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PROFESSOR: OK, let's start.

9
00:00:26,360 --> 00:00:32,360
So, we've moved onto the
two-dimensional xy model.

10
00:00:36,910 --> 00:00:43,070
This is a system
where, let's say

11
00:00:43,070 --> 00:00:49,410
on each side of a square lattice
you put a unit vector that

12
00:00:49,410 --> 00:00:53,030
has two components
and hence, can

13
00:00:53,030 --> 00:00:56,605
be described by an angle theta.

14
00:00:56,605 --> 00:01:01,130
So basically, at each side,
you have an angle theta

15
00:01:01,130 --> 00:01:02,950
for that side.

16
00:01:02,950 --> 00:01:09,450
And there is a tendency for a
neighboring spins to be aligned

17
00:01:09,450 --> 00:01:13,020
and the partition
function can be

18
00:01:13,020 --> 00:01:18,670
written as a sum over
all configurations, which

19
00:01:18,670 --> 00:01:22,780
is equivalent to integrating
over all of these angles.

20
00:01:25,700 --> 00:01:32,490
And the weight that wants
to make the two neighboring

21
00:01:32,490 --> 00:01:36,590
spins to be parallel
to each other.

22
00:01:36,590 --> 00:01:39,900
So we have a sum over
nearest neighbors.

23
00:01:39,900 --> 00:01:43,410
And the dot product of
the two spins amounts

24
00:01:43,410 --> 00:01:47,890
to looking at the cosine
of theta i minus theta j.

25
00:01:47,890 --> 00:01:51,390
And so, we have this
factor over here.

26
00:01:55,460 --> 00:02:05,850
Now, if we go to the
limit where k is large,

27
00:02:05,850 --> 00:02:10,210
then the cosine will tend
to keep the angles close

28
00:02:10,210 --> 00:02:12,180
to each other.

29
00:02:12,180 --> 00:02:20,080
And we are tempted to expand
this around the configurations

30
00:02:20,080 --> 00:02:22,680
where everybody's parallel.

31
00:02:22,680 --> 00:02:28,620
Let's call that nk
by the factor of 2.

32
00:02:28,620 --> 00:02:33,530
And then, expanding with the
cosine to the next order,

33
00:02:33,530 --> 00:02:40,550
you may want to replace
this-- let's call this k0--

34
00:02:40,550 --> 00:02:44,870
and have a factor of k,
which is proportional

35
00:02:44,870 --> 00:02:48,890
to k0 after some
lattice spacing.

36
00:02:48,890 --> 00:02:54,990
And integral of gradient
of theta squared.

37
00:02:54,990 --> 00:03:01,150
So basically, the difference
between the angles

38
00:03:01,150 --> 00:03:04,160
in the continuum version,
I want to replace

39
00:03:04,160 --> 00:03:11,000
with the term that tries to
make the gradient to be fixed.

40
00:03:11,000 --> 00:03:12,910
OK.

41
00:03:12,910 --> 00:03:18,940
Now the reason I put these
quotes around the gradient

42
00:03:18,940 --> 00:03:22,440
is something that we
noticed last time, which

43
00:03:22,440 --> 00:03:27,650
is that in principal,
theta is defined up

44
00:03:27,650 --> 00:03:30,330
to a multiple of 2 pi.

45
00:03:30,330 --> 00:03:36,290
So that if I were to take
a circuit along the lattice

46
00:03:36,290 --> 00:03:39,650
that comes back to itself.

47
00:03:39,650 --> 00:03:49,900
And all along, this circuit
integrate this gradient

48
00:03:49,900 --> 00:03:55,260
of theta, so basically, gradient
of theta would be a vector.

49
00:03:55,260 --> 00:03:59,330
I integrated along a circuit.

50
00:03:59,330 --> 00:04:03,370
And by the time I have come back
and close the circuit to where

51
00:04:03,370 --> 00:04:08,450
I started, the answer
may not come back to 0.

52
00:04:08,450 --> 00:04:16,985
It may be any integer
multiple of 2 pi.

53
00:04:16,985 --> 00:04:17,485
All right.

54
00:04:22,480 --> 00:04:25,690
So how do we account for this?

55
00:04:25,690 --> 00:04:29,060
The way we account
for this is that we

56
00:04:29,060 --> 00:04:38,280
note that this gradient of data
I can decompose into two parts.

57
00:04:38,280 --> 00:04:41,240
One, where I just
write it as a gradient

58
00:04:41,240 --> 00:04:44,440
of some regular function.

59
00:04:44,440 --> 00:04:46,950
And the characteristic
of gradient

60
00:04:46,950 --> 00:04:51,340
is that once you go over a
closed loop and you integrate,

61
00:04:51,340 --> 00:04:53,850
you essentially are
evaluating this field

62
00:04:53,850 --> 00:04:57,420
phi at the beginning
and the end.

63
00:04:57,420 --> 00:05:01,100
And for any regular
single valued phi,

64
00:05:01,100 --> 00:05:04,800
this would come back to zero.

65
00:05:04,800 --> 00:05:09,360
And to take care of
this fact the result

66
00:05:09,360 --> 00:05:11,770
does not have to come
to zero if I integrate

67
00:05:11,770 --> 00:05:19,900
this gradient of theta, I
introduce another field, u,

68
00:05:19,900 --> 00:05:26,190
that takes care of these
topological defects.

69
00:05:34,390 --> 00:05:36,820
OK?

70
00:05:36,820 --> 00:05:43,370
So that, really, I have to
include both configurations

71
00:05:43,370 --> 00:05:48,900
in order to correctly capture
the original model that

72
00:05:48,900 --> 00:05:51,380
had these angles.

73
00:05:51,380 --> 00:05:54,890
OK, so what can this u be?

74
00:05:54,890 --> 00:06:02,100
We already looked at what
u is for the case of one

75
00:06:02,100 --> 00:06:03,390
topological defect.

76
00:06:13,950 --> 00:06:17,560
And the idea here
was that maybe I

77
00:06:17,560 --> 00:06:23,530
had a configuration where
around a particular center,

78
00:06:23,530 --> 00:06:29,230
let's say all of the
spins were flowing out,

79
00:06:29,230 --> 00:06:32,900
or some other such
configuration such

80
00:06:32,900 --> 00:06:41,500
that when I go over a large
distance r from this center,

81
00:06:41,500 --> 00:06:51,940
and integrate this field u,
just like I did over there,

82
00:06:51,940 --> 00:06:55,620
the answer is going to
be, let's say, 2 pi n.

83
00:06:55,620 --> 00:06:58,940
So there's this u.

84
00:06:58,940 --> 00:07:03,960
And I integrated
along this circle.

85
00:07:03,960 --> 00:07:08,500
And the answer is
going to be 2 pi n.

86
00:07:08,500 --> 00:07:15,460
Well, clearly, the
magnitude of u times

87
00:07:15,460 --> 00:07:21,060
2 pi r, which is the
radius of the circle

88
00:07:21,060 --> 00:07:25,960
is going to be 2 pi times
some integer-- could

89
00:07:25,960 --> 00:07:29,890
be plus, minus 1, plus minus 2.

90
00:07:29,890 --> 00:07:34,010
And so, the magnitude
of u is n over r.

91
00:07:43,390 --> 00:07:51,446
The direction of u is orthogonal
to the direction of r.

92
00:07:51,446 --> 00:07:53,230
And how can I show that?

93
00:07:53,230 --> 00:07:55,850
Well, one way I
can show that is I

94
00:07:55,850 --> 00:08:05,170
can say that it is z
hat crossed with r hat,

95
00:08:05,170 --> 00:08:12,490
there z hat is the vector
that comes out of the plane.

96
00:08:12,490 --> 00:08:16,260
And r hat is the unit
vector in this direction.

97
00:08:16,260 --> 00:08:19,360
u is clearly orthogonal to r.

98
00:08:19,360 --> 00:08:24,530
The direction of the gradient of
this angle is orthogonal to r.

99
00:08:24,530 --> 00:08:29,290
It is in the plane so
it's orthogonal to this.

100
00:08:29,290 --> 00:08:36,730
And this I can also
write as z hat 3

101
00:08:36,730 --> 00:08:43,830
crossed with the gradient of
log of r, with some cut-off.

102
00:08:43,830 --> 00:08:47,800
Because the gradient of
log of r will give me,

103
00:08:47,800 --> 00:08:51,410
essentially, 1 over R in
the direction of r hat.

104
00:08:51,410 --> 00:08:53,540
And this is like
the potential that I

105
00:08:53,540 --> 00:08:57,450
would have for a charge
in two dimensions,

106
00:08:57,450 --> 00:09:01,640
except that I have
rotated it by somewhat.

107
00:09:01,640 --> 00:09:12,070
And this I can also write
as minus the curl of z hat

108
00:09:12,070 --> 00:09:16,655
log r over a with a factor of n.

109
00:09:20,120 --> 00:09:24,880
And essentially,
what you can see

110
00:09:24,880 --> 00:09:30,610
is that the gradient of
data for a field that

111
00:09:30,610 --> 00:09:33,860
has this topological
defect has a part

112
00:09:33,860 --> 00:09:39,240
can be written as a
potential gradient of some y,

113
00:09:39,240 --> 00:09:43,010
and a part that can be
written as curl to keep track

114
00:09:43,010 --> 00:09:45,310
of these vortices, if you like.

115
00:09:45,310 --> 00:09:48,140
If you were to think of
this gradient of data

116
00:09:48,140 --> 00:09:50,755
like the flow field that you
would have in two dimensions.

117
00:09:50,755 --> 00:09:52,510
It has a potential part.

118
00:09:52,510 --> 00:09:58,480
And it has a part that is due to
curvatures and vortices, which

119
00:09:58,480 --> 00:10:01,870
is what we have over here.

120
00:10:01,870 --> 00:10:03,950
OK.

121
00:10:03,950 --> 00:10:08,480
So this is, however, only
for one topological defect.

122
00:10:08,480 --> 00:10:11,935
What happens if I have
many such defects?

123
00:10:17,390 --> 00:10:22,940
What I can do is, rather
than having just one of them,

124
00:10:22,940 --> 00:10:27,690
I could have another topological
defect here, another one here,

125
00:10:27,690 --> 00:10:28,600
another one there.

126
00:10:28,600 --> 00:10:32,790
There should be a
combination of these things.

127
00:10:32,790 --> 00:10:39,840
And what I can do in order
to get the corresponding u

128
00:10:39,840 --> 00:10:45,080
is to superimpose solutions
that correspond to single ones.

129
00:10:45,080 --> 00:10:48,180
As you can see that
this is very much

130
00:10:48,180 --> 00:10:51,720
like the potential
that I would have

131
00:10:51,720 --> 00:10:54,260
for a charge at the
origin, and then

132
00:10:54,260 --> 00:10:57,220
taking the derivative
to create the field.

133
00:10:57,220 --> 00:11:00,930
And you know that as long
as things are linear,

134
00:11:00,930 --> 00:11:02,770
and there aren't
too many of them,

135
00:11:02,770 --> 00:11:07,560
you can superimpose solutions
for different charges.

136
00:11:07,560 --> 00:11:10,260
You could just add up
the electric fields.

137
00:11:10,260 --> 00:11:14,600
So what I'm claiming
is that I can write u

138
00:11:14,600 --> 00:11:25,280
as minus n curl of z hat, times
some potential u of r, where

139
00:11:25,280 --> 00:11:32,920
psi of r is essentially the
generalization of this log.

140
00:11:32,920 --> 00:11:39,000
I can write it as a sum over
all topological defects.

141
00:11:39,000 --> 00:11:45,565
And I will have the n of that
topological defect times log

142
00:11:45,565 --> 00:11:50,480
of r minus ri
divided by a, where

143
00:11:50,480 --> 00:11:53,960
ri are the locations of these.

144
00:11:53,960 --> 00:11:59,350
So there could be a vortex
here at r1 with charge n1,

145
00:11:59,350 --> 00:12:04,350
and other topological defect
here at r2 with charge n2,

146
00:12:04,350 --> 00:12:06,960
and so forth.

147
00:12:06,960 --> 00:12:12,270
And I can construct a
potential that basically looks

148
00:12:12,270 --> 00:12:17,460
at the log of r minus ri
for each individual one,

149
00:12:17,460 --> 00:12:20,790
and then do this.

150
00:12:20,790 --> 00:12:21,650
OK?

151
00:12:21,650 --> 00:12:27,100
I will sometimes write this in
a slightly different fashion.

152
00:12:27,100 --> 00:12:34,930
Recall that we had the
Coulomb potential, which

153
00:12:34,930 --> 00:12:41,550
was related to log by just
a factor of 1 over 2 pi.

154
00:12:41,550 --> 00:12:44,830
So the correct version of
defining the Coulomb potential

155
00:12:44,830 --> 00:12:46,030
is this.

156
00:12:46,030 --> 00:12:51,330
So this I can write as
the Coulomb potential,

157
00:12:51,330 --> 00:12:55,200
provided that I
multiply the 2 pi ni.

158
00:12:55,200 --> 00:12:58,190
And I sometimes
will call that qi.

159
00:12:58,190 --> 00:13:03,520
So essentially, qi
is 2 pi and i is

160
00:13:03,520 --> 00:13:06,770
the charge of the
topological defect.

161
00:13:06,770 --> 00:13:09,610
It can be plus or minus 2 pi.

162
00:13:09,610 --> 00:13:12,450
And then, the potential
is constructed

163
00:13:12,450 --> 00:13:18,160
by constructing superposition
of those charges divided

164
00:13:18,160 --> 00:13:21,611
or multiplied with
appropriate Coulomb potential.

165
00:13:21,611 --> 00:13:22,110
OK?

166
00:13:26,520 --> 00:13:27,020
All right.

167
00:13:29,980 --> 00:13:39,190
So I can construct a cost for
creating a configuration now.

168
00:13:39,190 --> 00:13:43,580
Previously, I had
this integral gradient

169
00:13:43,580 --> 00:13:47,376
of theta squared
in the continuum.

170
00:13:47,376 --> 00:13:54,400
And my gradient of
theta squared has now

171
00:13:54,400 --> 00:13:59,920
a part that is the gradient
of a regular, well-behaved

172
00:13:59,920 --> 00:14:04,890
potential, and a part
that is this field

173
00:14:04,890 --> 00:14:13,200
u, which is minus-- oops.

174
00:14:13,200 --> 00:14:15,560
Then I don't need
the ni's because I

175
00:14:15,560 --> 00:14:18,470
put the ni as part of the psi.

176
00:14:18,470 --> 00:14:25,760
Curl of z hat psi of r.

177
00:14:25,760 --> 00:14:28,070
So phi is a regular function.

178
00:14:28,070 --> 00:14:33,060
Psi with curl will give
me the contribution

179
00:14:33,060 --> 00:14:36,430
of the topological
defect involves

180
00:14:36,430 --> 00:14:39,500
both the charges
and the positions

181
00:14:39,500 --> 00:14:42,022
of these topological defects.

182
00:14:42,022 --> 00:14:44,990
OK?

183
00:14:44,990 --> 00:14:48,090
And this whole thing has
to be squared, of course.

184
00:14:48,090 --> 00:14:52,720
This is my gradient squared.

185
00:14:52,720 --> 00:14:55,810
And if I expand this, I
will have three terms.

186
00:14:59,040 --> 00:15:01,540
I have a gradient
of this 5 squared.

187
00:15:04,340 --> 00:15:09,980
I have a term, which is minus
2 gradient of phi dot producted

188
00:15:09,980 --> 00:15:14,540
with curl of z hat psi.

189
00:15:17,830 --> 00:15:25,800
And I have a term that is
curl of z hat psi squared.

190
00:15:29,180 --> 00:15:29,680
OK?

191
00:15:36,000 --> 00:15:39,470
Again, if you think
of this as vector,

192
00:15:39,470 --> 00:15:41,450
this is a vector
whose components

193
00:15:41,450 --> 00:15:49,740
are the xy and the
y phi, whereas this

194
00:15:49,740 --> 00:15:53,450
is a vector whose
components are, let's say,

195
00:15:53,450 --> 00:15:57,380
dy psi minus dx psi.

196
00:15:57,380 --> 00:16:01,340
Because of the curl operation--
the x and y components-- one

197
00:16:01,340 --> 00:16:02,470
of them gets a minus sign.

198
00:16:02,470 --> 00:16:05,950
Maybe I got the minus
wrong, but it's essentially

199
00:16:05,950 --> 00:16:06,650
that structure.

200
00:16:09,320 --> 00:16:14,880
Now you can see that if I were
to do the integration here,

201
00:16:14,880 --> 00:16:18,220
there is a dx psi, dy psi.

202
00:16:18,220 --> 00:16:22,310
I can do that integration
by parts and have,

203
00:16:22,310 --> 00:16:26,340
let's say phi dx dy psi.

204
00:16:26,340 --> 00:16:29,150
And then, I can do the same
integration by parts here.

205
00:16:29,150 --> 00:16:33,270
And I will have phi
minus dx dy psi.

206
00:16:33,270 --> 00:16:37,230
So if I do integration by
part, this will disappear.

207
00:16:37,230 --> 00:16:41,260
Another way of seeing
that is that the gradient

208
00:16:41,260 --> 00:16:42,990
will act on the curl.

209
00:16:42,990 --> 00:16:45,570
And the gradient of the
curl of a vector is 0,

210
00:16:45,570 --> 00:16:48,830
or otherwise, the curl
will act on the gradient

211
00:16:48,830 --> 00:16:50,150
with [INAUDIBLE].

212
00:16:50,150 --> 00:16:56,890
So basically, this term
does not contribute.

213
00:16:56,890 --> 00:17:03,360
And the contribution of
this part and the part

214
00:17:03,360 --> 00:17:08,069
from topological defects are
decoupled from each other.

215
00:17:08,069 --> 00:17:11,869
So there's essentially
the Gaussian type of stuff

216
00:17:11,869 --> 00:17:14,579
that we calculated
before is here.

217
00:17:14,579 --> 00:17:18,150
On top of that,
there is this part

218
00:17:18,150 --> 00:17:20,969
that is due to these
topological defects.

219
00:17:23,849 --> 00:17:29,170
Again, this vector
is this squared.

220
00:17:29,170 --> 00:17:30,950
You can see that
if I square it, I

221
00:17:30,950 --> 00:17:34,980
will get dy psi squared
plus dx psi squared.

222
00:17:34,980 --> 00:17:37,580
So to all intents and
purposes, this thing

223
00:17:37,580 --> 00:17:44,050
is the same thing as a
gradient of psi squared.

224
00:17:44,050 --> 00:17:46,510
Essentially, gradient
of psi and curl of psi

225
00:17:46,510 --> 00:17:49,980
are the same vector, just
rotated by 90 degrees.

226
00:17:49,980 --> 00:17:52,890
Integrating the square of
one over the whole space

227
00:17:52,890 --> 00:17:56,430
is the same as integrating
the square of the other.

228
00:17:56,430 --> 00:17:58,430
OK?

229
00:17:58,430 --> 00:18:04,020
So now, let's calculate this
contribution and what it is.

230
00:18:04,020 --> 00:18:11,250
Integral d2 x gradient of
psi squared with K over 2

231
00:18:11,250 --> 00:18:16,280
out front-- actually
you already know that.

232
00:18:16,280 --> 00:18:22,590
Because psi we see
is the potential

233
00:18:22,590 --> 00:18:23,945
due to a bunch of charges.

234
00:18:26,690 --> 00:18:29,630
So this is essentially
the electric field

235
00:18:29,630 --> 00:18:32,370
due to this
combination of charges

236
00:18:32,370 --> 00:18:34,760
integrated over
the entire space.

237
00:18:34,760 --> 00:18:37,530
It's the electrostatic image.

238
00:18:37,530 --> 00:18:41,090
But let's go through
that step by step.

239
00:18:41,090 --> 00:18:45,320
Let's do the
integration by parts.

240
00:18:45,320 --> 00:18:49,790
So this becomes minus k over 2.

241
00:18:49,790 --> 00:18:57,900
Integral d2 x psi the
gradient acting on this

242
00:18:57,900 --> 00:19:01,500
will give me Laplacian of psi.

243
00:19:01,500 --> 00:19:06,250
Of course, whenever I
do integration by parts,

244
00:19:06,250 --> 00:19:08,620
I have to worry
about boundary terms.

245
00:19:15,360 --> 00:19:17,550
And essentially, if
you think of what

246
00:19:17,550 --> 00:19:21,260
you will be seeing
at the boundary, far,

247
00:19:21,260 --> 00:19:26,260
far away from there all of
these charges are, let's say.

248
00:19:26,260 --> 00:19:30,120
Essentially, you will
see the electric field

249
00:19:30,120 --> 00:19:34,910
due to the combination
of all of those charges.

250
00:19:34,910 --> 00:19:42,170
So for a single one, I will
have a large electric field

251
00:19:42,170 --> 00:19:44,750
that will go as 1 over r.

252
00:19:44,750 --> 00:19:48,480
And we saw that integrating
that will give me the log.

253
00:19:48,480 --> 00:19:50,990
So that was not
particularly nice.

254
00:19:50,990 --> 00:19:54,390
So similarly, what
these boundary terms

255
00:19:54,390 --> 00:19:58,120
would amount to would
give you some kind

256
00:19:58,120 --> 00:20:01,470
of a logarithmic energy that
depends on the next charge

257
00:20:01,470 --> 00:20:03,470
that you have enclosed.

258
00:20:03,470 --> 00:20:17,070
And you can get rid of it
by setting the next charge

259
00:20:17,070 --> 00:20:18,225
to be zero.

260
00:20:23,050 --> 00:20:28,390
So essentially,
any configuration

261
00:20:28,390 --> 00:20:33,930
in which the sum total of our
topological charges is non-zero

262
00:20:33,930 --> 00:20:38,270
will get a huge energy cost
as we go to large distances

263
00:20:38,270 --> 00:20:40,140
from the self-energy,
if you like,

264
00:20:40,140 --> 00:20:42,980
of creating this huge monopole.

265
00:20:42,980 --> 00:20:45,600
So we are going to
use this condition

266
00:20:45,600 --> 00:20:48,140
and focus only on
configurations that

267
00:20:48,140 --> 00:20:52,390
are charged topological
charge neutral.

268
00:20:52,390 --> 00:20:53,530
OK.

269
00:20:53,530 --> 00:20:59,050
Now our psi is what
I have over here.

270
00:20:59,050 --> 00:21:03,950
It is sum over i qi.

271
00:21:03,950 --> 00:21:06,930
This Coulomb
interaction-- r minus ri.

272
00:21:10,740 --> 00:21:18,560
And therefore, Laplacian
of psi is essentially

273
00:21:18,560 --> 00:21:21,300
taking the Laplacian
of this expression

274
00:21:21,300 --> 00:21:25,590
is the sum over j qu.

275
00:21:25,590 --> 00:21:28,395
Laplacian of this is
the delta function.

276
00:21:33,150 --> 00:21:35,700
So basically, that
was the condition

277
00:21:35,700 --> 00:21:37,100
for the Coulomb potential.

278
00:21:37,100 --> 00:21:40,900
Or alternatively, you take
2 derivative of the log,

279
00:21:40,900 --> 00:21:45,390
and you will generate
the delta function.

280
00:21:45,390 --> 00:21:47,430
OK?

281
00:21:47,430 --> 00:21:56,990
So, what you see is that
you generate the following.

282
00:21:56,990 --> 00:22:12,210
You will get a minus k over 2
sum over pairs i and j, qi, qj.

283
00:22:12,210 --> 00:22:15,610
And then I have the
integral over x or r--

284
00:22:15,610 --> 00:22:17,240
they're basically
the same thing.

285
00:22:17,240 --> 00:22:18,985
Maybe I should have
written this as x.

286
00:22:23,860 --> 00:22:29,710
And the delta function
insures that x

287
00:22:29,710 --> 00:22:33,462
is set to be the other i.

288
00:22:33,462 --> 00:22:35,890
So I will get the
Coulomb interaction

289
00:22:35,890 --> 00:22:40,010
between ri minus rj.

290
00:22:40,010 --> 00:22:46,710
So basically, what you have is
that these topological defects

291
00:22:46,710 --> 00:22:51,730
that are characterized by these
integers n, or by the charges

292
00:22:51,730 --> 00:22:56,540
2 pi n, have exactly
this logarithmic Coulomb

293
00:22:56,540 --> 00:22:57,500
interaction.

294
00:22:57,500 --> 00:22:58,770
in two dimensions.

295
00:22:58,770 --> 00:23:01,750
And as I said, this
thing is none other

296
00:23:01,750 --> 00:23:03,990
than the electrostatic energy.

297
00:23:03,990 --> 00:23:06,180
The electrostatic energy
you can write either

298
00:23:06,180 --> 00:23:08,930
as an integral of the
electric field squared.

299
00:23:08,930 --> 00:23:11,640
Or you can write
as the interaction

300
00:23:11,640 --> 00:23:16,900
among the charges that give
rise to that electric field.

301
00:23:16,900 --> 00:23:19,620
OK?

302
00:23:19,620 --> 00:23:26,990
So, what I can do is I
can write this as follows.

303
00:23:26,990 --> 00:23:32,160
First of all, I can maybe
re-cast it in terms of the n's.

304
00:23:32,160 --> 00:23:36,820
So I will have 2 pi ni, 2 pi nj.

305
00:23:36,820 --> 00:23:43,330
So I will get minus
4pi squared k.

306
00:23:43,330 --> 00:23:46,010
There's a factor of one-half.

307
00:23:46,010 --> 00:23:49,870
But this is a sum over i
and j-- so every pair is now

308
00:23:49,870 --> 00:23:51,700
counted twice.

309
00:23:51,700 --> 00:23:54,080
So I get rid of that
factor of one-half

310
00:23:54,080 --> 00:24:00,180
by essentially counting
each pair only once.

311
00:24:00,180 --> 00:24:04,450
So I have the
Coulomb interaction

312
00:24:04,450 --> 00:24:08,610
between ri and rj, which
is this 1 over 2 pi

313
00:24:08,610 --> 00:24:13,390
log of ri minus rj
with some cut-off.

314
00:24:13,390 --> 00:24:17,320
And then, there's the term that
corresponds to i equals 2j.

315
00:24:17,320 --> 00:24:30,340
So I will have a minus, let's
say, 4pi squared k sum over i.

316
00:24:30,340 --> 00:24:33,040
And I forgot here
to put n, i, and j.

317
00:24:36,470 --> 00:24:39,730
I will have ni squared.

318
00:24:39,730 --> 00:24:43,334
The Coulomb interaction at zero.

319
00:24:43,334 --> 00:24:44,250
ri equals [INAUDIBLE].

320
00:24:46,970 --> 00:24:53,240
Now clearly, this expression
does not make sense.

321
00:24:53,240 --> 00:24:58,640
What it is trying to
tell me is that there

322
00:24:58,640 --> 00:25:06,340
is a cost to creating one of
these topological charges.

323
00:25:06,340 --> 00:25:11,810
And all of this theory--
again, in order to make sense,

324
00:25:11,810 --> 00:25:15,220
we should remember to put
some kind of a short distance

325
00:25:15,220 --> 00:25:16,670
cut-off a.

326
00:25:16,670 --> 00:25:18,550
All right?

327
00:25:18,550 --> 00:25:26,110
And basically, replacing this
original discrete lattice

328
00:25:26,110 --> 00:25:31,040
with a continuum will
only work as long as

329
00:25:31,040 --> 00:25:35,830
I keep in mind that I cannot
regard things at the level

330
00:25:35,830 --> 00:25:40,640
of lattice spacing, and replace
it by that formula, as we saw,

331
00:25:40,640 --> 00:25:41,800
for example, here.

332
00:25:41,800 --> 00:25:45,340
If I want to draw a
topological defect,

333
00:25:45,340 --> 00:25:48,430
I would need right at the
center to do something

334
00:25:48,430 --> 00:25:52,630
like this-- where replacing
the cosines with the gradient

335
00:25:52,630 --> 00:25:55,120
squared kind of
doesn't make sense.

336
00:25:55,120 --> 00:25:57,030
So basically, what
this theory is

337
00:25:57,030 --> 00:26:03,440
telling me is that once you
get to a very small distance,

338
00:26:03,440 --> 00:26:08,550
you have to keep
track of the existence

339
00:26:08,550 --> 00:26:13,520
of some underlying lattice
and the corresponding things.

340
00:26:13,520 --> 00:26:16,270
And what's really
this is describing

341
00:26:16,270 --> 00:26:22,720
for you is the core
energy of creating

342
00:26:22,720 --> 00:26:26,670
a defect that has object ni.

343
00:26:26,670 --> 00:26:27,470
OK.

344
00:26:27,470 --> 00:26:37,660
What do I mean by that
is that over here, I

345
00:26:37,660 --> 00:26:43,485
can calculate what the partition
function is for one defect.

346
00:26:43,485 --> 00:26:46,550
This we already did
last time around.

347
00:26:46,550 --> 00:26:50,780
And for that, I can
integrate out this energy

348
00:26:50,780 --> 00:26:56,650
that I have for the distortions.

349
00:26:56,650 --> 00:27:02,470
It's an integral of
n over r squared.

350
00:27:02,470 --> 00:27:07,980
And this integration
gave me this factor of e

351
00:27:07,980 --> 00:27:20,752
to the minus pi k log of r over
a-- actually, that was 2 pi k.

352
00:27:20,752 --> 00:27:25,380
Actually, let's do
this correctly once.

353
00:27:29,320 --> 00:27:32,100
I should have done it
earlier, and I forgot.

354
00:27:32,100 --> 00:27:37,130
So you have one defect.

355
00:27:37,130 --> 00:27:45,460
And we saw that for one defect,
the field at the distance r

356
00:27:45,460 --> 00:27:50,140
has n over r in magnitude.

357
00:27:53,830 --> 00:28:00,370
And then, the net energy cost
for one of these defects--

358
00:28:00,370 --> 00:28:05,120
if I say that I believe
this formula starting

359
00:28:05,120 --> 00:28:14,380
from a distance a is the
k over 2 integral from a.

360
00:28:14,380 --> 00:28:17,925
Let's say all the way up
to the size of my system.

361
00:28:20,780 --> 00:28:29,630
I have 2 pi r dr from
a shell at radius r

362
00:28:29,630 --> 00:28:32,880
magnitude of this u squared.

363
00:28:32,880 --> 00:28:37,830
So I have n squared
over r squared.

364
00:28:37,830 --> 00:28:41,860
But then, I have to worry
about all of the actual things

365
00:28:41,860 --> 00:28:45,320
that I have off the distance a.

366
00:28:45,320 --> 00:28:50,510
So on top of this,
there is a core energy

367
00:28:50,510 --> 00:28:56,760
for creating this object that
certainly explicitly depends on

368
00:28:56,760 --> 00:29:00,320
where I sit this parameter a.

369
00:29:00,320 --> 00:29:02,120
OK?

370
00:29:02,120 --> 00:29:06,740
This part is easy.

371
00:29:06,740 --> 00:29:13,510
It simply gives
me pi km squared.

372
00:29:13,510 --> 00:29:16,200
And then I have
the integral of 1

373
00:29:16,200 --> 00:29:21,340
over r, which gives
me log of L over a.

374
00:29:26,290 --> 00:29:31,790
So if I want to imagine what
the partition function of this

375
00:29:31,790 --> 00:29:38,560
is-- one defect in
a system of size L--

376
00:29:38,560 --> 00:29:46,160
I would say that z of one defect
is Boltzmann weight responding

377
00:29:46,160 --> 00:29:47,660
to creating this entity.

378
00:29:47,660 --> 00:29:54,570
So I have e to the minus pi
k m squared log of L over a.

379
00:29:54,570 --> 00:30:02,360
And then I have the core energy
that corresponds to this.

380
00:30:02,360 --> 00:30:07,190
And then, as we discussed,
I can place this anywhere

381
00:30:07,190 --> 00:30:10,750
in the system if I'm calculating
the partition function.

382
00:30:10,750 --> 00:30:14,130
So there's an integration
over the position

383
00:30:14,130 --> 00:30:16,570
of this that is implicit.

384
00:30:16,570 --> 00:30:19,480
And so, that's going
to give me the square

385
00:30:19,480 --> 00:30:27,770
of the size of my system,
except that I am unsure as

386
00:30:27,770 --> 00:30:33,330
to where I have placed
things up to this cut-off a.

387
00:30:33,330 --> 00:30:37,310
So really, the
number of distinct

388
00:30:37,310 --> 00:30:43,710
positions that I have scales
like L over a squared.

389
00:30:43,710 --> 00:30:48,870
So the whole thing we
can see scales like L

390
00:30:48,870 --> 00:30:52,790
over a 2 minus pi km squared.

391
00:30:55,470 --> 00:31:03,660
And then, there is this
factor of e to the minus

392
00:31:03,660 --> 00:31:08,180
this core energy
evaluated at the distance

393
00:31:08,180 --> 00:31:10,250
a that I will call y.

394
00:31:13,550 --> 00:31:15,480
Because again, in
some sense, there's

395
00:31:15,480 --> 00:31:19,280
some arbitrariness
in where I choose a.

396
00:31:19,280 --> 00:31:24,590
So this y would be a function of
a, if we depend on that choice.

397
00:31:24,590 --> 00:31:28,190
But the most important
thing is that if I

398
00:31:28,190 --> 00:31:34,830
have a huge system, whether or
not this partition function,

399
00:31:34,830 --> 00:31:38,630
as a function of the size of
the system, goes to infinity

400
00:31:38,630 --> 00:31:45,470
or goes to 0 is controlled by
this exponent 2 minus pi k.

401
00:31:45,470 --> 00:31:49,040
Let's say we focus on the
simplest of topological defects

402
00:31:49,040 --> 00:31:53,030
corresponding to n
equal 2 minus plus 1.

403
00:31:53,030 --> 00:31:56,650
You expect that there is some
potentially critical value

404
00:31:56,650 --> 00:32:02,940
of k, which is 2 over pi,
that distinguishes the two

405
00:32:02,940 --> 00:32:06,830
types of behavior.

406
00:32:06,830 --> 00:32:09,460
OK?

407
00:32:09,460 --> 00:32:16,060
But this picture is nice,
but certainly incomplete.

408
00:32:16,060 --> 00:32:22,270
Because who said that there's
any legitimacy in calculating

409
00:32:22,270 --> 00:32:24,390
the partition function
that corresponds

410
00:32:24,390 --> 00:32:27,700
to just a single
topological defect.

411
00:32:27,700 --> 00:32:31,680
If I integrate over all the
configurations of my angle

412
00:32:31,680 --> 00:32:38,280
field, I should really
be doing something

413
00:32:38,280 --> 00:32:43,700
that is analogous to this
and calculating a partition

414
00:32:43,700 --> 00:32:47,270
function that corresponds
to many defects.

415
00:32:47,270 --> 00:32:52,080
And actually what I calculated
over here was in some sense

416
00:32:52,080 --> 00:32:54,960
the configuration of
spins given that there

417
00:32:54,960 --> 00:32:59,170
is a topological defects
that has the lowest energy.

418
00:32:59,170 --> 00:33:01,900
Once I start with this
configuration-- let's

419
00:33:01,900 --> 00:33:04,270
say, with everybody
radiating out--

420
00:33:04,270 --> 00:33:07,620
I can start to distort
them a little bit which

421
00:33:07,620 --> 00:33:13,060
amounts to adding this
gradient of phi to that.

422
00:33:13,060 --> 00:33:15,940
So really, the
partition function

423
00:33:15,940 --> 00:33:19,900
that I want to calculate and
wrote down at the beginning--

424
00:33:19,900 --> 00:33:23,680
if I want to
calculate correctly,

425
00:33:23,680 --> 00:33:28,970
I have to include both
these fluctuations

426
00:33:28,970 --> 00:33:31,900
and these fluctuations
corresponding

427
00:33:31,900 --> 00:33:38,910
to an arbitrary set of
these topological defects.

428
00:33:38,910 --> 00:33:44,820
And what we see is that
actually, the partition

429
00:33:44,820 --> 00:33:49,460
functions and the energy
costs of the two components

430
00:33:49,460 --> 00:33:51,200
really separate out.

431
00:33:51,200 --> 00:33:53,360
And what we are
trying to calculate

432
00:33:53,360 --> 00:33:58,000
is the contribution that is
due to the topological defects.

433
00:33:58,000 --> 00:34:02,400
And what we see is
that once I tell you

434
00:34:02,400 --> 00:34:06,500
where the topological
defects are located,

435
00:34:06,500 --> 00:34:12,790
the partition function for them
has an energy component that

436
00:34:12,790 --> 00:34:16,560
is this Coulomb interaction
among the defect.

437
00:34:16,560 --> 00:34:21,320
But there is a
part that really is

438
00:34:21,320 --> 00:34:28,400
a remnant of this
core energy that we

439
00:34:28,400 --> 00:34:31,610
were calculating before.

440
00:34:31,610 --> 00:34:35,969
So when I was sort of
following my nose here,

441
00:34:35,969 --> 00:34:39,949
I had forgotten a little
bit about the short distance

442
00:34:39,949 --> 00:34:40,940
cut-off.

443
00:34:40,940 --> 00:34:44,739
And then, when I
encountered this C of zero,

444
00:34:44,739 --> 00:34:48,690
it told me that I have
to think about the limit

445
00:34:48,690 --> 00:34:51,380
when two things come
close to each other.

446
00:34:51,380 --> 00:34:56,360
And I know that that
limit is constrained

447
00:34:56,360 --> 00:34:59,190
by my original lattice,
and more importantly,

448
00:34:59,190 --> 00:35:02,560
by the place where
I am willing to do

449
00:35:02,560 --> 00:35:10,140
self-averaging and replace
this sum with a gradient.

450
00:35:10,140 --> 00:35:15,740
OK, so basically, this is
the explanation of this term.

451
00:35:15,740 --> 00:35:18,280
So the only thing that we
have established so far

452
00:35:18,280 --> 00:35:22,950
is that this partition
function that I wrote down

453
00:35:22,950 --> 00:35:28,530
at the beginning gets
decomposed into a part

454
00:35:28,530 --> 00:35:33,130
that we have calculated before,
which was the Gaussian term,

455
00:35:33,130 --> 00:35:38,540
and is caused, really, the
contribution due to spin waves.

456
00:35:38,540 --> 00:35:44,340
So this is when we just consider
these [INAUDIBLE] modes,

457
00:35:44,340 --> 00:35:47,470
we said that essentially
you can have an energy

458
00:35:47,470 --> 00:35:49,400
cost that is the
gradient squared.

459
00:35:49,400 --> 00:35:54,900
So this is the part that
corresponds to integral d phi

460
00:35:54,900 --> 00:36:00,150
into the minus k over 2 integral
d2 x gradient of phi squared,

461
00:36:00,150 --> 00:36:03,305
where phi is a well-behaved,
ordinary function.

462
00:36:11,300 --> 00:36:18,970
And what we find is that the
actual partition function also

463
00:36:18,970 --> 00:36:25,100
has the contribution from
the topological defects.

464
00:36:25,100 --> 00:36:31,670
And that I will indicate by ZQ.

465
00:36:31,670 --> 00:36:34,250
And Q stands for Coulomb gas.

466
00:36:39,260 --> 00:36:46,340
Because this partition
function Z sub Q

467
00:36:46,340 --> 00:36:53,550
is like I'm trying to calculate
this system of degrees

468
00:36:53,550 --> 00:37:00,080
of freedom that are
characterized by charges n that

469
00:37:00,080 --> 00:37:04,270
can be anywhere in this
two-dimensional space.

470
00:37:04,270 --> 00:37:08,300
And the interaction between
them is governed by the Coulomb

471
00:37:08,300 --> 00:37:10,990
interaction in two dimensions.

472
00:37:10,990 --> 00:37:15,640
So to calculate this,
I have to sum over

473
00:37:15,640 --> 00:37:19,040
all configuration of charges.

474
00:37:22,110 --> 00:37:26,520
The number of these charges
could be zero, could be two,

475
00:37:26,520 --> 00:37:30,590
could be four, could be
six, could be any number.

476
00:37:30,590 --> 00:37:33,540
But I say even
numbers because I want

477
00:37:33,540 --> 00:37:37,040
to maintain the
constraint of neutrality.

478
00:37:37,040 --> 00:37:39,450
Sum over ni should be zero.

479
00:37:39,450 --> 00:37:42,310
So I want to do that
constrained sum.

480
00:37:42,310 --> 00:37:47,410
So I only want to look at
neutral configurations.

481
00:37:47,410 --> 00:37:49,530
Once I have
specified-- let's say

482
00:37:49,530 --> 00:37:57,310
that I have eight charges--
four plus and four minus-- well,

483
00:37:57,310 --> 00:38:01,540
there is a term that is
going to come from here

484
00:38:01,540 --> 00:38:06,170
and I kind of said that the
exponential of this term I'm

485
00:38:06,170 --> 00:38:08,700
going to call y.

486
00:38:08,700 --> 00:38:13,110
So I have essentially
y raised to the power

487
00:38:13,110 --> 00:38:15,630
of the number of charges.

488
00:38:15,630 --> 00:38:19,190
Let's call this sum
over i ni squared.

489
00:38:19,190 --> 00:38:21,730
And I'm actually just
going to constrain ni

490
00:38:21,730 --> 00:38:23,480
to be minus plus 1.

491
00:38:23,480 --> 00:38:27,300
I'm going to only look
at these primary charges.

492
00:38:27,300 --> 00:38:29,930
So the sum over i
and i squared is just

493
00:38:29,930 --> 00:38:32,970
the total number of
charges irrespective

494
00:38:32,970 --> 00:38:35,135
of whether they
are plus or minus.

495
00:38:35,135 --> 00:38:39,390
It basically is replacing this.

496
00:38:39,390 --> 00:38:44,610
And then, I have
to integrate over

497
00:38:44,610 --> 00:38:46,880
the positions of these charges.

498
00:38:46,880 --> 00:38:49,570
Let's call this total number n.

499
00:38:49,570 --> 00:38:56,400
So I have to
integrate i1 2n d2 xi

500
00:38:56,400 --> 00:39:00,110
the position of
where this charge is

501
00:39:00,110 --> 00:39:05,440
and then interaction which
is exponential of minus

502
00:39:05,440 --> 00:39:11,610
4phi squared k sum
over i less than j.

503
00:39:11,610 --> 00:39:17,130
And i and j the Coulomb
interaction between location

504
00:39:17,130 --> 00:39:18,810
let's say xi and xj.

505
00:39:27,580 --> 00:39:32,240
Actually, I want to also
emphasize that throughout,

506
00:39:32,240 --> 00:39:33,770
I have this cut-off.

507
00:39:33,770 --> 00:39:36,790
So when I was
integrating over one,

508
00:39:36,790 --> 00:39:38,990
I said that the number
of positions that I had

509
00:39:38,990 --> 00:39:42,760
was not L squared, but L
over a squared to make it

510
00:39:42,760 --> 00:39:44,500
dimensionless.

511
00:39:44,500 --> 00:39:47,505
I will similarly make these
interactions dimensioned

512
00:39:47,505 --> 00:39:49,180
as I divide by a squared.

513
00:39:53,310 --> 00:40:00,580
And so basically, this is
the more interesting thing

514
00:40:00,580 --> 00:40:02,220
that we want to calculate.

515
00:40:08,172 --> 00:40:12,080
Also, again, remember I wrote
this a squared down here,

516
00:40:12,080 --> 00:40:19,750
also to emphasize that
within this expression,

517
00:40:19,750 --> 00:40:22,130
the minimal separation
that I'm going

518
00:40:22,130 --> 00:40:26,720
to allow between any pair of
charges is off the order of a.

519
00:40:26,720 --> 00:40:33,490
I have integrated out or moved
into some continuum description

520
00:40:33,490 --> 00:40:38,055
any configuration in which
the topological charges are

521
00:40:38,055 --> 00:40:39,908
less than distance a.

522
00:40:39,908 --> 00:40:40,408
OK?

523
00:40:43,800 --> 00:40:45,998
Yes?

524
00:40:45,998 --> 00:40:48,820
AUDIENCE: Essentially,
when we were

525
00:40:48,820 --> 00:40:53,656
during [INAUDIBLE] it
was canonical potential,

526
00:40:53,656 --> 00:40:57,002
[INAUDIBLE], canonical ensemble.

527
00:40:57,002 --> 00:41:00,030
And this is more like
grand canonical ensemble?

528
00:41:00,030 --> 00:41:00,660
PROFESSOR: Yes.

529
00:41:00,660 --> 00:41:07,225
So, as far as the original
two-dimensional xy model

530
00:41:07,225 --> 00:41:11,310
is concerned, I'm calculating
a canonical partition function

531
00:41:11,310 --> 00:41:14,820
for this spin or angle
degrees of freedom.

532
00:41:14,820 --> 00:41:20,590
And I find that that
integration over spin angle

533
00:41:20,590 --> 00:41:23,830
degrees of freedom
can be decomposed

534
00:41:23,830 --> 00:41:30,000
into a Gaussian part and a part
that as you correctly point out

535
00:41:30,000 --> 00:41:36,670
corresponds to a grand
canonical system of charges.

536
00:41:36,670 --> 00:41:38,720
So the number of
charges that are

537
00:41:38,720 --> 00:41:41,910
going to appear in
the system I have not

538
00:41:41,910 --> 00:41:45,630
specified whether it is
determined implicitly

539
00:41:45,630 --> 00:41:49,211
by how strong these
parameter was.

540
00:41:49,211 --> 00:41:55,175
AUDIENCE: [INAUDIBLE]
of canonical potential?

541
00:41:55,175 --> 00:41:58,080
PROFESSOR: y plays the
role of E to the beta mu.

542
00:42:02,700 --> 00:42:05,650
The quantity that
in 8333 we were

543
00:42:05,650 --> 00:42:09,750
writing as z-- E to
the beta mu small z.

544
00:42:12,743 --> 00:42:13,243
OK?

545
00:42:19,240 --> 00:42:26,990
So, we thought we were
solving the xy model.

546
00:42:26,990 --> 00:42:32,090
We ended up, indeed, with
this grand canonical system,

547
00:42:32,090 --> 00:42:36,240
which is currently
parametrized by two things.

548
00:42:36,240 --> 00:42:43,080
One is this k, which is this
strength of the potential.

549
00:42:43,080 --> 00:42:46,260
The other is this y.

550
00:42:46,260 --> 00:42:49,590
Of course, since this
system originally

551
00:42:49,590 --> 00:42:54,180
came from an xy model that
went only one parameter,

552
00:42:54,180 --> 00:42:59,300
I expect this y to
also be related to k.

553
00:42:59,300 --> 00:43:03,350
But just as an expression,
we can certainly

554
00:43:03,350 --> 00:43:06,770
regard it as a system
that is parametrized

555
00:43:06,770 --> 00:43:10,260
by two things-- the k and the y.

556
00:43:10,260 --> 00:43:12,120
For the case of
the xy model, there

557
00:43:12,120 --> 00:43:15,530
will be some additional
constraint between the two.

558
00:43:15,530 --> 00:43:19,900
But more generally, we can look
at this system with its two

559
00:43:19,900 --> 00:43:21,460
parameters.

560
00:43:21,460 --> 00:43:29,110
And essentially, we will try
to make an expansion in y.

561
00:43:29,110 --> 00:43:33,360
You'll say that,
OK, presumable, I

562
00:43:33,360 --> 00:43:38,700
know what is going to happen
when y is very, very small.

563
00:43:38,700 --> 00:43:47,210
Because then, in the system I
will create only a few charges.

564
00:43:47,210 --> 00:43:50,110
If I create many
charges, I'm going

565
00:43:50,110 --> 00:43:54,580
to penalize by more
and more factors of y.

566
00:43:54,580 --> 00:43:57,420
So maybe through leading
order, the system

567
00:43:57,420 --> 00:43:59,570
would be free of charge.

568
00:43:59,570 --> 00:44:01,410
And then, there
would be a few pairs

569
00:44:01,410 --> 00:44:03,800
that would appear
here and there.

570
00:44:03,800 --> 00:44:06,600
In fact, there should be
a small density of them,

571
00:44:06,600 --> 00:44:09,870
even no matter how
small I make y.

572
00:44:09,870 --> 00:44:12,170
There will be a
very small density

573
00:44:12,170 --> 00:44:14,310
of these things
that will appear.

574
00:44:14,310 --> 00:44:18,090
And presumably, these
things will always

575
00:44:18,090 --> 00:44:20,490
appear close to each other.

576
00:44:20,490 --> 00:44:26,270
So I will have lots and
lots of these pairs-- well,

577
00:44:26,270 --> 00:44:27,960
not lots and lots
of these pairs--

578
00:44:27,960 --> 00:44:33,270
a density of them that is
controlled by how big y is.

579
00:44:33,270 --> 00:44:42,470
And as I make y larger-- so
this is y becoming larger-- then

580
00:44:42,470 --> 00:44:46,930
presumably, I will generate
more and more of these pairs.

581
00:44:46,930 --> 00:44:50,170
And once I have more
and more of these pairs,

582
00:44:50,170 --> 00:44:54,480
they could, in principle,
get into each other's way.

583
00:44:54,480 --> 00:44:56,530
And when they get
into each other's way,

584
00:44:56,530 --> 00:44:59,890
then it's not clear who
is paired with whom.

585
00:44:59,890 --> 00:45:06,450
And at some point, I
should trade my picture

586
00:45:06,450 --> 00:45:11,240
of having a gas of
pairs of these objects

587
00:45:11,240 --> 00:45:14,340
to a plasma of charges,
plus and minus,

588
00:45:14,340 --> 00:45:18,380
that are moving
all over the place.

589
00:45:18,380 --> 00:45:23,650
So as I tune this parameter
y, I expect my system

590
00:45:23,650 --> 00:45:29,840
to go from a low density phase
of atoms of plus-minus bound

591
00:45:29,840 --> 00:45:34,470
to each other to a high
density phase where

592
00:45:34,470 --> 00:45:36,950
I have a plasma of
plus and minuses

593
00:45:36,950 --> 00:45:38,230
moving all over the place.

594
00:45:38,230 --> 00:45:39,030
Yes?

595
00:45:39,030 --> 00:45:41,965
AUDIENCE: So, y is related
to the core energy.

596
00:45:41,965 --> 00:45:42,800
PROFESSOR: Yes.

597
00:45:42,800 --> 00:45:46,370
AUDIENCE: And core energy is
defined through [INAUDIBLE]

598
00:45:46,370 --> 00:45:49,960
direction at zero separation--

599
00:45:49,960 --> 00:45:51,890
PROFESSOR: Well, no.

600
00:45:51,890 --> 00:45:54,590
Because the Coulomb
description is only

601
00:45:54,590 --> 00:45:57,290
valid large separations.

602
00:45:57,290 --> 00:46:02,550
When I get to short distances,
who knows what's going on?

603
00:46:02,550 --> 00:46:07,100
So there is some underlying
microscopic picture

604
00:46:07,100 --> 00:46:10,320
that determines what
the core energy is.

605
00:46:10,320 --> 00:46:12,800
Very roughly, yes,
you would expect

606
00:46:12,800 --> 00:46:16,850
it to have a form that
is of e to the minus k

607
00:46:16,850 --> 00:46:19,710
with some coefficient
that comes from adding

608
00:46:19,710 --> 00:46:21,040
all of those interactions here.

609
00:46:23,610 --> 00:46:24,718
Yes?

610
00:46:24,718 --> 00:46:26,570
AUDIENCE: Just based
on the sign convention,

611
00:46:26,570 --> 00:46:30,199
you're saying if you
increase or decrease y,

612
00:46:30,199 --> 00:46:32,011
that it will go
from a low density--

613
00:46:32,011 --> 00:46:36,200
PROFESSOR: OK, so y is the
exponential of something.

614
00:46:36,200 --> 00:46:42,010
y equals to zero means I will
not create any of these things.

615
00:46:42,010 --> 00:46:45,800
y approaching 1-- I will
create a lot of them.

616
00:46:45,800 --> 00:46:48,060
There's no cost at
y equals to one.

617
00:46:48,060 --> 00:46:50,360
There's no core energy.

618
00:46:50,360 --> 00:46:52,730
I can create them as I want.

619
00:46:52,730 --> 00:46:56,814
AUDIENCE: So this would be
like y equals minus epsilon c.

620
00:46:56,814 --> 00:46:57,730
Is that right?

621
00:46:57,730 --> 00:46:59,610
PROFESSOR: Yeah.

622
00:46:59,610 --> 00:47:00,510
Didn't I have that?

623
00:47:00,510 --> 00:47:03,650
You see in the exponential
it is with the minus.

624
00:47:10,290 --> 00:47:10,790
OK.

625
00:47:10,790 --> 00:47:13,190
But in any case, that
is the expectation.

626
00:47:13,190 --> 00:47:14,010
Right?

627
00:47:14,010 --> 00:47:17,330
So I expect that
when I calculate,

628
00:47:17,330 --> 00:47:19,020
I create one of these defects.

629
00:47:19,020 --> 00:47:23,730
There is an energy cost
which is mostly from outside.

630
00:47:23,730 --> 00:47:27,790
And then, there's an
additional piece on the inside.

631
00:47:27,790 --> 00:47:30,247
So the exponential of
that additional piece

632
00:47:30,247 --> 00:47:31,830
would be a number
that is less than 1.

633
00:47:31,830 --> 00:47:32,734
AUDIENCE: [INAUDIBLE]

634
00:47:43,684 --> 00:47:44,350
PROFESSOR: Yeah.

635
00:47:44,350 --> 00:47:48,260
I mean, the original model
has some particular form.

636
00:47:48,260 --> 00:47:51,240
And actually, the interactions
of the original model, I

637
00:47:51,240 --> 00:47:52,730
can make more complicated.

638
00:47:52,730 --> 00:47:57,030
I can add the full spin
interaction, for example.

639
00:47:57,030 --> 00:48:01,480
It doesn't affect the
overall form much,

640
00:48:01,480 --> 00:48:05,120
just modifies what
an effective k is,

641
00:48:05,120 --> 00:48:08,175
and what the core
energy is independent.

642
00:48:11,420 --> 00:48:11,920
OK?

643
00:48:14,500 --> 00:48:15,380
All right.

644
00:48:15,380 --> 00:48:19,850
But the key point is that
this system potentially

645
00:48:19,850 --> 00:48:25,520
has a phase transition as you
change the parameter of y.

646
00:48:25,520 --> 00:48:29,270
And another way of
looking at this transition

647
00:48:29,270 --> 00:48:35,460
is that what is happening
here in different languages,

648
00:48:35,460 --> 00:48:38,680
you can either call it
insulator or a dielectric.

649
00:48:42,840 --> 00:48:45,830
But what is happening here
in different languages,

650
00:48:45,830 --> 00:48:50,780
you can either call, say,
a metal or, as I said,

651
00:48:50,780 --> 00:48:51,550
maybe a plasma.

652
00:48:54,616 --> 00:48:59,960
The point is that here
you have free charges.

653
00:48:59,960 --> 00:49:03,670
Here you have bound
pairs of charges.

654
00:49:03,670 --> 00:49:08,090
And they respond
differently to, let's say,

655
00:49:08,090 --> 00:49:10,150
an external
electromagnetic field.

656
00:49:10,150 --> 00:49:15,850
So once we have this picture,
let's kind of expand our view.

657
00:49:15,850 --> 00:49:18,120
Forget about the xy model.

658
00:49:18,120 --> 00:49:20,490
Think of a system of charges.

659
00:49:20,490 --> 00:49:25,536
And notice that in
this low-density phase,

660
00:49:25,536 --> 00:49:29,220
it behaves like a
dielectric in the sense

661
00:49:29,220 --> 00:49:32,730
that there are no free charges.

662
00:49:32,730 --> 00:49:37,900
And here, there will be
lots of mobile charges.

663
00:49:37,900 --> 00:49:40,650
And it behaves like a metal.

664
00:49:40,650 --> 00:49:42,280
What do I mean by that?

665
00:49:42,280 --> 00:49:45,400
Well, here, if I,
let's say, bring

666
00:49:45,400 --> 00:49:48,180
in an external electric field.

667
00:49:48,180 --> 00:49:54,360
Or maybe if I put a huge
charge, what is going to happen

668
00:49:54,360 --> 00:50:00,290
is that opposite
charges will accumulate.

669
00:50:00,290 --> 00:50:04,880
Or there will be, essentially,
opposite charges for the field.

670
00:50:04,880 --> 00:50:08,590
So that once you
go inside, the fact

671
00:50:08,590 --> 00:50:13,160
that you have an external
electric field or a charge

672
00:50:13,160 --> 00:50:14,880
is completely screen.

673
00:50:14,880 --> 00:50:16,800
You won't see it.

674
00:50:16,800 --> 00:50:19,860
Whereas here, what
is going to happen

675
00:50:19,860 --> 00:50:23,380
is that if you put
in an electric field,

676
00:50:23,380 --> 00:50:26,170
it will penetrate
into the system

677
00:50:26,170 --> 00:50:28,970
although it will be
weakened a little bit

678
00:50:28,970 --> 00:50:32,220
by the re-orientation
of these charges.

679
00:50:32,220 --> 00:50:38,650
Now, if you put a plus charge,
the effect of that plus charge

680
00:50:38,650 --> 00:50:42,170
would be felt throughout,
although weakened a little bit.

681
00:50:42,170 --> 00:50:47,790
Because again, some of these
dipoles will re-orient in that.

682
00:50:47,790 --> 00:50:49,720
OK?

683
00:50:49,720 --> 00:50:56,380
So, this low-density
phase we can actually

684
00:50:56,380 --> 00:51:06,050
try to parametrize in terms of
a weakening of the interactions

685
00:51:06,050 --> 00:51:10,800
through a dielectric
constant epsilon.

686
00:51:10,800 --> 00:51:13,140
And so, what I'm going
to try to calculate

687
00:51:13,140 --> 00:51:18,580
for you is to imagine that I'm
in the limit of low density

688
00:51:18,580 --> 00:51:25,640
or small y and calculate
what the weakening is, what

689
00:51:25,640 --> 00:51:29,340
the dielectric function
is, perturbatively in y.

690
00:51:29,340 --> 00:51:29,840
Yes?

691
00:51:29,840 --> 00:51:33,644
AUDIENCE: If you were talking
about the real electric charges

692
00:51:33,644 --> 00:51:36,204
and the way to act
on that [INAUDIBLE]

693
00:51:36,204 --> 00:51:38,060
real electric field or charge.

694
00:51:38,060 --> 00:51:42,214
But if we are talking about
topological charges, what

695
00:51:42,214 --> 00:51:46,507
would be kind of
conjugate force to that?

696
00:51:46,507 --> 00:51:47,480
PROFESSOR: OK.

697
00:51:50,640 --> 00:51:52,280
It's not going to be easy.

698
00:51:52,280 --> 00:51:54,710
I have to do
something about say,

699
00:51:54,710 --> 00:51:58,530
re-orienting all of the spins
on the boundaries, et cetera.

700
00:51:58,530 --> 00:52:00,550
So let's forget about that.

701
00:52:00,550 --> 00:52:03,780
The point is that
mathematically, the problem

702
00:52:03,780 --> 00:52:06,620
is reduced to this system.

703
00:52:06,620 --> 00:52:11,040
And I can much more
easily do the mathematics

704
00:52:11,040 --> 00:52:15,950
if I change my perspective
and think about this picture.

705
00:52:15,950 --> 00:52:18,100
OK?

706
00:52:18,100 --> 00:52:22,890
And that's the thing you have
to do in theoretical physics.

707
00:52:22,890 --> 00:52:25,650
You basically take
advantage of mappings

708
00:52:25,650 --> 00:52:29,280
of one model to
another model in order

709
00:52:29,280 --> 00:52:35,900
to refine your intuition
using some other picture.

710
00:52:35,900 --> 00:52:38,370
So that's what we
are going to do.

711
00:52:38,370 --> 00:52:44,820
So completely different picture
from the original spin models--

712
00:52:44,820 --> 00:52:52,330
imagine that you have indeed
a box of this material.

713
00:52:52,330 --> 00:52:56,120
And this box of material
has, because you're wise,

714
00:52:56,120 --> 00:53:00,130
more some combination of these
plus and minus charges in it.

715
00:53:02,650 --> 00:53:07,320
And then, what I do is
that I bring externally

716
00:53:07,320 --> 00:53:10,655
a uniform electric
field in this direction.

717
00:53:13,870 --> 00:53:19,780
And I expect that once
inside the material,

718
00:53:19,780 --> 00:53:24,220
the electric field will be
reduced to a smaller value

719
00:53:24,220 --> 00:53:31,240
that I will call E prime because
of the dielectric function.

720
00:53:31,240 --> 00:53:34,790
Now, if you ever calculated
dielectric functions,

721
00:53:34,790 --> 00:53:37,250
that's exactly what
I'm going to do now.

722
00:53:37,250 --> 00:53:38,970
It's a simple process.

723
00:53:38,970 --> 00:53:42,310
What you do, for
example, is you do

724
00:53:42,310 --> 00:53:46,390
the analog of Gauss' theorem.

725
00:53:46,390 --> 00:53:53,150
Let's imagine that we draw
a circuit such as this

726
00:53:53,150 --> 00:53:59,930
that is partly on the inside,
and partly on the outside.

727
00:53:59,930 --> 00:54:06,070
So I can calculate what the
flux of the electric field

728
00:54:06,070 --> 00:54:12,070
is through this circuit, the
analog of the Gaussian pillbox.

729
00:54:12,070 --> 00:54:14,660
And so, what I have is
that what is going on

730
00:54:14,660 --> 00:54:20,140
is E. If I call this
distance to be L,

731
00:54:20,140 --> 00:54:26,710
the flux integrated
through the entire thing

732
00:54:26,710 --> 00:54:30,872
is E minus E prime times f.

733
00:54:30,872 --> 00:54:35,040
So this is the integral of
the divergence of the electric

734
00:54:35,040 --> 00:54:36,880
field .

735
00:54:36,880 --> 00:54:43,140
And by Gauss' theorem, this has
to be charge enclosed inside.

736
00:54:49,588 --> 00:54:52,570
OK.

737
00:54:52,570 --> 00:54:55,810
Now, why should there
be any charge enclosed

738
00:54:55,810 --> 00:55:01,045
inside when you have a
bunch of plus and minuses.

739
00:55:01,045 --> 00:55:02,630
I mean, there will
be some pluses

740
00:55:02,630 --> 00:55:05,190
and minuses out here
as I have indicated.

741
00:55:05,190 --> 00:55:08,920
There will be some pluses
and minuses that are inside.

742
00:55:08,920 --> 00:55:12,510
But the net of
these would be zero.

743
00:55:12,510 --> 00:55:17,560
So the only place that
you get a net charge

744
00:55:17,560 --> 00:55:20,934
is those dipoles that
happen to be sitting right

745
00:55:20,934 --> 00:55:21,600
at the boundary.

746
00:55:26,230 --> 00:55:32,860
And then, I have to count
how many of them are inside.

747
00:55:32,860 --> 00:55:35,270
And some of them will
have the plus inside.

748
00:55:35,270 --> 00:55:38,550
And some of them will
have the minus inside.

749
00:55:38,550 --> 00:55:41,640
And then, I have to
calculate the net.

750
00:55:41,640 --> 00:55:50,490
The thing is that my dipoles
do not have a fixed size.

751
00:55:50,490 --> 00:55:58,810
The size of these
plus/minus molecules r

752
00:55:58,810 --> 00:56:00,890
can be variable itself.

753
00:56:00,890 --> 00:56:02,820
OK?

754
00:56:02,820 --> 00:56:05,670
So there will be some that are
tightly bound to each other.

755
00:56:05,670 --> 00:56:10,160
There may be some that are
further apart, et cetera.

756
00:56:10,160 --> 00:56:16,530
So let's look at pairs
that are at the distance r

757
00:56:16,530 --> 00:56:20,350
and ask how many of them hit
this boundary so that one

758
00:56:20,350 --> 00:56:23,990
of them would be inside,
one of them will be outside.

759
00:56:23,990 --> 00:56:25,370
OK?

760
00:56:25,370 --> 00:56:30,270
So, that number has
to be proportional

761
00:56:30,270 --> 00:56:36,960
to essentially this area.

762
00:56:36,960 --> 00:56:39,100
What is that area?

763
00:56:39,100 --> 00:56:44,730
On one side, it is L. On
the other side, it is R.

764
00:56:44,730 --> 00:56:51,310
But if the dipole is
oriented at an angle theta,

765
00:56:51,310 --> 00:56:56,250
it is, in fact r cosine theta.

766
00:56:56,250 --> 00:56:56,750
OK?

767
00:56:59,310 --> 00:57:05,320
So that's the number.

768
00:57:05,320 --> 00:57:12,360
Now, what I will have here
would be the charge 2 pi.

769
00:57:12,360 --> 00:57:13,170
So this is qi.

770
00:57:19,880 --> 00:57:23,600
Actually, it could
be plus or minus.

771
00:57:23,600 --> 00:57:25,750
The reason that there's
going to be more

772
00:57:25,750 --> 00:57:31,500
plus as opposed to minus
is because the dipole

773
00:57:31,500 --> 00:57:35,880
gets oriented by
the electric field.

774
00:57:35,880 --> 00:57:45,090
So I will have a term here that
is E to the E prime times q

775
00:57:45,090 --> 00:57:50,660
ir-- so that's 2 pi r.

776
00:57:50,660 --> 00:57:57,280
So this is qr again,
times cosine of theta.

777
00:57:57,280 --> 00:58:00,150
So we can see that,
depending on cosine of theta

778
00:58:00,150 --> 00:58:04,150
being larger than
pi or less than pi,

779
00:58:04,150 --> 00:58:07,440
this number will be
positive or negative.

780
00:58:07,440 --> 00:58:11,075
And that's going to be
modified by this number also.

781
00:58:11,075 --> 00:58:14,300
And of course, the strength
of this whole thing

782
00:58:14,300 --> 00:58:16,870
is set by this parameter k.

783
00:58:21,230 --> 00:58:28,480
And also how likely it
is for me to have created

784
00:58:28,480 --> 00:58:36,840
a dipole of size r is controlled
by precisely this factor.

785
00:58:36,840 --> 00:58:42,030
A dipole is something
that has two cores.

786
00:58:42,030 --> 00:58:45,280
So it is something that will
appear at order of y squared.

787
00:58:48,570 --> 00:58:51,050
And there is the
energy, according

788
00:58:51,050 --> 00:58:54,810
to this formula, of
separating two things.

789
00:58:54,810 --> 00:58:57,675
And so you can see that
essentially, n of r--

790
00:58:57,675 --> 00:59:01,320
maybe I will write it
separately over here--

791
00:59:01,320 --> 00:59:10,600
is y squared times E to
the minus 4pi squared k.

792
00:59:10,600 --> 00:59:15,520
And from here, I have
log of r divided by a.

793
00:59:15,520 --> 00:59:17,430
And then there's
a factor of 2 pi

794
00:59:17,430 --> 00:59:19,610
because the Coulomb
potential is this.

795
00:59:22,240 --> 00:59:27,960
So, this is going
to be y squared

796
00:59:27,960 --> 00:59:33,430
a over r to the power of 2 pi k.

797
00:59:33,430 --> 00:59:38,720
The further you try to
separate these things, the more

798
00:59:38,720 --> 00:59:40,300
cost you have to pay.

799
00:59:46,024 --> 00:59:48,900
OK.

800
00:59:48,900 --> 00:59:52,700
So if you were
trying to calculate

801
00:59:52,700 --> 00:59:57,750
the contribution of,
say, polarizable atoms

802
00:59:57,750 --> 01:00:02,150
or dipoles to the dielectric
function of a solid,

803
01:00:02,150 --> 01:00:06,630
you would be doing exactly
this same calculation.

804
01:00:06,630 --> 01:00:13,090
The only difference is that
the size of your dipole

805
01:00:13,090 --> 01:00:16,580
would be set by the
size of your molecule

806
01:00:16,580 --> 01:00:20,590
and ultimately, related
to its polarizability.

807
01:00:20,590 --> 01:00:23,720
And rather than having
this Coulomb interaction,

808
01:00:23,720 --> 01:00:26,755
you would have some
dissociation energy or something

809
01:00:26,755 --> 01:00:31,800
else, or the density itself
would come over here.

810
01:00:31,800 --> 01:00:36,630
So, the only final
step is that I

811
01:00:36,630 --> 01:00:43,470
have to regard my system
having a composition

812
01:00:43,470 --> 01:00:46,110
of these things of
different sizes.

813
01:00:46,110 --> 01:00:52,000
So I have to do an integral
over r, as well as orientation.

814
01:00:52,000 --> 01:00:57,060
So I have to do an
integral over E theta.

815
01:00:57,060 --> 01:00:58,910
Of course, the
integration will go

816
01:00:58,910 --> 01:01:03,678
from a through essentially, the
size of the system or infinity.

817
01:01:07,670 --> 01:01:11,760
I forgot one other
thing, which is

818
01:01:11,760 --> 01:01:18,240
that when I'm calculating how
many places I can put this,

819
01:01:18,240 --> 01:01:21,570
again, I have been
calculating things

820
01:01:21,570 --> 01:01:26,020
per unit area of a squared.

821
01:01:26,020 --> 01:01:29,800
So I would have to divide
all of these places

822
01:01:29,800 --> 01:01:34,629
where r and L appear by
corresponding factors of a.

823
01:01:37,552 --> 01:01:38,052
OK?

824
01:01:42,950 --> 01:01:45,650
So, the last step
of the calculation

825
01:01:45,650 --> 01:01:48,145
is you expand this quantity.

826
01:01:48,145 --> 01:01:50,120
It is 1.

827
01:01:50,120 --> 01:01:52,670
For small values of
the electric field,

828
01:01:52,670 --> 01:02:01,000
it is 2 pi r E prime cosine
of theta k plus higher order

829
01:02:01,000 --> 01:02:03,240
terms.

830
01:02:03,240 --> 01:02:07,160
And then, you can do the
various integrations.

831
01:02:07,160 --> 01:02:12,570
First of all, 1 the
integration against 1

832
01:02:12,570 --> 01:02:14,960
will disappear because
you are integrating

833
01:02:14,960 --> 01:02:18,740
over all values of
cosine of theta.

834
01:02:18,740 --> 01:02:21,870
Integral of cosine of
theta gives you zero.

835
01:02:21,870 --> 01:02:25,220
Essentially, it says that if
there was no electric field,

836
01:02:25,220 --> 01:02:30,400
there was no reason for there
to be an additional net charge

837
01:02:30,400 --> 01:02:32,920
on one side or the other.

838
01:02:32,920 --> 01:02:35,780
So the first term
that will be non-zero

839
01:02:35,780 --> 01:02:38,930
is the average of cosine
theta squared, which

840
01:02:38,930 --> 01:02:41,160
will give you a
factor of one-half.

841
01:02:41,160 --> 01:02:50,970
And so, what you will get is
that E minus E prime times

842
01:02:50,970 --> 01:03:04,400
L is-- well, there's
going to be a factor of L.

843
01:03:04,400 --> 01:03:07,420
The integral of the
theta cosine of theta

844
01:03:07,420 --> 01:03:16,340
squared-- the integral of
cosine of theta squared

845
01:03:16,340 --> 01:03:20,780
is going to give you 2 pi,
which is the integration times

846
01:03:20,780 --> 01:03:21,900
one-half.

847
01:03:21,900 --> 01:03:24,790
So this is the integral
the theta cosine

848
01:03:24,790 --> 01:03:28,350
square theta will give you this.

849
01:03:28,350 --> 01:03:31,000
So we did this.

850
01:03:31,000 --> 01:03:33,700
We have two factors of y.

851
01:03:33,700 --> 01:03:36,330
y is our expansion parameter.

852
01:03:36,330 --> 01:03:38,510
We are at a low density limit.

853
01:03:38,510 --> 01:03:42,310
We've calculated things
assuming that essentially, I

854
01:03:42,310 --> 01:03:47,050
have to look at one
value of these diplodes.

855
01:03:47,050 --> 01:03:50,860
In principal, I can imagine that
there will be multiple dipoles.

856
01:03:50,860 --> 01:03:54,560
And you can see that ultimately,
therefore, potentially, I

857
01:03:54,560 --> 01:03:59,830
have order of y to the fourth
that I haven't calculated.

858
01:03:59,830 --> 01:04:02,325
OK, so we got rid
of the y squared.

859
01:04:05,340 --> 01:04:11,230
We have a factor of E prime
on the expansion here.

860
01:04:23,550 --> 01:04:27,595
This factor is bothering me
a little bit-- let me check.

861
01:04:27,595 --> 01:04:29,796
No, that's correct.

862
01:04:29,796 --> 01:04:31,940
OK, so I have the factor of k.

863
01:04:37,300 --> 01:04:41,010
I have a factor of 2 pi here
that came from the charge.

864
01:04:41,010 --> 01:04:44,435
I have another 2 pi here--
so I have 4pi squared.

865
01:04:51,470 --> 01:04:54,900
I think I got everything
except the integration

866
01:04:54,900 --> 01:05:04,110
over a to infinity dr.
There is this r dr, which

867
01:05:04,110 --> 01:05:07,650
is from the two
dimensional integration.

868
01:05:07,650 --> 01:05:11,495
There was another r
here, and another r here.

869
01:05:11,495 --> 01:05:15,360
So this becomes r to the three.

870
01:05:15,360 --> 01:05:18,400
From here, I have minus 2 pi k.

871
01:05:20,930 --> 01:05:23,650
And then, I have the
corresponding factors

872
01:05:23,650 --> 01:05:27,230
of a to the power
of 2 pi k minus 4.

873
01:05:34,200 --> 01:05:34,700
OK.

874
01:05:37,690 --> 01:05:44,690
So you can see that
the L's cancel.

875
01:05:44,690 --> 01:05:52,750
And what I get is
that E-- once I

876
01:05:52,750 --> 01:05:56,210
take the E prime
to the other side--

877
01:05:56,210 --> 01:06:10,010
becomes E prime 1 plus I have
4pi cubed k y squared-- again,

878
01:06:10,010 --> 01:06:16,290
y squared is my small
expansion parameter.

879
01:06:16,290 --> 01:06:22,460
And then, I have the integral
from a to infinity, the r, r

880
01:06:22,460 --> 01:06:26,140
to the power of
3 minus 2 pi k, a

881
01:06:26,140 --> 01:06:33,050
to the power of 2 pi k minus
4, and then order of y squared,

882
01:06:33,050 --> 01:06:34,366
y to the fourth.

883
01:06:37,852 --> 01:06:40,840
OK.

884
01:06:40,840 --> 01:06:51,450
So basically, you see that
the internal electric field

885
01:06:51,450 --> 01:06:57,700
is smaller than the
external electric field

886
01:06:57,700 --> 01:07:03,610
by this factor, which takes
into account the re-orientation

887
01:07:03,610 --> 01:07:08,080
of the dipoles in order to
screen the electric field.

888
01:07:08,080 --> 01:07:10,210
And it is proportional
in some sense

889
01:07:10,210 --> 01:07:15,000
to the density of these dipoles.

890
01:07:15,000 --> 01:07:18,610
And the twist is that
the dipoles that we have

891
01:07:18,610 --> 01:07:21,850
can have a range
of sizes that we

892
01:07:21,850 --> 01:07:24,940
have to integrate [INAUDIBLE].

893
01:07:24,940 --> 01:07:31,170
So typically, you would write
E prime to the E over epsilon.

894
01:07:31,170 --> 01:07:36,580
And so this is the
inverse of your epsilon.

895
01:07:36,580 --> 01:07:45,500
And essentially, this is
a reduction in everything

896
01:07:45,500 --> 01:07:49,110
that has to do with electric
interactions because

897
01:07:49,110 --> 01:07:53,280
of the screening
of other things.

898
01:07:53,280 --> 01:07:56,120
I can write it in the
following fashion.

899
01:07:56,120 --> 01:08:02,150
I can say that there is
an effective k-- that's

900
01:08:02,150 --> 01:08:08,870
called k effective-- which is
different from the original k

901
01:08:08,870 --> 01:08:11,950
that I have.

902
01:08:11,950 --> 01:08:14,250
It is reduced by a
factor of epsilon.

903
01:08:16,760 --> 01:08:23,960
So we were worried when we
were doing the nonlinear sigma

904
01:08:23,960 --> 01:08:26,090
model that for any
[INAUDIBLE], we

905
01:08:26,090 --> 01:08:31,689
saw that the parameter k was
not getting modified because

906
01:08:31,689 --> 01:08:37,090
of the interactions among the
spin modes, and that's correct.

907
01:08:37,090 --> 01:08:39,080
But really, at
high temperatures,

908
01:08:39,080 --> 01:08:40,460
it should disappear.

909
01:08:40,460 --> 01:08:42,870
We saw that the
correlations had to go away

910
01:08:42,870 --> 01:08:47,029
from power law form
to exponential form.

911
01:08:47,029 --> 01:08:50,220
And so, we needed some
mechanism for reducing

912
01:08:50,220 --> 01:08:52,399
the coupling constant.

913
01:08:52,399 --> 01:08:57,140
And what we find here is
that this topological defects

914
01:08:57,140 --> 01:09:00,910
and their screening provide
the right mechanism.

915
01:09:00,910 --> 01:09:03,540
So the effective
k that I have is

916
01:09:03,540 --> 01:09:06,660
going to be reduced
from the original k

917
01:09:06,660 --> 01:09:08,080
by the inverse of this.

918
01:09:08,080 --> 01:09:10,240
Since I'm doing
an expansion in y,

919
01:09:10,240 --> 01:09:14,704
it is simply minus
4pi cubed ky squared,

920
01:09:14,704 --> 01:09:20,870
integral a to infinity dr,
r to the 3 minus 2 pi k,

921
01:09:20,870 --> 01:09:28,210
a to the 2 pi k minus 4, plus
order or y to the fourteenth.

922
01:09:28,210 --> 01:09:28,710
OK.

923
01:09:31,920 --> 01:09:36,550
Now actually, in the lecture
notes that I have given you,

924
01:09:36,550 --> 01:09:42,210
I calculate this formula in
an entirely different way.

925
01:09:42,210 --> 01:09:49,080
What I do is I assume that I
have two topological defects--

926
01:09:49,080 --> 01:09:53,689
so there I sort of maintain the
picture of topological defect.

927
01:09:53,689 --> 01:09:58,290
And their interaction
between them

928
01:09:58,290 --> 01:10:03,440
is this logarithmic interaction
that has coefficient k.

929
01:10:03,440 --> 01:10:06,050
But then, we say that this
[INAUDIBLE] interaction

930
01:10:06,050 --> 01:10:13,640
is modified because I can create
pairs of topological defect,

931
01:10:13,640 --> 01:10:15,860
such as this, that
will partially

932
01:10:15,860 --> 01:10:19,060
screen the interaction.

933
01:10:19,060 --> 01:10:22,120
And in the notes,
we calculate what

934
01:10:22,120 --> 01:10:25,610
the effect of those
pairs at lowest order

935
01:10:25,610 --> 01:10:29,970
is on their interaction
that you have between them.

936
01:10:29,970 --> 01:10:32,860
And you find that the
effect is to modify

937
01:10:32,860 --> 01:10:35,370
the coefficient of
the logarithm, which

938
01:10:35,370 --> 01:10:38,670
is k, to a reduced k.

939
01:10:38,670 --> 01:10:42,230
And that reduced k is
given exactly by this form.

940
01:10:42,230 --> 01:10:44,771
So the same thing you
can get different ways.

941
01:10:44,771 --> 01:10:45,270
Yes?

942
01:10:45,270 --> 01:10:47,635
AUDIENCE: What if
k is too small--

943
01:10:47,635 --> 01:10:48,910
PROFESSOR: A-ha.

944
01:10:48,910 --> 01:10:50,280
Good.

945
01:10:50,280 --> 01:10:55,410
Because I framed the
entire thing as if I'm

946
01:10:55,410 --> 01:10:58,364
doing a preservation
theory for you

947
01:10:58,364 --> 01:11:02,140
in y being a small parameter.

948
01:11:02,140 --> 01:11:03,850
OK?

949
01:11:03,850 --> 01:11:07,720
But now, we see
that no matter how

950
01:11:07,720 --> 01:11:20,510
small y is, if k is in
fact less than 2 over pi--

951
01:11:20,510 --> 01:11:24,890
so this has dimensions of
r to the 4 minus 2 pi k.

952
01:11:24,890 --> 01:11:32,240
So if k is less than 2 over pi--
which incidentally is something

953
01:11:32,240 --> 01:11:41,920
that we saw earlier--
if k is less than that,

954
01:11:41,920 --> 01:11:43,940
this integral diverges.

955
01:11:46,750 --> 01:11:53,310
So I thought I was
controlling my expansion

956
01:11:53,310 --> 01:11:58,950
by making y arbitrarily
small, but what

957
01:11:58,950 --> 01:12:04,990
we see is that no matter
how small I make y,

958
01:12:04,990 --> 01:12:09,440
if k becomes too small,
the perturbation acuity

959
01:12:09,440 --> 01:12:12,600
blows up on me.

960
01:12:12,600 --> 01:12:16,990
So this is yet another example
of a singular perturbation

961
01:12:16,990 --> 01:12:20,810
theory, which is what we
had encountered when we were

962
01:12:20,810 --> 01:12:23,310
doing the Landau-Ginzburg model.

963
01:12:23,310 --> 01:12:27,070
We thought that our co-efficient
of phi to the fourth u

964
01:12:27,070 --> 01:12:29,020
was a small parameter.

965
01:12:29,020 --> 01:12:32,810
You are making an expansion
naively in powers of u.

966
01:12:32,810 --> 01:12:35,380
And then we found an
expression in which

967
01:12:35,380 --> 01:12:38,230
the coeffecient-- the thing
that was multiplying u

968
01:12:38,230 --> 01:12:41,240
at the critical point
was blowing up on us.

969
01:12:41,240 --> 01:12:44,490
And so the perturbation
theory inherently

970
01:12:44,490 --> 01:12:47,210
became singular,
despite your thinking

971
01:12:47,210 --> 01:12:50,840
that you had a small parameter.

972
01:12:50,840 --> 01:12:55,200
So we are going to
use the same trick

973
01:12:55,200 --> 01:13:04,300
that we used for the case of
the Landau-Ginzburg model--

974
01:13:04,300 --> 01:13:15,350
this is deal with
singular perturbations

975
01:13:15,350 --> 01:13:18,080
by renormalization group.

976
01:13:22,020 --> 01:13:28,110
So what we see is that
the origin of the problem

977
01:13:28,110 --> 01:13:31,520
is the divergence that
we get over here when

978
01:13:31,520 --> 01:13:34,670
we try to integrate
all the way to infinity

979
01:13:34,670 --> 01:13:37,880
or the size of the system.

980
01:13:37,880 --> 01:13:43,460
So what we do instead is
we said, OK, let's not

981
01:13:43,460 --> 01:13:45,890
integrate all the way.

982
01:13:45,890 --> 01:13:51,400
Let's replace the
short distance cut-off

983
01:13:51,400 --> 01:13:58,270
that we had with something
that is larger-- ba--

984
01:13:58,270 --> 01:14:03,470
and rather than integrating all
of a to infinity, we integrate

985
01:14:03,470 --> 01:14:10,450
only over short distance
fluctuations between a and ba.

986
01:14:10,450 --> 01:14:14,450
This is our usual [INAUDIBLE].

987
01:14:14,450 --> 01:14:19,790
So what we therefore get
is that the k effective is

988
01:14:19,790 --> 01:14:25,580
k 1 minus 4pi q ky
squared integral

989
01:14:25,580 --> 01:14:37,530
from a to ba br r to the 3 minus
2 pi k a to the 2 pi k minus 4.

990
01:14:37,530 --> 01:14:42,820
And then, I have to still
deal with 4pi cubed ky

991
01:14:42,820 --> 01:14:46,660
squared, integral
from ba infinity dr

992
01:14:46,660 --> 01:14:53,840
r to the 3 minus 2 pi k,
a to the 2 pi k minus 4,

993
01:14:53,840 --> 01:14:57,770
plus order of y to the fourth.

994
01:14:57,770 --> 01:15:00,728
OK?

995
01:15:00,728 --> 01:15:01,228
All right.

996
01:15:04,690 --> 01:15:14,990
So, you can see that the
effect of integrating this much

997
01:15:14,990 --> 01:15:22,500
is to modify the decoupling
to a new value which

998
01:15:22,500 --> 01:15:29,450
depends on b, which is just
k minus 4pi cubed k squared

999
01:15:29,450 --> 01:15:37,730
y squared, a to ba br r
to the 3 minus 2 pi k.

1000
01:15:37,730 --> 01:15:42,471
a to the 2 pi k minus 4.

1001
01:15:42,471 --> 01:15:42,971
OK?

1002
01:15:46,430 --> 01:15:52,060
And then, I can rewrite
the expression for k

1003
01:15:52,060 --> 01:16:00,130
effective to be this k tilde.

1004
01:16:00,130 --> 01:16:07,530
And then, whatever is
left, which is 4pi cubed k

1005
01:16:07,530 --> 01:16:14,220
squared y squared
integral ba to infinity dr

1006
01:16:14,220 --> 01:16:24,647
r to the 3 minus 2 pi k, a to
the 2 pi k minus 4, order of y

1007
01:16:24,647 --> 01:16:25,230
to the fourth.

1008
01:16:29,530 --> 01:16:35,480
You see that k has been shifted
through this transformation

1009
01:16:35,480 --> 01:16:39,700
by an amount that is
order of y squared.

1010
01:16:39,700 --> 01:16:45,320
So at order of y squared
in this new expression,

1011
01:16:45,320 --> 01:16:53,470
I can replace all the k's
that are carrying with k tilde

1012
01:16:53,470 --> 01:16:56,625
and it would still be
correct for this order.

1013
01:16:59,950 --> 01:17:05,320
Now I compare this expression
and the original expression

1014
01:17:05,320 --> 01:17:07,750
that I had.

1015
01:17:07,750 --> 01:17:12,510
And I see that they are pretty
much the same expression,

1016
01:17:12,510 --> 01:17:19,120
except that in this
one, the cut-off is ba.

1017
01:17:19,120 --> 01:17:21,290
So I do step two of origin.

1018
01:17:21,290 --> 01:17:32,480
I define my R prime to be dr
so that my new cut-off will

1019
01:17:32,480 --> 01:17:35,300
be back to a.

1020
01:17:35,300 --> 01:17:38,270
So then, this
whole thing becomes

1021
01:17:38,270 --> 01:17:45,850
k effective is k tilde
minus 4pi cubed k

1022
01:17:45,850 --> 01:17:49,740
tilde squared y squared.

1023
01:17:49,740 --> 01:17:53,430
Because of the transformation
that I did over here,

1024
01:17:53,430 --> 01:17:59,760
I will get a factor of b to
the 4 minus 2 pi k tilde,

1025
01:17:59,760 --> 01:18:05,040
integral from a' to infinity--
the r prime-- r prime to the 3

1026
01:18:05,040 --> 01:18:11,950
minus 2 pi k tilde, a to
the 2 pi k tilde, minus 4

1027
01:18:11,950 --> 01:18:13,629
plus order of y
to the fourteenth.

1028
01:18:20,120 --> 01:18:27,820
So we see that the same
effective interaction

1029
01:18:27,820 --> 01:18:32,810
can be obtained from
two theories that

1030
01:18:32,810 --> 01:18:39,780
have exactly the same cut-off,
a, except that in one case,

1031
01:18:39,780 --> 01:18:42,100
I had k and y.

1032
01:18:42,100 --> 01:18:47,082
In the new case, I have this
tilde or k prime at scale b.

1033
01:18:47,082 --> 01:18:49,390
And I have to
replace where I had

1034
01:18:49,390 --> 01:18:53,920
y with y with this
additional factor.

1035
01:18:53,920 --> 01:18:58,370
So the two theories
are equivalent provided

1036
01:18:58,370 --> 01:19:03,430
that I say that the new
interaction at scale b

1037
01:19:03,430 --> 01:19:11,440
is the old interaction minus
4pi cubed k squared y squared.

1038
01:19:11,440 --> 01:19:14,570
This integral is
easy to perform.

1039
01:19:14,570 --> 01:19:16,000
It is just the power law.

1040
01:19:16,000 --> 01:19:21,190
It is b to the fourth
minus 2 pi k minus 1.

1041
01:19:21,190 --> 01:19:28,820
And then, I have 4 minus 2 pi
k order of y to the fourth.

1042
01:19:28,820 --> 01:19:33,910
And my y prime is
y-- from here I

1043
01:19:33,910 --> 01:19:38,116
see b to the power
of 2 minus pi k.

1044
01:19:43,462 --> 01:19:44,434
AUDIENCE: [INAUDIBLE]

1045
01:19:48,322 --> 01:19:50,090
PROFESSOR: This k squared?

1046
01:19:50,090 --> 01:19:51,550
AUDIENCE: Oh, I see.

1047
01:19:51,550 --> 01:19:52,050
Sorry.

1048
01:19:58,525 --> 01:19:59,400
PROFESSOR: All right.

1049
01:20:03,820 --> 01:20:13,200
So, our theory is described
in terms of two parameters--

1050
01:20:13,200 --> 01:20:17,780
this y and this k.

1051
01:20:17,780 --> 01:20:22,970
or let's say, it's
inverse-- k inverse,

1052
01:20:22,970 --> 01:20:25,910
which is more like temperature.

1053
01:20:25,910 --> 01:20:29,100
And what we will
show next time is

1054
01:20:29,100 --> 01:20:34,950
that these recursion
relations, when I draw it here,

1055
01:20:34,950 --> 01:20:38,170
will give me two
types of behavior.

1056
01:20:38,170 --> 01:20:41,470
One set of behavior
that parameterizes

1057
01:20:41,470 --> 01:20:47,700
the low temperature dilute
limit that corresponds

1058
01:20:47,700 --> 01:20:51,890
to flows in which y
goes through zero.

1059
01:20:51,890 --> 01:20:55,400
So that when you look at the
system at larger and larger

1060
01:20:55,400 --> 01:20:59,830
lens scales, essentially it
becomes less and less depleted

1061
01:20:59,830 --> 01:21:04,030
of these excitations.

1062
01:21:04,030 --> 01:21:07,030
So, once you have integrated
the very essentially,

1063
01:21:07,030 --> 01:21:09,960
you don't see any excitations.

1064
01:21:09,960 --> 01:21:12,210
And then there's
another phase, which

1065
01:21:12,210 --> 01:21:18,810
as you do this removal of
short distance fluctuations.

1066
01:21:18,810 --> 01:21:22,550
You tend to flow to
high temperatures

1067
01:21:22,550 --> 01:21:24,690
and large densities.

1068
01:21:24,690 --> 01:21:29,451
And so, that corresponds
to this kind of face.

1069
01:21:29,451 --> 01:21:33,150
Now the beauty of
this whole thing

1070
01:21:33,150 --> 01:21:38,600
is that these recursion
relations are exact and allow

1071
01:21:38,600 --> 01:21:43,650
us to exactly determine
the behavior of these space

1072
01:21:43,650 --> 01:21:46,140
transition in two dimensions.

1073
01:21:46,140 --> 01:21:49,830
And that's actually one
of the other triumphs

1074
01:21:49,830 --> 01:21:54,040
of renormalization
group is to elucidate

1075
01:21:54,040 --> 01:21:56,950
exactly the critical
behavior of this transition,

1076
01:21:56,950 --> 01:22:00,000
as we will discuss next time.