47:53,060
This would be cosine
of phi from here.

634
00:47:53,060 --> 00:48:00,200
This becomes cosine of theta
0 minus theta 1 minus phi.

635
00:48:02,830 --> 00:48:09,520
And this I can expand as
cosine of a 0 minus theta 1

636
00:48:09,520 --> 00:48:17,350
cosine of phi minus sine
of theta 0 minus theta 1

637
00:48:17,350 --> 00:48:18,350
sine of phi.

638
00:48:22,900 --> 00:48:29,600
Then cosine integrated
against sine will give me 0.

639
00:48:29,600 --> 00:48:33,830
Cosine integrated against
cosine will give me 1/2.

640
00:48:33,830 --> 00:48:45,380
So this becomes 1/2 cosine of
theta 0 minus theta 1 theta r.

641
00:48:45,380 --> 00:49:03,020
What did I-- For theta 0, I
am writing phi plus theta 1.

642
00:49:03,020 --> 00:49:09,570
So this becomes phi plus
theta 1 minus theta r.

643
00:49:09,570 --> 00:49:16,640
This becomes cosine of
theta 1 minus theta r.

644
00:49:16,640 --> 00:49:20,200
This becomes theta
1 minus theta r.

645
00:49:20,200 --> 00:49:24,340
This becomes theta
1 minus theta r.

646
00:49:24,340 --> 00:49:33,380
OK, so essentially,
we had a term

647
00:49:33,380 --> 00:49:42,700
that was like a cosine of theta
0 minus theta r from here.

648
00:49:42,700 --> 00:49:46,540
Once we integrate
over this bond,

649
00:49:46,540 --> 00:49:49,120
then I get a factor
of 1/2, and it

650
00:49:49,120 --> 00:49:53,686
becomes like a connection
between these two.

651
00:49:53,686 --> 00:49:57,480
And you can see that
I can keep doing that

652
00:49:57,480 --> 00:50:04,260
and find the path that
connects from 0 to r.

653
00:50:04,260 --> 00:50:07,980
For each one of the
bonds along this path,

654
00:50:07,980 --> 00:50:12,000
I pick one of these factors.

655
00:50:12,000 --> 00:50:16,390
And this allows me to
get a finite value.

656
00:50:16,390 --> 00:50:21,570
And what I find once I do this
is that through the lowest

657
00:50:21,570 --> 00:50:27,660
order, I have to count the
shortest number of paths

658
00:50:27,660 --> 00:50:32,630
that I have between
the two, K. I

659
00:50:32,630 --> 00:50:35,470
will get a factor of K.
And then from the averaging

660
00:50:35,470 --> 00:50:38,290
over the angles, I will get 1/2.

661
00:50:38,290 --> 00:50:42,150
So it would b K
over 2 [INAUDIBLE].

662
00:50:42,150 --> 00:50:50,050
By this we indicate the
shortest path between the 2, OK?

663
00:50:50,050 --> 00:50:54,700
So the point is that
K is a small number.

664
00:50:54,700 --> 00:50:57,520
If I go further
and further away,

665
00:50:57,520 --> 00:51:01,265
this is going to be
exponentially small

666
00:51:01,265 --> 00:51:04,410
in the distance
between the 2 spaces,

667
00:51:04,410 --> 00:51:08,660
where [? c ?] can be
expressed in something that

668
00:51:08,660 --> 00:51:12,100
has to do with K.
So this is actually

669
00:51:12,100 --> 00:51:13,950
quite a general statement.

670
00:51:13,950 --> 00:51:16,110
We've already seen it
for the Ising model.

671
00:51:16,110 --> 00:51:19,510
We've now seen it
for the xy model.

672
00:51:19,510 --> 00:51:24,090
Quite generally, for systems
at high temperatures,

673
00:51:24,090 --> 00:51:27,466
once can show that correlations
decay exponentially

674
00:51:27,466 --> 00:51:32,570
in separation because
the information

675
00:51:32,570 --> 00:51:38,665
about the state of one variable
has to travel all the way

676
00:51:38,665 --> 00:51:41,620
to influence the other one.

677
00:51:41,620 --> 00:51:45,800
And the fidelity by which the
information is transmitted

678
00:51:45,800 --> 00:51:50,320
is very small at
high temperatures.

679
00:51:50,320 --> 00:51:51,310
So OK?

680
00:51:51,310 --> 00:51:54,130
So this is something that
you should have known.

681
00:51:54,130 --> 00:51:57,520
We are getting the answer.

682
00:51:57,520 --> 00:52:00,320
But now what happens if I go
and look at low temperatures?

683
00:52:03,020 --> 00:52:07,470
So for low temperatures,
what I need to do

684
00:52:07,470 --> 00:52:14,720
is to evaluate
something that has

685
00:52:14,720 --> 00:52:18,450
to do with the behavior
of these angles

686
00:52:18,450 --> 00:52:21,105
when I go to low temperatures.

687
00:52:21,105 --> 00:52:23,220
And when I go to
low temperatures,

688
00:52:23,220 --> 00:52:27,710
these angles tend to be very
much aligned to each other.

689
00:52:27,710 --> 00:52:31,290
And these factors of
cosine I can therefore

690
00:52:31,290 --> 00:52:36,080
start expanding around 1.

691
00:52:36,080 --> 00:52:40,790
So what I end up having to do is
something like a product over i

692
00:52:40,790 --> 00:52:46,400
theta i cosine of
theta 0 minus theta r.

693
00:52:46,400 --> 00:52:50,110
I have a product over
neighbors of factors

694
00:52:50,110 --> 00:52:54,110
such as K over 2 theta
i minus theta j squared,

695
00:52:54,110 --> 00:52:58,800
[? as ?] I expand the
Gaussian, expand the cosine.

696
00:52:58,800 --> 00:53:09,740
And in the denominator I would
have exactly the same thing

697
00:53:09,740 --> 00:53:13,260
without this.

698
00:53:13,260 --> 00:53:19,790
So essentially, we see
that since the cosine is

699
00:53:19,790 --> 00:53:25,620
the real part of e to the i
theta 0 minus theta r, what

700
00:53:25,620 --> 00:53:29,880
I need to do is to calculate
the average of this

701
00:53:29,880 --> 00:53:32,962
assuming the Gaussian weight.

702
00:53:32,962 --> 00:53:37,474
So the theta is Gaussian
distributed, OK?

703
00:53:40,350 --> 00:53:45,050
Now-- actually, this [INAUDIBLE]
I can take the outside also.

704
00:53:45,050 --> 00:53:46,592
It doesn't matter.

705
00:53:46,592 --> 00:53:50,830
I have to calculate
this expectation value.

706
00:53:50,830 --> 00:53:59,560
And this for any Gaussian
expectation value of e to the i

707
00:53:59,560 --> 00:54:06,960
some Gaussian
variable is minus 1/2

708
00:54:06,960 --> 00:54:19,111
the average of whatever
you have, weight.

709
00:54:19,111 --> 00:54:26,905
And again, in case you forgot
this, just insert the K here.

710
00:54:26,905 --> 00:54:32,180
You can see that this is
the characteristic function

711
00:54:32,180 --> 00:54:35,500
of the Gaussian
distributed variable, which

712
00:54:35,500 --> 00:54:37,950
is this difference.

713
00:54:37,950 --> 00:54:40,320
And the characteristic
function I

714
00:54:40,320 --> 00:54:45,080
can start expanding in
terms of the cumulants.

715
00:54:45,080 --> 00:54:49,680
The first cumulant, the
average is 0 by symmetry.

716
00:54:49,680 --> 00:54:51,650
So the first thing
that will appear,

717
00:54:51,650 --> 00:54:54,420
which would be at the
order of K squared,

718
00:54:54,420 --> 00:54:56,000
is going to be the
variance, which

719
00:54:56,000 --> 00:54:57,995
is what we have over here.

720
00:54:57,995 --> 00:55:01,170
And since it's a Gaussian,
all higher order terms

721
00:55:01,170 --> 00:55:05,150
in this series [? will. ?]
Another way to do

722
00:55:05,150 --> 00:55:09,270
is to of course just
complete the square.

723
00:55:09,270 --> 00:55:13,050
And this is what
would come out, OK?

724
00:55:13,050 --> 00:55:18,830
So all I need to do is to
calculate the expectation

725
00:55:18,830 --> 00:55:25,560
value of this quantity
where the thetas are

726
00:55:25,560 --> 00:55:28,470
Gaussian distributed.

727
00:55:28,470 --> 00:55:33,460
And the best way to do so
is to go to Fourier space.

728
00:55:33,460 --> 00:55:35,860
So I have integral.

729
00:55:35,860 --> 00:55:40,550
For each one of these factors
of theta 0 minus theta r,

730
00:55:40,550 --> 00:55:49,570
I will do an integral
d2 q 2 pi squared.

731
00:55:49,570 --> 00:55:53,570
I have 1 minus e
to the iq.r, which

732
00:55:53,570 --> 00:55:56,990
is from theta 0 minus theta r.

733
00:55:56,990 --> 00:56:01,260
And then I have a theta tilde q.

734
00:56:01,260 --> 00:56:02,715
I have two of those factors.

735
00:56:02,715 --> 00:56:06,647
I have d2 q prime 2 pi squared.

736
00:56:06,647 --> 00:56:18,900
I have e to the minus iq prime
dot r theta tilde q prime.

737
00:56:18,900 --> 00:56:22,935
And then this average
simply becomes this average.

738
00:56:25,460 --> 00:56:31,000
And the different modes are
independent of each other.

739
00:56:31,000 --> 00:56:34,560
So I will get a 2 pi
to the d-- actually,

740
00:56:34,560 --> 00:56:37,820
2 here, a delta
function q plus q prime.

741
00:56:40,540 --> 00:56:44,190
And for each mode,
I will get a factor

742
00:56:44,190 --> 00:56:47,200
of Kq squared because
after all, thetas

743
00:56:47,200 --> 00:56:52,885
are very much like the pi's that
I had written at the beginning,

744
00:56:52,885 --> 00:56:54,360
OK?

745
00:56:54,360 --> 00:57:01,660
So what I will have is that this
quantity is integral over one

746
00:57:01,660 --> 00:57:04,240
q's.

747
00:57:04,240 --> 00:57:06,180
Putting these two
factors together,

748
00:57:06,180 --> 00:57:08,663
realizing that q
prime is minus q

749
00:57:08,663 --> 00:57:15,180
will give me 2 minus 2 cosine
of q.r divided by Kq squared.

750
00:57:19,546 --> 00:57:22,160
So there is an
overall scale that

751
00:57:22,160 --> 00:57:26,521
is set by 1 over
K, by temperature.

752
00:57:26,521 --> 00:57:28,520
And then there's a function
[? on ?] [? from ?],

753
00:57:28,520 --> 00:57:40,100
which is the Fourier transform
of {} 1 over q squared, which,

754
00:57:40,100 --> 00:57:43,300
as usual, we call C. We
anticipate this to be like

755
00:57:43,300 --> 00:57:44,500
a Coulomb potential.

756
00:57:44,500 --> 00:57:48,840
Because if I take
a Laplacian of C,

757
00:57:48,840 --> 00:57:52,540
you can see that-- forget the
q-- the [? Laplacian ?] of C

758
00:57:52,540 --> 00:57:58,430
from the cosine, I will get a
minus q squared cancels that.

759
00:57:58,430 --> 00:58:03,870
I will have [INAUDIBLE]
d2q 2 pi squared.

760
00:58:03,870 --> 00:58:06,660
Cosine itself will be left.

761
00:58:06,660 --> 00:58:09,240
The 1 over q squared disappears.

762
00:58:09,240 --> 00:58:13,070
This is e to the iqr you plus
e to the minus iqr over 2.

763
00:58:13,070 --> 00:58:14,850
Each one of them gives
a delta function.

764
00:58:14,850 --> 00:58:18,980
So this is just
a delta function.

765
00:58:18,980 --> 00:58:23,770
So C is the potential that
you have from a unit charge

766
00:58:23,770 --> 00:58:26,040
in 2 dimensions.

767
00:58:26,040 --> 00:58:33,790
And again, you can perform
the usual Gaussian procedure

768
00:58:33,790 --> 00:58:43,810
to find that the gradient
of C times 2 pi r

769
00:58:43,810 --> 00:58:47,552
is the net charge that is
enclosed, which is [? unity ?].

770
00:58:47,552 --> 00:58:52,010
So gradient of C, which
points in the radial direction

771
00:58:52,010 --> 00:58:55,330
is going to be 1 over 2 pi r.

772
00:58:55,330 --> 00:59:00,485
And your C is going to be
log of r divided by 2 pi.

773
00:59:00,485 --> 00:59:08,326
So this is 1 over K log
of r divided by 2 pi.

774
00:59:08,326 --> 00:59:13,250
And I state that when
essentially the 2 angles are

775
00:59:13,250 --> 00:59:17,350
as close as some short distance
cut-off, fluctuations vanish.

776
00:59:17,350 --> 00:59:24,410
So that's how I set the
0 of my integration, OK?

777
00:59:24,410 --> 00:59:26,685
So again, you put
that over here.

778
00:59:26,685 --> 00:59:34,380
We find that s0.s of r in
the low temperature limit

779
00:59:34,380 --> 00:59:39,840
is the exponential
of minus 1/2 of this.

780
00:59:39,840 --> 00:59:46,570
So I have log of r over
a divided by 4 pi K.

781
00:59:46,570 --> 00:59:54,760
And I will get a over r to
the power of 1 over 4 pi K.

782
00:59:54,760 --> 00:59:59,210
And I kind of want to check
that I didn't lose a factor of 2

783
00:59:59,210 --> 01:00:01,282
somewhere, which I seem to have.

784
01:00:25,400 --> 01:00:32,400
Yeah, I lost a factor
of 2 right here.

785
01:00:32,400 --> 01:00:36,970
This should be 2
because of this 2,

786
01:00:36,970 --> 01:00:40,070
if I'm using this definition.

787
01:00:40,070 --> 01:00:42,960
So this should be 2.

788
01:00:42,960 --> 01:00:52,310
And this should be 2 pi K. OK.

789
01:00:59,390 --> 01:01:03,540
So what have we established?

790
01:01:03,540 --> 01:01:08,750
We have looked as a
function of temperature

791
01:01:08,750 --> 01:01:13,233
what the behavior of this
spin-spin correlation function

792
01:01:13,233 --> 01:01:13,733
is.

793
01:01:18,550 --> 01:01:23,410
We have established that in
the higher temperature limit,

794
01:01:23,410 --> 01:01:29,276
the behavior is something
that falls off exponentially

795
01:01:29,276 --> 01:01:30,105
with separation.

796
01:01:33,310 --> 01:01:36,655
We have also established
that at low temperature,

797
01:01:36,655 --> 01:01:42,962
it falls off as a power
law in separation, OK?

798
01:01:46,460 --> 01:01:51,890
So these two functional
behaviors are different.

799
01:01:51,890 --> 01:01:55,135
There is no way that you can
connect one to the other.

800
01:01:55,135 --> 01:01:59,545
So you pick two spins that
are sufficiently far apart

801
01:01:59,545 --> 01:02:03,190
and then move the separation
further and further away.

802
01:02:03,190 --> 01:02:05,900
And the functional form
of the correlations

803
01:02:05,900 --> 01:02:09,630
is either a power-law
decay, power-law decay,

804
01:02:09,630 --> 01:02:13,100
or an exponential decay.

805
01:02:13,100 --> 01:02:17,390
And in this form, you know
you have a high temperature.

806
01:02:17,390 --> 01:02:20,350
In this form, you know you
have a low temperature.

807
01:02:20,350 --> 01:02:25,520
So potentially, there could be
a phase transition separating

808
01:02:25,520 --> 01:02:29,390
the distinct behaviors of
the correlation function.

809
01:02:29,390 --> 01:02:33,220
And that could
potentially be underlying

810
01:02:33,220 --> 01:02:38,130
what is observed over here, OK?

811
01:02:38,130 --> 01:02:38,630
Yes?

812
01:02:38,630 --> 01:02:40,588
AUDIENCE: So where could
we make the assumption

813
01:02:40,588 --> 01:02:43,430
we're at a low temperature
in the second expansion?

814
01:02:43,430 --> 01:02:48,400
PROFESSOR: When we expanded
the cosines, right?

815
01:02:48,400 --> 01:02:53,790
So what I should really do is to
look at the terms such as this.

816
01:02:53,790 --> 01:02:56,780
But then I said that I'm low
enough temperature so that I

817
01:02:56,780 --> 01:03:00,980
look at near neighbors, and
they're almost parallel.

818
01:03:00,980 --> 01:03:04,120
So the cosine of the angle
difference between them

819
01:03:04,120 --> 01:03:06,803
is the square of
that small angle.

820
01:03:06,803 --> 01:03:07,636
AUDIENCE: Thank you.

821
01:03:07,636 --> 01:03:08,302
PROFESSOR: Yeah.

822
01:03:11,582 --> 01:03:12,082
OK?

823
01:03:18,010 --> 01:03:22,040
So actually, you
may have said that I

824
01:03:22,040 --> 01:03:26,430
could have done the same
analysis for small angle

825
01:03:26,430 --> 01:03:31,560
expansions not only
for n equals 2,

826
01:03:31,560 --> 01:03:34,950
but also for n
equals 3, et cetera.

827
01:03:34,950 --> 01:03:36,640
That would be correct.

828
01:03:36,640 --> 01:03:40,410
Because I could have made a
similar Gaussian analysis for n

829
01:03:40,410 --> 01:03:42,990
equals e also.

830
01:03:42,990 --> 01:03:46,130
And then I may have
concluded the same thing,

831
01:03:46,130 --> 01:03:49,960
except that I cannot conclude
the same thing because of this

832
01:03:49,960 --> 01:03:53,180
thing that we derived over here.

833
01:03:53,180 --> 01:03:59,870
What this shows is that the
expansion around 0 temperature

834
01:03:59,870 --> 01:04:04,610
regarded as Gaussian
is going to break down

835
01:04:04,610 --> 01:04:06,750
because of the
non-linear coupling

836
01:04:06,750 --> 01:04:09,170
that we have between modes.

837
01:04:09,170 --> 01:04:12,910
So although I may be tempted to
write something like this for n

838
01:04:12,910 --> 01:04:16,670
equals 3, I know
why it is wrong.

839
01:04:16,670 --> 01:04:19,358
And I know the
correlation length

840
01:04:19,358 --> 01:04:22,940
at which this kind
of behavior will

841
01:04:22,940 --> 01:04:26,520
need to be replaced with
this type of behavior

842
01:04:26,520 --> 01:04:30,410
because effectively, the
expansion parameter became

843
01:04:30,410 --> 01:04:32,780
of the order of 1.

844
01:04:32,780 --> 01:04:35,950
But I cannot do that
for the xy model.

845
01:04:35,950 --> 01:04:40,320
I don't have similar reason.

846
01:04:40,320 --> 01:04:44,490
So then the question
becomes, well,

847
01:04:44,490 --> 01:04:51,220
how does this expansion then
eventually break down so

848
01:04:51,220 --> 01:04:53,720
that I will have a phase
transition to a phase

849
01:04:53,720 --> 01:04:57,170
where the correlations are
decaying exponentially?

850
01:04:57,170 --> 01:04:59,080
And you may say,
well, I mean, it's

851
01:04:59,080 --> 01:05:00,850
really something
to do with having

852
01:05:00,850 --> 01:05:02,610
to go to higher and
higher ordered terms

853
01:05:02,610 --> 01:05:04,810
in the expansion of the cosine.

854
01:05:04,810 --> 01:05:08,080
And it's going to
be something which

855
01:05:08,080 --> 01:05:12,510
would be very difficult to
figure out, except that it

856
01:05:12,510 --> 01:05:16,250
turns out that there is a
much more elegant solution.

857
01:05:16,250 --> 01:05:18,380
And that was proposed
by [? Kastelitz ?]

858
01:05:18,380 --> 01:05:19,180
and [? Thales. ?]

859
01:05:26,975 --> 01:05:32,570
And they said that what you
have left out in the Gaussian

860
01:05:32,570 --> 01:05:43,796
analysis are
topological defects, OK?

861
01:05:43,796 --> 01:05:48,930
That is, when I did the
expansion of the cosine

862
01:05:48,930 --> 01:05:52,370
and we replaced the cosine with
the difference of the angles

863
01:05:52,370 --> 01:05:57,490
squared, that's
more or less fine,

864
01:05:57,490 --> 01:06:02,020
except that I should also
realize that cosine maintains

865
01:06:02,020 --> 01:06:07,051
its value if the angle
difference goes up by 2 pi.

866
01:06:07,051 --> 01:06:10,730
And you say, well,
neighboring spins are never

867
01:06:10,730 --> 01:06:14,060
going to be 2 pi
different or pi different

868
01:06:14,060 --> 01:06:16,980
because they're very
strongly coupled.

869
01:06:16,980 --> 01:06:18,420
Does it make any difference?

870
01:06:18,420 --> 01:06:21,200
Turns out that, OK, for
the neighboring spins,

871
01:06:21,200 --> 01:06:22,600
it doesn't make a difference.

872
01:06:22,600 --> 01:06:25,760
But what if you go far away?

873
01:06:25,760 --> 01:06:36,540
So let's imagine that this is,
let's say, our system of spins.

874
01:06:36,540 --> 01:06:41,350
And what I do is I look at a
configuration such as this.

875
01:07:02,960 --> 01:07:08,470
Essentially, I have
spins [? radiating ?] out

876
01:07:08,470 --> 01:07:11,810
from a center such as this, OK?

877
01:07:14,370 --> 01:07:17,200
There is, of course,
a lot of energy costs

878
01:07:17,200 --> 01:07:19,145
I have put over here.

879
01:07:22,070 --> 01:07:26,580
But when I go very
much further out,

880
01:07:26,580 --> 01:07:31,350
let's say, very far
away from this plus sign

881
01:07:31,350 --> 01:07:34,790
that I have indicated
over here, and I

882
01:07:34,790 --> 01:07:37,990
follow what the behavior
of the spins are,

883
01:07:37,990 --> 01:07:43,670
you see that as I go along
this circuit, the spins start

884
01:07:43,670 --> 01:07:46,400
by pointing this way,
they go point this way,

885
01:07:46,400 --> 01:07:49,920
this way, et cetera.

886
01:07:49,920 --> 01:07:54,680
And by the time I carry
a circuit such as this,

887
01:07:54,680 --> 01:08:02,017
I find that the angle theta
has also rotated by 2 pi, OK?

888
01:08:02,017 --> 01:08:04,760
Now, this is clearly
a configuration

889
01:08:04,760 --> 01:08:06,160
that is going to be costly.

890
01:08:06,160 --> 01:08:08,520
We'll calculate its cost.

891
01:08:08,520 --> 01:08:13,540
But the point is that there
is no continuous deformation

892
01:08:13,540 --> 01:08:17,880
that you can make
that will map this

893
01:08:17,880 --> 01:08:20,284
into what we were
expanding around here

894
01:08:20,284 --> 01:08:24,176
with all of the cosines
being parallel to each other.

895
01:08:24,176 --> 01:08:29,060
So this is a topologically
distinct contribution

896
01:08:29,060 --> 01:08:31,430
from the Gaussian
one, the Gaussian term

897
01:08:31,430 --> 01:08:34,250
that we've calculated.

898
01:08:34,250 --> 01:08:38,319
Since the direction of
the rotation of the spins

899
01:08:38,319 --> 01:08:42,720
is the same as the direction
of the circuit in this case,

900
01:08:42,720 --> 01:08:46,410
this is called a plus
topological defect.

901
01:08:46,410 --> 01:08:57,890
There is a corresponding
minus defect

902
01:08:57,890 --> 01:09:03,649
which is something like this.

903
01:09:42,259 --> 01:09:45,439
OK, and for this, you
can convince yourself

904
01:09:45,439 --> 01:09:48,724
that as you make a
circuit such as this

905
01:09:48,724 --> 01:09:51,779
that the direction
of the arrow actually

906
01:09:51,779 --> 01:09:55,705
goes in the opposite
direction, OK?

907
01:09:55,705 --> 01:10:02,830
This is called a negative
sign topological defect.

908
01:10:02,830 --> 01:10:06,480
Now, I said, well,
let's figure out

909
01:10:06,480 --> 01:10:11,160
what the energy cost of
one of these things is.

910
01:10:11,160 --> 01:10:16,740
If I'm away from the center
of one of these defects,

911
01:10:16,740 --> 01:10:22,985
then the change in angle
is small because the change

912
01:10:22,985 --> 01:10:30,130
in angle if I go all the way
around a circle of radius

913
01:10:30,130 --> 01:10:34,550
r should come back to be 2 pi.

914
01:10:34,550 --> 01:10:39,270
2 pi is the uncertainty that
I have from the cosines.

915
01:10:39,270 --> 01:10:42,030
So what I have is
that the gradient

916
01:10:42,030 --> 01:10:47,550
of theta [? between ?]
neighboring angles

917
01:10:47,550 --> 01:10:54,710
times 2 pi r, which is
this radius, is 2 pi.

918
01:10:54,710 --> 01:11:01,360
And this thing,
here it is plus 1.

919
01:11:01,360 --> 01:11:03,940
Here it is minus 1.

920
01:11:03,940 --> 01:11:07,920
And in general, you can
imagine possibilities

921
01:11:07,920 --> 01:11:12,180
where this is some integer
that is like 2 or minus 2

922
01:11:12,180 --> 01:11:16,050
or something else
that is allowed

923
01:11:16,050 --> 01:11:21,640
by this degeneracy
of this cosine, OK?

924
01:11:21,640 --> 01:11:23,910
So you can see that
when you are far away

925
01:11:23,910 --> 01:11:27,465
from the center of
whatever this defect is,

926
01:11:27,465 --> 01:11:34,800
the gradient of theta has
magnitude that is n over r, OK?

927
01:11:34,800 --> 01:11:36,630
And as you go
further and further,

928
01:11:36,630 --> 01:11:39,700
it becomes smaller and smaller.

929
01:11:39,700 --> 01:11:43,470
And the energy cost out here
you can obtain by essentially

930
01:11:43,470 --> 01:11:48,190
expanding the cosine is going
to be proportional to the change

931
01:11:48,190 --> 01:11:50,370
in angle squared.

932
01:11:50,370 --> 01:12:00,740
So the cost of the
defect is an integral

933
01:12:00,740 --> 01:12:13,200
of 2 pi rdr times this quantity
n over r squared multiplied

934
01:12:13,200 --> 01:12:16,050
by the coefficient of the
expansion of the cosine,

935
01:12:16,050 --> 01:12:17,175
or K over 2.

936
01:12:20,590 --> 01:12:24,130
And this integration I
have to go all the way to

937
01:12:24,130 --> 01:12:25,200
ends up of my system.

938
01:12:25,200 --> 01:12:26,750
Let's call it l.

939
01:12:29,780 --> 01:12:32,260
And then I can
bring it down, not

940
01:12:32,260 --> 01:12:36,110
necessary to the scale
of the lattice spacing,

941
01:12:36,110 --> 01:12:39,810
but maybe to scale of 5 lattice
spacing or something like this,

942
01:12:39,810 --> 01:12:44,040
where the approximations that
I have used of treating this

943
01:12:44,040 --> 01:12:45,850
as a continuum are still valid.

944
01:12:45,850 --> 01:12:50,550
So I will pick some kind of
a short distance cut-off a.

945
01:12:50,550 --> 01:12:57,366
And then whatever energy is
at scales that are below a,

946
01:12:57,366 --> 01:13:04,290
I will add to a core energy that
depends on whatever this a is.

947
01:13:04,290 --> 01:13:07,580
I don't know what that is.

948
01:13:07,580 --> 01:13:11,790
So basically, there
is some core energy

949
01:13:11,790 --> 01:13:16,010
depending on where I stop this.

950
01:13:16,010 --> 01:13:18,110
And the reason that
this is more important--

951
01:13:18,110 --> 01:13:22,490
because here I have an
integral of 1 over r.

952
01:13:22,490 --> 01:13:25,160
And an integral of 1
over r is something

953
01:13:25,160 --> 01:13:27,555
that is logarithmically
divergent.

954
01:13:27,555 --> 01:13:34,580
So I will get K. I
have 2 cancels the 2.

955
01:13:34,580 --> 01:13:49,634
I have pi en squared
log of l over a, OK?

956
01:13:58,990 --> 01:14:05,940
So you can see that creating
one of these defects

957
01:14:05,940 --> 01:14:08,180
is hugely expensive.

958
01:14:08,180 --> 01:14:12,450
And energy that as your system
becomes bigger and bigger

959
01:14:12,450 --> 01:14:13,860
and we're thinking
about infinite

960
01:14:13,860 --> 01:14:19,220
sized systems is
logarithmically large.

961
01:14:19,220 --> 01:14:21,110
So you would say these
things will never

962
01:14:21,110 --> 01:14:25,960
occur because they cost an
infinite amount of energy.

963
01:14:25,960 --> 01:14:30,090
Well, the thing is that
entropy is also important.

964
01:14:30,090 --> 01:14:33,560
So if I were to calculate the
partition function that I would

965
01:14:33,560 --> 01:14:39,330
assign to one of these
defects, part of it

966
01:14:39,330 --> 01:14:41,610
would be exponential
of this energy.

967
01:14:41,610 --> 01:14:48,530
So I would have this e to
the minus this core energy.

968
01:14:48,530 --> 01:15:00,480
I would have this exponential of
K pi n squared log of l over a.

969
01:15:00,480 --> 01:15:06,040
So that's the [INAUDIBLE]
weight for this.

970
01:15:06,040 --> 01:15:10,990
But then I realize that
I can put this anywhere

971
01:15:10,990 --> 01:15:16,360
on the lattice so there is
an entropy [? gain ?] factor.

972
01:15:16,360 --> 01:15:19,810
And since I have assigned this
to have some kind of a bulk

973
01:15:19,810 --> 01:15:23,550
to it as some
characteristic size a,

974
01:15:23,550 --> 01:15:26,832
the number of distinct
places that I can put it

975
01:15:26,832 --> 01:15:30,280
is over the order
of l over a squared.

976
01:15:30,280 --> 01:15:33,830
So basically, I take my huge
lattice and I partition it

977
01:15:33,830 --> 01:15:35,640
into sizes of a's.

978
01:15:35,640 --> 01:15:39,630
And I say I can put it in any
one of these configurations.

979
01:15:39,630 --> 01:15:42,030
You can see that
the whole thing is

980
01:15:42,030 --> 01:15:45,920
going to be e to the
minus this core energy.

981
01:15:45,920 --> 01:15:51,930
And then I have l over
a to the power of 2

982
01:15:51,930 --> 01:15:57,980
minus pi K n squared, OK?

983
01:15:57,980 --> 01:16:04,210
So the logarithmic energy
cost is the same form

984
01:16:04,210 --> 01:16:09,560
as the logarithmic entropy
gain that you have over here.

985
01:16:09,560 --> 01:16:15,890
And this precise balance will
give you a value of K such

986
01:16:15,890 --> 01:16:20,790
that if K is larger than
2 over pi n squared,

987
01:16:20,790 --> 01:16:26,180
this is going to be an
exponentially large cost.

988
01:16:26,180 --> 01:16:29,700
There is a huge
negative power of l here

989
01:16:29,700 --> 01:16:33,440
that says, no, you don't
want to create this.

990
01:16:33,440 --> 01:16:40,200
But if K becomes weak such
that the 2, the entropy factor,

991
01:16:40,200 --> 01:16:44,490
[? wins ?], then you will
start creating [INAUDIBLE].

992
01:16:44,490 --> 01:16:48,970
So you can see that
over here, suddenly we

993
01:16:48,970 --> 01:16:53,070
have a mechanism along
our picture over here,

994
01:16:53,070 --> 01:16:59,520
maybe something like K,
which is 2 over pi, such

995
01:16:59,520 --> 01:17:03,980
that on one side you
would say, I will not

996
01:17:03,980 --> 01:17:09,906
have topological defect and
I can use the Gaussian model.

997
01:17:09,906 --> 01:17:13,830
And on the other side, you
say that I will spontaneously

998
01:17:13,830 --> 01:17:16,790
create these
topological defects.

999
01:17:16,790 --> 01:17:19,370
And then the
Gaussian description

1000
01:17:19,370 --> 01:17:23,430
is no longer valid because
I have to now really take

1001
01:17:23,430 --> 01:17:29,250
care of the angular nature
of these variables, OK?

1002
01:17:29,250 --> 01:17:38,290
So this is a nice picture, which
is only a zeroed order picture.

1003
01:17:38,290 --> 01:17:41,600
And to zeroed order,
it is correct.

1004
01:17:41,600 --> 01:17:44,400
But this is not fully correct.

1005
01:17:44,400 --> 01:17:48,910
Because even at
low temperatures,

1006
01:17:48,910 --> 01:18:02,540
you can certainly create hairs
of plus minus defects, OK?

1007
01:18:02,540 --> 01:18:12,650
And whereas the
field for one of them

1008
01:18:12,650 --> 01:18:16,140
will fall off at
large distances,

1009
01:18:16,140 --> 01:18:19,730
the gradient of
theta is 1 over r,

1010
01:18:19,730 --> 01:18:34,380
if you superimpose what is
happening for two of these,

1011
01:18:34,380 --> 01:18:38,480
what you will convince
yourself that if you have

1012
01:18:38,480 --> 01:18:44,160
a pair of defects of
opposite sign at the distance

1013
01:18:44,160 --> 01:18:54,150
d that the distortion that they
generate at large distances

1014
01:18:54,150 --> 01:18:58,850
falls off not the as 1 over
r, which is, if you like,

1015
01:18:58,850 --> 01:19:05,140
a monopole field, but
as d over r squared,

1016
01:19:05,140 --> 01:19:07,200
which is a dipole field.

1017
01:19:07,200 --> 01:19:10,770
So whenever you
have a dipole, you

1018
01:19:10,770 --> 01:19:13,530
will have to multiply
by the separation

1019
01:19:13,530 --> 01:19:15,400
of the charges in the dipole.

1020
01:19:15,400 --> 01:19:17,910
And that is compensated
by a factor of 1

1021
01:19:17,910 --> 01:19:19,580
over r in the denominator.

1022
01:19:19,580 --> 01:19:20,965
There is some
angular dependence,

1023
01:19:20,965 --> 01:19:24,560
but we are not so
interested in that.

1024
01:19:24,560 --> 01:19:32,570
Now, if I were to
integrate this square,

1025
01:19:32,570 --> 01:19:34,260
we can see that it
is something that

1026
01:19:34,260 --> 01:19:38,460
is convergent at
large distances.

1027
01:19:38,460 --> 01:19:41,860
And so this is
going to be finite.

1028
01:19:41,860 --> 01:19:48,670
It is not going to diverge as
the size of the system, which

1029
01:19:48,670 --> 01:19:53,860
means that whereas individual
defects there was no way

1030
01:19:53,860 --> 01:19:58,030
that I could create
in my system,

1031
01:19:58,030 --> 01:20:01,242
I can always create
pairs of these defects.

1032
01:20:04,140 --> 01:20:08,300
So the correct picture
that we should have

1033
01:20:08,300 --> 01:20:12,960
is not that at low temperatures
you don't have defects,

1034
01:20:12,960 --> 01:20:15,850
at high temperatures you have
these defects spontaneously

1035
01:20:15,850 --> 01:20:17,390
appearing.

1036
01:20:17,390 --> 01:20:20,740
The correct picture is
that at low temperatures

1037
01:20:20,740 --> 01:20:28,510
what you have is lots and lots
of dipoles that are pretty much

1038
01:20:28,510 --> 01:20:32,130
bound to each other.

1039
01:20:32,130 --> 01:20:41,270
And when you go to high
temperatures, what happens

1040
01:20:41,270 --> 01:20:46,990
is that you will have these
pluses and minuses unbound

1041
01:20:46,990 --> 01:20:48,890
from each other.

1042
01:20:48,890 --> 01:20:51,950
So if you like,
the transition is

1043
01:20:51,950 --> 01:20:59,080
between molecules to a plasma
as temperature is changed.

1044
01:20:59,080 --> 01:21:05,440
Or if you like, it is between
an insulator and a conductor.

1045
01:21:05,440 --> 01:21:09,530
And how to mathematically
describe this phase transition

1046
01:21:09,530 --> 01:21:13,932
in 2 dimensions, which
we can rigorously,

1047
01:21:13,932 --> 01:21:18,174
will be what we will do in
the next couple of lectures.