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PROFESSOR: OK, let's start.
00:00:26.360 --> 00:00:32.360
So, we've moved onto the
two-dimensional xy model.
00:00:36.910 --> 00:00:43.070
This is a system
where, let's say
00:00:43.070 --> 00:00:49.410
on each side of a square lattice
you put a unit vector that
00:00:49.410 --> 00:00:53.030
has two components
and hence, can
00:00:53.030 --> 00:00:56.605
be described by an angle theta.
00:00:56.605 --> 00:01:01.130
So basically, at each side,
you have an angle theta
00:01:01.130 --> 00:01:02.950
for that side.
00:01:02.950 --> 00:01:09.450
And there is a tendency for a
neighboring spins to be aligned
00:01:09.450 --> 00:01:13.020
and the partition
function can be
00:01:13.020 --> 00:01:18.670
written as a sum over
all configurations, which
00:01:18.670 --> 00:01:22.780
is equivalent to integrating
over all of these angles.
00:01:25.700 --> 00:01:32.490
And the weight that wants
to make the two neighboring
00:01:32.490 --> 00:01:36.590
spins to be parallel
to each other.
00:01:36.590 --> 00:01:39.900
So we have a sum over
nearest neighbors.
00:01:39.900 --> 00:01:43.410
And the dot product of
the two spins amounts
00:01:43.410 --> 00:01:47.890
to looking at the cosine
of theta i minus theta j.
00:01:47.890 --> 00:01:51.390
And so, we have this
factor over here.
00:01:55.460 --> 00:02:05.850
Now, if we go to the
limit where k is large,
00:02:05.850 --> 00:02:10.210
then the cosine will tend
to keep the angles close
00:02:10.210 --> 00:02:12.180
to each other.
00:02:12.180 --> 00:02:20.080
And we are tempted to expand
this around the configurations
00:02:20.080 --> 00:02:22.680
where everybody's parallel.
00:02:22.680 --> 00:02:28.620
Let's call that nk
by the factor of 2.
00:02:28.620 --> 00:02:33.530
And then, expanding with the
cosine to the next order,
00:02:33.530 --> 00:02:40.550
you may want to replace
this-- let's call this k0--
00:02:40.550 --> 00:02:44.870
and have a factor of k,
which is proportional
00:02:44.870 --> 00:02:48.890
to k0 after some
lattice spacing.
00:02:48.890 --> 00:02:54.990
And integral of gradient
of theta squared.
00:02:54.990 --> 00:03:01.150
So basically, the difference
between the angles
00:03:01.150 --> 00:03:04.160
in the continuum version,
I want to replace
00:03:04.160 --> 00:03:11.000
with the term that tries to
make the gradient to be fixed.
00:03:11.000 --> 00:03:12.910
OK.
00:03:12.910 --> 00:03:18.940
Now the reason I put these
quotes around the gradient
00:03:18.940 --> 00:03:22.440
is something that we
noticed last time, which
00:03:22.440 --> 00:03:27.650
is that in principal,
theta is defined up
00:03:27.650 --> 00:03:30.330
to a multiple of 2 pi.
00:03:30.330 --> 00:03:36.290
So that if I were to take
a circuit along the lattice
00:03:36.290 --> 00:03:39.650
that comes back to itself.
00:03:39.650 --> 00:03:49.900
And all along, this circuit
integrate this gradient
00:03:49.900 --> 00:03:55.260
of theta, so basically, gradient
of theta would be a vector.
00:03:55.260 --> 00:03:59.330
I integrated along a circuit.
00:03:59.330 --> 00:04:03.370
And by the time I have come back
and close the circuit to where
00:04:03.370 --> 00:04:08.450
I started, the answer
may not come back to 0.
00:04:08.450 --> 00:04:16.985
It may be any integer
multiple of 2 pi.
00:04:16.985 --> 00:04:17.485
All right.
00:04:22.480 --> 00:04:25.690
So how do we account for this?
00:04:25.690 --> 00:04:29.060
The way we account
for this is that we
00:04:29.060 --> 00:04:38.280
note that this gradient of data
I can decompose into two parts.
00:04:38.280 --> 00:04:41.240
One, where I just
write it as a gradient
00:04:41.240 --> 00:04:44.440
of some regular function.
00:04:44.440 --> 00:04:46.950
And the characteristic
of gradient
00:04:46.950 --> 00:04:51.340
is that once you go over a
closed loop and you integrate,
00:04:51.340 --> 00:04:53.850
you essentially are
evaluating this field
00:04:53.850 --> 00:04:57.420
phi at the beginning
and the end.
00:04:57.420 --> 00:05:01.100
And for any regular
single valued phi,
00:05:01.100 --> 00:05:04.800
this would come back to zero.
00:05:04.800 --> 00:05:09.360
And to take care of
this fact the result
00:05:09.360 --> 00:05:11.770
does not have to come
to zero if I integrate
00:05:11.770 --> 00:05:19.900
this gradient of theta, I
introduce another field, u,
00:05:19.900 --> 00:05:26.190
that takes care of these
topological defects.
00:05:34.390 --> 00:05:36.820
OK?
00:05:36.820 --> 00:05:43.370
So that, really, I have to
include both configurations
00:05:43.370 --> 00:05:48.900
in order to correctly capture
the original model that
00:05:48.900 --> 00:05:51.380
had these angles.
00:05:51.380 --> 00:05:54.890
OK, so what can this u be?
00:05:54.890 --> 00:06:02.100
We already looked at what
u is for the case of one
00:06:02.100 --> 00:06:03.390
topological defect.
00:06:13.950 --> 00:06:17.560
And the idea here
was that maybe I
00:06:17.560 --> 00:06:23.530
had a configuration where
around a particular center,
00:06:23.530 --> 00:06:29.230
let's say all of the
spins were flowing out,
00:06:29.230 --> 00:06:32.900
or some other such
configuration such
00:06:32.900 --> 00:06:41.500
that when I go over a large
distance r from this center,
00:06:41.500 --> 00:06:51.940
and integrate this field u,
just like I did over there,
00:06:51.940 --> 00:06:55.620
the answer is going to
be, let's say, 2 pi n.
00:06:55.620 --> 00:06:58.940
So there's this u.
00:06:58.940 --> 00:07:03.960
And I integrated
along this circle.
00:07:03.960 --> 00:07:08.500
And the answer is
going to be 2 pi n.
00:07:08.500 --> 00:07:15.460
Well, clearly, the
magnitude of u times
00:07:15.460 --> 00:07:21.060
2 pi r, which is the
radius of the circle
00:07:21.060 --> 00:07:25.960
is going to be 2 pi times
some integer-- could
00:07:25.960 --> 00:07:29.890
be plus, minus 1, plus minus 2.
00:07:29.890 --> 00:07:34.010
And so, the magnitude
of u is n over r.
00:07:43.390 --> 00:07:51.446
The direction of u is orthogonal
to the direction of r.
00:07:51.446 --> 00:07:53.230
And how can I show that?
00:07:53.230 --> 00:07:55.850
Well, one way I
can show that is I
00:07:55.850 --> 00:08:05.170
can say that it is z
hat crossed with r hat,
00:08:05.170 --> 00:08:12.490
there z hat is the vector
that comes out of the plane.
00:08:12.490 --> 00:08:16.260
And r hat is the unit
vector in this direction.
00:08:16.260 --> 00:08:19.360
u is clearly orthogonal to r.
00:08:19.360 --> 00:08:24.530
The direction of the gradient of
this angle is orthogonal to r.
00:08:24.530 --> 00:08:29.290
It is in the plane so
it's orthogonal to this.
00:08:29.290 --> 00:08:36.730
And this I can also
write as z hat 3
00:08:36.730 --> 00:08:43.830
crossed with the gradient of
log of r, with some cut-off.
00:08:43.830 --> 00:08:47.800
Because the gradient of
log of r will give me,
00:08:47.800 --> 00:08:51.410
essentially, 1 over R in
the direction of r hat.
00:08:51.410 --> 00:08:53.540
And this is like
the potential that I
00:08:53.540 --> 00:08:57.450
would have for a charge
in two dimensions,
00:08:57.450 --> 00:09:01.640
except that I have
rotated it by somewhat.
00:09:01.640 --> 00:09:12.070
And this I can also write
as minus the curl of z hat
00:09:12.070 --> 00:09:16.655
log r over a with a factor of n.
00:09:20.120 --> 00:09:24.880
And essentially,
what you can see
00:09:24.880 --> 00:09:30.610
is that the gradient of
data for a field that
00:09:30.610 --> 00:09:33.860
has this topological
defect has a part
00:09:33.860 --> 00:09:39.240
can be written as a
potential gradient of some y,
00:09:39.240 --> 00:09:43.010
and a part that can be
written as curl to keep track
00:09:43.010 --> 00:09:45.310
of these vortices, if you like.
00:09:45.310 --> 00:09:48.140
If you were to think of
this gradient of data
00:09:48.140 --> 00:09:50.755
like the flow field that you
would have in two dimensions.
00:09:50.755 --> 00:09:52.510
It has a potential part.
00:09:52.510 --> 00:09:58.480
And it has a part that is due to
curvatures and vortices, which
00:09:58.480 --> 00:10:01.870
is what we have over here.
00:10:01.870 --> 00:10:03.950
OK.
00:10:03.950 --> 00:10:08.480
So this is, however, only
for one topological defect.
00:10:08.480 --> 00:10:11.935
What happens if I have
many such defects?
00:10:17.390 --> 00:10:22.940
What I can do is, rather
than having just one of them,
00:10:22.940 --> 00:10:27.690
I could have another topological
defect here, another one here,
00:10:27.690 --> 00:10:28.600
another one there.
00:10:28.600 --> 00:10:32.790
There should be a
combination of these things.
00:10:32.790 --> 00:10:39.840
And what I can do in order
to get the corresponding u
00:10:39.840 --> 00:10:45.080
is to superimpose solutions
that correspond to single ones.
00:10:45.080 --> 00:10:48.180
As you can see that
this is very much
00:10:48.180 --> 00:10:51.720
like the potential
that I would have
00:10:51.720 --> 00:10:54.260
for a charge at the
origin, and then
00:10:54.260 --> 00:10:57.220
taking the derivative
to create the field.
00:10:57.220 --> 00:11:00.930
And you know that as long
as things are linear,
00:11:00.930 --> 00:11:02.770
and there aren't
too many of them,
00:11:02.770 --> 00:11:07.560
you can superimpose solutions
for different charges.
00:11:07.560 --> 00:11:10.260
You could just add up
the electric fields.
00:11:10.260 --> 00:11:14.600
So what I'm claiming
is that I can write u
00:11:14.600 --> 00:11:25.280
as minus n curl of z hat, times
some potential u of r, where
00:11:25.280 --> 00:11:32.920
psi of r is essentially the
generalization of this log.
00:11:32.920 --> 00:11:39.000
I can write it as a sum over
all topological defects.
00:11:39.000 --> 00:11:45.565
And I will have the n of that
topological defect times log
00:11:45.565 --> 00:11:50.480
of r minus ri
divided by a, where
00:11:50.480 --> 00:11:53.960
ri are the locations of these.
00:11:53.960 --> 00:11:59.350
So there could be a vortex
here at r1 with charge n1,
00:11:59.350 --> 00:12:04.350
and other topological defect
here at r2 with charge n2,
00:12:04.350 --> 00:12:06.960
and so forth.
00:12:06.960 --> 00:12:12.270
And I can construct a
potential that basically looks
00:12:12.270 --> 00:12:17.460
at the log of r minus ri
for each individual one,
00:12:17.460 --> 00:12:20.790
and then do this.
00:12:20.790 --> 00:12:21.650
OK?
00:12:21.650 --> 00:12:27.100
I will sometimes write this in
a slightly different fashion.
00:12:27.100 --> 00:12:34.930
Recall that we had the
Coulomb potential, which
00:12:34.930 --> 00:12:41.550
was related to log by just
a factor of 1 over 2 pi.
00:12:41.550 --> 00:12:44.830
So the correct version of
defining the Coulomb potential
00:12:44.830 --> 00:12:46.030
is this.
00:12:46.030 --> 00:12:51.330
So this I can write as
the Coulomb potential,
00:12:51.330 --> 00:12:55.200
provided that I
multiply the 2 pi ni.
00:12:55.200 --> 00:12:58.190
And I sometimes
will call that qi.
00:12:58.190 --> 00:13:03.520
So essentially, qi
is 2 pi and i is
00:13:03.520 --> 00:13:06.770
the charge of the
topological defect.
00:13:06.770 --> 00:13:09.610
It can be plus or minus 2 pi.
00:13:09.610 --> 00:13:12.450
And then, the potential
is constructed
00:13:12.450 --> 00:13:18.160
by constructing superposition
of those charges divided
00:13:18.160 --> 00:13:21.611
or multiplied with
appropriate Coulomb potential.
00:13:21.611 --> 00:13:22.110
OK?
00:13:26.520 --> 00:13:27.020
All right.
00:13:29.980 --> 00:13:39.190
So I can construct a cost for
creating a configuration now.
00:13:39.190 --> 00:13:43.580
Previously, I had
this integral gradient
00:13:43.580 --> 00:13:47.376
of theta squared
in the continuum.
00:13:47.376 --> 00:13:54.400
And my gradient of
theta squared has now
00:13:54.400 --> 00:13:59.920
a part that is the gradient
of a regular, well-behaved
00:13:59.920 --> 00:14:04.890
potential, and a part
that is this field
00:14:04.890 --> 00:14:13.200
u, which is minus-- oops.
00:14:13.200 --> 00:14:15.560
Then I don't need
the ni's because I
00:14:15.560 --> 00:14:18.470
put the ni as part of the psi.
00:14:18.470 --> 00:14:25.760
Curl of z hat psi of r.
00:14:25.760 --> 00:14:28.070
So phi is a regular function.
00:14:28.070 --> 00:14:33.060
Psi with curl will give
me the contribution
00:14:33.060 --> 00:14:36.430
of the topological
defect involves
00:14:36.430 --> 00:14:39.500
both the charges
and the positions
00:14:39.500 --> 00:14:42.022
of these topological defects.
00:14:42.022 --> 00:14:44.990
OK?
00:14:44.990 --> 00:14:48.090
And this whole thing has
to be squared, of course.
00:14:48.090 --> 00:14:52.720
This is my gradient squared.
00:14:52.720 --> 00:14:55.810
And if I expand this, I
will have three terms.
00:14:59.040 --> 00:15:01.540
I have a gradient
of this 5 squared.
00:15:04.340 --> 00:15:09.980
I have a term, which is minus
2 gradient of phi dot producted
00:15:09.980 --> 00:15:14.540
with curl of z hat psi.
00:15:17.830 --> 00:15:25.800
And I have a term that is
curl of z hat psi squared.
00:15:29.180 --> 00:15:29.680
OK?
00:15:36.000 --> 00:15:39.470
Again, if you think
of this as vector,
00:15:39.470 --> 00:15:41.450
this is a vector
whose components
00:15:41.450 --> 00:15:49.740
are the xy and the
y phi, whereas this
00:15:49.740 --> 00:15:53.450
is a vector whose
components are, let's say,
00:15:53.450 --> 00:15:57.380
dy psi minus dx psi.
00:15:57.380 --> 00:16:01.340
Because of the curl operation--
the x and y components-- one
00:16:01.340 --> 00:16:02.470
of them gets a minus sign.
00:16:02.470 --> 00:16:05.950
Maybe I got the minus
wrong, but it's essentially
00:16:05.950 --> 00:16:06.650
that structure.
00:16:09.320 --> 00:16:14.880
Now you can see that if I were
to do the integration here,
00:16:14.880 --> 00:16:18.220
there is a dx psi, dy psi.
00:16:18.220 --> 00:16:22.310
I can do that integration
by parts and have,
00:16:22.310 --> 00:16:26.340
let's say phi dx dy psi.
00:16:26.340 --> 00:16:29.150
And then, I can do the same
integration by parts here.
00:16:29.150 --> 00:16:33.270
And I will have phi
minus dx dy psi.
00:16:33.270 --> 00:16:37.230
So if I do integration by
part, this will disappear.
00:16:37.230 --> 00:16:41.260
Another way of seeing
that is that the gradient
00:16:41.260 --> 00:16:42.990
will act on the curl.
00:16:42.990 --> 00:16:45.570
And the gradient of the
curl of a vector is 0,
00:16:45.570 --> 00:16:48.830
or otherwise, the curl
will act on the gradient
00:16:48.830 --> 00:16:50.150
with [INAUDIBLE].
00:16:50.150 --> 00:16:56.890
So basically, this term
does not contribute.
00:16:56.890 --> 00:17:03.360
And the contribution of
this part and the part
00:17:03.360 --> 00:17:08.069
from topological defects are
decoupled from each other.
00:17:08.069 --> 00:17:11.869
So there's essentially
the Gaussian type of stuff
00:17:11.869 --> 00:17:14.579
that we calculated
before is here.
00:17:14.579 --> 00:17:18.150
On top of that,
there is this part
00:17:18.150 --> 00:17:20.969
that is due to these
topological defects.
00:17:23.849 --> 00:17:29.170
Again, this vector
is this squared.
00:17:29.170 --> 00:17:30.950
You can see that
if I square it, I
00:17:30.950 --> 00:17:34.980
will get dy psi squared
plus dx psi squared.
00:17:34.980 --> 00:17:37.580
So to all intents and
purposes, this thing
00:17:37.580 --> 00:17:44.050
is the same thing as a
gradient of psi squared.
00:17:44.050 --> 00:17:46.510
Essentially, gradient
of psi and curl of psi
00:17:46.510 --> 00:17:49.980
are the same vector, just
rotated by 90 degrees.
00:17:49.980 --> 00:17:52.890
Integrating the square of
one over the whole space
00:17:52.890 --> 00:17:56.430
is the same as integrating
the square of the other.
00:17:56.430 --> 00:17:58.430
OK?
00:17:58.430 --> 00:18:04.020
So now, let's calculate this
contribution and what it is.
00:18:04.020 --> 00:18:11.250
Integral d2 x gradient of
psi squared with K over 2
00:18:11.250 --> 00:18:16.280
out front-- actually
you already know that.
00:18:16.280 --> 00:18:22.590
Because psi we see
is the potential
00:18:22.590 --> 00:18:23.945
due to a bunch of charges.
00:18:26.690 --> 00:18:29.630
So this is essentially
the electric field
00:18:29.630 --> 00:18:32.370
due to this
combination of charges
00:18:32.370 --> 00:18:34.760
integrated over
the entire space.
00:18:34.760 --> 00:18:37.530
It's the electrostatic image.
00:18:37.530 --> 00:18:41.090
But let's go through
that step by step.
00:18:41.090 --> 00:18:45.320
Let's do the
integration by parts.
00:18:45.320 --> 00:18:49.790
So this becomes minus k over 2.
00:18:49.790 --> 00:18:57.900
Integral d2 x psi the
gradient acting on this
00:18:57.900 --> 00:19:01.500
will give me Laplacian of psi.
00:19:01.500 --> 00:19:06.250
Of course, whenever I
do integration by parts,
00:19:06.250 --> 00:19:08.620
I have to worry
about boundary terms.
00:19:15.360 --> 00:19:17.550
And essentially, if
you think of what
00:19:17.550 --> 00:19:21.260
you will be seeing
at the boundary, far,
00:19:21.260 --> 00:19:26.260
far away from there all of
these charges are, let's say.
00:19:26.260 --> 00:19:30.120
Essentially, you will
see the electric field
00:19:30.120 --> 00:19:34.910
due to the combination
of all of those charges.
00:19:34.910 --> 00:19:42.170
So for a single one, I will
have a large electric field
00:19:42.170 --> 00:19:44.750
that will go as 1 over r.
00:19:44.750 --> 00:19:48.480
And we saw that integrating
that will give me the log.
00:19:48.480 --> 00:19:50.990
So that was not
particularly nice.
00:19:50.990 --> 00:19:54.390
So similarly, what
these boundary terms
00:19:54.390 --> 00:19:58.120
would amount to would
give you some kind
00:19:58.120 --> 00:20:01.470
of a logarithmic energy that
depends on the next charge
00:20:01.470 --> 00:20:03.470
that you have enclosed.
00:20:03.470 --> 00:20:17.070
And you can get rid of it
by setting the next charge
00:20:17.070 --> 00:20:18.225
to be zero.
00:20:23.050 --> 00:20:28.390
So essentially,
any configuration
00:20:28.390 --> 00:20:33.930
in which the sum total of our
topological charges is non-zero
00:20:33.930 --> 00:20:38.270
will get a huge energy cost
as we go to large distances
00:20:38.270 --> 00:20:40.140
from the self-energy,
if you like,
00:20:40.140 --> 00:20:42.980
of creating this huge monopole.
00:20:42.980 --> 00:20:45.600
So we are going to
use this condition
00:20:45.600 --> 00:20:48.140
and focus only on
configurations that
00:20:48.140 --> 00:20:52.390
are charged topological
charge neutral.
00:20:52.390 --> 00:20:53.530
OK.
00:20:53.530 --> 00:20:59.050
Now our psi is what
I have over here.
00:20:59.050 --> 00:21:03.950
It is sum over i qi.
00:21:03.950 --> 00:21:06.930
This Coulomb
interaction-- r minus ri.
00:21:10.740 --> 00:21:18.560
And therefore, Laplacian
of psi is essentially
00:21:18.560 --> 00:21:21.300
taking the Laplacian
of this expression
00:21:21.300 --> 00:21:25.590
is the sum over j qu.
00:21:25.590 --> 00:21:28.395
Laplacian of this is
the delta function.
00:21:33.150 --> 00:21:35.700
So basically, that
was the condition
00:21:35.700 --> 00:21:37.100
for the Coulomb potential.
00:21:37.100 --> 00:21:40.900
Or alternatively, you take
2 derivative of the log,
00:21:40.900 --> 00:21:45.390
and you will generate
the delta function.
00:21:45.390 --> 00:21:47.430
OK?
00:21:47.430 --> 00:21:56.990
So, what you see is that
you generate the following.
00:21:56.990 --> 00:22:12.210
You will get a minus k over 2
sum over pairs i and j, qi, qj.
00:22:12.210 --> 00:22:15.610
And then I have the
integral over x or r--
00:22:15.610 --> 00:22:17.240
they're basically
the same thing.
00:22:17.240 --> 00:22:18.985
Maybe I should have
written this as x.
00:22:23.860 --> 00:22:29.710
And the delta function
insures that x
00:22:29.710 --> 00:22:33.462
is set to be the other i.
00:22:33.462 --> 00:22:35.890
So I will get the
Coulomb interaction
00:22:35.890 --> 00:22:40.010
between ri minus rj.
00:22:40.010 --> 00:22:46.710
So basically, what you have is
that these topological defects
00:22:46.710 --> 00:22:51.730
that are characterized by these
integers n, or by the charges
00:22:51.730 --> 00:22:56.540
2 pi n, have exactly
this logarithmic Coulomb
00:22:56.540 --> 00:22:57.500
interaction.
00:22:57.500 --> 00:22:58.770
in two dimensions.
00:22:58.770 --> 00:23:01.750
And as I said, this
thing is none other
00:23:01.750 --> 00:23:03.990
than the electrostatic energy.
00:23:03.990 --> 00:23:06.180
The electrostatic energy
you can write either
00:23:06.180 --> 00:23:08.930
as an integral of the
electric field squared.
00:23:08.930 --> 00:23:11.640
Or you can write
as the interaction
00:23:11.640 --> 00:23:16.900
among the charges that give
rise to that electric field.
00:23:16.900 --> 00:23:19.620
OK?
00:23:19.620 --> 00:23:26.990
So, what I can do is I
can write this as follows.
00:23:26.990 --> 00:23:32.160
First of all, I can maybe
re-cast it in terms of the n's.
00:23:32.160 --> 00:23:36.820
So I will have 2 pi ni, 2 pi nj.
00:23:36.820 --> 00:23:43.330
So I will get minus
4pi squared k.
00:23:43.330 --> 00:23:46.010
There's a factor of one-half.
00:23:46.010 --> 00:23:49.870
But this is a sum over i
and j-- so every pair is now
00:23:49.870 --> 00:23:51.700
counted twice.
00:23:51.700 --> 00:23:54.080
So I get rid of that
factor of one-half
00:23:54.080 --> 00:24:00.180
by essentially counting
each pair only once.
00:24:00.180 --> 00:24:04.450
So I have the
Coulomb interaction
00:24:04.450 --> 00:24:08.610
between ri and rj, which
is this 1 over 2 pi
00:24:08.610 --> 00:24:13.390
log of ri minus rj
with some cut-off.
00:24:13.390 --> 00:24:17.320
And then, there's the term that
corresponds to i equals 2j.
00:24:17.320 --> 00:24:30.340
So I will have a minus, let's
say, 4pi squared k sum over i.
00:24:30.340 --> 00:24:33.040
And I forgot here
to put n, i, and j.
00:24:36.470 --> 00:24:39.730
I will have ni squared.
00:24:39.730 --> 00:24:43.334
The Coulomb interaction at zero.
00:24:43.334 --> 00:24:44.250
ri equals [INAUDIBLE].
00:24:46.970 --> 00:24:53.240
Now clearly, this expression
does not make sense.
00:24:53.240 --> 00:24:58.640
What it is trying to
tell me is that there
00:24:58.640 --> 00:25:06.340
is a cost to creating one of
these topological charges.
00:25:06.340 --> 00:25:11.810
And all of this theory--
again, in order to make sense,
00:25:11.810 --> 00:25:15.220
we should remember to put
some kind of a short distance
00:25:15.220 --> 00:25:16.670
cut-off a.
00:25:16.670 --> 00:25:18.550
All right?
00:25:18.550 --> 00:25:26.110
And basically, replacing this
original discrete lattice
00:25:26.110 --> 00:25:31.040
with a continuum will
only work as long as
00:25:31.040 --> 00:25:35.830
I keep in mind that I cannot
regard things at the level
00:25:35.830 --> 00:25:40.640
of lattice spacing, and replace
it by that formula, as we saw,
00:25:40.640 --> 00:25:41.800
for example, here.
00:25:41.800 --> 00:25:45.340
If I want to draw a
topological defect,
00:25:45.340 --> 00:25:48.430
I would need right at the
center to do something
00:25:48.430 --> 00:25:52.630
like this-- where replacing
the cosines with the gradient
00:25:52.630 --> 00:25:55.120
squared kind of
doesn't make sense.
00:25:55.120 --> 00:25:57.030
So basically, what
this theory is
00:25:57.030 --> 00:26:03.440
telling me is that once you
get to a very small distance,
00:26:03.440 --> 00:26:08.550
you have to keep
track of the existence
00:26:08.550 --> 00:26:13.520
of some underlying lattice
and the corresponding things.
00:26:13.520 --> 00:26:16.270
And what's really
this is describing
00:26:16.270 --> 00:26:22.720
for you is the core
energy of creating
00:26:22.720 --> 00:26:26.670
a defect that has object ni.
00:26:26.670 --> 00:26:27.470
OK.
00:26:27.470 --> 00:26:37.660
What do I mean by that
is that over here, I
00:26:37.660 --> 00:26:43.485
can calculate what the partition
function is for one defect.
00:26:43.485 --> 00:26:46.550
This we already did
last time around.
00:26:46.550 --> 00:26:50.780
And for that, I can
integrate out this energy
00:26:50.780 --> 00:26:56.650
that I have for the distortions.
00:26:56.650 --> 00:27:02.470
It's an integral of
n over r squared.
00:27:02.470 --> 00:27:07.980
And this integration
gave me this factor of e
00:27:07.980 --> 00:27:20.752
to the minus pi k log of r over
a-- actually, that was 2 pi k.
00:27:20.752 --> 00:27:25.380
Actually, let's do
this correctly once.
00:27:29.320 --> 00:27:32.100
I should have done it
earlier, and I forgot.
00:27:32.100 --> 00:27:37.130
So you have one defect.
00:27:37.130 --> 00:27:45.460
And we saw that for one defect,
the field at the distance r
00:27:45.460 --> 00:27:50.140
has n over r in magnitude.
00:27:53.830 --> 00:28:00.370
And then, the net energy cost
for one of these defects--
00:28:00.370 --> 00:28:05.120
if I say that I believe
this formula starting
00:28:05.120 --> 00:28:14.380
from a distance a is the
k over 2 integral from a.
00:28:14.380 --> 00:28:17.925
Let's say all the way up
to the size of my system.
00:28:20.780 --> 00:28:29.630
I have 2 pi r dr from
a shell at radius r
00:28:29.630 --> 00:28:32.880
magnitude of this u squared.
00:28:32.880 --> 00:28:37.830
So I have n squared
over r squared.
00:28:37.830 --> 00:28:41.860
But then, I have to worry
about all of the actual things
00:28:41.860 --> 00:28:45.320
that I have off the distance a.
00:28:45.320 --> 00:28:50.510
So on top of this,
there is a core energy
00:28:50.510 --> 00:28:56.760
for creating this object that
certainly explicitly depends on
00:28:56.760 --> 00:29:00.320
where I sit this parameter a.
00:29:00.320 --> 00:29:02.120
OK?
00:29:02.120 --> 00:29:06.740
This part is easy.
00:29:06.740 --> 00:29:13.510
It simply gives
me pi km squared.
00:29:13.510 --> 00:29:16.200
And then I have
the integral of 1
00:29:16.200 --> 00:29:21.340
over r, which gives
me log of L over a.
00:29:26.290 --> 00:29:31.790
So if I want to imagine what
the partition function of this
00:29:31.790 --> 00:29:38.560
is-- one defect in
a system of size L--
00:29:38.560 --> 00:29:46.160
I would say that z of one defect
is Boltzmann weight responding
00:29:46.160 --> 00:29:47.660
to creating this entity.
00:29:47.660 --> 00:29:54.570
So I have e to the minus pi
k m squared log of L over a.
00:29:54.570 --> 00:30:02.360
And then I have the core energy
that corresponds to this.
00:30:02.360 --> 00:30:07.190
And then, as we discussed,
I can place this anywhere
00:30:07.190 --> 00:30:10.750
in the system if I'm calculating
the partition function.
00:30:10.750 --> 00:30:14.130
So there's an integration
over the position
00:30:14.130 --> 00:30:16.570
of this that is implicit.
00:30:16.570 --> 00:30:19.480
And so, that's going
to give me the square
00:30:19.480 --> 00:30:27.770
of the size of my system,
except that I am unsure as
00:30:27.770 --> 00:30:33.330
to where I have placed
things up to this cut-off a.
00:30:33.330 --> 00:30:37.310
So really, the
number of distinct
00:30:37.310 --> 00:30:43.710
positions that I have scales
like L over a squared.
00:30:43.710 --> 00:30:48.870
So the whole thing we
can see scales like L
00:30:48.870 --> 00:30:52.790
over a 2 minus pi km squared.
00:30:55.470 --> 00:31:03.660
And then, there is this
factor of e to the minus
00:31:03.660 --> 00:31:08.180
this core energy
evaluated at the distance
00:31:08.180 --> 00:31:10.250
a that I will call y.
00:31:13.550 --> 00:31:15.480
Because again, in
some sense, there's
00:31:15.480 --> 00:31:19.280
some arbitrariness
in where I choose a.
00:31:19.280 --> 00:31:24.590
So this y would be a function of
a, if we depend on that choice.
00:31:24.590 --> 00:31:28.190
But the most important
thing is that if I
00:31:28.190 --> 00:31:34.830
have a huge system, whether or
not this partition function,
00:31:34.830 --> 00:31:38.630
as a function of the size of
the system, goes to infinity
00:31:38.630 --> 00:31:45.470
or goes to 0 is controlled by
this exponent 2 minus pi k.
00:31:45.470 --> 00:31:49.040
Let's say we focus on the
simplest of topological defects
00:31:49.040 --> 00:31:53.030
corresponding to n
equal 2 minus plus 1.
00:31:53.030 --> 00:31:56.650
You expect that there is some
potentially critical value
00:31:56.650 --> 00:32:02.940
of k, which is 2 over pi,
that distinguishes the two
00:32:02.940 --> 00:32:06.830
types of behavior.
00:32:06.830 --> 00:32:09.460
OK?
00:32:09.460 --> 00:32:16.060
But this picture is nice,
but certainly incomplete.
00:32:16.060 --> 00:32:22.270
Because who said that there's
any legitimacy in calculating
00:32:22.270 --> 00:32:24.390
the partition function
that corresponds
00:32:24.390 --> 00:32:27.700
to just a single
topological defect.
00:32:27.700 --> 00:32:31.680
If I integrate over all the
configurations of my angle
00:32:31.680 --> 00:32:38.280
field, I should really
be doing something
00:32:38.280 --> 00:32:43.700
that is analogous to this
and calculating a partition
00:32:43.700 --> 00:32:47.270
function that corresponds
to many defects.
00:32:47.270 --> 00:32:52.080
And actually what I calculated
over here was in some sense
00:32:52.080 --> 00:32:54.960
the configuration of
spins given that there
00:32:54.960 --> 00:32:59.170
is a topological defects
that has the lowest energy.
00:32:59.170 --> 00:33:01.900
Once I start with this
configuration-- let's
00:33:01.900 --> 00:33:04.270
say, with everybody
radiating out--
00:33:04.270 --> 00:33:07.620
I can start to distort
them a little bit which
00:33:07.620 --> 00:33:13.060
amounts to adding this
gradient of phi to that.
00:33:13.060 --> 00:33:15.940
So really, the
partition function
00:33:15.940 --> 00:33:19.900
that I want to calculate and
wrote down at the beginning--
00:33:19.900 --> 00:33:23.680
if I want to
calculate correctly,
00:33:23.680 --> 00:33:28.970
I have to include both
these fluctuations
00:33:28.970 --> 00:33:31.900
and these fluctuations
corresponding
00:33:31.900 --> 00:33:38.910
to an arbitrary set of
these topological defects.
00:33:38.910 --> 00:33:44.820
And what we see is that
actually, the partition
00:33:44.820 --> 00:33:49.460
functions and the energy
costs of the two components
00:33:49.460 --> 00:33:51.200
really separate out.
00:33:51.200 --> 00:33:53.360
And what we are
trying to calculate
00:33:53.360 --> 00:33:58.000
is the contribution that is
due to the topological defects.
00:33:58.000 --> 00:34:02.400
And what we see is
that once I tell you
00:34:02.400 --> 00:34:06.500
where the topological
defects are located,
00:34:06.500 --> 00:34:12.790
the partition function for them
has an energy component that
00:34:12.790 --> 00:34:16.560
is this Coulomb interaction
among the defect.
00:34:16.560 --> 00:34:21.320
But there is a
part that really is
00:34:21.320 --> 00:34:28.400
a remnant of this
core energy that we
00:34:28.400 --> 00:34:31.610
were calculating before.
00:34:31.610 --> 00:34:35.969
So when I was sort of
following my nose here,
00:34:35.969 --> 00:34:39.949
I had forgotten a little
bit about the short distance
00:34:39.949 --> 00:34:40.940
cut-off.
00:34:40.940 --> 00:34:44.739
And then, when I
encountered this C of zero,
00:34:44.739 --> 00:34:48.690
it told me that I have
to think about the limit
00:34:48.690 --> 00:34:51.380
when two things come
close to each other.
00:34:51.380 --> 00:34:56.360
And I know that that
limit is constrained
00:34:56.360 --> 00:34:59.190
by my original lattice,
and more importantly,
00:34:59.190 --> 00:35:02.560
by the place where
I am willing to do
00:35:02.560 --> 00:35:10.140
self-averaging and replace
this sum with a gradient.
00:35:10.140 --> 00:35:15.740
OK, so basically, this is
the explanation of this term.
00:35:15.740 --> 00:35:18.280
So the only thing that we
have established so far
00:35:18.280 --> 00:35:22.950
is that this partition
function that I wrote down
00:35:22.950 --> 00:35:28.530
at the beginning gets
decomposed into a part
00:35:28.530 --> 00:35:33.130
that we have calculated before,
which was the Gaussian term,
00:35:33.130 --> 00:35:38.540
and is caused, really, the
contribution due to spin waves.
00:35:38.540 --> 00:35:44.340
So this is when we just consider
these [INAUDIBLE] modes,
00:35:44.340 --> 00:35:47.470
we said that essentially
you can have an energy
00:35:47.470 --> 00:35:49.400
cost that is the
gradient squared.
00:35:49.400 --> 00:35:54.900
So this is the part that
corresponds to integral d phi
00:35:54.900 --> 00:36:00.150
into the minus k over 2 integral
d2 x gradient of phi squared,
00:36:00.150 --> 00:36:03.305
where phi is a well-behaved,
ordinary function.
00:36:11.300 --> 00:36:18.970
And what we find is that the
actual partition function also
00:36:18.970 --> 00:36:25.100
has the contribution from
the topological defects.
00:36:25.100 --> 00:36:31.670
And that I will indicate by ZQ.
00:36:31.670 --> 00:36:34.250
And Q stands for Coulomb gas.
00:36:39.260 --> 00:36:46.340
Because this partition
function Z sub Q
00:36:46.340 --> 00:36:53.550
is like I'm trying to calculate
this system of degrees
00:36:53.550 --> 00:37:00.080
of freedom that are
characterized by charges n that
00:37:00.080 --> 00:37:04.270
can be anywhere in this
two-dimensional space.
00:37:04.270 --> 00:37:08.300
And the interaction between
them is governed by the Coulomb
00:37:08.300 --> 00:37:10.990
interaction in two dimensions.
00:37:10.990 --> 00:37:15.640
So to calculate this,
I have to sum over
00:37:15.640 --> 00:37:19.040
all configuration of charges.
00:37:22.110 --> 00:37:26.520
The number of these charges
could be zero, could be two,
00:37:26.520 --> 00:37:30.590
could be four, could be
six, could be any number.
00:37:30.590 --> 00:37:33.540
But I say even
numbers because I want
00:37:33.540 --> 00:37:37.040
to maintain the
constraint of neutrality.
00:37:37.040 --> 00:37:39.450
Sum over ni should be zero.
00:37:39.450 --> 00:37:42.310
So I want to do that
constrained sum.
00:37:42.310 --> 00:37:47.410
So I only want to look at
neutral configurations.
00:37:47.410 --> 00:37:49.530
Once I have
specified-- let's say
00:37:49.530 --> 00:37:57.310
that I have eight charges--
four plus and four minus-- well,
00:37:57.310 --> 00:38:01.540
there is a term that is
going to come from here
00:38:01.540 --> 00:38:06.170
and I kind of said that the
exponential of this term I'm
00:38:06.170 --> 00:38:08.700
going to call y.
00:38:08.700 --> 00:38:13.110
So I have essentially
y raised to the power
00:38:13.110 --> 00:38:15.630
of the number of charges.
00:38:15.630 --> 00:38:19.190
Let's call this sum
over i ni squared.
00:38:19.190 --> 00:38:21.730
And I'm actually just
going to constrain ni
00:38:21.730 --> 00:38:23.480
to be minus plus 1.
00:38:23.480 --> 00:38:27.300
I'm going to only look
at these primary charges.
00:38:27.300 --> 00:38:29.930
So the sum over i
and i squared is just
00:38:29.930 --> 00:38:32.970
the total number of
charges irrespective
00:38:32.970 --> 00:38:35.135
of whether they
are plus or minus.
00:38:35.135 --> 00:38:39.390
It basically is replacing this.
00:38:39.390 --> 00:38:44.610
And then, I have
to integrate over
00:38:44.610 --> 00:38:46.880
the positions of these charges.
00:38:46.880 --> 00:38:49.570
Let's call this total number n.
00:38:49.570 --> 00:38:56.400
So I have to
integrate i1 2n d2 xi
00:38:56.400 --> 00:39:00.110
the position of
where this charge is
00:39:00.110 --> 00:39:05.440
and then interaction which
is exponential of minus
00:39:05.440 --> 00:39:11.610
4phi squared k sum
over i less than j.
00:39:11.610 --> 00:39:17.130
And i and j the Coulomb
interaction between location
00:39:17.130 --> 00:39:18.810
let's say xi and xj.
00:39:27.580 --> 00:39:32.240
Actually, I want to also
emphasize that throughout,
00:39:32.240 --> 00:39:33.770
I have this cut-off.
00:39:33.770 --> 00:39:36.790
So when I was
integrating over one,
00:39:36.790 --> 00:39:38.990
I said that the number
of positions that I had
00:39:38.990 --> 00:39:42.760
was not L squared, but L
over a squared to make it
00:39:42.760 --> 00:39:44.500
dimensionless.
00:39:44.500 --> 00:39:47.505
I will similarly make these
interactions dimensioned
00:39:47.505 --> 00:39:49.180
as I divide by a squared.
00:39:53.310 --> 00:40:00.580
And so basically, this is
the more interesting thing
00:40:00.580 --> 00:40:02.220
that we want to calculate.
00:40:08.172 --> 00:40:12.080
Also, again, remember I wrote
this a squared down here,
00:40:12.080 --> 00:40:19.750
also to emphasize that
within this expression,
00:40:19.750 --> 00:40:22.130
the minimal separation
that I'm going
00:40:22.130 --> 00:40:26.720
to allow between any pair of
charges is off the order of a.
00:40:26.720 --> 00:40:33.490
I have integrated out or moved
into some continuum description
00:40:33.490 --> 00:40:38.055
any configuration in which
the topological charges are
00:40:38.055 --> 00:40:39.908
less than distance a.
00:40:39.908 --> 00:40:40.408
OK?
00:40:43.800 --> 00:40:45.998
Yes?
00:40:45.998 --> 00:40:48.820
AUDIENCE: Essentially,
when we were
00:40:48.820 --> 00:40:53.656
during [INAUDIBLE] it
was canonical potential,
00:40:53.656 --> 00:40:57.002
[INAUDIBLE], canonical ensemble.
00:40:57.002 --> 00:41:00.030
And this is more like
grand canonical ensemble?
00:41:00.030 --> 00:41:00.660
PROFESSOR: Yes.
00:41:00.660 --> 00:41:07.225
So, as far as the original
two-dimensional xy model
00:41:07.225 --> 00:41:11.310
is concerned, I'm calculating
a canonical partition function
00:41:11.310 --> 00:41:14.820
for this spin or angle
degrees of freedom.
00:41:14.820 --> 00:41:20.590
And I find that that
integration over spin angle
00:41:20.590 --> 00:41:23.830
degrees of freedom
can be decomposed
00:41:23.830 --> 00:41:30.000
into a Gaussian part and a part
that as you correctly point out
00:41:30.000 --> 00:41:36.670
corresponds to a grand
canonical system of charges.
00:41:36.670 --> 00:41:38.720
So the number of
charges that are
00:41:38.720 --> 00:41:41.910
going to appear in
the system I have not
00:41:41.910 --> 00:41:45.630
specified whether it is
determined implicitly
00:41:45.630 --> 00:41:49.211
by how strong these
parameter was.
00:41:49.211 --> 00:41:55.175
AUDIENCE: [INAUDIBLE]
of canonical potential?
00:41:55.175 --> 00:41:58.080
PROFESSOR: y plays the
role of E to the beta mu.
00:42:02.700 --> 00:42:05.650
The quantity that
in 8333 we were
00:42:05.650 --> 00:42:09.750
writing as z-- E to
the beta mu small z.
00:42:12.743 --> 00:42:13.243
OK?
00:42:19.240 --> 00:42:26.990
So, we thought we were
solving the xy model.
00:42:26.990 --> 00:42:32.090
We ended up, indeed, with
this grand canonical system,
00:42:32.090 --> 00:42:36.240
which is currently
parametrized by two things.
00:42:36.240 --> 00:42:43.080
One is this k, which is this
strength of the potential.
00:42:43.080 --> 00:42:46.260
The other is this y.
00:42:46.260 --> 00:42:49.590
Of course, since this
system originally
00:42:49.590 --> 00:42:54.180
came from an xy model that
went only one parameter,
00:42:54.180 --> 00:42:59.300
I expect this y to
also be related to k.
00:42:59.300 --> 00:43:03.350
But just as an expression,
we can certainly
00:43:03.350 --> 00:43:06.770
regard it as a system
that is parametrized
00:43:06.770 --> 00:43:10.260
by two things-- the k and the y.
00:43:10.260 --> 00:43:12.120
For the case of
the xy model, there
00:43:12.120 --> 00:43:15.530
will be some additional
constraint between the two.
00:43:15.530 --> 00:43:19.900
But more generally, we can look
at this system with its two
00:43:19.900 --> 00:43:21.460
parameters.
00:43:21.460 --> 00:43:29.110
And essentially, we will try
to make an expansion in y.
00:43:29.110 --> 00:43:33.360
You'll say that,
OK, presumable, I
00:43:33.360 --> 00:43:38.700
know what is going to happen
when y is very, very small.
00:43:38.700 --> 00:43:47.210
Because then, in the system I
will create only a few charges.
00:43:47.210 --> 00:43:50.110
If I create many
charges, I'm going
00:43:50.110 --> 00:43:54.580
to penalize by more
and more factors of y.
00:43:54.580 --> 00:43:57.420
So maybe through leading
order, the system
00:43:57.420 --> 00:43:59.570
would be free of charge.
00:43:59.570 --> 00:44:01.410
And then, there
would be a few pairs
00:44:01.410 --> 00:44:03.800
that would appear
here and there.
00:44:03.800 --> 00:44:06.600
In fact, there should be
a small density of them,
00:44:06.600 --> 00:44:09.870
even no matter how
small I make y.
00:44:09.870 --> 00:44:12.170
There will be a
very small density
00:44:12.170 --> 00:44:14.310
of these things
that will appear.
00:44:14.310 --> 00:44:18.090
And presumably, these
things will always
00:44:18.090 --> 00:44:20.490
appear close to each other.
00:44:20.490 --> 00:44:26.270
So I will have lots and
lots of these pairs-- well,
00:44:26.270 --> 00:44:27.960
not lots and lots
of these pairs--
00:44:27.960 --> 00:44:33.270
a density of them that is
controlled by how big y is.
00:44:33.270 --> 00:44:42.470
And as I make y larger-- so
this is y becoming larger-- then
00:44:42.470 --> 00:44:46.930
presumably, I will generate
more and more of these pairs.
00:44:46.930 --> 00:44:50.170
And once I have more
and more of these pairs,
00:44:50.170 --> 00:44:54.480
they could, in principle,
get into each other's way.
00:44:54.480 --> 00:44:56.530
And when they get
into each other's way,
00:44:56.530 --> 00:44:59.890
then it's not clear who
is paired with whom.
00:44:59.890 --> 00:45:06.450
And at some point, I
should trade my picture
00:45:06.450 --> 00:45:11.240
of having a gas of
pairs of these objects
00:45:11.240 --> 00:45:14.340
to a plasma of charges,
plus and minus,
00:45:14.340 --> 00:45:18.380
that are moving
all over the place.
00:45:18.380 --> 00:45:23.650
So as I tune this parameter
y, I expect my system
00:45:23.650 --> 00:45:29.840
to go from a low density phase
of atoms of plus-minus bound
00:45:29.840 --> 00:45:34.470
to each other to a high
density phase where
00:45:34.470 --> 00:45:36.950
I have a plasma of
plus and minuses
00:45:36.950 --> 00:45:38.230
moving all over the place.
00:45:38.230 --> 00:45:39.030
Yes?
00:45:39.030 --> 00:45:41.965
AUDIENCE: So, y is related
to the core energy.
00:45:41.965 --> 00:45:42.800
PROFESSOR: Yes.
00:45:42.800 --> 00:45:46.370
AUDIENCE: And core energy is
defined through [INAUDIBLE]
00:45:46.370 --> 00:45:49.960
direction at zero separation--
00:45:49.960 --> 00:45:51.890
PROFESSOR: Well, no.
00:45:51.890 --> 00:45:54.590
Because the Coulomb
description is only
00:45:54.590 --> 00:45:57.290
valid large separations.
00:45:57.290 --> 00:46:02.550
When I get to short distances,
who knows what's going on?
00:46:02.550 --> 00:46:07.100
So there is some underlying
microscopic picture
00:46:07.100 --> 00:46:10.320
that determines what
the core energy is.
00:46:10.320 --> 00:46:12.800
Very roughly, yes,
you would expect
00:46:12.800 --> 00:46:16.850
it to have a form that
is of e to the minus k
00:46:16.850 --> 00:46:19.710
with some coefficient
that comes from adding
00:46:19.710 --> 00:46:21.040
all of those interactions here.
00:46:23.610 --> 00:46:24.718
Yes?
00:46:24.718 --> 00:46:26.570
AUDIENCE: Just based
on the sign convention,
00:46:26.570 --> 00:46:30.199
you're saying if you
increase or decrease y,
00:46:30.199 --> 00:46:32.011
that it will go
from a low density--
00:46:32.011 --> 00:46:36.200
PROFESSOR: OK, so y is the
exponential of something.
00:46:36.200 --> 00:46:42.010
y equals to zero means I will
not create any of these things.
00:46:42.010 --> 00:46:45.800
y approaching 1-- I will
create a lot of them.
00:46:45.800 --> 00:46:48.060
There's no cost at
y equals to one.
00:46:48.060 --> 00:46:50.360
There's no core energy.
00:46:50.360 --> 00:46:52.730
I can create them as I want.
00:46:52.730 --> 00:46:56.814
AUDIENCE: So this would be
like y equals minus epsilon c.
00:46:56.814 --> 00:46:57.730
Is that right?
00:46:57.730 --> 00:46:59.610
PROFESSOR: Yeah.
00:46:59.610 --> 00:47:00.510
Didn't I have that?
00:47:00.510 --> 00:47:03.650
You see in the exponential
it is with the minus.
00:47:10.290 --> 00:47:10.790
OK.
00:47:10.790 --> 00:47:13.190
But in any case, that
is the expectation.
00:47:13.190 --> 00:47:14.010
Right?
00:47:14.010 --> 00:47:17.330
So I expect that
when I calculate,
00:47:17.330 --> 00:47:19.020
I create one of these defects.
00:47:19.020 --> 00:47:23.730
There is an energy cost
which is mostly from outside.
00:47:23.730 --> 00:47:27.790
And then, there's an
additional piece on the inside.
00:47:27.790 --> 00:47:30.247
So the exponential of
that additional piece
00:47:30.247 --> 00:47:31.830
would be a number
that is less than 1.
00:47:31.830 --> 00:47:32.734
AUDIENCE: [INAUDIBLE]
00:47:43.684 --> 00:47:44.350
PROFESSOR: Yeah.
00:47:44.350 --> 00:47:48.260
I mean, the original model
has some particular form.
00:47:48.260 --> 00:47:51.240
And actually, the interactions
of the original model, I
00:47:51.240 --> 00:47:52.730
can make more complicated.
00:47:52.730 --> 00:47:57.030
I can add the full spin
interaction, for example.
00:47:57.030 --> 00:48:01.480
It doesn't affect the
overall form much,
00:48:01.480 --> 00:48:05.120
just modifies what
an effective k is,
00:48:05.120 --> 00:48:08.175
and what the core
energy is independent.
00:48:11.420 --> 00:48:11.920
OK?
00:48:14.500 --> 00:48:15.380
All right.
00:48:15.380 --> 00:48:19.850
But the key point is that
this system potentially
00:48:19.850 --> 00:48:25.520
has a phase transition as you
change the parameter of y.
00:48:25.520 --> 00:48:29.270
And another way of
looking at this transition
00:48:29.270 --> 00:48:35.460
is that what is happening
here in different languages,
00:48:35.460 --> 00:48:38.680
you can either call it
insulator or a dielectric.
00:48:42.840 --> 00:48:45.830
But what is happening here
in different languages,
00:48:45.830 --> 00:48:50.780
you can either call, say,
a metal or, as I said,
00:48:50.780 --> 00:48:51.550
maybe a plasma.
00:48:54.616 --> 00:48:59.960
The point is that here
you have free charges.
00:48:59.960 --> 00:49:03.670
Here you have bound
pairs of charges.
00:49:03.670 --> 00:49:08.090
And they respond
differently to, let's say,
00:49:08.090 --> 00:49:10.150
an external
electromagnetic field.
00:49:10.150 --> 00:49:15.850
So once we have this picture,
let's kind of expand our view.
00:49:15.850 --> 00:49:18.120
Forget about the xy model.
00:49:18.120 --> 00:49:20.490
Think of a system of charges.
00:49:20.490 --> 00:49:25.536
And notice that in
this low-density phase,
00:49:25.536 --> 00:49:29.220
it behaves like a
dielectric in the sense
00:49:29.220 --> 00:49:32.730
that there are no free charges.
00:49:32.730 --> 00:49:37.900
And here, there will be
lots of mobile charges.
00:49:37.900 --> 00:49:40.650
And it behaves like a metal.
00:49:40.650 --> 00:49:42.280
What do I mean by that?
00:49:42.280 --> 00:49:45.400
Well, here, if I,
let's say, bring
00:49:45.400 --> 00:49:48.180
in an external electric field.
00:49:48.180 --> 00:49:54.360
Or maybe if I put a huge
charge, what is going to happen
00:49:54.360 --> 00:50:00.290
is that opposite
charges will accumulate.
00:50:00.290 --> 00:50:04.880
Or there will be, essentially,
opposite charges for the field.
00:50:04.880 --> 00:50:08.590
So that once you
go inside, the fact
00:50:08.590 --> 00:50:13.160
that you have an external
electric field or a charge
00:50:13.160 --> 00:50:14.880
is completely screen.
00:50:14.880 --> 00:50:16.800
You won't see it.
00:50:16.800 --> 00:50:19.860
Whereas here, what
is going to happen
00:50:19.860 --> 00:50:23.380
is that if you put
in an electric field,
00:50:23.380 --> 00:50:26.170
it will penetrate
into the system
00:50:26.170 --> 00:50:28.970
although it will be
weakened a little bit
00:50:28.970 --> 00:50:32.220
by the re-orientation
of these charges.
00:50:32.220 --> 00:50:38.650
Now, if you put a plus charge,
the effect of that plus charge
00:50:38.650 --> 00:50:42.170
would be felt throughout,
although weakened a little bit.
00:50:42.170 --> 00:50:47.790
Because again, some of these
dipoles will re-orient in that.
00:50:47.790 --> 00:50:49.720
OK?
00:50:49.720 --> 00:50:56.380
So, this low-density
phase we can actually
00:50:56.380 --> 00:51:06.050
try to parametrize in terms of
a weakening of the interactions
00:51:06.050 --> 00:51:10.800
through a dielectric
constant epsilon.
00:51:10.800 --> 00:51:13.140
And so, what I'm going
to try to calculate
00:51:13.140 --> 00:51:18.580
for you is to imagine that I'm
in the limit of low density
00:51:18.580 --> 00:51:25.640
or small y and calculate
what the weakening is, what
00:51:25.640 --> 00:51:29.340
the dielectric function
is, perturbatively in y.
00:51:29.340 --> 00:51:29.840
Yes?
00:51:29.840 --> 00:51:33.644
AUDIENCE: If you were talking
about the real electric charges
00:51:33.644 --> 00:51:36.204
and the way to act
on that [INAUDIBLE]
00:51:36.204 --> 00:51:38.060
real electric field or charge.
00:51:38.060 --> 00:51:42.214
But if we are talking about
topological charges, what
00:51:42.214 --> 00:51:46.507
would be kind of
conjugate force to that?
00:51:46.507 --> 00:51:47.480
PROFESSOR: OK.
00:51:50.640 --> 00:51:52.280
It's not going to be easy.
00:51:52.280 --> 00:51:54.710
I have to do
something about say,
00:51:54.710 --> 00:51:58.530
re-orienting all of the spins
on the boundaries, et cetera.
00:51:58.530 --> 00:52:00.550
So let's forget about that.
00:52:00.550 --> 00:52:03.780
The point is that
mathematically, the problem
00:52:03.780 --> 00:52:06.620
is reduced to this system.
00:52:06.620 --> 00:52:11.040
And I can much more
easily do the mathematics
00:52:11.040 --> 00:52:15.950
if I change my perspective
and think about this picture.
00:52:15.950 --> 00:52:18.100
OK?
00:52:18.100 --> 00:52:22.890
And that's the thing you have
to do in theoretical physics.
00:52:22.890 --> 00:52:25.650
You basically take
advantage of mappings
00:52:25.650 --> 00:52:29.280
of one model to
another model in order
00:52:29.280 --> 00:52:35.900
to refine your intuition
using some other picture.
00:52:35.900 --> 00:52:38.370
So that's what we
are going to do.
00:52:38.370 --> 00:52:44.820
So completely different picture
from the original spin models--
00:52:44.820 --> 00:52:52.330
imagine that you have indeed
a box of this material.
00:52:52.330 --> 00:52:56.120
And this box of material
has, because you're wise,
00:52:56.120 --> 00:53:00.130
more some combination of these
plus and minus charges in it.
00:53:02.650 --> 00:53:07.320
And then, what I do is
that I bring externally
00:53:07.320 --> 00:53:10.655
a uniform electric
field in this direction.
00:53:13.870 --> 00:53:19.780
And I expect that once
inside the material,
00:53:19.780 --> 00:53:24.220
the electric field will be
reduced to a smaller value
00:53:24.220 --> 00:53:31.240
that I will call E prime because
of the dielectric function.
00:53:31.240 --> 00:53:34.790
Now, if you ever calculated
dielectric functions,
00:53:34.790 --> 00:53:37.250
that's exactly what
I'm going to do now.
00:53:37.250 --> 00:53:38.970
It's a simple process.
00:53:38.970 --> 00:53:42.310
What you do, for
example, is you do
00:53:42.310 --> 00:53:46.390
the analog of Gauss' theorem.
00:53:46.390 --> 00:53:53.150
Let's imagine that we draw
a circuit such as this
00:53:53.150 --> 00:53:59.930
that is partly on the inside,
and partly on the outside.
00:53:59.930 --> 00:54:06.070
So I can calculate what the
flux of the electric field
00:54:06.070 --> 00:54:12.070
is through this circuit, the
analog of the Gaussian pillbox.
00:54:12.070 --> 00:54:14.660
And so, what I have is
that what is going on
00:54:14.660 --> 00:54:20.140
is E. If I call this
distance to be L,
00:54:20.140 --> 00:54:26.710
the flux integrated
through the entire thing
00:54:26.710 --> 00:54:30.872
is E minus E prime times f.
00:54:30.872 --> 00:54:35.040
So this is the integral of
the divergence of the electric
00:54:35.040 --> 00:54:36.880
field .
00:54:36.880 --> 00:54:43.140
And by Gauss' theorem, this has
to be charge enclosed inside.
00:54:49.588 --> 00:54:52.570
OK.
00:54:52.570 --> 00:54:55.810
Now, why should there
be any charge enclosed
00:54:55.810 --> 00:55:01.045
inside when you have a
bunch of plus and minuses.
00:55:01.045 --> 00:55:02.630
I mean, there will
be some pluses
00:55:02.630 --> 00:55:05.190
and minuses out here
as I have indicated.
00:55:05.190 --> 00:55:08.920
There will be some pluses
and minuses that are inside.
00:55:08.920 --> 00:55:12.510
But the net of
these would be zero.
00:55:12.510 --> 00:55:17.560
So the only place that
you get a net charge
00:55:17.560 --> 00:55:20.934
is those dipoles that
happen to be sitting right
00:55:20.934 --> 00:55:21.600
at the boundary.
00:55:26.230 --> 00:55:32.860
And then, I have to count
how many of them are inside.
00:55:32.860 --> 00:55:35.270
And some of them will
have the plus inside.
00:55:35.270 --> 00:55:38.550
And some of them will
have the minus inside.
00:55:38.550 --> 00:55:41.640
And then, I have to
calculate the net.
00:55:41.640 --> 00:55:50.490
The thing is that my dipoles
do not have a fixed size.
00:55:50.490 --> 00:55:58.810
The size of these
plus/minus molecules r
00:55:58.810 --> 00:56:00.890
can be variable itself.
00:56:00.890 --> 00:56:02.820
OK?
00:56:02.820 --> 00:56:05.670
So there will be some that are
tightly bound to each other.
00:56:05.670 --> 00:56:10.160
There may be some that are
further apart, et cetera.
00:56:10.160 --> 00:56:16.530
So let's look at pairs
that are at the distance r
00:56:16.530 --> 00:56:20.350
and ask how many of them hit
this boundary so that one
00:56:20.350 --> 00:56:23.990
of them would be inside,
one of them will be outside.
00:56:23.990 --> 00:56:25.370
OK?
00:56:25.370 --> 00:56:30.270
So, that number has
to be proportional
00:56:30.270 --> 00:56:36.960
to essentially this area.
00:56:36.960 --> 00:56:39.100
What is that area?
00:56:39.100 --> 00:56:44.730
On one side, it is L. On
the other side, it is R.
00:56:44.730 --> 00:56:51.310
But if the dipole is
oriented at an angle theta,
00:56:51.310 --> 00:56:56.250
it is, in fact r cosine theta.
00:56:56.250 --> 00:56:56.750
OK?
00:56:59.310 --> 00:57:05.320
So that's the number.
00:57:05.320 --> 00:57:12.360
Now, what I will have here
would be the charge 2 pi.
00:57:12.360 --> 00:57:13.170
So this is qi.
00:57:19.880 --> 00:57:23.600
Actually, it could
be plus or minus.
00:57:23.600 --> 00:57:25.750
The reason that there's
going to be more
00:57:25.750 --> 00:57:31.500
plus as opposed to minus
is because the dipole
00:57:31.500 --> 00:57:35.880
gets oriented by
the electric field.
00:57:35.880 --> 00:57:45.090
So I will have a term here that
is E to the E prime times q
00:57:45.090 --> 00:57:50.660
ir-- so that's 2 pi r.
00:57:50.660 --> 00:57:57.280
So this is qr again,
times cosine of theta.
00:57:57.280 --> 00:58:00.150
So we can see that,
depending on cosine of theta
00:58:00.150 --> 00:58:04.150
being larger than
pi or less than pi,
00:58:04.150 --> 00:58:07.440
this number will be
positive or negative.
00:58:07.440 --> 00:58:11.075
And that's going to be
modified by this number also.
00:58:11.075 --> 00:58:14.300
And of course, the strength
of this whole thing
00:58:14.300 --> 00:58:16.870
is set by this parameter k.
00:58:21.230 --> 00:58:28.480
And also how likely it
is for me to have created
00:58:28.480 --> 00:58:36.840
a dipole of size r is controlled
by precisely this factor.
00:58:36.840 --> 00:58:42.030
A dipole is something
that has two cores.
00:58:42.030 --> 00:58:45.280
So it is something that will
appear at order of y squared.
00:58:48.570 --> 00:58:51.050
And there is the
energy, according
00:58:51.050 --> 00:58:54.810
to this formula, of
separating two things.
00:58:54.810 --> 00:58:57.675
And so you can see that
essentially, n of r--
00:58:57.675 --> 00:59:01.320
maybe I will write it
separately over here--
00:59:01.320 --> 00:59:10.600
is y squared times E to
the minus 4pi squared k.
00:59:10.600 --> 00:59:15.520
And from here, I have
log of r divided by a.
00:59:15.520 --> 00:59:17.430
And then there's
a factor of 2 pi
00:59:17.430 --> 00:59:19.610
because the Coulomb
potential is this.
00:59:22.240 --> 00:59:27.960
So, this is going
to be y squared
00:59:27.960 --> 00:59:33.430
a over r to the power of 2 pi k.
00:59:33.430 --> 00:59:38.720
The further you try to
separate these things, the more
00:59:38.720 --> 00:59:40.300
cost you have to pay.
00:59:46.024 --> 00:59:48.900
OK.
00:59:48.900 --> 00:59:52.700
So if you were
trying to calculate
00:59:52.700 --> 00:59:57.750
the contribution of,
say, polarizable atoms
00:59:57.750 --> 01:00:02.150
or dipoles to the dielectric
function of a solid,
01:00:02.150 --> 01:00:06.630
you would be doing exactly
this same calculation.
01:00:06.630 --> 01:00:13.090
The only difference is that
the size of your dipole
01:00:13.090 --> 01:00:16.580
would be set by the
size of your molecule
01:00:16.580 --> 01:00:20.590
and ultimately, related
to its polarizability.
01:00:20.590 --> 01:00:23.720
And rather than having
this Coulomb interaction,
01:00:23.720 --> 01:00:26.755
you would have some
dissociation energy or something
01:00:26.755 --> 01:00:31.800
else, or the density itself
would come over here.
01:00:31.800 --> 01:00:36.630
So, the only final
step is that I
01:00:36.630 --> 01:00:43.470
have to regard my system
having a composition
01:00:43.470 --> 01:00:46.110
of these things of
different sizes.
01:00:46.110 --> 01:00:52.000
So I have to do an integral
over r, as well as orientation.
01:00:52.000 --> 01:00:57.060
So I have to do an
integral over E theta.
01:00:57.060 --> 01:00:58.910
Of course, the
integration will go
01:00:58.910 --> 01:01:03.678
from a through essentially, the
size of the system or infinity.
01:01:07.670 --> 01:01:11.760
I forgot one other
thing, which is
01:01:11.760 --> 01:01:18.240
that when I'm calculating how
many places I can put this,
01:01:18.240 --> 01:01:21.570
again, I have been
calculating things
01:01:21.570 --> 01:01:26.020
per unit area of a squared.
01:01:26.020 --> 01:01:29.800
So I would have to divide
all of these places
01:01:29.800 --> 01:01:34.629
where r and L appear by
corresponding factors of a.
01:01:37.552 --> 01:01:38.052
OK?
01:01:42.950 --> 01:01:45.650
So, the last step
of the calculation
01:01:45.650 --> 01:01:48.145
is you expand this quantity.
01:01:48.145 --> 01:01:50.120
It is 1.
01:01:50.120 --> 01:01:52.670
For small values of
the electric field,
01:01:52.670 --> 01:02:01.000
it is 2 pi r E prime cosine
of theta k plus higher order
01:02:01.000 --> 01:02:03.240
terms.
01:02:03.240 --> 01:02:07.160
And then, you can do the
various integrations.
01:02:07.160 --> 01:02:12.570
First of all, 1 the
integration against 1
01:02:12.570 --> 01:02:14.960
will disappear because
you are integrating
01:02:14.960 --> 01:02:18.740
over all values of
cosine of theta.
01:02:18.740 --> 01:02:21.870
Integral of cosine of
theta gives you zero.
01:02:21.870 --> 01:02:25.220
Essentially, it says that if
there was no electric field,
01:02:25.220 --> 01:02:30.400
there was no reason for there
to be an additional net charge
01:02:30.400 --> 01:02:32.920
on one side or the other.
01:02:32.920 --> 01:02:35.780
So the first term
that will be non-zero
01:02:35.780 --> 01:02:38.930
is the average of cosine
theta squared, which
01:02:38.930 --> 01:02:41.160
will give you a
factor of one-half.
01:02:41.160 --> 01:02:50.970
And so, what you will get is
that E minus E prime times
01:02:50.970 --> 01:03:04.400
L is-- well, there's
going to be a factor of L.
01:03:04.400 --> 01:03:07.420
The integral of the
theta cosine of theta
01:03:07.420 --> 01:03:16.340
squared-- the integral of
cosine of theta squared
01:03:16.340 --> 01:03:20.780
is going to give you 2 pi,
which is the integration times
01:03:20.780 --> 01:03:21.900
one-half.
01:03:21.900 --> 01:03:24.790
So this is the integral
the theta cosine
01:03:24.790 --> 01:03:28.350
square theta will give you this.
01:03:28.350 --> 01:03:31.000
So we did this.
01:03:31.000 --> 01:03:33.700
We have two factors of y.
01:03:33.700 --> 01:03:36.330
y is our expansion parameter.
01:03:36.330 --> 01:03:38.510
We are at a low density limit.
01:03:38.510 --> 01:03:42.310
We've calculated things
assuming that essentially, I
01:03:42.310 --> 01:03:47.050
have to look at one
value of these diplodes.
01:03:47.050 --> 01:03:50.860
In principal, I can imagine that
there will be multiple dipoles.
01:03:50.860 --> 01:03:54.560
And you can see that ultimately,
therefore, potentially, I
01:03:54.560 --> 01:03:59.830
have order of y to the fourth
that I haven't calculated.
01:03:59.830 --> 01:04:02.325
OK, so we got rid
of the y squared.
01:04:05.340 --> 01:04:11.230
We have a factor of E prime
on the expansion here.
01:04:23.550 --> 01:04:27.595
This factor is bothering me
a little bit-- let me check.
01:04:27.595 --> 01:04:29.796
No, that's correct.
01:04:29.796 --> 01:04:31.940
OK, so I have the factor of k.
01:04:37.300 --> 01:04:41.010
I have a factor of 2 pi here
that came from the charge.
01:04:41.010 --> 01:04:44.435
I have another 2 pi here--
so I have 4pi squared.
01:04:51.470 --> 01:04:54.900
I think I got everything
except the integration
01:04:54.900 --> 01:05:04.110
over a to infinity dr.
There is this r dr, which
01:05:04.110 --> 01:05:07.650
is from the two
dimensional integration.
01:05:07.650 --> 01:05:11.495
There was another r
here, and another r here.
01:05:11.495 --> 01:05:15.360
So this becomes r to the three.
01:05:15.360 --> 01:05:18.400
From here, I have minus 2 pi k.
01:05:20.930 --> 01:05:23.650
And then, I have the
corresponding factors
01:05:23.650 --> 01:05:27.230
of a to the power
of 2 pi k minus 4.
01:05:34.200 --> 01:05:34.700
OK.
01:05:37.690 --> 01:05:44.690
So you can see that
the L's cancel.
01:05:44.690 --> 01:05:52.750
And what I get is
that E-- once I
01:05:52.750 --> 01:05:56.210
take the E prime
to the other side--
01:05:56.210 --> 01:06:10.010
becomes E prime 1 plus I have
4pi cubed k y squared-- again,
01:06:10.010 --> 01:06:16.290
y squared is my small
expansion parameter.
01:06:16.290 --> 01:06:22.460
And then, I have the integral
from a to infinity, the r, r
01:06:22.460 --> 01:06:26.140
to the power of
3 minus 2 pi k, a
01:06:26.140 --> 01:06:33.050
to the power of 2 pi k minus
4, and then order of y squared,
01:06:33.050 --> 01:06:34.366
y to the fourth.
01:06:37.852 --> 01:06:40.840
OK.
01:06:40.840 --> 01:06:51.450
So basically, you see that
the internal electric field
01:06:51.450 --> 01:06:57.700
is smaller than the
external electric field
01:06:57.700 --> 01:07:03.610
by this factor, which takes
into account the re-orientation
01:07:03.610 --> 01:07:08.080
of the dipoles in order to
screen the electric field.
01:07:08.080 --> 01:07:10.210
And it is proportional
in some sense
01:07:10.210 --> 01:07:15.000
to the density of these dipoles.
01:07:15.000 --> 01:07:18.610
And the twist is that
the dipoles that we have
01:07:18.610 --> 01:07:21.850
can have a range
of sizes that we
01:07:21.850 --> 01:07:24.940
have to integrate [INAUDIBLE].
01:07:24.940 --> 01:07:31.170
So typically, you would write
E prime to the E over epsilon.
01:07:31.170 --> 01:07:36.580
And so this is the
inverse of your epsilon.
01:07:36.580 --> 01:07:45.500
And essentially, this is
a reduction in everything
01:07:45.500 --> 01:07:49.110
that has to do with electric
interactions because
01:07:49.110 --> 01:07:53.280
of the screening
of other things.
01:07:53.280 --> 01:07:56.120
I can write it in the
following fashion.
01:07:56.120 --> 01:08:02.150
I can say that there is
an effective k-- that's
01:08:02.150 --> 01:08:08.870
called k effective-- which is
different from the original k
01:08:08.870 --> 01:08:11.950
that I have.
01:08:11.950 --> 01:08:14.250
It is reduced by a
factor of epsilon.
01:08:16.760 --> 01:08:23.960
So we were worried when we
were doing the nonlinear sigma
01:08:23.960 --> 01:08:26.090
model that for any
[INAUDIBLE], we
01:08:26.090 --> 01:08:31.689
saw that the parameter k was
not getting modified because
01:08:31.689 --> 01:08:37.090
of the interactions among the
spin modes, and that's correct.
01:08:37.090 --> 01:08:39.080
But really, at
high temperatures,
01:08:39.080 --> 01:08:40.460
it should disappear.
01:08:40.460 --> 01:08:42.870
We saw that the
correlations had to go away
01:08:42.870 --> 01:08:47.029
from power law form
to exponential form.
01:08:47.029 --> 01:08:50.220
And so, we needed some
mechanism for reducing
01:08:50.220 --> 01:08:52.399
the coupling constant.
01:08:52.399 --> 01:08:57.140
And what we find here is
that this topological defects
01:08:57.140 --> 01:09:00.910
and their screening provide
the right mechanism.
01:09:00.910 --> 01:09:03.540
So the effective
k that I have is
01:09:03.540 --> 01:09:06.660
going to be reduced
from the original k
01:09:06.660 --> 01:09:08.080
by the inverse of this.
01:09:08.080 --> 01:09:10.240
Since I'm doing
an expansion in y,
01:09:10.240 --> 01:09:14.704
it is simply minus
4pi cubed ky squared,
01:09:14.704 --> 01:09:20.870
integral a to infinity dr,
r to the 3 minus 2 pi k,
01:09:20.870 --> 01:09:28.210
a to the 2 pi k minus 4, plus
order or y to the fourteenth.
01:09:28.210 --> 01:09:28.710
OK.
01:09:31.920 --> 01:09:36.550
Now actually, in the lecture
notes that I have given you,
01:09:36.550 --> 01:09:42.210
I calculate this formula in
an entirely different way.
01:09:42.210 --> 01:09:49.080
What I do is I assume that I
have two topological defects--
01:09:49.080 --> 01:09:53.689
so there I sort of maintain the
picture of topological defect.
01:09:53.689 --> 01:09:58.290
And their interaction
between them
01:09:58.290 --> 01:10:03.440
is this logarithmic interaction
that has coefficient k.
01:10:03.440 --> 01:10:06.050
But then, we say that this
[INAUDIBLE] interaction
01:10:06.050 --> 01:10:13.640
is modified because I can create
pairs of topological defect,
01:10:13.640 --> 01:10:15.860
such as this, that
will partially
01:10:15.860 --> 01:10:19.060
screen the interaction.
01:10:19.060 --> 01:10:22.120
And in the notes,
we calculate what
01:10:22.120 --> 01:10:25.610
the effect of those
pairs at lowest order
01:10:25.610 --> 01:10:29.970
is on their interaction
that you have between them.
01:10:29.970 --> 01:10:32.860
And you find that the
effect is to modify
01:10:32.860 --> 01:10:35.370
the coefficient of
the logarithm, which
01:10:35.370 --> 01:10:38.670
is k, to a reduced k.
01:10:38.670 --> 01:10:42.230
And that reduced k is
given exactly by this form.
01:10:42.230 --> 01:10:44.771
So the same thing you
can get different ways.
01:10:44.771 --> 01:10:45.270
Yes?
01:10:45.270 --> 01:10:47.635
AUDIENCE: What if
k is too small--
01:10:47.635 --> 01:10:48.910
PROFESSOR: A-ha.
01:10:48.910 --> 01:10:50.280
Good.
01:10:50.280 --> 01:10:55.410
Because I framed the
entire thing as if I'm
01:10:55.410 --> 01:10:58.364
doing a preservation
theory for you
01:10:58.364 --> 01:11:02.140
in y being a small parameter.
01:11:02.140 --> 01:11:03.850
OK?
01:11:03.850 --> 01:11:07.720
But now, we see
that no matter how
01:11:07.720 --> 01:11:20.510
small y is, if k is in
fact less than 2 over pi--
01:11:20.510 --> 01:11:24.890
so this has dimensions of
r to the 4 minus 2 pi k.
01:11:24.890 --> 01:11:32.240
So if k is less than 2 over pi--
which incidentally is something
01:11:32.240 --> 01:11:41.920
that we saw earlier--
if k is less than that,
01:11:41.920 --> 01:11:43.940
this integral diverges.
01:11:46.750 --> 01:11:53.310
So I thought I was
controlling my expansion
01:11:53.310 --> 01:11:58.950
by making y arbitrarily
small, but what
01:11:58.950 --> 01:12:04.990
we see is that no matter
how small I make y,
01:12:04.990 --> 01:12:09.440
if k becomes too small,
the perturbation acuity
01:12:09.440 --> 01:12:12.600
blows up on me.
01:12:12.600 --> 01:12:16.990
So this is yet another example
of a singular perturbation
01:12:16.990 --> 01:12:20.810
theory, which is what we
had encountered when we were
01:12:20.810 --> 01:12:23.310
doing the Landau-Ginzburg model.
01:12:23.310 --> 01:12:27.070
We thought that our co-efficient
of phi to the fourth u
01:12:27.070 --> 01:12:29.020
was a small parameter.
01:12:29.020 --> 01:12:32.810
You are making an expansion
naively in powers of u.
01:12:32.810 --> 01:12:35.380
And then we found an
expression in which
01:12:35.380 --> 01:12:38.230
the coeffecient-- the thing
that was multiplying u
01:12:38.230 --> 01:12:41.240
at the critical point
was blowing up on us.
01:12:41.240 --> 01:12:44.490
And so the perturbation
theory inherently
01:12:44.490 --> 01:12:47.210
became singular,
despite your thinking
01:12:47.210 --> 01:12:50.840
that you had a small parameter.
01:12:50.840 --> 01:12:55.200
So we are going to
use the same trick
01:12:55.200 --> 01:13:04.300
that we used for the case of
the Landau-Ginzburg model--
01:13:04.300 --> 01:13:15.350
this is deal with
singular perturbations
01:13:15.350 --> 01:13:18.080
by renormalization group.
01:13:22.020 --> 01:13:28.110
So what we see is that
the origin of the problem
01:13:28.110 --> 01:13:31.520
is the divergence that
we get over here when
01:13:31.520 --> 01:13:34.670
we try to integrate
all the way to infinity
01:13:34.670 --> 01:13:37.880
or the size of the system.
01:13:37.880 --> 01:13:43.460
So what we do instead is
we said, OK, let's not
01:13:43.460 --> 01:13:45.890
integrate all the way.
01:13:45.890 --> 01:13:51.400
Let's replace the
short distance cut-off
01:13:51.400 --> 01:13:58.270
that we had with something
that is larger-- ba--
01:13:58.270 --> 01:14:03.470
and rather than integrating all
of a to infinity, we integrate
01:14:03.470 --> 01:14:10.450
only over short distance
fluctuations between a and ba.
01:14:10.450 --> 01:14:14.450
This is our usual [INAUDIBLE].
01:14:14.450 --> 01:14:19.790
So what we therefore get
is that the k effective is
01:14:19.790 --> 01:14:25.580
k 1 minus 4pi q ky
squared integral
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.
01:14:37.530 --> 01:14:42.820
And then, I have to still
deal with 4pi cubed ky
01:14:42.820 --> 01:14:46.660
squared, integral
from ba infinity dr
01:14:46.660 --> 01:14:53.840
r to the 3 minus 2 pi k,
a to the 2 pi k minus 4,
01:14:53.840 --> 01:14:57.770
plus order of y to the fourth.
01:14:57.770 --> 01:15:00.728
OK?
01:15:00.728 --> 01:15:01.228
All right.
01:15:04.690 --> 01:15:14.990
So, you can see that the
effect of integrating this much
01:15:14.990 --> 01:15:22.500
is to modify the decoupling
to a new value which
01:15:22.500 --> 01:15:29.450
depends on b, which is just
k minus 4pi cubed k squared
01:15:29.450 --> 01:15:37.730
y squared, a to ba br r
to the 3 minus 2 pi k.
01:15:37.730 --> 01:15:42.471
a to the 2 pi k minus 4.
01:15:42.471 --> 01:15:42.971
OK?
01:15:46.430 --> 01:15:52.060
And then, I can rewrite
the expression for k
01:15:52.060 --> 01:16:00.130
effective to be this k tilde.
01:16:00.130 --> 01:16:07.530
And then, whatever is
left, which is 4pi cubed k
01:16:07.530 --> 01:16:14.220
squared y squared
integral ba to infinity dr
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
01:16:24.647 --> 01:16:25.230
to the fourth.
01:16:29.530 --> 01:16:35.480
You see that k has been shifted
through this transformation
01:16:35.480 --> 01:16:39.700
by an amount that is
order of y squared.
01:16:39.700 --> 01:16:45.320
So at order of y squared
in this new expression,
01:16:45.320 --> 01:16:53.470
I can replace all the k's
that are carrying with k tilde
01:16:53.470 --> 01:16:56.625
and it would still be
correct for this order.
01:16:59.950 --> 01:17:05.320
Now I compare this expression
and the original expression
01:17:05.320 --> 01:17:07.750
that I had.
01:17:07.750 --> 01:17:12.510
And I see that they are pretty
much the same expression,
01:17:12.510 --> 01:17:19.120
except that in this
one, the cut-off is ba.
01:17:19.120 --> 01:17:21.290
So I do step two of origin.
01:17:21.290 --> 01:17:32.480
I define my R prime to be dr
so that my new cut-off will
01:17:32.480 --> 01:17:35.300
be back to a.
01:17:35.300 --> 01:17:38.270
So then, this
whole thing becomes
01:17:38.270 --> 01:17:45.850
k effective is k tilde
minus 4pi cubed k
01:17:45.850 --> 01:17:49.740
tilde squared y squared.
01:17:49.740 --> 01:17:53.430
Because of the transformation
that I did over here,
01:17:53.430 --> 01:17:59.760
I will get a factor of b to
the 4 minus 2 pi k tilde,
01:17:59.760 --> 01:18:05.040
integral from a' to infinity--
the r prime-- r prime to the 3
01:18:05.040 --> 01:18:11.950
minus 2 pi k tilde, a to
the 2 pi k tilde, minus 4
01:18:11.950 --> 01:18:13.629
plus order of y
to the fourteenth.
01:18:20.120 --> 01:18:27.820
So we see that the same
effective interaction
01:18:27.820 --> 01:18:32.810
can be obtained from
two theories that
01:18:32.810 --> 01:18:39.780
have exactly the same cut-off,
a, except that in one case,
01:18:39.780 --> 01:18:42.100
I had k and y.
01:18:42.100 --> 01:18:47.082
In the new case, I have this
tilde or k prime at scale b.
01:18:47.082 --> 01:18:49.390
And I have to
replace where I had
01:18:49.390 --> 01:18:53.920
y with y with this
additional factor.
01:18:53.920 --> 01:18:58.370
So the two theories
are equivalent provided
01:18:58.370 --> 01:19:03.430
that I say that the new
interaction at scale b
01:19:03.430 --> 01:19:11.440
is the old interaction minus
4pi cubed k squared y squared.
01:19:11.440 --> 01:19:14.570
This integral is
easy to perform.
01:19:14.570 --> 01:19:16.000
It is just the power law.
01:19:16.000 --> 01:19:21.190
It is b to the fourth
minus 2 pi k minus 1.
01:19:21.190 --> 01:19:28.820
And then, I have 4 minus 2 pi
k order of y to the fourth.
01:19:28.820 --> 01:19:33.910
And my y prime is
y-- from here I
01:19:33.910 --> 01:19:38.116
see b to the power
of 2 minus pi k.
01:19:43.462 --> 01:19:44.434
AUDIENCE: [INAUDIBLE]
01:19:48.322 --> 01:19:50.090
PROFESSOR: This k squared?
01:19:50.090 --> 01:19:51.550
AUDIENCE: Oh, I see.
01:19:51.550 --> 01:19:52.050
Sorry.
01:19:58.525 --> 01:19:59.400
PROFESSOR: All right.
01:20:03.820 --> 01:20:13.200
So, our theory is described
in terms of two parameters--
01:20:13.200 --> 01:20:17.780
this y and this k.
01:20:17.780 --> 01:20:22.970
or let's say, it's
inverse-- k inverse,
01:20:22.970 --> 01:20:25.910
which is more like temperature.
01:20:25.910 --> 01:20:29.100
And what we will
show next time is
01:20:29.100 --> 01:20:34.950
that these recursion
relations, when I draw it here,
01:20:34.950 --> 01:20:38.170
will give me two
types of behavior.
01:20:38.170 --> 01:20:41.470
One set of behavior
that parameterizes
01:20:41.470 --> 01:20:47.700
the low temperature dilute
limit that corresponds
01:20:47.700 --> 01:20:51.890
to flows in which y
goes through zero.
01:20:51.890 --> 01:20:55.400
So that when you look at the
system at larger and larger
01:20:55.400 --> 01:20:59.830
lens scales, essentially it
becomes less and less depleted
01:20:59.830 --> 01:21:04.030
of these excitations.
01:21:04.030 --> 01:21:07.030
So, once you have integrated
the very essentially,
01:21:07.030 --> 01:21:09.960
you don't see any excitations.
01:21:09.960 --> 01:21:12.210
And then there's
another phase, which
01:21:12.210 --> 01:21:18.810
as you do this removal of
short distance fluctuations.
01:21:18.810 --> 01:21:22.550
You tend to flow to
high temperatures
01:21:22.550 --> 01:21:24.690
and large densities.
01:21:24.690 --> 01:21:29.451
And so, that corresponds
to this kind of face.
01:21:29.451 --> 01:21:33.150
Now the beauty of
this whole thing
01:21:33.150 --> 01:21:38.600
is that these recursion
relations are exact and allow
01:21:38.600 --> 01:21:43.650
us to exactly determine
the behavior of these space
01:21:43.650 --> 01:21:46.140
transition in two dimensions.
01:21:46.140 --> 01:21:49.830
And that's actually one
of the other triumphs
01:21:49.830 --> 01:21:54.040
of renormalization
group is to elucidate
01:21:54.040 --> 01:21:56.950
exactly the critical
behavior of this transition,
01:21:56.950 --> 01:22:00.000
as we will discuss next time.