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PROFESSOR: So last
time, we started
00:00:27.970 --> 00:00:32.032
looking at the system
of [? spins. ?] So there
00:00:32.032 --> 00:00:36.640
was a field S of
x on the lattice.
00:00:36.640 --> 00:00:41.840
And the energy cost was
proportional to differences
00:00:41.840 --> 00:00:45.280
of space on two
neighboring sites,
00:00:45.280 --> 00:00:50.290
which if we go to the continuum,
became something like gradient
00:00:50.290 --> 00:00:55.110
of the vector S. You have to
integrate this, of course,
00:00:55.110 --> 00:00:58.372
over all space.
00:00:58.372 --> 00:01:01.580
We gave this a rate of K over 2.
00:01:01.580 --> 00:01:08.200
There was some energy costs
[INAUDIBLE] of this [? form ?]
00:01:08.200 --> 00:01:10.760
so that the particular
configuration
00:01:10.760 --> 00:01:13.630
was weighted by this factor.
00:01:13.630 --> 00:01:16.830
And to calculate the
partition function,
00:01:16.830 --> 00:01:21.940
we had to integrate over
all configurations of this
00:01:21.940 --> 00:01:26.980
over this field S.
00:01:26.980 --> 00:01:31.670
And the constraint that we had
was that this was a unit vector
00:01:31.670 --> 00:01:39.522
so that this was an n component
field whose magnitude was 1,
00:01:39.522 --> 00:01:40.380
OK?
00:01:40.380 --> 00:01:43.460
So this is what we
want to calculate.
00:01:43.460 --> 00:01:48.200
Again, whenever we are writing
an expression such as this,
00:01:48.200 --> 00:01:52.760
thinking that we started with
some average system [INAUDIBLE]
00:01:52.760 --> 00:01:55.780
some kind of a coarse graining.
00:01:55.780 --> 00:01:58.400
There is a short distance
[INAUDIBLE] [? replacing ?]
00:01:58.400 --> 00:02:01.570
all of these tiers.
00:02:01.570 --> 00:02:07.360
Now what we can do is
imagine that this vector
00:02:07.360 --> 00:02:10.300
S in its ground
state, let's say,
00:02:10.300 --> 00:02:15.490
is pointing in some particular
direction throughout the system
00:02:15.490 --> 00:02:18.030
and that fluctuations
around this ground
00:02:18.030 --> 00:02:21.950
state in the
transverse direction
00:02:21.950 --> 00:02:24.670
are characterized by
some vector of pi that
00:02:24.670 --> 00:02:27.690
is n minus 1 dimensional.
00:02:27.690 --> 00:02:32.660
And so this partition function
can be written entirely
00:02:32.660 --> 00:02:37.360
[? rather ?] than fluctuations
of the unit vector S in terms
00:02:37.360 --> 00:02:40.950
of the fluctuations of these
transverse [? coordinates ?]
00:02:40.950 --> 00:02:43.110
[INAUDIBLE] pi.
00:02:43.110 --> 00:02:46.310
And we saw that the
appropriate weight
00:02:46.310 --> 00:02:51.410
for this n minus 1
component vector of pi
00:02:51.410 --> 00:02:54.950
has within it a factor of
something like square root of 1
00:02:54.950 --> 00:02:56.140
minus pi squared.
00:02:56.140 --> 00:03:00.520
There's an overall factor
of 2, but it doesn't matter.
00:03:00.520 --> 00:03:05.500
And essentially, this says that
because you have a unit vector,
00:03:05.500 --> 00:03:09.316
this pi cannot get too big.
00:03:09.316 --> 00:03:11.700
You have to pay
[? a cost ?] here,
00:03:11.700 --> 00:03:14.760
certainly not larger than 1.
00:03:14.760 --> 00:03:20.960
And then the expression
for the energy costs
00:03:20.960 --> 00:03:24.680
can be written in terms of
two parts [? where it ?]
00:03:24.680 --> 00:03:31.730
is the gradient
of this vector pi.
00:03:31.730 --> 00:03:33.830
But then there's
also the gradient
00:03:33.830 --> 00:03:42.140
in the other direction
which has magnitude
00:03:42.140 --> 00:03:45.030
square root of 1
minus pi squared.
00:03:45.030 --> 00:03:48.830
So we have this
gradient squared.
00:03:48.830 --> 00:03:52.400
And so basically,
these are two ways
00:03:52.400 --> 00:03:56.115
of writing the same thing, OK?
00:03:59.680 --> 00:04:05.340
So we looked at this
and we said that once
00:04:05.340 --> 00:04:09.770
we include all of these
terms, what we have here is
00:04:09.770 --> 00:04:14.410
a non-linear [? activity ?]
that includes,
00:04:14.410 --> 00:04:19.940
for example, interactions
among the various modes.
00:04:19.940 --> 00:04:23.915
And one particular
leading order term
00:04:23.915 --> 00:04:29.000
is if we expand this square
root of 1 minus pi squared,
00:04:29.000 --> 00:04:32.880
if it would be something
like pi square root of pi,
00:04:32.880 --> 00:04:37.260
so a particular term in this
expansion as the from pi
00:04:37.260 --> 00:04:42.938
[? grad ?] pi multiplied
with pi [? grad ?] pi.
00:04:42.938 --> 00:04:46.802
[INAUDIBLE], OK?
00:04:49.700 --> 00:04:57.626
So a particular way of dealing
these kinds of theories
00:04:57.626 --> 00:05:03.652
is to regard all of these
things as interactions
00:05:03.652 --> 00:05:08.500
and perturbations with respect
to a Gaussian weight which
00:05:08.500 --> 00:05:12.970
we can compute easily.
00:05:12.970 --> 00:05:18.860
And then you can either do that
perturbation straightforwardly
00:05:18.860 --> 00:05:22.280
or from the beginning to a
perturbative [? origin, ?]
00:05:22.280 --> 00:05:24.080
which is the route
that we chose.
00:05:30.440 --> 00:05:37.270
And this amount to
changing the short distance
00:05:37.270 --> 00:05:40.220
cut off that we
have here that is
00:05:40.220 --> 00:05:46.220
a to be b times a and
averaging over all [? nodes ?]
00:05:46.220 --> 00:05:53.330
within that distance short
wavelength between a and ba.
00:05:53.330 --> 00:06:01.680
And once we do that, we
arrive at a new interaction.
00:06:01.680 --> 00:06:08.214
So the first step is
to do a coarse graining
00:06:08.214 --> 00:06:10.504
between the range a and ba.
00:06:13.250 --> 00:06:18.750
But then steps two
and three amount
00:06:18.750 --> 00:06:27.170
to a rescaling in position space
so that the cut-off comes back
00:06:27.170 --> 00:06:38.480
to ba and the corresponding
thing in the spin space
00:06:38.480 --> 00:06:42.600
so that we start with a
partition function that
00:06:42.600 --> 00:06:44.610
describes unit vectors.
00:06:44.610 --> 00:06:47.660
And after this
transformation, we
00:06:47.660 --> 00:06:50.200
end up with a new
partition function
00:06:50.200 --> 00:06:53.360
that also describes
unit vectors so
00:06:53.360 --> 00:06:57.120
that after all of
these three procedures,
00:06:57.120 --> 00:07:00.290
we hope that we are
back to exactly the form
00:07:00.290 --> 00:07:05.230
that we had at the beginning
with the same cut-off,
00:07:05.230 --> 00:07:09.510
with the same unit
vector constraint,
00:07:09.510 --> 00:07:15.660
but potentially with a new
interaction parameter K
00:07:15.660 --> 00:07:19.240
And calculating
what this new K is
00:07:19.240 --> 00:07:24.550
after we've scaled by a factor
of b, the parts that correspond
00:07:24.550 --> 00:07:29.010
to 2 and 3 are
immediately obvious.
00:07:29.010 --> 00:07:35.960
Because whenever I see x, I
have to replace it wit bx prime.
00:07:35.960 --> 00:07:38.250
And so from
integration, I then get
00:07:38.250 --> 00:07:41.740
a factor of b to the d
from the two gradients,
00:07:41.740 --> 00:07:44.830
I would get a factor of minus 2.
00:07:44.830 --> 00:07:52.410
So the step that corresponds
to this is trivial.
00:07:52.410 --> 00:07:57.130
The step that corresponds to
replacing S with zeta S prime
00:07:57.130 --> 00:07:58.020
is also trivial.
00:07:58.020 --> 00:08:00.680
And it will give
you zeta squared.
00:08:00.680 --> 00:08:03.680
Or do I have yet to
tell you what is?
00:08:03.680 --> 00:08:07.000
We'll do that shortly.
00:08:07.000 --> 00:08:11.840
And finally, the first step,
which was the coarse graining,
00:08:11.840 --> 00:08:17.710
we found that what it did
was that it replaced K
00:08:17.710 --> 00:08:21.055
by a strongly-- sorry.
00:08:21.055 --> 00:08:22.540
I didn't expect this.
00:08:28.432 --> 00:08:41.200
All right-- the factor K, which
is larger by a certain amount.
00:08:41.200 --> 00:08:45.660
And the mathematical
justification
00:08:45.660 --> 00:08:53.070
that I gave for this is we
look at this expression,
00:08:53.070 --> 00:08:55.410
and we see that in
this expression,
00:08:55.410 --> 00:08:59.690
each one of these pi's
can be a long wavelength
00:08:59.690 --> 00:09:03.670
fluctuation or a short
wavelength fluctuation.
00:09:03.670 --> 00:09:08.960
Among the many possibilities
is when these 2 pi's
00:09:08.960 --> 00:09:12.480
that are sitting
out front correspond
00:09:12.480 --> 00:09:15.800
to the short wavelength
fluctuations.
00:09:15.800 --> 00:09:19.040
These correspond to the long
wavelength fluctuations.
00:09:24.450 --> 00:09:29.710
And you can see that
averaging over these two
00:09:29.710 --> 00:09:33.390
will generate an
interaction that
00:09:33.390 --> 00:09:36.740
looks like gradient
of pi lesser squared.
00:09:36.740 --> 00:09:42.380
And that will change the
coefficient over here
00:09:42.380 --> 00:09:46.620
by an amount that is clearly
proportional to the average
00:09:46.620 --> 00:09:48.540
of pi greater squared.
00:09:54.070 --> 00:09:56.440
And that we can see
in Fourier space
00:09:56.440 --> 00:10:00.350
is simply 1 over
Kq squared over KK
00:10:00.350 --> 00:10:06.740
squared for modes that
have wave number K.
00:10:06.740 --> 00:10:09.790
So if I, rather than
write this in real space,
00:10:09.790 --> 00:10:11.620
I write it in
Fourier space, this
00:10:11.620 --> 00:10:14.030
is what I would get
for the average.
00:10:14.030 --> 00:10:18.790
And in real space, I have
to integrate over this K
00:10:18.790 --> 00:10:23.050
appropriately [INAUDIBLE] within
the wave numbers lambda over
00:10:23.050 --> 00:10:33.320
[? real ?] [? light. ?] And this
is clearly something that is
00:10:33.320 --> 00:10:40.960
inversely proportional to K. And
the result of this integration
00:10:40.960 --> 00:10:44.570
of 1 over K squared-- we
simply gave a [? 9 ?],
00:10:44.570 --> 00:10:49.530
which was i sub d of b.
00:10:49.530 --> 00:10:51.415
Because it depends
on the dimension.
00:10:51.415 --> 00:10:53.586
It depends on
[? et ?] [? cetera. ?]
00:10:53.586 --> 00:10:54.265
AUDIENCE: Sir?
00:10:54.265 --> 00:10:54.890
PROFESSOR: Yes?
00:10:54.890 --> 00:10:55.698
AUDIENCE: Shouldn't
there be an exponential
00:10:55.698 --> 00:10:56.674
inside the integral?
00:10:59.040 --> 00:11:00.915
PROFESSOR: Why should
there be an exponential
00:11:00.915 --> 00:11:01.740
inside the integral?
00:11:01.740 --> 00:11:03.948
AUDIENCE: Oh, I thought we
were Fourier transforming.
00:11:03.948 --> 00:11:06.640
PROFESSOR: OK, it is true,
when we Fourier transform
00:11:06.640 --> 00:11:10.630
for each pi, we will have
a factor of e to the IK.
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.
00:11:16.190 --> 00:11:18.420
But the averaging
[INAUDIBLE] set
00:11:18.420 --> 00:11:20.940
K and K prime can be
opposite each other.
00:11:20.940 --> 00:11:22.710
So the exponentials disappear.
00:11:22.710 --> 00:11:28.190
So always remember the
integral of any field squared
00:11:28.190 --> 00:11:32.300
in real space is the same
thing as the integral
00:11:32.300 --> 00:11:34.700
of that field squared
in Fourier space.
00:11:34.700 --> 00:11:36.710
This is one of
the first theorems
00:11:36.710 --> 00:11:38.030
of Fourier transformation.
00:11:41.190 --> 00:11:46.430
OK, so this is a correction
that goes like [INAUDIBLE].
00:11:49.890 --> 00:11:57.210
And last time, to give you a
kind of visual demonstration
00:11:57.210 --> 00:12:01.960
of what this factor is, I
said that it is similar,
00:12:01.960 --> 00:12:06.250
but by no means
identical to something
00:12:06.250 --> 00:12:16.940
like this, which is that a mode
by itself has very low energy.
00:12:16.940 --> 00:12:21.170
But because we have coupling
among different modes,
00:12:21.170 --> 00:12:24.220
here for the Goldstone
modes of the surface,
00:12:24.220 --> 00:12:27.740
but here for the Goldstone
modes of the spin,
00:12:27.740 --> 00:12:33.980
the presence of a certain amount
of short range fluctuations
00:12:33.980 --> 00:12:40.390
will stiffen the modes that you
have for longer wavelengths.
00:12:40.390 --> 00:12:43.740
Now, I'm not saying that these
two problems are mathematically
00:12:43.740 --> 00:12:45.040
identical.
00:12:45.040 --> 00:12:49.390
All I'm showing you
is that the coupling
00:12:49.390 --> 00:12:52.390
between the short and
long wavelength modes
00:12:52.390 --> 00:12:59.000
can lead to a stiffening of
the modes over long distances
00:12:59.000 --> 00:13:03.390
because they have to fight
off the [? rails ?] that
00:13:03.390 --> 00:13:06.440
have been established by
shorter wavelength modes.
00:13:06.440 --> 00:13:08.410
You have to try to undo them.
00:13:08.410 --> 00:13:12.870
And that's an
additional cost, OK?
00:13:12.870 --> 00:13:21.570
Now, that stiffening over here
is opposed by a factor of zeta
00:13:21.570 --> 00:13:23.180
over here.
00:13:23.180 --> 00:13:24.860
Essentially, we
said that we have
00:13:24.860 --> 00:13:30.240
to ensure that what we
are seeing after the three
00:13:30.240 --> 00:13:34.500
steps of RG is a
description of a [? TOD ?]
00:13:34.500 --> 00:13:39.640
that has the same short distance
cut-off and the same length
00:13:39.640 --> 00:13:42.860
so that the two partition
functions can map on
00:13:42.860 --> 00:13:44.390
to each other.
00:13:44.390 --> 00:13:48.820
And again, another
visual demonstration
00:13:48.820 --> 00:13:54.556
is that you can decompose
the spin over here
00:13:54.556 --> 00:13:59.510
to a superposition of short
and long wavelength modes.
00:13:59.510 --> 00:14:03.085
And we are averaging over
these short wavelength modes.
00:14:03.085 --> 00:14:07.090
And because of that, we will
see that the effective length
00:14:07.090 --> 00:14:12.210
once that averaging has been
performed has been reduced.
00:14:12.210 --> 00:14:17.600
It has been reduced because I
will write this as 1 minus pi
00:14:17.600 --> 00:14:20.920
squared over 2 to
the lowest order.
00:14:20.920 --> 00:14:25.443
And pi squared has n
minus 1 components.
00:14:25.443 --> 00:14:28.670
So this is n minus 1 over 2.
00:14:28.670 --> 00:14:31.790
And then I have
to integrate over
00:14:31.790 --> 00:14:38.720
all of the modes pi
alpha of K in this range.
00:14:38.720 --> 00:14:43.330
So I'm performing exactly
the same integral as above.
00:14:43.330 --> 00:14:53.410
So the reduction is precisely
the same integral as above, OK?
00:14:53.410 --> 00:15:00.612
So the three steps of RG
performed for this model
00:15:00.612 --> 00:15:04.940
to lowest order
in this inverse K,
00:15:04.940 --> 00:15:09.310
our [? temperature-like ?]
variable is given by this one
00:15:09.310 --> 00:15:13.590
[INAUDIBLE] once I substitute
the value of zeta over there.
00:15:13.590 --> 00:15:18.070
So you can see that
the answer K prime of b
00:15:18.070 --> 00:15:21.590
is going to b to
the b minus 2 zeta
00:15:21.590 --> 00:15:24.470
squared-- ah, that's right.
00:15:24.470 --> 00:15:26.322
For zeta squared,
essentially the square
00:15:26.322 --> 00:15:31.180
of that, 1 minus
n minus 1 over K,
00:15:31.180 --> 00:15:33.797
the 2 disappears
once I squared it.
00:15:33.797 --> 00:15:39.120
Id of b divided by K-- that
comes from zeta squared.
00:15:39.120 --> 00:15:44.990
And from here I can get
the plus Id of be over K.
00:15:44.990 --> 00:15:48.960
And the whole thing gets
multiplied by this K.
00:15:48.960 --> 00:15:52.955
And there is still terms at
the order of temperature of 1
00:15:52.955 --> 00:15:59.020
over K squared, OK?
00:15:59.020 --> 00:16:04.960
And finally, we are going
to do the same choice
00:16:04.960 --> 00:16:09.090
that we were doing for
our epsilon expansion.
00:16:09.090 --> 00:16:12.800
That is, choose a
rescaling factor that
00:16:12.800 --> 00:16:15.631
is just slightly larger than 1.
00:16:15.631 --> 00:16:16.130
Yes?
00:16:16.130 --> 00:16:20.580
AUDIENCE: Sir, you're at n
minus 1 over K times Id over K.
00:16:20.580 --> 00:16:24.490
PROFESSOR: Yeah, I
gave this too much.
00:16:24.490 --> 00:16:25.424
Thank you.
00:16:28.230 --> 00:16:31.100
OK, thank You.
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.
00:16:41.570 --> 00:16:46.980
And we note that for
calculating this Id of b, when
00:16:46.980 --> 00:16:52.010
b goes to 1 plus delta
l, all I need to do
00:16:52.010 --> 00:16:58.240
is to evaluate this integrand
essentially on the shell.
00:16:58.240 --> 00:17:04.310
So what I will get is lambda
to the power of d minus 2.
00:17:04.310 --> 00:17:10.270
The surface area of a unit
sphere divided by 2 pi
00:17:10.270 --> 00:17:12.329
to the d, which is the
combination that we
00:17:12.329 --> 00:17:16.700
have been calling K sub d, OK?
00:17:19.849 --> 00:17:27.400
So once I do that, I will
get that the dK by dl, OK?
00:17:27.400 --> 00:17:29.050
What do I get?
00:17:29.050 --> 00:17:36.980
I will get a d minus
2 here times K.
00:17:36.980 --> 00:17:39.700
And then I will get
these two factors.
00:17:39.700 --> 00:17:41.560
There's n minus 1.
00:17:41.560 --> 00:17:42.930
And then there's 1 here.
00:17:42.930 --> 00:17:44.810
So that becomes n minus 2.
00:17:47.678 --> 00:17:56.170
I have a Idb, which is Kd
lambda to the d minus 2.
00:17:56.170 --> 00:17:59.120
And then the 1 over
K and K disappear.
00:17:59.120 --> 00:18:02.080
And so that's the
expression that we have.
00:18:07.730 --> 00:18:08.380
Yes?
00:18:08.380 --> 00:18:13.170
AUDIENCE: Sorry, is the Kd
[? some ?] angle factor again?
00:18:13.170 --> 00:18:18.820
PROFESSOR: OK, so you have
to do this integration, which
00:18:18.820 --> 00:18:24.140
is written as the surface
area inside an angle,
00:18:24.140 --> 00:18:29.590
K to the d minus 1 dK
divided by 2 pi to the d.
00:18:29.590 --> 00:18:33.009
This is the combination that
we have always called K sub d.
00:18:33.009 --> 00:18:33.550
AUDIENCE: OK.
00:18:40.410 --> 00:18:43.070
PROFESSOR: OK?
00:18:43.070 --> 00:18:46.343
Now, it actually
makes more sense
00:18:46.343 --> 00:18:50.490
since we are making a
low temperature expansion
00:18:50.490 --> 00:18:57.734
to define a T that is
simply 1 over K. Its
00:18:57.734 --> 00:19:00.120
again, dimensionless.
00:19:00.120 --> 00:19:04.330
And then clearly
dT by dl is going
00:19:04.330 --> 00:19:12.759
to be minus K squared dK by
dl minus 1 over K squared.
00:19:17.450 --> 00:19:24.840
Minus 1 over K squared becomes
minus T squared dK by dl.
00:19:24.840 --> 00:19:27.730
So I just have to
multiply the expression
00:19:27.730 --> 00:19:32.790
that I have up here
with minus T squared,
00:19:32.790 --> 00:19:35.253
recognizing that TK is 1.
00:19:35.253 --> 00:19:39.210
So I end up with the
[? recursion ?] convention
00:19:39.210 --> 00:19:45.640
for T, which is
minus d minus 2T.
00:19:45.640 --> 00:19:53.800
And then it becomes plus n minus
2 Kd lambda to the d minus 2 T
00:19:53.800 --> 00:19:54.550
squared.
00:19:54.550 --> 00:19:57.390
And presumably, there
are high order terms
00:19:57.390 --> 00:20:01.800
that we have not
bothered to calculate.
00:20:01.800 --> 00:20:05.980
So this is the
[INAUDIBLE] we focused on.
00:20:11.646 --> 00:20:12.146
OK?
00:20:15.130 --> 00:20:20.640
So let's see whether this
expression makes sense.
00:20:20.640 --> 00:20:27.930
So if I'm looking at dimensions
that are less than 2,
00:20:27.930 --> 00:20:32.760
then the linear term
in the expression
00:20:32.760 --> 00:20:41.920
is positive, which means that if
I'm looking at the temperature
00:20:41.920 --> 00:20:48.560
axis and this is 0, and
I start with a value that
00:20:48.560 --> 00:20:55.770
is slightly positive,
because of this term,
00:20:55.770 --> 00:21:00.700
it will be pushed larger
and larger values.
00:21:00.700 --> 00:21:03.890
So you think that
you have a system
00:21:03.890 --> 00:21:05.965
at very low temperatures.
00:21:05.965 --> 00:21:08.600
You look at it at larger
and larger scales,
00:21:08.600 --> 00:21:10.930
and you find that it
becomes effectively
00:21:10.930 --> 00:21:13.380
something that has
higher temperature
00:21:13.380 --> 00:21:16.260
and becomes more
and more disordered.
00:21:16.260 --> 00:21:23.190
So basically, this is a
manifestation of something
00:21:23.190 --> 00:21:31.010
that we had said before,
Mermin-Wagner theorem, which
00:21:31.010 --> 00:21:42.440
is no [? long ?] range
order in d less than 2, OK?
00:21:42.440 --> 00:21:51.470
Now, if I go to the other
limit, d greater than 2,
00:21:51.470 --> 00:21:57.660
then something
interesting happens,
00:21:57.660 --> 00:22:04.700
in that the linear
term is negative.
00:22:04.700 --> 00:22:09.700
So if I start with a
sufficiently small temperature
00:22:09.700 --> 00:22:13.360
or a large enough coupling,
it will get stronger
00:22:13.360 --> 00:22:17.450
as we go towards
an ordered phase,
00:22:17.450 --> 00:22:21.540
whereas the quadratic
term for n greater than 2
00:22:21.540 --> 00:22:25.545
has the opposite sign--
this is n greater than 2--
00:22:25.545 --> 00:22:32.010
and pushes me towards disorder,
which means that there should
00:22:32.010 --> 00:22:39.570
be a fixed point that
separates the two behaviors.
00:22:39.570 --> 00:22:45.090
Any temperature lower than this
will give me an ordered phase.
00:22:45.090 --> 00:22:47.800
Any temperature higher
than this will give me
00:22:47.800 --> 00:22:50.250
a disordered phase.
00:22:50.250 --> 00:22:55.150
And suddenly, we see that
we have potentially a way
00:22:55.150 --> 00:23:00.430
of figuring out what the phase
transition is because this T
00:23:00.430 --> 00:23:06.630
star is a location that we
can perturbatively access.
00:23:06.630 --> 00:23:10.280
Because we set this
to 0, and we find
00:23:10.280 --> 00:23:16.250
that T star is
equal to d minus 2
00:23:16.250 --> 00:23:26.430
divided by n minus 2 Kd
lambda to the d minus 2.
00:23:26.430 --> 00:23:33.830
So now in order to have
a theory that makes sense
00:23:33.830 --> 00:23:38.130
in the sense of the perturbation
that we have carried out,
00:23:38.130 --> 00:23:41.570
we have to make sure
that this is small.
00:23:41.570 --> 00:23:47.706
So we can do that by assuming
that this quantity d minus 2
00:23:47.706 --> 00:23:52.853
is a small quantity in making
an expansion in d minus 2, OK?
00:23:56.280 --> 00:24:00.150
So in particular,
T star itself we
00:24:00.150 --> 00:24:05.590
expect to be related to
transition temperature, not
00:24:05.590 --> 00:24:07.490
something that is universal.
00:24:07.490 --> 00:24:10.700
But exponents are universal.
00:24:10.700 --> 00:24:14.950
So what we do is we
look at d by dl of delta
00:24:14.950 --> 00:24:18.390
T. Delta T is,
let's say, T minus T
00:24:18.390 --> 00:24:21.060
star in one direction
or the other.
00:24:21.060 --> 00:24:22.860
[INAUDIBLE]
00:24:22.860 --> 00:24:26.490
And for that, what
I need to do it
00:24:26.490 --> 00:24:29.240
to linearize this expression.
00:24:29.240 --> 00:24:33.046
So I will get a minus
epsilon from her.
00:24:33.046 --> 00:24:39.334
And from here, I will
get 2 n minus 2 Kd lambda
00:24:39.334 --> 00:24:49.570
to the d minus 2 T
star times delta T.
00:24:49.570 --> 00:24:53.315
I just took the derivative,
evaluated the T star.
00:24:53.315 --> 00:24:58.530
And we can see that this
combination is precisely
00:24:58.530 --> 00:25:02.510
the combination that I
have to solve for T star.
00:25:02.510 --> 00:25:05.780
So this really becomes
another factor of epsilon.
00:25:05.780 --> 00:25:09.370
I have minus epsilon
plus [? 2 ?] epsilon.
00:25:09.370 --> 00:25:13.960
So this is epsilon delta
T. So that tells me
00:25:13.960 --> 00:25:25.520
that my thermal
eigenvalue is epsilon,
00:25:25.520 --> 00:25:29.300
a disorder clearly
independent of n.
00:25:32.480 --> 00:25:36.170
Now, we've seen that in
order to fully characterize
00:25:36.170 --> 00:25:42.440
the exponent, including things
like magnetization, et cetera,
00:25:42.440 --> 00:25:47.690
it makes sense to also put
a magnetic field direction
00:25:47.690 --> 00:25:52.900
and figure out how rapidly you
go along the magnetic field
00:25:52.900 --> 00:25:55.060
direction.
00:25:55.060 --> 00:25:59.530
So for that, one
way of doing this
00:25:59.530 --> 00:26:06.698
is to go and add to this term,
which is h integral S of x.
00:26:10.040 --> 00:26:12.950
And you can see very easily
that under these steps
00:26:12.950 --> 00:26:15.580
of the transformation,
essentially the only thing that
00:26:15.580 --> 00:26:20.240
happens is that I will
get h prime at scale
00:26:20.240 --> 00:26:23.380
d is h from the integration.
00:26:23.380 --> 00:26:26.490
I will get a factor
of b to the d.
00:26:26.490 --> 00:26:29.050
From the replacement
of s with s prime,
00:26:29.050 --> 00:26:30.560
I will get a factor of zeta.
00:26:34.230 --> 00:26:39.290
So this combination
is simply my yh.
00:26:39.290 --> 00:26:42.666
And just bringing a
little bit of manipulation
00:26:42.666 --> 00:26:47.730
will tell you that yh
is d minus the part that
00:26:47.730 --> 00:26:53.280
comes from zeta,
which is n minus 1
00:26:53.280 --> 00:27:00.980
over 2 Id of b, which is
lambda to the d minus 2 Kd.
00:27:00.980 --> 00:27:04.280
And then we have T star.
00:27:07.380 --> 00:27:11.200
And again, you
substitute for lambda
00:27:11.200 --> 00:27:16.520
to the d minus 2 Kd T star
on what we have over here.
00:27:16.520 --> 00:27:22.750
And you get this to
be d minus n minus 1
00:27:22.750 --> 00:27:25.360
over 2n minus 2 epsilon.
00:27:28.040 --> 00:27:31.290
And again, to be consistent
to order of epsilon,
00:27:31.290 --> 00:27:36.650
this d you will have to
replace with 2 plus epsilon.
00:27:36.650 --> 00:27:39.480
And a little bit of
manipulation will give you
00:27:39.480 --> 00:27:50.240
yh, which is 2 minus n minus 3
divided by 2n minus 2 epsilon.
00:27:59.930 --> 00:28:07.270
I did this calculation of the
two exponents rather rapidly.
00:28:07.270 --> 00:28:12.036
The reason for that is they
are not particularly useful.
00:28:12.036 --> 00:28:15.730
That is, whereas
we saw that coming
00:28:15.730 --> 00:28:20.080
from four dimensions the
epsilon expansion was very
00:28:20.080 --> 00:28:23.600
useful to give us
corrections to the [? mean ?]
00:28:23.600 --> 00:28:29.390
field values of 1/2, for
example, for mu to order of 10%
00:28:29.390 --> 00:28:32.780
or so already by setting
epsilon equals to 1.
00:28:32.780 --> 00:28:36.670
If I, for example, put
here epsilon equal to 1
00:28:36.670 --> 00:28:39.610
to [? access ?] 3
dimensions, I will
00:28:39.610 --> 00:28:42.310
conclude that mu, which
is the inverse of yT
00:28:42.310 --> 00:28:45.990
is 1 in 3 dimensions
independent of n.
00:28:45.990 --> 00:28:50.570
And let's say for super
fluid it is closer to 2/3.
00:28:50.570 --> 00:28:56.000
And so essentially, this
expansion in some sense
00:28:56.000 --> 00:29:01.030
is much further away from 3
dimensions than the 4 minus
00:29:01.030 --> 00:29:03.610
epsilon coming
from 4 dimensions,
00:29:03.610 --> 00:29:06.030
although numerically,
we would have
00:29:06.030 --> 00:29:09.730
said epsilon equaled
1 to both of them.
00:29:09.730 --> 00:29:20.485
So nobody really has taken
much advantage of this 2 plus
00:29:20.485 --> 00:29:23.810
epsilon expansion.
00:29:23.810 --> 00:29:26.440
So why is it useful?
00:29:26.440 --> 00:29:30.200
The reason that this is
useful is the third case
00:29:30.200 --> 00:29:32.276
that I have not
explained, yet, which
00:29:32.276 --> 00:29:38.150
is, what happens if you sit
exactly in 2 dimensions, OK?
00:29:38.150 --> 00:29:41.620
So if we sit exactly
in 2 dimensions,
00:29:41.620 --> 00:29:43.870
this first term disappears.
00:29:43.870 --> 00:29:46.940
And you can see
that the behavioral
00:29:46.940 --> 00:29:50.180
is determined by
the second order
00:29:50.180 --> 00:29:54.360
and [? then ?] depends
on the value of n.
00:29:54.360 --> 00:29:59.320
So if I look at n, let's
say, that is less than 2,
00:29:59.320 --> 00:30:04.210
then what I will see is
that along the temperature
00:30:04.210 --> 00:30:09.180
axis, the quadratic
term-- the linear term
00:30:09.180 --> 00:30:12.580
is absent-- the quadratic
term, let's say, for n
00:30:12.580 --> 00:30:16.080
equals 1 is negative.
00:30:16.080 --> 00:30:19.540
And you're being
pushed quadratically
00:30:19.540 --> 00:30:21.550
very slowly towards 0.
00:30:21.550 --> 00:30:25.165
The one example that
we know is indeed n
00:30:25.165 --> 00:30:27.130
equals 1, the Ising model.
00:30:27.130 --> 00:30:31.560
And we know that the Ising
model in 2 dimensions
00:30:31.560 --> 00:30:32.850
has an ordered phase.
00:30:32.850 --> 00:30:35.305
It shouldn't really even
be described by this
00:30:35.305 --> 00:30:38.990
because there are
no Goldstone modes.
00:30:38.990 --> 00:30:43.941
But n greater than 2, like
n equals 3-- the Heisenberg
00:30:43.941 --> 00:30:48.270
model, is interesting.
00:30:48.270 --> 00:30:54.435
And what we see is that here the
second order term is positive.
00:30:54.435 --> 00:30:59.690
And it is pushing you
towards high temperatures.
00:30:59.690 --> 00:31:02.928
So you can see a
disordered behavior.
00:31:02.928 --> 00:31:06.230
And what this calculation
tells you that is useful
00:31:06.230 --> 00:31:09.740
that you wouldn't
have known otherwise
00:31:09.740 --> 00:31:12.800
is, what is the
correlation [? length? ?]
00:31:12.800 --> 00:31:19.090
Because the recursion relation
is now dT by dl is, let's say,
00:31:19.090 --> 00:31:20.270
n minus 2.
00:31:23.150 --> 00:31:24.480
And my d equals to 2.
00:31:24.480 --> 00:31:27.280
The lambda to the d
minus 2, I can ignore.
00:31:27.280 --> 00:31:34.160
Kd is 2 pi from the [? solid ?]
angle divided by 2 pi squared.
00:31:34.160 --> 00:31:38.980
So that's 1 over 2 pi
times T squared, OK?
00:31:38.980 --> 00:31:43.010
So that's the recursion relation
that we are dealing with.
00:31:43.010 --> 00:31:46.576
I can divide by
1 over T squared.
00:31:46.576 --> 00:31:53.640
And then this becomes d
by dl of minus 1 over T
00:31:53.640 --> 00:31:57.670
equals n minus 2
divided by 2 pi.
00:32:00.600 --> 00:32:09.330
I can integrate this from, say,
some initial value minus 1 over
00:32:09.330 --> 00:32:16.254
some initial temperature
to some temperature where
00:32:16.254 --> 00:32:18.775
I'm at [? length ?]
[? scale ?] l.
00:32:18.775 --> 00:32:25.582
What I would have on the right
hand side would be n minus 2
00:32:25.582 --> 00:32:29.940
over [? 2yl ?], OK?
00:32:29.940 --> 00:32:35.800
So I start very, very close
to the origin T equals 0.
00:32:35.800 --> 00:32:40.700
So I have a very strong
coupling at the beginning.
00:32:40.700 --> 00:32:42.310
So this factor is huge.
00:32:42.310 --> 00:32:44.440
T is [? more than ?]
1 over T is huge.
00:32:44.440 --> 00:32:47.930
I have a huge coupling.
00:32:47.930 --> 00:32:54.390
And then I rescale to a point
where the coupling has become
00:32:54.390 --> 00:33:00.470
weak, let's say some number
of order of 1, order of 1
00:33:00.470 --> 00:33:01.970
or order of 0.
00:33:01.970 --> 00:33:04.840
In any case, it is
overwhelmingly smaller
00:33:04.840 --> 00:33:07.550
than this.
00:33:07.550 --> 00:33:09.470
How far did I have to go?
00:33:09.470 --> 00:33:12.294
I had to rescale
by a factor of l
00:33:12.294 --> 00:33:13.710
that is related
to the temperature
00:33:13.710 --> 00:33:18.150
that I started with
by this factor,
00:33:18.150 --> 00:33:20.443
except that I forgot
the minus that I
00:33:20.443 --> 00:33:23.470
had in front of the whole thing.
00:33:23.470 --> 00:33:31.430
So the resulting l will
be large and positive.
00:33:31.430 --> 00:33:37.100
And the correlation length--
the length scale at which
00:33:37.100 --> 00:33:39.020
we arrived at the
coupling, which
00:33:39.020 --> 00:33:45.070
is of the order of 1 or 0,
is whatever my initial length
00:33:45.070 --> 00:33:51.090
scale was times this factor
that I have rescaled by,
00:33:51.090 --> 00:33:54.290
b, which is e to the l.
00:33:54.290 --> 00:34:02.268
And so this is a exponential
of n minus 1 over 2 pi
00:34:02.268 --> 00:34:09.270
times 1 over T. The
statement is that if you're
00:34:09.270 --> 00:34:13.914
having 2 dimensions, a
system of, let's say,
00:34:13.914 --> 00:34:17.679
3 component spins-- and
that is something that
00:34:17.679 --> 00:34:21.690
has a lot of experimental
realizations--
00:34:21.690 --> 00:34:25.790
you find that as you go
towards low temperature,
00:34:25.790 --> 00:34:28.520
the size of domains
that are ordered
00:34:28.520 --> 00:34:34.580
diverges according to
this nice universal form.
00:34:34.580 --> 00:34:41.489
And let's say around
1995 or so, when
00:34:41.489 --> 00:34:44.530
people had these high
temperature superconductors
00:34:44.530 --> 00:34:49.280
which are effectively 2
dimensional layers of magnets--
00:34:49.280 --> 00:34:50.949
they're actually
antiferromagnets,
00:34:50.949 --> 00:34:53.730
but they are still described
by this [INAUDIBLE] with n
00:34:53.730 --> 00:34:57.450
equals 3-- there were
lots of x-ray studies
00:34:57.450 --> 00:35:01.930
of what happens to the ordering
of these antiferromagnetic
00:35:01.930 --> 00:35:04.730
copper oxide layers as you
go to low temperatures.
00:35:04.730 --> 00:35:10.566
And this form was very
much used and confirmed.
00:35:10.566 --> 00:35:16.610
OK, so that's really one thing
that one can get from this
00:35:16.610 --> 00:35:19.344
analysis that has been
explicitly [? confirmed ?]
00:35:19.344 --> 00:35:20.260
[? for ?] experiments.
00:35:23.290 --> 00:35:25.526
And finally, there's
one case in this
00:35:25.526 --> 00:35:32.060
that I have not mentioned
so far, which is n equals 2.
00:35:32.060 --> 00:35:37.780
And when I am at n
equals 2, what I have is
00:35:37.780 --> 00:35:42.350
that the first and second
order terms in this series
00:35:42.350 --> 00:35:46.400
are both vanishing.
00:35:46.400 --> 00:35:53.710
And I really at this stage don't
quite know what is happening.
00:35:53.710 --> 00:35:55.800
But we can think
about it a little bit.
00:35:55.800 --> 00:36:00.160
And you can see that if you are
n equals 2, then essentially,
00:36:00.160 --> 00:36:03.830
you have a 1 component angle.
00:36:03.830 --> 00:36:07.510
And if I write the theory
in terms of the angle theta,
00:36:07.510 --> 00:36:11.000
let's say, between
neighboring spins,
00:36:11.000 --> 00:36:15.960
then the expansions would simply
be gradient of theta squared.
00:36:15.960 --> 00:36:21.560
And there isn't any other
mode to couple with.
00:36:21.560 --> 00:36:24.430
You may worry a little bit
about gradient of theta
00:36:24.430 --> 00:36:26.290
to the 4th and such things.
00:36:26.290 --> 00:36:28.640
But a little bit of
thinking will convince you
00:36:28.640 --> 00:36:31.150
that all of those
terms are irrelevant.
00:36:31.150 --> 00:36:35.280
So as far as we can
show, there is reason
00:36:35.280 --> 00:36:38.820
that essentially
this series for n
00:36:38.820 --> 00:36:44.460
equals 2 is 0 at
all orders, which
00:36:44.460 --> 00:36:50.902
means that as far as this
analysis is concerned,
00:36:50.902 --> 00:36:55.550
there is a kind of a
line of fixed points.
00:36:55.550 --> 00:36:58.610
You start with any
temperature, and you
00:36:58.610 --> 00:37:04.100
will stay at that
temperature, OK?
00:37:04.100 --> 00:37:08.490
Still you would say that even if
you have this gradient of theta
00:37:08.490 --> 00:37:12.260
squared type of theory,
the fluctuations that you
00:37:12.260 --> 00:37:18.630
have are solutions
of 1 over q squared.
00:37:18.630 --> 00:37:22.690
And the integral of 1 over
2 squared in 2 dimensions
00:37:22.690 --> 00:37:25.480
is logarithmically divergent.
00:37:25.480 --> 00:37:30.760
So the more correct statement
of the Mermin-Wagner's theorem
00:37:30.760 --> 00:37:32.830
is that there should
be no long range
00:37:32.830 --> 00:37:37.310
order in d less
than or equal to 2.
00:37:37.310 --> 00:37:40.750
Because for d equals
2 also, you have
00:37:40.750 --> 00:37:44.790
these logarithmic
divergence of fluctuations.
00:37:44.790 --> 00:37:47.930
So you may have thought that you
are pointing along, say, the y
00:37:47.930 --> 00:37:49.140
direction.
00:37:49.140 --> 00:37:51.380
But you average more
and more, and you
00:37:51.380 --> 00:37:54.730
see that the extent of
the fluctuations in angle
00:37:54.730 --> 00:37:56.560
are growing logarithmically.
00:37:56.560 --> 00:38:00.470
You say that once that logarithm
becomes of the order of pi,
00:38:00.470 --> 00:38:02.620
I have no idea
where my angle is.
00:38:02.620 --> 00:38:05.200
There should be no
true long range order.
00:38:05.200 --> 00:38:10.110
And I'm not going to try
to interpret this too much.
00:38:10.110 --> 00:38:12.920
I just say that
Mermin-Wagner's theorem
00:38:12.920 --> 00:38:16.260
says that there should
be no true long range
00:38:16.260 --> 00:38:20.890
order in systems that have
continuous symmetry in 2
00:38:20.890 --> 00:38:24.850
dimensions and below, OK?
00:38:30.530 --> 00:38:36.810
And that statement is
correct, except that
00:38:36.810 --> 00:38:47.180
around that same time, Stanley
and Kaplan did low temperature
00:38:47.180 --> 00:38:55.780
series analysis-- actually,
no, I'm incorrect--
00:38:55.780 --> 00:39:06.370
did high temperatures
series of these spin models
00:39:06.370 --> 00:39:08.740
in 2 dimensions.
00:39:08.740 --> 00:39:12.625
And what they found was,
OK, let's [? re-plot ?]
00:39:12.625 --> 00:39:16.840
susceptibility as a
function of temperature.
00:39:16.840 --> 00:39:19.950
We calculate our best
estimate of susceptibility
00:39:19.950 --> 00:39:22.300
from the high
temperature series.
00:39:22.300 --> 00:39:25.440
And what they do is, let's
say, they look at the system
00:39:25.440 --> 00:39:29.170
that corresponds to n equals 3.
00:39:29.170 --> 00:39:35.420
And they see that the
susceptibility diverges only
00:39:35.420 --> 00:39:39.210
when you get in the
vicinity of 0 temperature,
00:39:39.210 --> 00:39:43.390
which is consistent with all
of these statements that first
00:39:43.390 --> 00:39:46.240
of all, this is a [? direct ?]
correlation [? when it ?] only
00:39:46.240 --> 00:39:47.760
diverges at 0 temperature.
00:39:47.760 --> 00:39:51.790
And divergence of susceptibility
has to be coupled through that.
00:39:51.790 --> 00:39:55.090
And therefore, really,
the only exciting thing
00:39:55.090 --> 00:39:58.315
is right at 0 temperature,
there is no region where
00:39:58.315 --> 00:40:03.020
this is long range
order, except that when
00:40:03.020 --> 00:40:09.050
they did the analysis
for n equals 2,
00:40:09.050 --> 00:40:14.590
they kept getting
signature that there
00:40:14.590 --> 00:40:18.390
is a phase transition at
a finite temperature in d
00:40:18.390 --> 00:40:23.750
equals 2 for this xy
model that described
00:40:23.750 --> 00:40:26.680
by just an [INAUDIBLE], OK?
00:40:26.680 --> 00:40:41.200
So there is lots of numerical
evidence of phase transition
00:40:41.200 --> 00:40:47.225
for n equals 2 in d
equals [? 2, ?] OK?
00:40:51.040 --> 00:40:56.350
So this is another
one of those puzzles
00:40:56.350 --> 00:41:02.120
which [INAUDIBLE]
if we interpret
00:41:02.120 --> 00:41:08.350
the existence of a diverging
susceptibility in the way
00:41:08.350 --> 00:41:10.770
that we are used to,
let's say the Ising model
00:41:10.770 --> 00:41:13.130
and all the models
that we have discussed
00:41:13.130 --> 00:41:18.050
so far, in all cases that
we have seen, essentially,
00:41:18.050 --> 00:41:25.170
the divergence of
the susceptibility
00:41:25.170 --> 00:41:28.970
was an indicator of the onset
of true long range order
00:41:28.970 --> 00:41:34.050
so that on the other side, you
had something like a magnet.
00:41:34.050 --> 00:41:38.570
But that is rigorously
ruled out by Mermin-Wagner.
00:41:38.570 --> 00:41:42.820
So the question is, can
we have a phase transition
00:41:42.820 --> 00:41:47.200
in the absence of
symmetry breaking?
00:41:47.200 --> 00:41:49.080
All right?
00:41:49.080 --> 00:41:55.980
And we already saw one example
of that a couple of lectures
00:41:55.980 --> 00:41:59.770
back when we were doing the
dual of the 3-dimensional Ising
00:41:59.770 --> 00:42:01.330
model.
00:42:01.330 --> 00:42:05.550
We saw that the 3-dimensional
Ising model, its dual
00:42:05.550 --> 00:42:10.110
had a phase transition but
was rigorously prevented
00:42:10.110 --> 00:42:13.090
from having true
long range order.
00:42:13.090 --> 00:42:17.090
So there, how did we distinguish
the different phases?
00:42:17.090 --> 00:42:20.580
We found some appropriate
correlation function.
00:42:20.580 --> 00:42:23.100
And we showed that that
correlation function
00:42:23.100 --> 00:42:26.730
had different behaviors at
high and low temperature.
00:42:26.730 --> 00:42:30.650
And these two different
behaviors could not be matched.
00:42:30.650 --> 00:42:34.350
And so the phase
transition was an indicator
00:42:34.350 --> 00:42:37.850
of the switch-over in the
behavior of the correlation
00:42:37.850 --> 00:42:40.190
functions.
00:42:40.190 --> 00:42:45.270
So here, let's examine
the correlation functions
00:42:45.270 --> 00:42:46.400
of our model.
00:42:52.320 --> 00:42:54.460
And the simplest
correlation that we
00:42:54.460 --> 00:42:59.830
can think of for a system that
is described by unit spins
00:42:59.830 --> 00:43:04.060
is to look at the spin at
some location and the spin
00:43:04.060 --> 00:43:07.230
at some far away
location and ask
00:43:07.230 --> 00:43:10.730
how correlated they
are to each other?
00:43:10.730 --> 00:43:15.080
And so basically, there is
some kind of, let's say,
00:43:15.080 --> 00:43:16.170
underlying lattice.
00:43:20.510 --> 00:43:27.060
And we pick 2 points
at 0 and at r.
00:43:29.980 --> 00:43:34.352
And we ask, what
is the dot product
00:43:34.352 --> 00:43:37.225
of the spins that we have
at these 2 locations?
00:43:40.760 --> 00:43:44.033
And clearly, this is invariant
under the [? global ?]
00:43:44.033 --> 00:43:45.870
rotation.
00:43:45.870 --> 00:43:49.040
What I can do is I can
pick some kind of axis
00:43:49.040 --> 00:43:53.040
and define angles with
respect to some axis.
00:43:53.040 --> 00:43:55.590
Let's say with respect
to the x direction,
00:43:55.590 --> 00:43:58.200
I define an angle theta.
00:43:58.200 --> 00:44:02.120
And then clearly, this
is the expectation value
00:44:02.120 --> 00:44:05.960
of cosine of theta
0 minus theta r.
00:44:10.720 --> 00:44:14.800
Now, this quantity
I can asymptotically
00:44:14.800 --> 00:44:16.950
calculate both at
high temperatures
00:44:16.950 --> 00:44:20.190
and low temperatures
and compare them.
00:44:20.190 --> 00:44:22.116
So let's do a high T expansion.
00:44:25.380 --> 00:44:28.820
For the high T expansion,
I sort of go back
00:44:28.820 --> 00:44:37.470
to the discrete model and
say that what I have here
00:44:37.470 --> 00:44:42.340
is a system that
is characterized
00:44:42.340 --> 00:44:52.020
by a bunch of angles that I
have to integrate in theta i.
00:44:52.020 --> 00:44:58.690
I have the cosine of
theta 0 minus theta r.
00:44:58.690 --> 00:45:03.910
And I have a weight that wants
to make near neighbours to be
00:45:03.910 --> 00:45:05.400
parallel.
00:45:05.400 --> 00:45:09.140
And so I will write it as
product over nearest neighbors,
00:45:09.140 --> 00:45:15.826
p to the K cosine of
theta i minus theta j.
00:45:15.826 --> 00:45:19.750
OK, so the dot
product of 2 spins
00:45:19.750 --> 00:45:23.160
I have written as the
cosine between [? nearest ?]
00:45:23.160 --> 00:45:24.410
neighbors.
00:45:24.410 --> 00:45:33.344
And of course, I have to
then divide by [INAUDIBLE].
00:45:41.790 --> 00:45:47.820
Now, if I'm doing the high
temperature expansion,
00:45:47.820 --> 00:45:50.450
that means that this
coupling constant K
00:45:50.450 --> 00:45:53.050
scaled by temperature is known.
00:45:53.050 --> 00:45:59.320
And I can expand this as 1 plus
K cosine of theta i minus theta
00:45:59.320 --> 00:46:08.160
j [? plus ?] [? higher ?] orders
in powers of K of course, OK?
00:46:08.160 --> 00:46:12.110
Now, this looks to
have the same structure
00:46:12.110 --> 00:46:15.470
as we had for the Ising model.
00:46:15.470 --> 00:46:18.920
In the Ising model, I had
something like sigma i sigma j.
00:46:18.920 --> 00:46:22.120
And if I had a sigma
by itself and I
00:46:22.120 --> 00:46:26.130
summed over the possible
values, I would get 0.
00:46:26.130 --> 00:46:31.630
Here, I have something
like a cosine of an angle.
00:46:31.630 --> 00:46:38.570
And if I integrate, let's say, d
theta 0 cosine of theta 0 minus
00:46:38.570 --> 00:46:44.570
something, just because
theta 0 can be both positive
00:46:44.570 --> 00:46:46.520
as [? it just ?] goes
over the entire angle,
00:46:46.520 --> 00:46:47.670
this will give me 0.
00:46:50.400 --> 00:46:56.480
So this cosine I
better get rid of.
00:46:56.480 --> 00:47:01.380
And the way that I
can do that is let's
00:47:01.380 --> 00:47:05.606
say I multiply cosine
of theta 0 minus theta r
00:47:05.606 --> 00:47:07.530
with one of the
terms that I would
00:47:07.530 --> 00:47:11.560
get in the expansion, such
as, let's say, a factor of K
00:47:11.560 --> 00:47:15.154
cosine of theta 0 minus theta 1.
00:47:15.154 --> 00:47:17.830
So if I call the
next one theta 1,
00:47:17.830 --> 00:47:20.510
I will have a term
in the expansion that
00:47:20.510 --> 00:47:28.710
is cosine of theta
0 minus theta 1, OK?
00:47:28.710 --> 00:47:34.140
Then this will be non-zero
because I can certainly
00:47:34.140 --> 00:47:35.320
change the origin.
00:47:35.320 --> 00:47:46.610
I can write this as integral
d theta 0 minus theta 1--
00:47:46.610 --> 00:47:49.270
I can call phi.
00:47:49.270 --> 00:47:53.060
This would be cosine
of phi from here.
00:47:53.060 --> 00:48:00.200
This becomes cosine of theta
0 minus theta 1 minus phi.
00:48:02.830 --> 00:48:09.520
And this I can expand as
cosine of a 0 minus theta 1
00:48:09.520 --> 00:48:17.350
cosine of phi minus sine
of theta 0 minus theta 1
00:48:17.350 --> 00:48:18.350
sine of phi.
00:48:22.900 --> 00:48:29.600
Then cosine integrated
against sine will give me 0.
00:48:29.600 --> 00:48:33.830
Cosine integrated against
cosine will give me 1/2.
00:48:33.830 --> 00:48:45.380
So this becomes 1/2 cosine of
theta 0 minus theta 1 theta r.
00:48:45.380 --> 00:49:03.020
What did I-- For theta 0, I
am writing phi plus theta 1.
00:49:03.020 --> 00:49:09.570
So this becomes phi plus
theta 1 minus theta r.
00:49:09.570 --> 00:49:16.640
This becomes cosine of
theta 1 minus theta r.
00:49:16.640 --> 00:49:20.200
This becomes theta
1 minus theta r.
00:49:20.200 --> 00:49:24.340
This becomes theta
1 minus theta r.
00:49:24.340 --> 00:49:33.380
OK, so essentially,
we had a term
00:49:33.380 --> 00:49:42.700
that was like a cosine of theta
0 minus theta r from here.
00:49:42.700 --> 00:49:46.540
Once we integrate
over this bond,
00:49:46.540 --> 00:49:49.120
then I get a factor
of 1/2, and it
00:49:49.120 --> 00:49:53.686
becomes like a connection
between these two.
00:49:53.686 --> 00:49:57.480
And you can see that
I can keep doing that
00:49:57.480 --> 00:50:04.260
and find the path that
connects from 0 to r.
00:50:04.260 --> 00:50:07.980
For each one of the
bonds along this path,
00:50:07.980 --> 00:50:12.000
I pick one of these factors.
00:50:12.000 --> 00:50:16.390
And this allows me to
get a finite value.
00:50:16.390 --> 00:50:21.570
And what I find once I do this
is that through the lowest
00:50:21.570 --> 00:50:27.660
order, I have to count the
shortest number of paths
00:50:27.660 --> 00:50:32.630
that I have between
the two, K. I
00:50:32.630 --> 00:50:35.470
will get a factor of K.
And then from the averaging
00:50:35.470 --> 00:50:38.290
over the angles, I will get 1/2.
00:50:38.290 --> 00:50:42.150
So it would b K
over 2 [INAUDIBLE].
00:50:42.150 --> 00:50:50.050
By this we indicate the
shortest path between the 2, OK?
00:50:50.050 --> 00:50:54.700
So the point is that
K is a small number.
00:50:54.700 --> 00:50:57.520
If I go further
and further away,
00:50:57.520 --> 00:51:01.265
this is going to be
exponentially small
00:51:01.265 --> 00:51:04.410
in the distance
between the 2 spaces,
00:51:04.410 --> 00:51:08.660
where [? c ?] can be
expressed in something that
00:51:08.660 --> 00:51:12.100
has to do with K.
So this is actually
00:51:12.100 --> 00:51:13.950
quite a general statement.
00:51:13.950 --> 00:51:16.110
We've already seen it
for the Ising model.
00:51:16.110 --> 00:51:19.510
We've now seen it
for the xy model.
00:51:19.510 --> 00:51:24.090
Quite generally, for systems
at high temperatures,
00:51:24.090 --> 00:51:27.466
once can show that correlations
decay exponentially
00:51:27.466 --> 00:51:32.570
in separation because
the information
00:51:32.570 --> 00:51:38.665
about the state of one variable
has to travel all the way
00:51:38.665 --> 00:51:41.620
to influence the other one.
00:51:41.620 --> 00:51:45.800
And the fidelity by which the
information is transmitted
00:51:45.800 --> 00:51:50.320
is very small at
high temperatures.
00:51:50.320 --> 00:51:51.310
So OK?
00:51:51.310 --> 00:51:54.130
So this is something that
you should have known.
00:51:54.130 --> 00:51:57.520
We are getting the answer.
00:51:57.520 --> 00:52:00.320
But now what happens if I go
and look at low temperatures?
00:52:03.020 --> 00:52:07.470
So for low temperatures,
what I need to do
00:52:07.470 --> 00:52:14.720
is to evaluate
something that has
00:52:14.720 --> 00:52:18.450
to do with the behavior
of these angles
00:52:18.450 --> 00:52:21.105
when I go to low temperatures.
00:52:21.105 --> 00:52:23.220
And when I go to
low temperatures,
00:52:23.220 --> 00:52:27.710
these angles tend to be very
much aligned to each other.
00:52:27.710 --> 00:52:31.290
And these factors of
cosine I can therefore
00:52:31.290 --> 00:52:36.080
start expanding around 1.
00:52:36.080 --> 00:52:40.790
So what I end up having to do is
something like a product over i
00:52:40.790 --> 00:52:46.400
theta i cosine of
theta 0 minus theta r.
00:52:46.400 --> 00:52:50.110
I have a product over
neighbors of factors
00:52:50.110 --> 00:52:54.110
such as K over 2 theta
i minus theta j squared,
00:52:54.110 --> 00:52:58.800
[? as ?] I expand the
Gaussian, expand the cosine.
00:52:58.800 --> 00:53:09.740
And in the denominator I would
have exactly the same thing
00:53:09.740 --> 00:53:13.260
without this.
00:53:13.260 --> 00:53:19.790
So essentially, we see
that since the cosine is
00:53:19.790 --> 00:53:25.620
the real part of e to the i
theta 0 minus theta r, what
00:53:25.620 --> 00:53:29.880
I need to do is to calculate
the average of this
00:53:29.880 --> 00:53:32.962
assuming the Gaussian weight.
00:53:32.962 --> 00:53:37.474
So the theta is Gaussian
distributed, OK?
00:53:40.350 --> 00:53:45.050
Now-- actually, this [INAUDIBLE]
I can take the outside also.
00:53:45.050 --> 00:53:46.592
It doesn't matter.
00:53:46.592 --> 00:53:50.830
I have to calculate
this expectation value.
00:53:50.830 --> 00:53:59.560
And this for any Gaussian
expectation value of e to the i
00:53:59.560 --> 00:54:06.960
some Gaussian
variable is minus 1/2
00:54:06.960 --> 00:54:19.111
the average of whatever
you have, weight.
00:54:19.111 --> 00:54:26.905
And again, in case you forgot
this, just insert the K here.
00:54:26.905 --> 00:54:32.180
You can see that this is
the characteristic function
00:54:32.180 --> 00:54:35.500
of the Gaussian
distributed variable, which
00:54:35.500 --> 00:54:37.950
is this difference.
00:54:37.950 --> 00:54:40.320
And the characteristic
function I
00:54:40.320 --> 00:54:45.080
can start expanding in
terms of the cumulants.
00:54:45.080 --> 00:54:49.680
The first cumulant, the
average is 0 by symmetry.
00:54:49.680 --> 00:54:51.650
So the first thing
that will appear,
00:54:51.650 --> 00:54:54.420
which would be at the
order of K squared,
00:54:54.420 --> 00:54:56.000
is going to be the
variance, which
00:54:56.000 --> 00:54:57.995
is what we have over here.
00:54:57.995 --> 00:55:01.170
And since it's a Gaussian,
all higher order terms
00:55:01.170 --> 00:55:05.150
in this series [? will. ?]
Another way to do
00:55:05.150 --> 00:55:09.270
is to of course just
complete the square.
00:55:09.270 --> 00:55:13.050
And this is what
would come out, OK?
00:55:13.050 --> 00:55:18.830
So all I need to do is to
calculate the expectation
00:55:18.830 --> 00:55:25.560
value of this quantity
where the thetas are
00:55:25.560 --> 00:55:28.470
Gaussian distributed.
00:55:28.470 --> 00:55:33.460
And the best way to do so
is to go to Fourier space.
00:55:33.460 --> 00:55:35.860
So I have integral.
00:55:35.860 --> 00:55:40.550
For each one of these factors
of theta 0 minus theta r,
00:55:40.550 --> 00:55:49.570
I will do an integral
d2 q 2 pi squared.
00:55:49.570 --> 00:55:53.570
I have 1 minus e
to the iq.r, which
00:55:53.570 --> 00:55:56.990
is from theta 0 minus theta r.
00:55:56.990 --> 00:56:01.260
And then I have a theta tilde q.
00:56:01.260 --> 00:56:02.715
I have two of those factors.
00:56:02.715 --> 00:56:06.647
I have d2 q prime 2 pi squared.
00:56:06.647 --> 00:56:18.900
I have e to the minus iq prime
dot r theta tilde q prime.
00:56:18.900 --> 00:56:22.935
And then this average
simply becomes this average.
00:56:25.460 --> 00:56:31.000
And the different modes are
independent of each other.
00:56:31.000 --> 00:56:34.560
So I will get a 2 pi
to the d-- actually,
00:56:34.560 --> 00:56:37.820
2 here, a delta
function q plus q prime.
00:56:40.540 --> 00:56:44.190
And for each mode,
I will get a factor
00:56:44.190 --> 00:56:47.200
of Kq squared because
after all, thetas
00:56:47.200 --> 00:56:52.885
are very much like the pi's that
I had written at the beginning,
00:56:52.885 --> 00:56:54.360
OK?
00:56:54.360 --> 00:57:01.660
So what I will have is that this
quantity is integral over one
00:57:01.660 --> 00:57:04.240
q's.
00:57:04.240 --> 00:57:06.180
Putting these two
factors together,
00:57:06.180 --> 00:57:08.663
realizing that q
prime is minus q
00:57:08.663 --> 00:57:15.180
will give me 2 minus 2 cosine
of q.r divided by Kq squared.
00:57:19.546 --> 00:57:22.160
So there is an
overall scale that
00:57:22.160 --> 00:57:26.521
is set by 1 over
K, by temperature.
00:57:26.521 --> 00:57:28.520
And then there's a function
[? on ?] [? from ?],
00:57:28.520 --> 00:57:40.100
which is the Fourier transform
of {} 1 over q squared, which,
00:57:40.100 --> 00:57:43.300
as usual, we call C. We
anticipate this to be like
00:57:43.300 --> 00:57:44.500
a Coulomb potential.
00:57:44.500 --> 00:57:48.840
Because if I take
a Laplacian of C,
00:57:48.840 --> 00:57:52.540
you can see that-- forget the
q-- the [? Laplacian ?] of C
00:57:52.540 --> 00:57:58.430
from the cosine, I will get a
minus q squared cancels that.
00:57:58.430 --> 00:58:03.870
I will have [INAUDIBLE]
d2q 2 pi squared.
00:58:03.870 --> 00:58:06.660
Cosine itself will be left.
00:58:06.660 --> 00:58:09.240
The 1 over q squared disappears.
00:58:09.240 --> 00:58:13.070
This is e to the iqr you plus
e to the minus iqr over 2.
00:58:13.070 --> 00:58:14.850
Each one of them gives
a delta function.
00:58:14.850 --> 00:58:18.980
So this is just
a delta function.
00:58:18.980 --> 00:58:23.770
So C is the potential that
you have from a unit charge
00:58:23.770 --> 00:58:26.040
in 2 dimensions.
00:58:26.040 --> 00:58:33.790
And again, you can perform
the usual Gaussian procedure
00:58:33.790 --> 00:58:43.810
to find that the gradient
of C times 2 pi r
00:58:43.810 --> 00:58:47.552
is the net charge that is
enclosed, which is [? unity ?].
00:58:47.552 --> 00:58:52.010
So gradient of C, which
points in the radial direction
00:58:52.010 --> 00:58:55.330
is going to be 1 over 2 pi r.
00:58:55.330 --> 00:59:00.485
And your C is going to be
log of r divided by 2 pi.
00:59:00.485 --> 00:59:08.326
So this is 1 over K log
of r divided by 2 pi.
00:59:08.326 --> 00:59:13.250
And I state that when
essentially the 2 angles are
00:59:13.250 --> 00:59:17.350
as close as some short distance
cut-off, fluctuations vanish.
00:59:17.350 --> 00:59:24.410
So that's how I set the
0 of my integration, OK?
00:59:24.410 --> 00:59:26.685
So again, you put
that over here.
00:59:26.685 --> 00:59:34.380
We find that s0.s of r in
the low temperature limit
00:59:34.380 --> 00:59:39.840
is the exponential
of minus 1/2 of this.
00:59:39.840 --> 00:59:46.570
So I have log of r over
a divided by 4 pi K.
00:59:46.570 --> 00:59:54.760
And I will get a over r to
the power of 1 over 4 pi K.
00:59:54.760 --> 00:59:59.210
And I kind of want to check
that I didn't lose a factor of 2
00:59:59.210 --> 01:00:01.282
somewhere, which I seem to have.
01:00:25.400 --> 01:00:32.400
Yeah, I lost a factor
of 2 right here.
01:00:32.400 --> 01:00:36.970
This should be 2
because of this 2,
01:00:36.970 --> 01:00:40.070
if I'm using this definition.
01:00:40.070 --> 01:00:42.960
So this should be 2.
01:00:42.960 --> 01:00:52.310
And this should be 2 pi K. OK.
01:00:59.390 --> 01:01:03.540
So what have we established?
01:01:03.540 --> 01:01:08.750
We have looked as a
function of temperature
01:01:08.750 --> 01:01:13.233
what the behavior of this
spin-spin correlation function
01:01:13.233 --> 01:01:13.733
is.
01:01:18.550 --> 01:01:23.410
We have established that in
the higher temperature limit,
01:01:23.410 --> 01:01:29.276
the behavior is something
that falls off exponentially
01:01:29.276 --> 01:01:30.105
with separation.
01:01:33.310 --> 01:01:36.655
We have also established
that at low temperature,
01:01:36.655 --> 01:01:42.962
it falls off as a power
law in separation, OK?
01:01:46.460 --> 01:01:51.890
So these two functional
behaviors are different.
01:01:51.890 --> 01:01:55.135
There is no way that you can
connect one to the other.
01:01:55.135 --> 01:01:59.545
So you pick two spins that
are sufficiently far apart
01:01:59.545 --> 01:02:03.190
and then move the separation
further and further away.
01:02:03.190 --> 01:02:05.900
And the functional form
of the correlations
01:02:05.900 --> 01:02:09.630
is either a power-law
decay, power-law decay,
01:02:09.630 --> 01:02:13.100
or an exponential decay.
01:02:13.100 --> 01:02:17.390
And in this form, you know
you have a high temperature.
01:02:17.390 --> 01:02:20.350
In this form, you know you
have a low temperature.
01:02:20.350 --> 01:02:25.520
So potentially, there could be
a phase transition separating
01:02:25.520 --> 01:02:29.390
the distinct behaviors of
the correlation function.
01:02:29.390 --> 01:02:33.220
And that could
potentially be underlying
01:02:33.220 --> 01:02:38.130
what is observed over here, OK?
01:02:38.130 --> 01:02:38.630
Yes?
01:02:38.630 --> 01:02:40.588
AUDIENCE: So where could
we make the assumption
01:02:40.588 --> 01:02:43.430
we're at a low temperature
in the second expansion?
01:02:43.430 --> 01:02:48.400
PROFESSOR: When we expanded
the cosines, right?
01:02:48.400 --> 01:02:53.790
So what I should really do is to
look at the terms such as this.
01:02:53.790 --> 01:02:56.780
But then I said that I'm low
enough temperature so that I
01:02:56.780 --> 01:03:00.980
look at near neighbors, and
they're almost parallel.
01:03:00.980 --> 01:03:04.120
So the cosine of the angle
difference between them
01:03:04.120 --> 01:03:06.803
is the square of
that small angle.
01:03:06.803 --> 01:03:07.636
AUDIENCE: Thank you.
01:03:07.636 --> 01:03:08.302
PROFESSOR: Yeah.
01:03:11.582 --> 01:03:12.082
OK?
01:03:18.010 --> 01:03:22.040
So actually, you
may have said that I
01:03:22.040 --> 01:03:26.430
could have done the same
analysis for small angle
01:03:26.430 --> 01:03:31.560
expansions not only
for n equals 2,
01:03:31.560 --> 01:03:34.950
but also for n
equals 3, et cetera.
01:03:34.950 --> 01:03:36.640
That would be correct.
01:03:36.640 --> 01:03:40.410
Because I could have made a
similar Gaussian analysis for n
01:03:40.410 --> 01:03:42.990
equals e also.
01:03:42.990 --> 01:03:46.130
And then I may have
concluded the same thing,
01:03:46.130 --> 01:03:49.960
except that I cannot conclude
the same thing because of this
01:03:49.960 --> 01:03:53.180
thing that we derived over here.
01:03:53.180 --> 01:03:59.870
What this shows is that the
expansion around 0 temperature
01:03:59.870 --> 01:04:04.610
regarded as Gaussian
is going to break down
01:04:04.610 --> 01:04:06.750
because of the
non-linear coupling
01:04:06.750 --> 01:04:09.170
that we have between modes.
01:04:09.170 --> 01:04:12.910
So although I may be tempted to
write something like this for n
01:04:12.910 --> 01:04:16.670
equals 3, I know
why it is wrong.
01:04:16.670 --> 01:04:19.358
And I know the
correlation length
01:04:19.358 --> 01:04:22.940
at which this kind
of behavior will
01:04:22.940 --> 01:04:26.520
need to be replaced with
this type of behavior
01:04:26.520 --> 01:04:30.410
because effectively, the
expansion parameter became
01:04:30.410 --> 01:04:32.780
of the order of 1.
01:04:32.780 --> 01:04:35.950
But I cannot do that
for the xy model.
01:04:35.950 --> 01:04:40.320
I don't have similar reason.
01:04:40.320 --> 01:04:44.490
So then the question
becomes, well,
01:04:44.490 --> 01:04:51.220
how does this expansion then
eventually break down so
01:04:51.220 --> 01:04:53.720
that I will have a phase
transition to a phase
01:04:53.720 --> 01:04:57.170
where the correlations are
decaying exponentially?
01:04:57.170 --> 01:04:59.080
And you may say,
well, I mean, it's
01:04:59.080 --> 01:05:00.850
really something
to do with having
01:05:00.850 --> 01:05:02.610
to go to higher and
higher ordered terms
01:05:02.610 --> 01:05:04.810
in the expansion of the cosine.
01:05:04.810 --> 01:05:08.080
And it's going to
be something which
01:05:08.080 --> 01:05:12.510
would be very difficult to
figure out, except that it
01:05:12.510 --> 01:05:16.250
turns out that there is a
much more elegant solution.
01:05:16.250 --> 01:05:18.380
And that was proposed
by [? Kastelitz ?]
01:05:18.380 --> 01:05:19.180
and [? Thales. ?]
01:05:26.975 --> 01:05:32.570
And they said that what you
have left out in the Gaussian
01:05:32.570 --> 01:05:43.796
analysis are
topological defects, OK?
01:05:43.796 --> 01:05:48.930
That is, when I did the
expansion of the cosine
01:05:48.930 --> 01:05:52.370
and we replaced the cosine with
the difference of the angles
01:05:52.370 --> 01:05:57.490
squared, that's
more or less fine,
01:05:57.490 --> 01:06:02.020
except that I should also
realize that cosine maintains
01:06:02.020 --> 01:06:07.051
its value if the angle
difference goes up by 2 pi.
01:06:07.051 --> 01:06:10.730
And you say, well,
neighboring spins are never
01:06:10.730 --> 01:06:14.060
going to be 2 pi
different or pi different
01:06:14.060 --> 01:06:16.980
because they're very
strongly coupled.
01:06:16.980 --> 01:06:18.420
Does it make any difference?
01:06:18.420 --> 01:06:21.200
Turns out that, OK, for
the neighboring spins,
01:06:21.200 --> 01:06:22.600
it doesn't make a difference.
01:06:22.600 --> 01:06:25.760
But what if you go far away?
01:06:25.760 --> 01:06:36.540
So let's imagine that this is,
let's say, our system of spins.
01:06:36.540 --> 01:06:41.350
And what I do is I look at a
configuration such as this.
01:07:02.960 --> 01:07:08.470
Essentially, I have
spins [? radiating ?] out
01:07:08.470 --> 01:07:11.810
from a center such as this, OK?
01:07:14.370 --> 01:07:17.200
There is, of course,
a lot of energy costs
01:07:17.200 --> 01:07:19.145
I have put over here.
01:07:22.070 --> 01:07:26.580
But when I go very
much further out,
01:07:26.580 --> 01:07:31.350
let's say, very far
away from this plus sign
01:07:31.350 --> 01:07:34.790
that I have indicated
over here, and I
01:07:34.790 --> 01:07:37.990
follow what the behavior
of the spins are,
01:07:37.990 --> 01:07:43.670
you see that as I go along
this circuit, the spins start
01:07:43.670 --> 01:07:46.400
by pointing this way,
they go point this way,
01:07:46.400 --> 01:07:49.920
this way, et cetera.
01:07:49.920 --> 01:07:54.680
And by the time I carry
a circuit such as this,
01:07:54.680 --> 01:08:02.017
I find that the angle theta
has also rotated by 2 pi, OK?
01:08:02.017 --> 01:08:04.760
Now, this is clearly
a configuration
01:08:04.760 --> 01:08:06.160
that is going to be costly.
01:08:06.160 --> 01:08:08.520
We'll calculate its cost.
01:08:08.520 --> 01:08:13.540
But the point is that there
is no continuous deformation
01:08:13.540 --> 01:08:17.880
that you can make
that will map this
01:08:17.880 --> 01:08:20.284
into what we were
expanding around here
01:08:20.284 --> 01:08:24.176
with all of the cosines
being parallel to each other.
01:08:24.176 --> 01:08:29.060
So this is a topologically
distinct contribution
01:08:29.060 --> 01:08:31.430
from the Gaussian
one, the Gaussian term
01:08:31.430 --> 01:08:34.250
that we've calculated.
01:08:34.250 --> 01:08:38.319
Since the direction of
the rotation of the spins
01:08:38.319 --> 01:08:42.720
is the same as the direction
of the circuit in this case,
01:08:42.720 --> 01:08:46.410
this is called a plus
topological defect.
01:08:46.410 --> 01:08:57.890
There is a corresponding
minus defect
01:08:57.890 --> 01:09:03.649
which is something like this.
01:09:42.259 --> 01:09:45.439
OK, and for this, you
can convince yourself
01:09:45.439 --> 01:09:48.724
that as you make a
circuit such as this
01:09:48.724 --> 01:09:51.779
that the direction
of the arrow actually
01:09:51.779 --> 01:09:55.705
goes in the opposite
direction, OK?
01:09:55.705 --> 01:10:02.830
This is called a negative
sign topological defect.
01:10:02.830 --> 01:10:06.480
Now, I said, well,
let's figure out
01:10:06.480 --> 01:10:11.160
what the energy cost of
one of these things is.
01:10:11.160 --> 01:10:16.740
If I'm away from the center
of one of these defects,
01:10:16.740 --> 01:10:22.985
then the change in angle
is small because the change
01:10:22.985 --> 01:10:30.130
in angle if I go all the way
around a circle of radius
01:10:30.130 --> 01:10:34.550
r should come back to be 2 pi.
01:10:34.550 --> 01:10:39.270
2 pi is the uncertainty that
I have from the cosines.
01:10:39.270 --> 01:10:42.030
So what I have is
that the gradient
01:10:42.030 --> 01:10:47.550
of theta [? between ?]
neighboring angles
01:10:47.550 --> 01:10:54.710
times 2 pi r, which is
this radius, is 2 pi.
01:10:54.710 --> 01:11:01.360
And this thing,
here it is plus 1.
01:11:01.360 --> 01:11:03.940
Here it is minus 1.
01:11:03.940 --> 01:11:07.920
And in general, you can
imagine possibilities
01:11:07.920 --> 01:11:12.180
where this is some integer
that is like 2 or minus 2
01:11:12.180 --> 01:11:16.050
or something else
that is allowed
01:11:16.050 --> 01:11:21.640
by this degeneracy
of this cosine, OK?
01:11:21.640 --> 01:11:23.910
So you can see that
when you are far away
01:11:23.910 --> 01:11:27.465
from the center of
whatever this defect is,
01:11:27.465 --> 01:11:34.800
the gradient of theta has
magnitude that is n over r, OK?
01:11:34.800 --> 01:11:36.630
And as you go
further and further,
01:11:36.630 --> 01:11:39.700
it becomes smaller and smaller.
01:11:39.700 --> 01:11:43.470
And the energy cost out here
you can obtain by essentially
01:11:43.470 --> 01:11:48.190
expanding the cosine is going
to be proportional to the change
01:11:48.190 --> 01:11:50.370
in angle squared.
01:11:50.370 --> 01:12:00.740
So the cost of the
defect is an integral
01:12:00.740 --> 01:12:13.200
of 2 pi rdr times this quantity
n over r squared multiplied
01:12:13.200 --> 01:12:16.050
by the coefficient of the
expansion of the cosine,
01:12:16.050 --> 01:12:17.175
or K over 2.
01:12:20.590 --> 01:12:24.130
And this integration I
have to go all the way to
01:12:24.130 --> 01:12:25.200
ends up of my system.
01:12:25.200 --> 01:12:26.750
Let's call it l.
01:12:29.780 --> 01:12:32.260
And then I can
bring it down, not
01:12:32.260 --> 01:12:36.110
necessary to the scale
of the lattice spacing,
01:12:36.110 --> 01:12:39.810
but maybe to scale of 5 lattice
spacing or something like this,
01:12:39.810 --> 01:12:44.040
where the approximations that
I have used of treating this
01:12:44.040 --> 01:12:45.850
as a continuum are still valid.
01:12:45.850 --> 01:12:50.550
So I will pick some kind of
a short distance cut-off a.
01:12:50.550 --> 01:12:57.366
And then whatever energy is
at scales that are below a,
01:12:57.366 --> 01:13:04.290
I will add to a core energy that
depends on whatever this a is.
01:13:04.290 --> 01:13:07.580
I don't know what that is.
01:13:07.580 --> 01:13:11.790
So basically, there
is some core energy
01:13:11.790 --> 01:13:16.010
depending on where I stop this.
01:13:16.010 --> 01:13:18.110
And the reason that
this is more important--
01:13:18.110 --> 01:13:22.490
because here I have an
integral of 1 over r.
01:13:22.490 --> 01:13:25.160
And an integral of 1
over r is something
01:13:25.160 --> 01:13:27.555
that is logarithmically
divergent.
01:13:27.555 --> 01:13:34.580
So I will get K. I
have 2 cancels the 2.
01:13:34.580 --> 01:13:49.634
I have pi en squared
log of l over a, OK?
01:13:58.990 --> 01:14:05.940
So you can see that creating
one of these defects
01:14:05.940 --> 01:14:08.180
is hugely expensive.
01:14:08.180 --> 01:14:12.450
And energy that as your system
becomes bigger and bigger
01:14:12.450 --> 01:14:13.860
and we're thinking
about infinite
01:14:13.860 --> 01:14:19.220
sized systems is
logarithmically large.
01:14:19.220 --> 01:14:21.110
So you would say these
things will never
01:14:21.110 --> 01:14:25.960
occur because they cost an
infinite amount of energy.
01:14:25.960 --> 01:14:30.090
Well, the thing is that
entropy is also important.
01:14:30.090 --> 01:14:33.560
So if I were to calculate the
partition function that I would
01:14:33.560 --> 01:14:39.330
assign to one of these
defects, part of it
01:14:39.330 --> 01:14:41.610
would be exponential
of this energy.
01:14:41.610 --> 01:14:48.530
So I would have this e to
the minus this core energy.
01:14:48.530 --> 01:15:00.480
I would have this exponential of
K pi n squared log of l over a.
01:15:00.480 --> 01:15:06.040
So that's the [INAUDIBLE]
weight for this.
01:15:06.040 --> 01:15:10.990
But then I realize that
I can put this anywhere
01:15:10.990 --> 01:15:16.360
on the lattice so there is
an entropy [? gain ?] factor.
01:15:16.360 --> 01:15:19.810
And since I have assigned this
to have some kind of a bulk
01:15:19.810 --> 01:15:23.550
to it as some
characteristic size a,
01:15:23.550 --> 01:15:26.832
the number of distinct
places that I can put it
01:15:26.832 --> 01:15:30.280
is over the order
of l over a squared.
01:15:30.280 --> 01:15:33.830
So basically, I take my huge
lattice and I partition it
01:15:33.830 --> 01:15:35.640
into sizes of a's.
01:15:35.640 --> 01:15:39.630
And I say I can put it in any
one of these configurations.
01:15:39.630 --> 01:15:42.030
You can see that
the whole thing is
01:15:42.030 --> 01:15:45.920
going to be e to the
minus this core energy.
01:15:45.920 --> 01:15:51.930
And then I have l over
a to the power of 2
01:15:51.930 --> 01:15:57.980
minus pi K n squared, OK?
01:15:57.980 --> 01:16:04.210
So the logarithmic energy
cost is the same form
01:16:04.210 --> 01:16:09.560
as the logarithmic entropy
gain that you have over here.
01:16:09.560 --> 01:16:15.890
And this precise balance will
give you a value of K such
01:16:15.890 --> 01:16:20.790
that if K is larger than
2 over pi n squared,
01:16:20.790 --> 01:16:26.180
this is going to be an
exponentially large cost.
01:16:26.180 --> 01:16:29.700
There is a huge
negative power of l here
01:16:29.700 --> 01:16:33.440
that says, no, you don't
want to create this.
01:16:33.440 --> 01:16:40.200
But if K becomes weak such
that the 2, the entropy factor,
01:16:40.200 --> 01:16:44.490
[? wins ?], then you will
start creating [INAUDIBLE].
01:16:44.490 --> 01:16:48.970
So you can see that
over here, suddenly we
01:16:48.970 --> 01:16:53.070
have a mechanism along
our picture over here,
01:16:53.070 --> 01:16:59.520
maybe something like K,
which is 2 over pi, such
01:16:59.520 --> 01:17:03.980
that on one side you
would say, I will not
01:17:03.980 --> 01:17:09.906
have topological defect and
I can use the Gaussian model.
01:17:09.906 --> 01:17:13.830
And on the other side, you
say that I will spontaneously
01:17:13.830 --> 01:17:16.790
create these
topological defects.
01:17:16.790 --> 01:17:19.370
And then the
Gaussian description
01:17:19.370 --> 01:17:23.430
is no longer valid because
I have to now really take
01:17:23.430 --> 01:17:29.250
care of the angular nature
of these variables, OK?
01:17:29.250 --> 01:17:38.290
So this is a nice picture, which
is only a zeroed order picture.
01:17:38.290 --> 01:17:41.600
And to zeroed order,
it is correct.
01:17:41.600 --> 01:17:44.400
But this is not fully correct.
01:17:44.400 --> 01:17:48.910
Because even at
low temperatures,
01:17:48.910 --> 01:18:02.540
you can certainly create hairs
of plus minus defects, OK?
01:18:02.540 --> 01:18:12.650
And whereas the
field for one of them
01:18:12.650 --> 01:18:16.140
will fall off at
large distances,
01:18:16.140 --> 01:18:19.730
the gradient of
theta is 1 over r,
01:18:19.730 --> 01:18:34.380
if you superimpose what is
happening for two of these,
01:18:34.380 --> 01:18:38.480
what you will convince
yourself that if you have
01:18:38.480 --> 01:18:44.160
a pair of defects of
opposite sign at the distance
01:18:44.160 --> 01:18:54.150
d that the distortion that they
generate at large distances
01:18:54.150 --> 01:18:58.850
falls off not the as 1 over
r, which is, if you like,
01:18:58.850 --> 01:19:05.140
a monopole field, but
as d over r squared,
01:19:05.140 --> 01:19:07.200
which is a dipole field.
01:19:07.200 --> 01:19:10.770
So whenever you
have a dipole, you
01:19:10.770 --> 01:19:13.530
will have to multiply
by the separation
01:19:13.530 --> 01:19:15.400
of the charges in the dipole.
01:19:15.400 --> 01:19:17.910
And that is compensated
by a factor of 1
01:19:17.910 --> 01:19:19.580
over r in the denominator.
01:19:19.580 --> 01:19:20.965
There is some
angular dependence,
01:19:20.965 --> 01:19:24.560
but we are not so
interested in that.
01:19:24.560 --> 01:19:32.570
Now, if I were to
integrate this square,
01:19:32.570 --> 01:19:34.260
we can see that it
is something that
01:19:34.260 --> 01:19:38.460
is convergent at
large distances.
01:19:38.460 --> 01:19:41.860
And so this is
going to be finite.
01:19:41.860 --> 01:19:48.670
It is not going to diverge as
the size of the system, which
01:19:48.670 --> 01:19:53.860
means that whereas individual
defects there was no way
01:19:53.860 --> 01:19:58.030
that I could create
in my system,
01:19:58.030 --> 01:20:01.242
I can always create
pairs of these defects.
01:20:04.140 --> 01:20:08.300
So the correct picture
that we should have
01:20:08.300 --> 01:20:12.960
is not that at low temperatures
you don't have defects,
01:20:12.960 --> 01:20:15.850
at high temperatures you have
these defects spontaneously
01:20:15.850 --> 01:20:17.390
appearing.
01:20:17.390 --> 01:20:20.740
The correct picture is
that at low temperatures
01:20:20.740 --> 01:20:28.510
what you have is lots and lots
of dipoles that are pretty much
01:20:28.510 --> 01:20:32.130
bound to each other.
01:20:32.130 --> 01:20:41.270
And when you go to high
temperatures, what happens
01:20:41.270 --> 01:20:46.990
is that you will have these
pluses and minuses unbound
01:20:46.990 --> 01:20:48.890
from each other.
01:20:48.890 --> 01:20:51.950
So if you like,
the transition is
01:20:51.950 --> 01:20:59.080
between molecules to a plasma
as temperature is changed.
01:20:59.080 --> 01:21:05.440
Or if you like, it is between
an insulator and a conductor.
01:21:05.440 --> 01:21:09.530
And how to mathematically
describe this phase transition
01:21:09.530 --> 01:21:13.932
in 2 dimensions, which
we can rigorously,
01:21:13.932 --> 01:21:18.174
will be what we will do in
the next couple of lectures.