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PROFESSOR: OK, let's start.
00:00:25.180 --> 00:00:31.820
So we've been looking at the
xy model in two dimensions.
00:00:34.910 --> 00:00:40.060
It's a collection of
units spins located
00:00:40.060 --> 00:00:44.890
on a site of potentially a
lattice in two dimensions.
00:00:44.890 --> 00:00:48.345
Since they are unit
vectors, each one of them
00:00:48.345 --> 00:00:52.550
is characterized by
an angle theta i.
00:00:52.550 --> 00:00:56.190
And the partition
function would be
00:00:56.190 --> 00:01:00.450
obtained by integrating
over all of the angles.
00:01:03.490 --> 00:01:09.790
And the weight that we said had
the form k e cosine of theta
00:01:09.790 --> 00:01:12.252
i minus theta j.
00:01:12.252 --> 00:01:15.380
So there's a coupling
that corresponds
00:01:15.380 --> 00:01:18.480
to the dot per dot of
neighboring spins, which
00:01:18.480 --> 00:01:22.500
can be written as
this cosine form.
00:01:22.500 --> 00:01:33.290
If we go to low temperatures
where k is large,
00:01:33.290 --> 00:01:36.610
then this was
roughly an integral
00:01:36.610 --> 00:01:40.150
over all configurations of
the continuous field, theta
00:01:40.150 --> 00:01:48.800
of x, with a weight that
is the appropriate choice
00:01:48.800 --> 00:01:54.015
of the lattice spacing, the same
k integral of gradient of theta
00:01:54.015 --> 00:01:54.515
squared.
00:02:00.860 --> 00:02:07.200
Now if we look at
this weight and ask
00:02:07.200 --> 00:02:10.990
what is happening as
a function of changing
00:02:10.990 --> 00:02:17.270
temperature or the
inverse of this parameter
00:02:17.270 --> 00:02:22.670
so that we are close
to zero temperature,
00:02:22.670 --> 00:02:26.430
if we just work
with this Gaussian,
00:02:26.430 --> 00:02:28.240
the conclusion
would be that if we
00:02:28.240 --> 00:02:33.330
look at the correlation between
two spins that are located
00:02:33.330 --> 00:02:42.540
at a distance r from each other,
just from this Gaussian weight,
00:02:42.540 --> 00:02:49.000
we found that these correlations
decay as something like 1/r,
00:02:49.000 --> 00:02:53.250
or maybe we put some kind
for a lattice spacing,
00:02:53.250 --> 00:02:57.270
to the power of an
exponent theta that
00:02:57.270 --> 00:02:59.064
was related to this k.
00:03:03.330 --> 00:03:08.380
So the conclusion was that if
that is the appropriate weight,
00:03:08.380 --> 00:03:13.460
we have power-law
decay correlations.
00:03:13.460 --> 00:03:20.510
There is no long range order for
correlations decay very weakly.
00:03:20.510 --> 00:03:24.220
On the other hand, if we
start with the full cosine
00:03:24.220 --> 00:03:28.580
and don't do this low
temperature expansion,
00:03:28.580 --> 00:03:33.700
just go and do the typical
high temperature expansion,
00:03:33.700 --> 00:03:35.530
from the high
temperature expansion
00:03:35.530 --> 00:03:37.690
we conclude that these
collections decay
00:03:37.690 --> 00:03:46.160
exponentially, which are
two totally different forms.
00:03:46.160 --> 00:03:48.520
And presumably, there
should be some kind
00:03:48.520 --> 00:03:52.370
of a critical value
of k or temperature
00:03:52.370 --> 00:03:56.169
that separates a low temperature
formed with power-law decay
00:03:56.169 --> 00:03:58.335
and a high temperature
formed with exponential decay
00:03:58.335 --> 00:03:59.147
of correlations.
00:04:01.950 --> 00:04:07.080
Now there was no sign
of such a kc when
00:04:07.080 --> 00:04:12.710
we tried to do a low
temperature expansion
00:04:12.710 --> 00:04:16.890
like the non-linear sigma
model in this particular case
00:04:16.890 --> 00:04:21.480
of the xy model that
corresponds to n equals to 2.
00:04:21.480 --> 00:04:23.980
Again, we said that
the reason for that
00:04:23.980 --> 00:04:28.120
is that the only high order
terms that I can write down
00:04:28.120 --> 00:04:30.990
in this theory close to low
temperature gradient of theta
00:04:30.990 --> 00:04:35.680
to the fourth, sixth, et cetera
are explicitly irrelevant.
00:04:35.680 --> 00:04:38.055
And unlike n equals
3, et cetera,
00:04:38.055 --> 00:04:40.860
it cannot cause any change.
00:04:40.860 --> 00:04:46.590
So as far as that theory
was concerned, all of these
00:04:46.590 --> 00:04:48.640
corresponded to
being fixed points.
00:04:52.450 --> 00:04:58.650
Then we said that there
is a twist that is not
00:04:58.650 --> 00:05:02.490
taken into account when I
make that transformation
00:05:02.490 --> 00:05:06.610
in the first line, as pointed
out by Kosterlitz and Thouless.
00:05:13.140 --> 00:05:16.820
What you have are topological
defects that are left out.
00:05:24.360 --> 00:05:28.346
And an example of
such a defect would
00:05:28.346 --> 00:05:35.620
be a configuration of spins
that are kind of radiating out
00:05:35.620 --> 00:05:41.160
from some particular point,
such that when I complete
00:05:41.160 --> 00:05:44.920
a circuit going
around, the value
00:05:44.920 --> 00:05:55.260
of spin changes by 2 pi for
the gradient at the distance r,
00:05:55.260 --> 00:06:00.210
gradient of the change in angle
should fall off as 1 over r.
00:06:00.210 --> 00:06:06.050
So if we put one
of these defects
00:06:06.050 --> 00:06:11.680
and calculate what the partition
function is for one defect,
00:06:11.680 --> 00:06:15.080
z of one defect,
we would say, OK,
00:06:15.080 --> 00:06:20.130
what I have to do is
to calculate the energy
00:06:20.130 --> 00:06:22.520
cost of this distortion.
00:06:22.520 --> 00:06:26.830
If I use that theory that I
have over there, I have k/2.
00:06:26.830 --> 00:06:34.580
I have integral over all space
which, because of the symmetry
00:06:34.580 --> 00:06:38.330
here, I can write
as 2 pi r to the r.
00:06:41.160 --> 00:06:44.540
And then I have the
gradient squared.
00:06:44.540 --> 00:06:48.650
And as I said, the gradient
for a topological defect
00:06:48.650 --> 00:06:51.025
such as this goes as 1/r.
00:06:51.025 --> 00:06:55.540
It's square goes like
1 over r squared.
00:06:55.540 --> 00:07:01.940
So this is an integral that
is logarithmically divergent.
00:07:01.940 --> 00:07:04.320
It's a an integral of 1/r.
00:07:04.320 --> 00:07:08.610
It's logarithmically divergent
both at large distances--
00:07:08.610 --> 00:07:12.500
let's say of the order of
the size of the system--
00:07:12.500 --> 00:07:15.250
and has difficulty
at short distances.
00:07:15.250 --> 00:07:18.680
But at short distances, we
know that there is some lattice
00:07:18.680 --> 00:07:23.200
structure, and this
approximation will break down
00:07:23.200 --> 00:07:27.540
when I get to some order of some
multiple lattice spacing-- so
00:07:27.540 --> 00:07:31.780
let's call that a-- and that
the additional energy that
00:07:31.780 --> 00:07:36.260
comes from all of the
interactions that are smaller
00:07:36.260 --> 00:07:40.970
than this core value of a I
have to separately calculate,
00:07:40.970 --> 00:07:45.820
and I'm going to call
them beta epsilon that
00:07:45.820 --> 00:07:50.120
depends on the distance a
that I choose for the core.
00:07:53.330 --> 00:07:58.510
This is just a energy term that
goes into the Boltzmann factor.
00:07:58.510 --> 00:08:02.670
But this defect can
be placed any place
00:08:02.670 --> 00:08:10.070
in a lattice that has size
L up to this factor of 1/a
00:08:10.070 --> 00:08:12.300
that I call the
size of the core.
00:08:12.300 --> 00:08:17.450
So the number of places goes
like L/a squared, the area
00:08:17.450 --> 00:08:19.870
that I'm looking at.
00:08:19.870 --> 00:08:23.940
And so we see that this
expression actually
00:08:23.940 --> 00:08:26.820
goes like L/a.
00:08:26.820 --> 00:08:29.620
I have the power 2 here.
00:08:29.620 --> 00:08:31.920
And from this
logarithmic interaction,
00:08:31.920 --> 00:08:37.059
I will get here power
of 2 pi k log L/a,
00:08:37.059 --> 00:08:39.119
which I can absorb into this.
00:08:42.030 --> 00:08:46.840
And I define the exponential
of the core energy
00:08:46.840 --> 00:08:50.220
to be some parameter y.
00:08:50.220 --> 00:08:55.805
That clearly depends on the
choice of what I call my core.
00:08:59.260 --> 00:09:03.650
So if I look at just that
expression by itself,
00:09:03.650 --> 00:09:08.030
I would say that
there is the chance
00:09:08.030 --> 00:09:14.500
to see one defect or no
defects as L/a becomes
00:09:14.500 --> 00:09:17.460
large in the thermodynamic
limit depending
00:09:17.460 --> 00:09:21.790
on whether this exponent
is positive or negative.
00:09:21.790 --> 00:09:27.590
So we are kind of-- oops, I
forgot the factor of 2 here.
00:09:27.590 --> 00:09:30.960
It's 1 over pi k.
00:09:30.960 --> 00:09:35.760
We are led from this expression
that something interesting
00:09:35.760 --> 00:09:40.920
should happen at some
kc for one defect, which
00:09:40.920 --> 00:09:44.450
is let's call it the
inverse of the kc.
00:09:44.450 --> 00:09:49.090
That is like a
temperature, which is pi/2.
00:09:49.090 --> 00:09:52.870
So maybe around pi/2
here there should
00:09:52.870 --> 00:09:56.050
be something interesting
that happens.
00:09:56.050 --> 00:10:01.750
In fact, if the theory
was independent vertices,
00:10:01.750 --> 00:10:06.280
I would predict there would
be no vertices up to here,
00:10:06.280 --> 00:10:09.100
and then there would be
whole bunches vertices
00:10:09.100 --> 00:10:12.940
that would be
appearing later on.
00:10:12.940 --> 00:10:16.710
But the point is
that these are not
00:10:16.710 --> 00:10:20.000
vertices that don't
interact with each other.
00:10:20.000 --> 00:10:23.150
There are interactions
between them.
00:10:23.150 --> 00:10:30.440
And if I look at a situation
with many vertices, what
00:10:30.440 --> 00:10:36.680
I need to do is to calculate
the partition function
00:10:36.680 --> 00:10:40.100
for the defects, for
the vertices that
00:10:40.100 --> 00:10:45.700
has a resemblance to the Coulomb
system, so we call it z sub q.
00:10:45.700 --> 00:10:50.620
And this z sub q is
obtained by summing over
00:10:50.620 --> 00:10:56.650
all numbers of these defects
that can appear in the system.
00:10:56.650 --> 00:10:59.370
So let's do a sum
over n, starting
00:10:59.370 --> 00:11:02.170
from 0 to however many.
00:11:02.170 --> 00:11:04.102
And if I have a
situation in which there
00:11:04.102 --> 00:11:08.030
are n vertices in
the system, I clearly
00:11:08.030 --> 00:11:15.290
have to pay a cost of y per
core of each one of them.
00:11:15.290 --> 00:11:20.060
So there's a factor of y
raised to the power of n.
00:11:20.060 --> 00:11:24.040
These vertices then can be
placed anywhere on the lattice,
00:11:24.040 --> 00:11:27.660
in the same way that I
had this factor of L/a
00:11:27.660 --> 00:11:29.880
for a single vertex.
00:11:29.880 --> 00:11:36.860
I will have ability to put
each one of these vertices
00:11:36.860 --> 00:11:38.150
at some point on the lattice.
00:11:41.080 --> 00:11:47.070
And then we found that when
you do the calculation,
00:11:47.070 --> 00:11:50.870
the distortions that are caused
by the independent vertices
00:11:50.870 --> 00:11:53.520
clearly add up on
top of each other.
00:11:53.520 --> 00:11:56.260
And when we add it
up and superimpose
00:11:56.260 --> 00:11:58.460
the gradient of
thetas that correspond
00:11:58.460 --> 00:12:02.190
to the different
vertices and calculated
00:12:02.190 --> 00:12:05.800
the integral of gradient of
theta squared, what we found
00:12:05.800 --> 00:12:09.530
was that there was an
interaction between them
00:12:09.530 --> 00:12:14.800
that I could write
as 4 pi squared
00:12:14.800 --> 00:12:23.760
k sum over distinct pairs,
i less than j, qi qj.
00:12:23.760 --> 00:12:27.150
The Coulomb interaction
[? it fits inside ?] xi and xj.
00:12:31.820 --> 00:12:38.490
And actually, the form of
this came about as follows.
00:12:38.490 --> 00:12:45.670
Those That basically the charges
of these topological defects
00:12:45.670 --> 00:12:50.530
are multiples of 2 pi, but
these qi's I have written
00:12:50.530 --> 00:12:52.790
are minus plus 1.
00:12:52.790 --> 00:12:56.480
So the two pis are
absorbed in the charges.
00:12:56.480 --> 00:13:02.310
The actual charges being 2 pi
is what makes this 4 pi squared.
00:13:02.310 --> 00:13:05.190
k is the [INAUDIBLE]
of interaction.
00:13:05.190 --> 00:13:08.320
Furthermore, we have
to require the system
00:13:08.320 --> 00:13:15.910
to be overall neutral
because, otherwise, there
00:13:15.910 --> 00:13:18.740
would be a large
energy for creating
00:13:18.740 --> 00:13:22.860
the monopole in a large system.
00:13:22.860 --> 00:13:26.590
And again, just as a
matter of notation,
00:13:26.590 --> 00:13:32.380
our z of x has a
1 over 2 pi itself
00:13:32.380 --> 00:13:37.330
log of the displacement
in units of this a
00:13:37.330 --> 00:13:39.920
because we can't allow
these things to come
00:13:39.920 --> 00:13:41.340
very close to each other.
00:13:44.070 --> 00:13:53.650
So our task was to
calculate properties encoded
00:13:53.650 --> 00:13:58.890
in this partition function,
which is, in some sense,
00:13:58.890 --> 00:14:01.965
a grand canonical
system of charges
00:14:01.965 --> 00:14:04.570
that can appear and disappear.
00:14:04.570 --> 00:14:11.620
And our expectation is that
at the low temperatures,
00:14:11.620 --> 00:14:19.930
essentially all I have are a few
dipoles that are kind of small.
00:14:19.930 --> 00:14:23.600
As I go to higher
temperature, the two monopoles
00:14:23.600 --> 00:14:28.710
making the dipole can fluctuate
and go further from each other.
00:14:28.710 --> 00:14:32.670
And eventually at some
point, they will be all mixed
00:14:32.670 --> 00:14:34.810
up together and
the picture should
00:14:34.810 --> 00:14:38.617
be regarded as a mixture
of plus and minus charges
00:14:38.617 --> 00:14:39.200
in the plasma.
00:14:39.200 --> 00:14:40.156
Yes?
00:14:40.156 --> 00:14:43.396
AUDIENCE: [INAUDIBLE] if
we have an external field,
00:14:43.396 --> 00:14:48.760
would this also be
fixed [INAUDIBLE], like
00:14:48.760 --> 00:14:51.230
and edge or something external?
00:14:56.190 --> 00:14:58.130
PROFESSOR: It is
very hard to imagine
00:14:58.130 --> 00:15:01.930
what that external field has
to be in the language of the xy
00:15:01.930 --> 00:15:04.370
model, because
what you can do is
00:15:04.370 --> 00:15:06.360
you can put a field
that, let's say,
00:15:06.360 --> 00:15:11.410
rotates the spins on one side--
let's say to point down--
00:15:11.410 --> 00:15:14.260
spins on the other
side to point up.
00:15:14.260 --> 00:15:17.080
But then what happens
is that the angles
00:15:17.080 --> 00:15:20.750
would adjust themselves
so that at 0 temperature,
00:15:20.750 --> 00:15:22.580
you would have a
configuration that
00:15:22.580 --> 00:15:27.510
would go from plus to minus, and
all topological charges would
00:15:27.510 --> 00:15:31.170
be on top of that
base configuration.
00:15:31.170 --> 00:15:35.830
So that kind of field certainly
does not have any effect
00:15:35.830 --> 00:15:38.420
that I can ascribe over here.
00:15:38.420 --> 00:15:40.035
If I change my
picture completely
00:15:40.035 --> 00:15:42.860
and say forget
about the xy model,
00:15:42.860 --> 00:15:46.270
think about this as a
system of point charges,
00:15:46.270 --> 00:15:49.290
then I can certainly,
like I did last time,
00:15:49.290 --> 00:15:53.682
put an electric field on the
system and see what happens.
00:15:53.682 --> 00:15:54.650
Yes?
00:15:54.650 --> 00:15:56.206
AUDIENCE: [INAUDIBLE]
saying that you
00:15:56.206 --> 00:15:58.550
can create even
bigger defects, where
00:15:58.550 --> 00:16:00.410
q would be [INAUDIBLE]
plus minus 1,
00:16:00.410 --> 00:16:02.120
plus minus bigger integer?
00:16:02.120 --> 00:16:05.166
But that's [? discounted ?]
as high [INAUDIBLE] effect.
00:16:05.166 --> 00:16:05.790
PROFESSOR: Yes.
00:16:05.790 --> 00:16:08.710
So intrinsically, we
could go beyond that.
00:16:08.710 --> 00:16:12.930
We would have a fugacity
for a creation of cores
00:16:12.930 --> 00:16:15.560
of singular charge,
another for cores
00:16:15.560 --> 00:16:17.770
of double charge, et cetera.
00:16:17.770 --> 00:16:20.710
You expect those y's
or double charges
00:16:20.710 --> 00:16:23.475
to be much larger because
the configuration is going
00:16:23.475 --> 00:16:25.015
to be more difficult
at the core.
00:16:27.870 --> 00:16:30.230
And in some sense,
you can imagine
00:16:30.230 --> 00:16:33.550
that we are including
something similar to that
00:16:33.550 --> 00:16:37.700
because we can create
two single charges that
00:16:37.700 --> 00:16:40.820
are closing off to each other.
00:16:40.820 --> 00:16:41.320
Yes?
00:16:41.320 --> 00:16:44.470
AUDIENCE: So why is a
Coulomb state [INAUDIBLE]?
00:16:47.490 --> 00:16:48.308
Is that high order?
00:16:51.000 --> 00:16:52.800
PROFESSOR: Well,
you would expect
00:16:52.800 --> 00:16:56.800
that if y is a
small parameter, you
00:16:56.800 --> 00:17:00.120
would like to create as
few things as possible.
00:17:00.120 --> 00:17:05.310
The reason you create any is
because you have entropy gain.
00:17:05.310 --> 00:17:07.800
So I would say
energetically, even
00:17:07.800 --> 00:17:10.329
creating a pair is unfavorable.
00:17:10.329 --> 00:17:13.180
But the pair has lots of
places that it can go,
00:17:13.180 --> 00:17:15.230
so because of the
gain in entropy,
00:17:15.230 --> 00:17:17.109
it is willing to accept that.
00:17:17.109 --> 00:17:19.400
If I create a
quadrupole, you say,
00:17:19.400 --> 00:17:21.940
well, I break the
quadrupole into two dipoles
00:17:21.940 --> 00:17:23.819
and then I have
much more entropy.
00:17:23.819 --> 00:17:28.119
So that's why it's not--
we can have that term,
00:17:28.119 --> 00:17:30.772
but it is going to
have much less weight.
00:17:41.220 --> 00:17:45.310
So I won't repeat
the calculation,
00:17:45.310 --> 00:17:50.980
but last time we indeed asked
what happens if we have,
00:17:50.980 --> 00:17:55.500
let's say, some kind
of an electric field.
00:17:55.500 --> 00:17:58.240
And because of the presence
of the electric field,
00:17:58.240 --> 00:18:02.280
dipoles are going to be aligned.
00:18:02.280 --> 00:18:08.560
And the effect of that is to
reduce the effective strength
00:18:08.560 --> 00:18:11.550
of all kinds of
Coulomb interactions.
00:18:11.550 --> 00:18:18.190
We found that the effective
strength was reduced by from k
00:18:18.190 --> 00:18:25.150
by an amount that was related
to the likelihood of creating
00:18:25.150 --> 00:18:28.430
the dipole of size r.
00:18:28.430 --> 00:18:32.660
And that was clearly
proportional to y squared
00:18:32.660 --> 00:18:44.710
into the minus from there,
4 pi squared q log of r/a.
00:18:44.710 --> 00:18:48.553
That's the co-ability we
create a dipole of this size.
00:18:51.480 --> 00:18:57.580
And then I have
to, in principle,
00:18:57.580 --> 00:19:00.260
integrate over all dipole sizes.
00:19:03.470 --> 00:19:08.160
But this writing this as an
orientationally independent
00:19:08.160 --> 00:19:11.730
result is not correct
because in the presence
00:19:11.730 --> 00:19:14.250
of an electric field,
you have more likelihood
00:19:14.250 --> 00:19:16.760
to be oriented in one
direction as opposed
00:19:16.760 --> 00:19:18.450
to the other direction.
00:19:18.450 --> 00:19:21.340
So this factor of e
to the cosine theta,
00:19:21.340 --> 00:19:24.940
et cetera, that we
expanded, first of all,
00:19:24.940 --> 00:19:27.820
gave us an average of
cosine of theta squared.
00:19:27.820 --> 00:19:30.300
So there was a
factor of 1/2 here.
00:19:30.300 --> 00:19:33.660
Rather than full rotation,
I was doing an average
00:19:33.660 --> 00:19:38.320
of cosine squared
to be that factor.
00:19:38.320 --> 00:19:42.910
Expanding that actually
gave me a factor of 4 pi
00:19:42.910 --> 00:19:48.220
squared k because of the Coulomb
term that I had up there.
00:19:48.220 --> 00:19:54.320
And then I had essentially
the polarizability of one
00:19:54.320 --> 00:19:59.410
of these objects that
goes like r squared.
00:19:59.410 --> 00:20:03.320
Again, coming from
expanding this factor
00:20:03.320 --> 00:20:05.800
that we had in the exponents.
00:20:05.800 --> 00:20:12.006
Actually, I calculated
everything in units of A,
00:20:12.006 --> 00:20:14.250
so I should really do this.
00:20:14.250 --> 00:20:17.960
And this was correct
to order of y squared.
00:20:17.960 --> 00:20:20.820
And in principle,
one can imagine
00:20:20.820 --> 00:20:24.120
that there are configurations
of four charges, quadrupole,
00:20:24.120 --> 00:20:28.730
like things, et cetera,
that further modify this.
00:20:28.730 --> 00:20:33.830
And that this is a result
that I have to-- oops,
00:20:33.830 --> 00:20:37.120
this was 1 minus.
00:20:37.120 --> 00:20:38.990
It was an overall factor of k.
00:20:38.990 --> 00:20:42.740
This was the correction
term that we calculated.
00:20:42.740 --> 00:20:45.990
And then the size
of these dipoles,
00:20:45.990 --> 00:20:50.430
we have to integrate from
A to the size of the system
00:20:50.430 --> 00:20:52.340
or, if you like, infinity.
00:20:52.340 --> 00:20:58.100
And although we were attempting
to make an expansion in powers
00:20:58.100 --> 00:21:02.875
of y, what we see is that
because this is giving me
00:21:02.875 --> 00:21:07.760
a factor of k/r, the
r has to be integrated
00:21:07.760 --> 00:21:11.000
against these three
factors of rdr.
00:21:11.000 --> 00:21:15.770
Whether or not this integral is
dominated by its upper cut-off,
00:21:15.770 --> 00:21:19.350
and hence divergent,
depends on value of k
00:21:19.350 --> 00:21:22.560
that is related to
the same divergence
00:21:22.560 --> 00:21:25.010
that we have for
a single vertex.
00:21:25.010 --> 00:21:31.370
So this perturbation theory
is, in principle, not valid
00:21:31.370 --> 00:21:35.580
no matter how much y I
try to make small as one
00:21:35.580 --> 00:21:40.710
long as my k inverse
is greater than pi/2.
00:21:40.710 --> 00:21:47.130
So what we decided to do was not
to do this entire integration
00:21:47.130 --> 00:21:52.150
that gives us infinity,
but rather to recast this
00:21:52.150 --> 00:22:02.160
as a re-normalization
group in which core size is
00:22:02.160 --> 00:22:08.044
changed from a to beta.
00:22:13.890 --> 00:22:17.310
Now one way to see the
effect of it-- last time,
00:22:17.310 --> 00:22:19.400
I did this slightly
differently--
00:22:19.400 --> 00:22:26.240
is to ensure that the result for
the partition function of one
00:22:26.240 --> 00:22:29.430
charge is unmodified.
00:22:29.430 --> 00:22:32.590
If I simply do this
change, the weight
00:22:32.590 --> 00:22:34.820
should not change
for one defect.
00:22:34.820 --> 00:22:37.060
And so clearly, you
can see that there's
00:22:37.060 --> 00:22:39.090
a change in power
of beta that I would
00:22:39.090 --> 00:22:45.200
that I need to
compensate by changing
00:22:45.200 --> 00:22:52.012
the core energy by a factor of
b to the power of 2 minus pi k.
00:22:56.930 --> 00:23:01.310
So the statement that I have
for z 1, in order for z 1
00:23:01.310 --> 00:23:06.010
to be left invariant, I have
to rescale the core energies
00:23:06.010 --> 00:23:08.360
by this factor.
00:23:08.360 --> 00:23:13.270
And then over
here, I essentially
00:23:13.270 --> 00:23:21.480
just integrate up
to a factor of da.
00:23:21.480 --> 00:23:24.930
Just get rid of
those interactions.
00:23:24.930 --> 00:23:28.990
And so this becomes
minus 4 pi q k
00:23:28.990 --> 00:23:34.130
squared y squared--
as we're looking
00:23:34.130 --> 00:23:43.455
at dipole contribution--
integral from a to ba of dr
00:23:43.455 --> 00:23:47.820
r cubed divided a to the fourth.
00:23:47.820 --> 00:23:54.060
a/r to the power of 2 pi k.
00:23:59.428 --> 00:24:03.243
It means that I probably
made a mistake somewhere.
00:24:08.180 --> 00:24:11.165
Yeah, this has a 2 pi.
00:24:11.165 --> 00:24:16.076
I forgot the 2 pi from
the definition of the log.
00:24:25.870 --> 00:24:30.070
So these are the
recursion relations.
00:24:30.070 --> 00:24:38.290
So basically, this same results
at large scale for the Coulomb
00:24:38.290 --> 00:24:41.600
gas can be obtained
either by the theory
00:24:41.600 --> 00:24:46.620
that is parametrized
by y and original k,
00:24:46.620 --> 00:24:51.110
or after going through this
removal of short distance
00:24:51.110 --> 00:24:54.420
degrees of freedom by theory
in which y is modified
00:24:54.420 --> 00:24:56.946
by this factor and k is
modified by this factor.
00:25:00.520 --> 00:25:07.110
And as usual, we can change
these recursion relations
00:25:07.110 --> 00:25:13.530
into flow equations by
choosing the value of b
00:25:13.530 --> 00:25:18.190
that is very close to 1, and
then essentially converting
00:25:18.190 --> 00:25:26.990
these things to y evaluated
at a slightly larger than 1,
00:25:26.990 --> 00:25:30.146
and from that,
constructing the y by dl.
00:25:32.920 --> 00:25:40.730
And y by dl simply becomes
2 minus pi k times y.
00:25:40.730 --> 00:25:42.860
And I can do the
same thing here.
00:25:42.860 --> 00:25:45.912
This is k plus dk dl.
00:25:45.912 --> 00:25:48.690
The k part cancels.
00:25:48.690 --> 00:25:54.120
And what I will get
is that dk by dl
00:25:54.120 --> 00:26:01.430
is minus 4 pi cubed
k squared y squared.
00:26:01.430 --> 00:26:06.250
And actually, all I need to do
is evaluate this on the shell
00:26:06.250 --> 00:26:08.470
where r equals to
a, and you can see
00:26:08.470 --> 00:26:11.920
that the integral
essentially gives me 1.
00:26:11.920 --> 00:26:16.090
It gives you a delta l,
basically goes over here.
00:26:16.090 --> 00:26:17.950
So these are order of y squared.
00:26:23.130 --> 00:26:30.820
Actually, it is kind of
better to cast results
00:26:30.820 --> 00:26:35.340
rather in terms of k, in
terms of k inverse, which
00:26:35.340 --> 00:26:39.270
is kind of like a
temperature variable.
00:26:39.270 --> 00:26:45.990
And then what we get is
that d by dl of k inverse
00:26:45.990 --> 00:26:48.790
essentially is
going to be minus 1
00:26:48.790 --> 00:26:52.330
over k squared by
dl of k squared.
00:26:52.330 --> 00:26:55.770
So the minus k squared
cancels, and it simply
00:26:55.770 --> 00:27:04.760
becomes 4 pi q y squared,
order of y to the fourth.
00:27:04.760 --> 00:27:13.491
And divide by dl, it's actually
2 minus pi k y plus O of y.
00:27:21.290 --> 00:27:26.890
So these are the equations
that describe the changing
00:27:26.890 --> 00:27:32.150
parameters under rescaling
for this Coulomb gas.
00:27:32.150 --> 00:27:34.490
And so we can plot them.
00:27:34.490 --> 00:27:41.440
Essentially, we have
two paramters-- y,
00:27:41.440 --> 00:27:46.170
and we have k inverse.
00:27:46.170 --> 00:27:53.650
And what we see
is that k inverse,
00:27:53.650 --> 00:27:57.790
its change is always positive.
00:27:57.790 --> 00:28:00.740
So the flow should
always be to the right.
00:28:03.950 --> 00:28:10.100
y, whether y decreases
or decreases depends on
00:28:10.100 --> 00:28:16.970
whether I am above or below
this critical value of 2/pi
00:28:16.970 --> 00:28:20.110
that we keep encountering.
00:28:20.110 --> 00:28:23.552
And in particular,
what we will find
00:28:23.552 --> 00:28:31.920
is that there is a trajectory
that goes into this point.
00:28:31.920 --> 00:28:37.410
And if you are to the
left of that trajectory,
00:28:37.410 --> 00:28:44.140
y is getting smaller, k
inverse is getting larger.
00:28:44.140 --> 00:28:46.230
And so you go like this.
00:28:46.230 --> 00:28:52.855
Eventually, you land on
a point down here where
00:28:52.855 --> 00:28:54.360
y has gone to 0.
00:28:54.360 --> 00:28:58.370
And if y has gone to 0, then
k inverse does not change.
00:28:58.370 --> 00:29:06.150
So you have a structure where
you have a line of fixed points
00:29:06.150 --> 00:29:12.120
so any point over
here is a fixed point,
00:29:12.120 --> 00:29:14.210
but it is also a
stable fixed point.
00:29:14.210 --> 00:29:18.275
It is true that points
that are over here,
00:29:18.275 --> 00:29:23.170
if you are exactly at y
equals to 0 are fixed points.
00:29:23.170 --> 00:29:26.330
But as soon as you
have a little bit of y,
00:29:26.330 --> 00:29:28.054
then they start flowing away.
00:29:31.930 --> 00:29:35.536
And essentially, the
general pattern of flow
00:29:35.536 --> 00:29:36.880
is something like this.
00:29:43.400 --> 00:29:51.170
So I go back to my
original xy model,
00:29:51.170 --> 00:29:55.530
and I'm at some value at low
temperatures-- means that I'm
00:29:55.530 --> 00:29:57.790
down here-- but
presumably, there's
00:29:57.790 --> 00:30:04.040
a finite cost for creating the
core, so I may be over here.
00:30:04.040 --> 00:30:08.120
And when I go to slightly
higher temperature,
00:30:08.120 --> 00:30:10.920
k inverse becomes larger.
00:30:10.920 --> 00:30:14.170
But the core energy typically
becomes smaller also
00:30:14.170 --> 00:30:16.180
at lower temperatures
because everything
00:30:16.180 --> 00:30:19.160
is scaled by scale by 1/kt.
00:30:19.160 --> 00:30:21.850
So as I go to higher
and higher temperatures,
00:30:21.850 --> 00:30:28.960
my xy model presumably goes
through some trajectory.
00:30:28.960 --> 00:30:32.780
The trajectory of changing
the xy model as temperature
00:30:32.780 --> 00:30:36.460
is modified has nothing
to do with [? rg ?].
00:30:36.460 --> 00:30:48.180
So basically is xy
model on increasing T.
00:30:48.180 --> 00:30:50.770
And what is happening in
the xy model of increasing
00:30:50.770 --> 00:30:55.900
T is that at low temperatures,
I'm at some point
00:30:55.900 --> 00:30:59.510
here, which if I look at
larger and larger scales,
00:30:59.510 --> 00:31:03.320
I find that eventually
I go to a place
00:31:03.320 --> 00:31:07.940
where the effective whole
cost for creating vertices
00:31:07.940 --> 00:31:11.960
is so large that they
are not created at all.
00:31:11.960 --> 00:31:16.950
So then I'm back to that theory
that has no vertices and simply
00:31:16.950 --> 00:31:20.770
gradient squared, and I
expect that correlations
00:31:20.770 --> 00:31:25.630
will be given by this
power law type of form.
00:31:25.630 --> 00:31:30.180
However, at some point,
I am in this region.
00:31:30.180 --> 00:31:34.606
And when I'm in this region, I
find that maybe even initially
00:31:34.606 --> 00:31:40.190
the core energy goes
down or y goes down.
00:31:40.190 --> 00:31:44.580
But eventually, I end up
going to a regime where
00:31:44.580 --> 00:31:49.150
both y is large and effective
temperature-- the inverse-- are
00:31:49.150 --> 00:31:49.980
large.
00:31:49.980 --> 00:31:55.940
So essentially anywhere here
eventually at large scales,
00:31:55.940 --> 00:32:00.490
I will see that I will be
creating vertices pretty much
00:32:00.490 --> 00:32:04.910
at ease and at sufficiently
long, large scale.
00:32:04.910 --> 00:32:07.235
My picture should
be that of a plasma
00:32:07.235 --> 00:32:09.200
in which the [? plas ?]
plus and minus
00:32:09.200 --> 00:32:12.440
charges are moving around.
00:32:12.440 --> 00:32:15.510
And so then there is
this transition line
00:32:15.510 --> 00:32:17.150
that separates the two regimes.
00:32:19.800 --> 00:32:23.300
So let's find the
behavior of that.
00:32:23.300 --> 00:32:26.650
And clearly, what
I need to do is
00:32:26.650 --> 00:32:31.180
to focus in the vicinity
of this fixed point that
00:32:31.180 --> 00:32:33.602
controls the transition.
00:32:33.602 --> 00:32:36.610
That is, anything that
undergoes the transition
00:32:36.610 --> 00:32:41.810
eventually comes and flows to
the vicinity of this point.
00:32:41.810 --> 00:32:47.550
So what we can do is we
can construct, if you like,
00:32:47.550 --> 00:32:51.910
a two dimensional
blow-up of that.
00:32:51.910 --> 00:32:55.780
And what I'm going
to do is to introduce
00:32:55.780 --> 00:33:04.810
a variable of x, which
is k inverse minus 2/pi.
00:33:04.810 --> 00:33:09.430
Essentially, how far I have gone
from this in this direction--
00:33:09.430 --> 00:33:12.200
y, I can use as y [INAUDIBLE].
00:33:12.200 --> 00:33:18.770
And so what we see
is that my k inverse
00:33:18.770 --> 00:33:23.386
is 2/pi, my critical value.
00:33:23.386 --> 00:33:25.052
AUDIENCE: I think
that should be a pi/2.
00:33:28.370 --> 00:33:31.415
PROFESSOR: k inverse
this is pi/2.
00:33:31.415 --> 00:33:32.259
Thank you.
00:33:38.746 --> 00:33:44.894
Which means that
this has to be pi/2.
00:33:44.894 --> 00:33:46.040
This has to be pi/2.
00:33:48.930 --> 00:33:54.195
And this I can write
as pi/2, 1 plus 2x/pi.
00:33:58.000 --> 00:34:04.750
So that to lowest
order in x, k is 2/pi.
00:34:04.750 --> 00:34:07.511
The inverse of
this factor, which
00:34:07.511 --> 00:34:13.290
is 1 minus 2x/pi plus
order of x squared.
00:34:13.290 --> 00:34:16.350
I'm expanding for small x.
00:34:16.350 --> 00:34:22.940
I put that value in here
and I find that my dy by dl
00:34:22.940 --> 00:34:30.170
is now 2 minus pi times
what I have over there.
00:34:30.170 --> 00:34:40.845
So it is pi times 2/pi-- so
it becomes 2-- plus 4/pi x.
00:34:40.845 --> 00:34:43.730
So essentially, I have
minus 4/pi squared.
00:34:43.730 --> 00:34:45.355
I multiple by pi.
00:34:45.355 --> 00:34:48.366
It becomes plus 4/pi x.
00:34:48.366 --> 00:34:50.630
Multiply by y.
00:34:50.630 --> 00:34:56.480
So this is simply a 4/pi xy.
00:34:56.480 --> 00:35:00.040
Now the point is
that typically we
00:35:00.040 --> 00:35:05.000
are used to expanding
in the vicinity
00:35:05.000 --> 00:35:08.160
of that important fixed point.
00:35:08.160 --> 00:35:10.750
And all the cases that
we had seen so far,
00:35:10.750 --> 00:35:14.210
once we did that
expression, we ended up
00:35:14.210 --> 00:35:19.510
with a linear behavior-- divide
by dl plus something times y.
00:35:19.510 --> 00:35:23.040
Here we see that the
vicinity of this point
00:35:23.040 --> 00:35:27.350
is clearly a quadratic
type of behavior.
00:35:27.350 --> 00:35:30.810
And this quadratic
behavior leads
00:35:30.810 --> 00:35:34.240
to some unusual and
interesting critical behavior
00:35:34.240 --> 00:35:37.578
that we are going to explore.
00:35:37.578 --> 00:35:40.790
So let's stick with this
a little bit longer.
00:35:40.790 --> 00:35:45.666
We can see that if I look
at d by dl of y squared,
00:35:45.666 --> 00:35:49.700
it is going to be
2y divided by dl,
00:35:49.700 --> 00:35:52.720
so I have to
multiply this by 2y,
00:35:52.720 --> 00:35:56.620
so I will get 8/pi
pi xy squared.
00:35:59.510 --> 00:36:01.290
Why did I do that?
00:36:01.290 --> 00:36:12.570
It's because you can recognize
his xy squared shortly.
00:36:15.580 --> 00:36:19.278
Let's go and do the x by dl.
00:36:23.400 --> 00:36:29.440
The x by dl is simply
dk inverse by dl,
00:36:29.440 --> 00:36:35.170
so that is 4 pi cubed y squared.
00:36:35.170 --> 00:36:42.150
And now you can see that if
I do d by dl of x squared,
00:36:42.150 --> 00:36:51.460
I will have 2x dx by dl, so
I will have 8 pi cubed xy
00:36:51.460 --> 00:36:52.040
squared.
00:36:52.040 --> 00:36:58.220
So now we can recognize
that these two quantities
00:36:58.220 --> 00:37:01.340
up to some factor
of pi to the fourth
00:37:01.340 --> 00:37:03.960
are really the same thing.
00:37:03.960 --> 00:37:10.540
So from here, we
conclude that d by dl
00:37:10.540 --> 00:37:21.070
of x squared minus pi
to the fourth y squared.
00:37:21.070 --> 00:37:24.330
Essentially once I do
that, I will get 0.
00:37:31.750 --> 00:37:37.120
So as I go along
these trajectories,
00:37:37.120 --> 00:37:41.890
x and y are changing,
but the combination
00:37:41.890 --> 00:37:46.900
x squared minus pi to the fourth
y squared is not changing.
00:37:46.900 --> 00:37:50.030
So all of the trajectories
that I have drawn-- at least
00:37:50.030 --> 00:37:55.690
sufficiently close at this point
around which I am expanding--
00:37:55.690 --> 00:37:58.340
correspond to lines
that are x squared
00:37:58.340 --> 00:38:03.570
minus pi to the fourth y squared
is some constant I'll call c.
00:38:08.190 --> 00:38:11.640
And that constant must meet
whatever you started with.
00:38:11.640 --> 00:38:15.460
So if I call the trajectory
here to be the combination
00:38:15.460 --> 00:38:20.150
x0 by x0-- your
original values--
00:38:20.150 --> 00:38:24.270
I can figure out what my
x0 to the fourth minus pi
00:38:24.270 --> 00:38:27.030
the fourth x0 to the fourth is.
00:38:27.030 --> 00:38:30.189
And that's going to
be staying constant
00:38:30.189 --> 00:38:31.355
along the entire trajectory.
00:38:33.970 --> 00:38:41.980
So these trajectories are, in
fact, portions of a hyperbole.
00:38:41.980 --> 00:38:44.420
And this is the
equation that you
00:38:44.420 --> 00:38:46.610
would have for a
hyperbola in xy.
00:38:49.260 --> 00:38:53.340
Now clearly there are
two types of hyperbole--
00:38:53.340 --> 00:38:58.520
the ones that go like this and
the ones that go like that.
00:38:58.520 --> 00:39:02.880
In fact, this one and this one
are pretty much the same thing.
00:39:02.880 --> 00:39:09.400
And what distinguishes this
pattern versus that pattern
00:39:09.400 --> 00:39:14.080
is whether this constant
c is positive or negative
00:39:14.080 --> 00:39:18.960
because you can
see that out here,
00:39:18.960 --> 00:39:24.910
ultimately you end up at the
point where y has gone to 0.
00:39:24.910 --> 00:39:28.880
So depending on x positive or
negative, it doesn't matter.
00:39:28.880 --> 00:39:31.680
This combination
will be positive.
00:39:31.680 --> 00:39:36.277
So throughout here, what I
have is that c is positive.
00:39:39.140 --> 00:39:44.060
Whereas what I have up
here is c that is negative.
00:39:44.060 --> 00:39:46.335
Presumably, there is
other trajectory here
00:39:46.335 --> 00:39:48.520
and down here, c
is again positive.
00:39:54.170 --> 00:39:58.660
And again, if you want to ensure
y over here c is negative,
00:39:58.660 --> 00:40:03.550
because over here you can see
you crossed the line where
00:40:03.550 --> 00:40:06.110
x is 0, but you have
some value of y.
00:40:13.890 --> 00:40:21.935
So if I were to blow up that
region both as a function of x
00:40:21.935 --> 00:40:27.030
and y, well, first
of all, I will
00:40:27.030 --> 00:40:31.320
have a particular set of
trajectories-- the ones
00:40:31.320 --> 00:40:36.350
that end up at this
important fix point, which
00:40:36.350 --> 00:40:40.340
correspond clearly
to c cos to 0.
00:40:40.340 --> 00:40:44.710
So that c cos to 0 will
give me two straight lines.
00:40:44.710 --> 00:40:50.030
So presumably there
is this straight line,
00:40:50.030 --> 00:40:56.670
and then there is another
straight line goes out there.
00:40:56.670 --> 00:41:01.320
And then I have this
bunch of trajectories
00:41:01.320 --> 00:41:05.920
that are these hyperbole
that end up over here.
00:41:05.920 --> 00:41:10.440
I can have hyperboles
that will be going out.
00:41:10.440 --> 00:41:14.540
And then I will have
hyperboles that are like this.
00:41:17.240 --> 00:41:19.230
This are all in the
high temperature phase,
00:41:19.230 --> 00:41:23.038
so let's [INAUDIBLE] like this.
00:41:28.290 --> 00:41:32.500
So one thing that
you immediately see
00:41:32.500 --> 00:41:36.540
is that the location
of the transition that
00:41:36.540 --> 00:41:43.320
is given by this critical
line when c equals to 0.
00:41:43.320 --> 00:41:47.465
So statement number
one that we can get
00:41:47.465 --> 00:41:55.900
is that the transition line
corresponds to c equals to 0.
00:41:55.900 --> 00:41:59.790
So solving for x
as a function of y,
00:41:59.790 --> 00:42:05.580
I will get that x critical is
either minus or plus pi squared
00:42:05.580 --> 00:42:06.750
y.
00:42:06.750 --> 00:42:11.060
Clearly from the figure,
the solution that I want
00:42:11.060 --> 00:42:13.488
is the one that
corresponds to minus.
00:42:16.260 --> 00:42:18.820
My x was k inverse minus pi/2.
00:42:21.390 --> 00:42:25.920
This is kc inverse minus pi/2.
00:42:25.920 --> 00:42:31.130
And so what I see
is that kc inverse--
00:42:31.130 --> 00:42:35.360
the correct transition
temperature-- is,
00:42:35.360 --> 00:42:42.630
in fact, lower than the value
of pi/2 that we had deduced,
00:42:42.630 --> 00:42:45.500
assuming that there is
only a single vertex
00:42:45.500 --> 00:42:52.290
in the entire system by an
amount that to lowest order
00:42:52.290 --> 00:42:56.860
is related to the core
energy or core fugacity.
00:42:56.860 --> 00:42:59.680
And presumably, there
are higher order terms
00:42:59.680 --> 00:43:03.420
that I haven't calculated.
00:43:03.420 --> 00:43:09.430
So this number that
we have calculated
00:43:09.430 --> 00:43:11.510
by looking at a
single defect, we
00:43:11.510 --> 00:43:14.140
can see that in the presence
of multiple defects,
00:43:14.140 --> 00:43:17.680
starts to get lower.
00:43:17.680 --> 00:43:21.350
And this is precisely
correct in the limit
00:43:21.350 --> 00:43:23.644
where y is a small quantity.
00:43:28.860 --> 00:43:32.320
Now that's a transition line.
00:43:32.320 --> 00:43:35.120
We can look to the
left or to the right.
00:43:35.120 --> 00:43:37.395
Let us just look at the
low temperature phase.
00:43:44.470 --> 00:43:48.820
So for the low T phase, we
expect c to be negative.
00:43:53.360 --> 00:44:02.850
And I can, for example, make
that explicit by writing
00:44:02.850 --> 00:44:11.950
it as-- OK, so c was x0 squared
minus pi to the fourth y0
00:44:11.950 --> 00:44:14.620
squared-- what the starting
parameters of the system
00:44:14.620 --> 00:44:16.180
dictate.
00:44:16.180 --> 00:44:20.585
If you are under low temperature
phase such that c is negative,
00:44:20.585 --> 00:44:24.040
it means that you
are at temperatures
00:44:24.040 --> 00:44:27.350
that are smaller than Tc.
00:44:27.350 --> 00:44:31.840
So let's see, T minus
Tc, let's write it
00:44:31.840 --> 00:44:36.570
as Tc minus T--
would be positive.
00:44:36.570 --> 00:44:39.150
But this has to be
negative, so let me just
00:44:39.150 --> 00:44:41.340
introduce some parameter b.
00:44:41.340 --> 00:44:43.010
It's not the same b as here.
00:44:43.010 --> 00:44:47.030
Just some coefficient
that has to be squared.
00:44:47.030 --> 00:44:54.420
So I know that as I hit
Tc, this c goes to 0.
00:44:54.420 --> 00:44:58.820
If I'm slightly away from
Tc along the trajectory
00:44:58.820 --> 00:45:04.290
that I have indicated over
here, right here I'm 0,
00:45:04.290 --> 00:45:07.200
so right I'm slightly negative.
00:45:07.200 --> 00:45:11.230
And there's no reason why
the value that I calculated
00:45:11.230 --> 00:45:15.330
from x0 squared minus pi
to the fourth y0 squared
00:45:15.330 --> 00:45:18.010
should not be an
analytical function.
00:45:18.010 --> 00:45:21.130
So I have expanded that
analytical function,
00:45:21.130 --> 00:45:24.640
knowing that that
Tc is equal to 0.
00:45:24.640 --> 00:45:26.680
There will be higher
order terms for sure,
00:45:26.680 --> 00:45:28.910
but this is the
lowest order term
00:45:28.910 --> 00:45:30.430
that I would have
in that expansion.
00:45:33.100 --> 00:45:37.030
And as we said,
this is preserved
00:45:37.030 --> 00:45:40.460
all along the trajectory
that ends on this point.
00:45:40.460 --> 00:45:43.390
So along that trajectory,
this is the same
00:45:43.390 --> 00:45:49.660
as x squared minus pi
to the fourth y squared,
00:45:49.660 --> 00:45:58.330
which means that I can write
y squared to be 1 over pi
00:45:58.330 --> 00:46:01.130
to the fourth.
00:46:01.130 --> 00:46:10.260
x squared plus b
squared Tc minus T. So
00:46:10.260 --> 00:46:14.350
if I want to solve
for this curve, that's
00:46:14.350 --> 00:46:18.520
what I will have for some
value of this quantity.
00:46:18.520 --> 00:46:24.940
And what I do is I look at what
that implies for the xy dl.
00:46:24.940 --> 00:46:31.700
The xy dl is 4 pi
cubed y squared.
00:46:31.700 --> 00:46:35.456
I substitute the y squared
that I have over there.
00:46:35.456 --> 00:46:47.760
I will get 4/pi times x squared
plus b squared Tc minus T.
00:46:47.760 --> 00:46:53.920
So under rescaling, this tells
me what is happening to x.
00:46:53.920 --> 00:46:59.280
And in particular, what I can do
is to integrate this equation.
00:46:59.280 --> 00:47:06.730
I have dx divided by x squared
plus b squared Tc minus T is
00:47:06.730 --> 00:47:12.950
4/pi dl-- just rearranging
this differential equation.
00:47:12.950 --> 00:47:18.890
And this I can certainly
integrate out to l.
00:47:18.890 --> 00:47:23.810
This you should recognize
as the differential form
00:47:23.810 --> 00:47:32.890
of the inverse tangent
up to a factor of 1
00:47:32.890 --> 00:47:37.650
over b square root
of Tc minus T.
00:47:37.650 --> 00:47:39.140
So I integrate this.
00:47:39.140 --> 00:47:42.549
And on the other
side, I have 4/pi l.
00:47:47.510 --> 00:48:04.970
So eventually, I know that--
that's what I wanted to do?
00:48:04.970 --> 00:48:10.060
I needed to do this later on,
but we'll use it later on.
00:48:10.060 --> 00:48:16.090
What I needed to get is
what is the eventual fate
00:48:16.090 --> 00:48:19.170
of this differential equation.
00:48:19.170 --> 00:48:22.530
Eventually, we see that this
differential equation arrives
00:48:22.530 --> 00:48:25.715
at the point that I
will call x infinity.
00:48:28.400 --> 00:48:33.380
When it arrives at x infinity,
this is 0 and y is 0,
00:48:33.380 --> 00:48:35.880
so I immediately know
that x infinity--
00:48:35.880 --> 00:48:40.520
I didn't need to do any of that
calculation-- this expression
00:48:40.520 --> 00:48:50.000
has to be 0, is minus the square
root of Tc minus T. So let
00:48:50.000 --> 00:48:54.347
me figure out what I did with
the signs that is incorrect.
00:48:54.347 --> 00:48:56.255
AUDIENCE: [INAUDIBLE]
temperature [INAUDIBLE]
00:48:56.255 --> 00:48:57.209
positive [INAUDIBLE].
00:49:00.444 --> 00:49:02.110
PROFESSOR: In the low
temperature phase,
00:49:02.110 --> 00:49:07.340
I have indeed stated that
c has to be positive,
00:49:07.340 --> 00:49:11.600
which means that this
coefficient better be positive,
00:49:11.600 --> 00:49:16.100
which means that I would
have a minus sign here.
00:49:16.100 --> 00:49:26.790
And then x would be
b times Tc minus T.
00:49:26.790 --> 00:49:30.220
Right, so this would
be plus or minus.
00:49:30.220 --> 00:49:34.460
The plus solution is
somewhere out here,
00:49:34.460 --> 00:49:36.460
which I'm not interested.
00:49:36.460 --> 00:49:40.690
The solution that I'm interested
corresponds to this value.
00:49:48.260 --> 00:49:51.230
You say, well, what is
important about that?
00:49:56.545 --> 00:50:04.460
You see that various properties
of this low temperature phase
00:50:04.460 --> 00:50:08.580
are characterized by
this power law as opposed
00:50:08.580 --> 00:50:12.380
to exponential behavior.
00:50:12.380 --> 00:50:17.250
The power law is determined
by the value of k
00:50:17.250 --> 00:50:20.070
where the description in
terms of this gradient
00:50:20.070 --> 00:50:23.340
squared theory is correct.
00:50:23.340 --> 00:50:26.560
Now out here, the
description is not correct
00:50:26.560 --> 00:50:30.190
because I still have the
topological difference.
00:50:30.190 --> 00:50:33.360
But if I look at
sufficiently large distances,
00:50:33.360 --> 00:50:37.090
I see that the topology
defects have disappeared.
00:50:37.090 --> 00:50:40.420
But by the time the topological
defects have disappeared,
00:50:40.420 --> 00:50:43.180
I don't have the
original value of k.
00:50:43.180 --> 00:50:46.950
I have a slightly
different value of k.
00:50:46.950 --> 00:50:51.340
So presumably,
the properties are
00:50:51.340 --> 00:50:56.615
going to be described by
what value of this k inverse
00:50:56.615 --> 00:50:59.420
is of large behaviors.
00:50:59.420 --> 00:51:07.630
And so what I expect is that
the effective behavior of this k
00:51:07.630 --> 00:51:22.450
inverse-- actually, the
effective behavior of k--
00:51:22.450 --> 00:51:29.130
as a function of whatever the
temperature of the system is.
00:51:29.130 --> 00:51:35.020
We expect that in the original
xy model, or any system that
00:51:35.020 --> 00:51:37.950
is described by
this behavior, there
00:51:37.950 --> 00:51:43.410
is a critical temperature,
Tc, such that at higher
00:51:43.410 --> 00:51:47.880
temperatures, correlations
are decaying exponentially.
00:51:47.880 --> 00:51:51.915
So essentially, the effective
value of k has gone to 0.
00:51:51.915 --> 00:51:53.730
There is no stiffness parameter.
00:51:53.730 --> 00:51:57.120
So basically at
high temperatures,
00:51:57.120 --> 00:51:59.000
you should be over here.
00:52:02.180 --> 00:52:10.450
What I see is that the effective
value of k, however, other
00:52:10.450 --> 00:52:15.050
is meaningful all the way
to the inverse of 2/pi.
00:52:18.890 --> 00:52:26.220
So there is a value
here at 2/pi which
00:52:26.220 --> 00:52:36.330
corresponds to the largest
temperature or the smallest
00:52:36.330 --> 00:52:38.930
k that is acceptable.
00:52:38.930 --> 00:52:41.870
Now what I see is
that on approaching
00:52:41.870 --> 00:52:52.940
the transition, the value of k--
I have it up there-- is 2/pi.
00:52:52.940 --> 00:52:57.470
This limiting value
that we have over here.
00:52:57.470 --> 00:53:05.210
And then there is a correction
that is 4 over pi squared x.
00:53:05.210 --> 00:53:09.380
And presumably here, I have
to put the x in infinity.
00:53:09.380 --> 00:53:12.270
And what I have
for the x infinity
00:53:12.270 --> 00:53:16.720
is something like that,
so I will get 2/pi plus 4
00:53:16.720 --> 00:53:23.110
e over pi squared square
root of Tc minus T.
00:53:23.110 --> 00:53:32.120
So the prediction is that
the effective value of k
00:53:32.120 --> 00:53:35.910
comes to its limiting
value of 2/pi
00:53:35.910 --> 00:53:37.620
with a square root singularity.
00:53:42.840 --> 00:53:53.620
So we can replace
the theory that
00:53:53.620 --> 00:53:58.130
describes anything that is
in this universality class
00:53:58.130 --> 00:54:02.865
in the temperature phase
by an effective value of k.
00:54:02.865 --> 00:54:07.680
If we then ask how does that
effective value of k change
00:54:07.680 --> 00:54:10.890
as a function of
temperature, the prediction
00:54:10.890 --> 00:54:13.500
is that, well, at
very low temperature,
00:54:13.500 --> 00:54:17.240
it's presumably inversely
related to temperature.
00:54:17.240 --> 00:54:19.780
It will come down,
but [INAUDIBLE]
00:54:19.780 --> 00:54:24.280
it will change its behavior,
come with a square root
00:54:24.280 --> 00:54:28.880
singularity to a number that
is 2/pi, and then jump to 0.
00:54:32.950 --> 00:54:39.390
Now you are justified in saying,
well, this is all very obscure.
00:54:39.390 --> 00:54:42.650
Is there any way to see this?
00:54:42.650 --> 00:54:45.730
And the answer is that
people have experimentally
00:54:45.730 --> 00:54:50.120
verified this, and
I'll tell you how.
00:54:50.120 --> 00:55:00.470
So a system that belongs to this
universality class and we've
00:55:00.470 --> 00:55:02.895
mentioned all the way in
the class is the superfluid.
00:55:06.670 --> 00:55:11.310
We've said that the superfluid
transition is characterized
00:55:11.310 --> 00:55:16.940
by a quantum order parameter
that applies a magnitude,
00:55:16.940 --> 00:55:19.880
but then it has a phase theta.
00:55:19.880 --> 00:55:22.350
And roughly, we would say
that the phase theta should
00:55:22.350 --> 00:55:27.050
be described by this kind of
theory at low temperatures.
00:55:27.050 --> 00:55:31.270
So if we want basically a
two dimensional system, what
00:55:31.270 --> 00:55:35.900
we need to do is to look
at the superfluid field.
00:55:35.900 --> 00:55:47.250
And this is something that
Bishop and Reppy did in 1978
00:55:47.250 --> 00:55:54.760
where they constructed the
analog of the Andronikashvili
00:55:54.760 --> 00:56:01.480
experiment that we mentioned in
8333, applied it to the field.
00:56:01.480 --> 00:56:06.770
So let me remind you what the
Andronikashvili experiment was.
00:56:06.770 --> 00:56:14.590
Basically, you will have
a torsional oscillator.
00:56:14.590 --> 00:56:23.890
This torsional
oscillator was connected
00:56:23.890 --> 00:56:28.510
to a vat that had helium in it.
00:56:28.510 --> 00:56:33.880
So basically, this
thing was oscillating,
00:56:33.880 --> 00:56:36.780
and the frequency
of oscillations
00:56:36.780 --> 00:56:46.090
was related to some kind of a
effective torsion of constant k
00:56:46.090 --> 00:56:49.880
divided by some
huge mass which is
00:56:49.880 --> 00:56:52.550
contained within the cylinder.
00:56:52.550 --> 00:56:58.700
So basically you can
probe classically-- you
00:56:58.700 --> 00:57:00.900
would say there's some
kind of a density here.
00:57:00.900 --> 00:57:02.660
You can calculate
what the mass is
00:57:02.660 --> 00:57:07.236
if you know what this is,
you know what the omega is.
00:57:07.236 --> 00:57:12.630
Now what he noticed was that
if this thing was filled
00:57:12.630 --> 00:57:19.330
with liquid helium and you
went below Tc of helium, then
00:57:19.330 --> 00:57:22.660
suddenly this frequency changed.
00:57:22.660 --> 00:57:28.010
And the reason was that the
mass that was rotating along
00:57:28.010 --> 00:57:32.060
with this whole thing was
changed because the part that
00:57:32.060 --> 00:57:35.960
was superfluid
was sitting still,
00:57:35.960 --> 00:57:39.800
and the normal part was the
part that was oscillating.
00:57:39.800 --> 00:57:44.510
So the mass that was
oscillating was reduced,
00:57:44.510 --> 00:57:46.500
frequency would go up.
00:57:46.500 --> 00:57:48.830
And from the change
in frequency,
00:57:48.830 --> 00:57:54.070
he could figure out the change
in the density of the part that
00:57:54.070 --> 00:57:57.210
was oscillating,
and hence calculate
00:57:57.210 --> 00:58:01.350
what the density of
the normal part was.
00:58:01.350 --> 00:58:07.430
So what Bishop and Reppy did was
to make this two dimensional.
00:58:07.430 --> 00:58:10.220
How did they make
it two dimensional?
00:58:10.220 --> 00:58:15.980
Rather than having a container
of helium, what they did was
00:58:15.980 --> 00:58:19.250
they made, if you like,
some kind of a toilet paper.
00:58:19.250 --> 00:58:21.770
They call it a
jelly roll my Mylar.
00:58:21.770 --> 00:58:26.630
So it was Mylar that was
wrapped in a cylinder.
00:58:26.630 --> 00:58:32.666
And then the helium was absorbed
between the surfaces of Mylar.
00:58:32.666 --> 00:58:36.390
So effectively, it was
a two dimensional system
00:58:36.390 --> 00:58:37.520
in this very setup.
00:58:40.860 --> 00:58:43.620
So for that two
dimensional system,
00:58:43.620 --> 00:58:46.380
they-- again, with
the same thing--
00:58:46.380 --> 00:58:49.100
they measure the
change in frequency.
00:58:49.100 --> 00:58:51.940
They found that if they go
to low enough temperatures,
00:58:51.940 --> 00:58:54.730
suddenly there is a
change in frequency.
00:58:54.730 --> 00:58:58.080
Of course, the temperature that
they were seeing in this case
00:58:58.080 --> 00:59:01.180
was something like 1
degrees Kelvin or a fraction
00:59:01.180 --> 00:59:04.580
of 1 degree Kelvin, whereas when
you have the full superfluid,
00:59:04.580 --> 00:59:09.550
it's 2.8 degrees Kelvin clearly
because of that dimensionality
00:59:09.550 --> 00:59:11.500
the critical
temperature changes.
00:59:11.500 --> 00:59:15.880
But you would say that's not
particularly the inverse.
00:59:15.880 --> 00:59:21.760
So they could measure
the change in frequency
00:59:21.760 --> 00:59:24.350
and relate the
change in frequency
00:59:24.350 --> 00:59:27.845
to the density that
became superfluid.
00:59:32.310 --> 00:59:36.880
Now how does the
superfluid density
00:59:36.880 --> 00:59:39.750
tell us anything
about this curve?
00:59:39.750 --> 00:59:43.130
Well, the answer
is that everything
00:59:43.130 --> 00:59:46.300
is going to be
weighted by something
00:59:46.300 --> 00:59:51.720
like e the minus beta
times some energy.
00:59:51.720 --> 00:59:55.898
The one part of the energy that
is associated with oscillations
00:59:55.898 --> 00:59:59.360
is certainly the kinetic energy.
00:59:59.360 --> 01:00:03.384
So let's see what we would
write down for beta times
01:00:03.384 --> 01:00:10.340
the kinetic energy of
superfluid or superfluid film.
01:00:10.340 --> 01:00:15.790
What I have to do is
beta will give me 1/kt.
01:00:18.690 --> 01:00:22.990
The kinetic energy is
obtained by integrating
01:00:22.990 --> 01:00:27.060
mass times velocity
squared, or density
01:00:27.060 --> 01:00:30.725
integrated against velocity.
01:00:34.170 --> 01:00:36.400
It's a two dimensional
film, so we sort of
01:00:36.400 --> 01:00:39.530
integrate as we
go along the film.
01:00:39.530 --> 01:00:46.450
The superfluid
velocity can be related
01:00:46.450 --> 01:00:55.140
to the mass of helium
h bar and the gradient
01:00:55.140 --> 01:00:58.050
of this phase of the
superconducting order
01:00:58.050 --> 01:00:59.500
parameter.
01:00:59.500 --> 01:01:04.870
So you can, for example, write
your weight function as sidebar
01:01:04.870 --> 01:01:08.810
into the i theta of x, calculate
what the current is using
01:01:08.810 --> 01:01:13.430
the usual formula of h bar
over m psi star [? grat ?] psi
01:01:13.430 --> 01:01:15.030
[? minus ?] psi
[? grat ?] psi star,
01:01:15.030 --> 01:01:19.830
and you would see that effective
mass is something like this.
01:01:19.830 --> 01:01:26.753
So this is going to give
me rho over kt h bar over n
01:01:26.753 --> 01:01:34.390
helium 4 squared integral
gradient of theta
01:01:34.390 --> 01:01:41.580
squared, which you can see
is identical to the very
01:01:41.580 --> 01:01:44.570
first line that I
wrote down for you.
01:01:44.570 --> 01:01:54.314
And we can see that k can be
interpreted as rho-- kt h bar
01:01:54.314 --> 01:01:57.660
over n squared.
01:02:00.210 --> 01:02:05.020
So all of these quantities--
h bar m, you know.
01:02:05.020 --> 01:02:08.080
T is the temperature
that you're measuring.
01:02:08.080 --> 01:02:13.232
Rho you get through the
change in this frequency.
01:02:13.232 --> 01:02:18.360
And so then they can
plot what [INAUDIBLE]
01:02:18.360 --> 01:02:23.976
rho is as a function
of temperature.
01:02:23.976 --> 01:02:27.490
And we see that it's
very much related to k.
01:02:27.490 --> 01:02:33.550
And indeed, they find that
the rho that they measure
01:02:33.550 --> 01:02:37.140
has some kind of
behavior such as this.
01:02:40.110 --> 01:02:45.000
And then they go and change
their Mylar, make the films
01:02:45.000 --> 01:02:46.860
thicker or whatever.
01:02:46.860 --> 01:02:49.100
They find that the
transition temperature
01:02:49.100 --> 01:02:55.870
changes so that a
different type of film
01:02:55.870 --> 01:02:58.295
would show a behavior
such as this.
01:02:58.295 --> 01:03:02.630
And thicker films will have a
higher critical temperature.
01:03:02.630 --> 01:03:06.520
They do it for a number
of film thicknesses,
01:03:06.520 --> 01:03:11.860
and they got things'
behavior such as this,
01:03:11.860 --> 01:03:16.390
and found that this behavior
followed this gray line, which
01:03:16.390 --> 01:03:18.940
is exactly what is
predicted from here.
01:03:18.940 --> 01:03:23.000
It was predicted
that rho c over Tc
01:03:23.000 --> 01:03:31.690
should kb m over h
bar squared times what
01:03:31.690 --> 01:03:43.090
the critical value of k is that
we've calculated to be 2/pi
01:03:43.090 --> 01:03:47.725
So they could precisely
check these 2/pi
01:03:47.725 --> 01:03:49.400
that we've calculated.
01:03:49.400 --> 01:03:53.210
They could more or less see
this square root approach
01:03:53.210 --> 01:03:54.840
to this singularity.
01:03:54.840 --> 01:03:57.150
I'm not sure the data
at that point were good
01:03:57.150 --> 01:04:00.849
enough so that they could say
this exponent is precisely 1/2.
01:04:15.520 --> 01:04:20.090
So this was for the
low temperature phase.
01:04:20.090 --> 01:04:22.210
What can I say about the
high temperature phase?
01:04:29.830 --> 01:04:35.470
So in the high temperature
phase is where my c is negative.
01:04:39.070 --> 01:04:46.480
So there I can write
x0 squared minus pi
01:04:46.480 --> 01:04:52.483
to the fourth y0 squared
as being a negative number,
01:04:52.483 --> 01:04:58.740
which I will write as
minus b squared T minus Tc.
01:04:58.740 --> 01:05:02.710
So I have now T that is
greater that Tc, multiply
01:05:02.710 --> 01:05:07.550
with some constant,
and I get this.
01:05:07.550 --> 01:05:10.450
And this is the same all
along the trajectory.
01:05:13.880 --> 01:05:17.190
So as I go further
and x and y change,
01:05:17.190 --> 01:05:19.430
they will change
in a manner that
01:05:19.430 --> 01:05:23.300
is consistent with
this, which implies
01:05:23.300 --> 01:05:31.890
that as x changes with l and y
changes with l, the two of them
01:05:31.890 --> 01:05:45.551
will be related by y squared is
being x squared plus b squared
01:05:45.551 --> 01:05:53.190
T minus Tc divided
by pi to the fourth.
01:05:53.190 --> 01:05:55.770
So this is where
I don't really see
01:05:55.770 --> 01:05:57.730
the endpoint of the trajectory.
01:05:57.730 --> 01:06:01.300
I just want to see how the
trajectory is behaving.
01:06:01.300 --> 01:06:06.300
So I go back to this
equation, dx by dl
01:06:06.300 --> 01:06:11.700
is 4 pi cube y squared.
01:06:11.700 --> 01:06:17.120
Substitute that y
squared, I will get 4/pi 1
01:06:17.120 --> 01:06:21.330
over x squared plus
b squared T minus Tc.
01:06:35.040 --> 01:06:38.920
And then I rearrange
this in a form
01:06:38.920 --> 01:06:40.850
I can see how to integrate.
01:06:40.850 --> 01:06:50.340
dx x squared plus b squared
T minus Tc is 4/pi dl.
01:06:50.340 --> 01:06:52.660
I integrate the left-hand side.
01:06:52.660 --> 01:06:55.620
And as I already
jumped ahead, it
01:06:55.620 --> 01:07:02.670
is the inverse
tangent of x divided
01:07:02.670 --> 01:07:10.291
by b square root of T
minus Tc is 4/pi times l.
01:07:15.190 --> 01:07:20.660
So what do I want to do
with this expression?
01:07:23.570 --> 01:07:30.390
So what I want to do is to
see the trajectories that
01:07:30.390 --> 01:07:34.150
just cross to the
high temperature side.
01:07:34.150 --> 01:07:38.880
So I start with a point
that is just slightly
01:07:38.880 --> 01:07:42.690
to the right of this
transition line.
01:07:42.690 --> 01:07:46.140
Presumably, what is
happening is that I
01:07:46.140 --> 01:07:51.880
will follow the transition
trajectory for a long while,
01:07:51.880 --> 01:07:57.300
then I will start
to head out, which
01:07:57.300 --> 01:08:02.580
means that for this
trajectory, if I look
01:08:02.580 --> 01:08:07.080
at the system over larger
and larger scales, initially
01:08:07.080 --> 01:08:10.500
I find that it becomes
harder and harder
01:08:10.500 --> 01:08:13.730
to create these
topological defects.
01:08:13.730 --> 01:08:16.920
The core energy for
them becomes large.
01:08:16.920 --> 01:08:19.729
The fugacity for
them becomes small.
01:08:19.729 --> 01:08:27.430
But ultimately I manage to
break that, and I go to a regime
01:08:27.430 --> 01:08:30.250
where it becomes
easier and easier
01:08:30.250 --> 01:08:33.189
to create these
topological defects.
01:08:33.189 --> 01:08:36.969
And presumably at some
point out here, everything
01:08:36.969 --> 01:08:40.540
that I have said I have to
throw out because I'm making
01:08:40.540 --> 01:08:45.340
an expansion, assuming that y
is small, x is small, et cetera.
01:08:45.340 --> 01:08:52.080
So presumably, as I integrate,
I come to a point where I say,
01:08:52.080 --> 01:08:55.220
OK, differential
equations break down.
01:08:55.220 --> 01:09:00.170
But my intuition tells me that
I have reached the regime where
01:09:00.170 --> 01:09:09.189
I can create pretty much plus
and minus charges at ease.
01:09:09.189 --> 01:09:14.479
So I would say that once I
have reached that region where
01:09:14.479 --> 01:09:18.880
x and y have managed to escape
the region where they are
01:09:18.880 --> 01:09:22.920
small, they have become
of the order of 1.
01:09:22.920 --> 01:09:26.580
Maybe you can put them 1/3,
1/4-- it doesn't matter.
01:09:26.580 --> 01:09:31.890
Once they have become something
that is not infinitesimal,
01:09:31.890 --> 01:09:35.740
then I can create these
charges more or less at will.
01:09:35.740 --> 01:09:38.300
I will have a
system where I have
01:09:38.300 --> 01:09:41.490
lots of charges that
can be created at ease.
01:09:41.490 --> 01:09:45.740
And my intuition tells
me that in that system,
01:09:45.740 --> 01:09:52.229
I shall have this kind of
decay-- exponential decay.
01:09:52.229 --> 01:09:59.890
So how far did I have to go in
order to reach that value of l?
01:09:59.890 --> 01:10:03.090
I have to go to a
correlation length
01:10:03.090 --> 01:10:07.270
for a size that is
larger than 1 what
01:10:07.270 --> 01:10:15.210
I started with by a factor of
e to the l where the value of x
01:10:15.210 --> 01:10:19.240
became something that
is of the order of 1.
01:10:19.240 --> 01:10:21.170
And actually, you
can see from here
01:10:21.170 --> 01:10:24.000
that if I'm very close
to the transition,
01:10:24.000 --> 01:10:26.150
it doesn't matter
whether I choose here
01:10:26.150 --> 01:10:31.300
to be 1/10, 1/2, even 1/100.
01:10:31.300 --> 01:10:34.070
As long as I'm
close enough to Tc,
01:10:34.070 --> 01:10:38.180
I'm dividing something by
something that is close to 0.
01:10:38.180 --> 01:10:43.250
And this is tan inverse
of a large number.
01:10:43.250 --> 01:10:45.810
And tan inverse
of a large number
01:10:45.810 --> 01:10:48.730
is tan inverse of 90 degrees.
01:10:48.730 --> 01:10:51.590
So essentially, I
go to some value
01:10:51.590 --> 01:10:58.230
of x where I can approximate
this by pi over 2.
01:10:58.230 --> 01:11:01.660
And you can see that that is
really insensitive to what
01:11:01.660 --> 01:11:05.600
I choose to be my x's as
long as I'm sufficiently
01:11:05.600 --> 01:11:07.640
close to the critical point.
01:11:07.640 --> 01:11:09.728
So you can see that
once I have done
01:11:09.728 --> 01:11:13.840
that, I have figured out what
my l star is, if you like.
01:11:13.840 --> 01:11:16.360
And if I substituted
that over there,
01:11:16.360 --> 01:11:27.320
I will get a behavior that is
about pi/4 times pi/2 times
01:11:27.320 --> 01:11:30.060
1 over the square
root of T minus Tc.
01:11:32.700 --> 01:11:39.210
Now these coefficients out
front are not that important.
01:11:39.210 --> 01:11:44.080
What you see is that, indeed,
we get a correlation length
01:11:44.080 --> 01:11:48.080
that, as we approach
Tc, diverges,
01:11:48.080 --> 01:11:50.820
but it is not that at
all of any of the forms
01:11:50.820 --> 01:11:52.090
that we had seen before.
01:11:52.090 --> 01:11:55.540
So typically, we wrote that
the correlation lens diverges
01:11:55.540 --> 01:11:59.110
T minus Tc to some
exponent minus mu.
01:11:59.110 --> 01:12:01.350
This is not that
type of divergence.
01:12:01.350 --> 01:12:04.870
It's a very different
type of divergence.
01:12:04.870 --> 01:12:09.430
And again, it's root is
in the non-linear version
01:12:09.430 --> 01:12:13.300
of the recursion
undulations that we have.
01:12:13.300 --> 01:12:16.420
The closest thing
to this that we have
01:12:16.420 --> 01:12:19.540
is when we were calculating
the correlation length
01:12:19.540 --> 01:12:23.340
for the non-linear sigma model
where we had something that
01:12:23.340 --> 01:12:26.980
had a 1 over temperature
type of behavior.
01:12:26.980 --> 01:12:30.770
This is even more complicated.
01:12:30.770 --> 01:12:33.848
Now once you know the singular
behavior of the correlation
01:12:33.848 --> 01:12:38.660
length, you would say that in
the two dimensional system,
01:12:38.660 --> 01:12:40.880
the singular part
of the free energy
01:12:40.880 --> 01:12:45.510
should scale like
c to the minus 2.
01:12:45.510 --> 01:12:48.900
Essentially, you
break your system
01:12:48.900 --> 01:12:52.490
into pieces that are of the
size correlation length.
01:12:52.490 --> 01:12:57.090
The number of those pieces
is l over xi squared,
01:12:57.090 --> 01:12:58.580
because you are
in two dimensions.
01:12:58.580 --> 01:12:59.970
So you would get this.
01:12:59.970 --> 01:13:04.270
So that says that your
singularity of the free energy
01:13:04.270 --> 01:13:08.080
is something like, I
don't know, pi squared
01:13:08.080 --> 01:13:14.640
8 4b square root of T minus Tc.
01:13:14.640 --> 01:13:17.370
Again, not a
popular singularity.
01:13:17.370 --> 01:13:18.620
It's an essential singularity.
01:13:27.052 --> 01:13:32.140
And essential singularity is
a kind of singular function
01:13:32.140 --> 01:13:34.860
that no matter how
many derivatives
01:13:34.860 --> 01:13:40.100
you take, as T comes to Tc,
there is no singularity.
01:13:40.100 --> 01:13:42.940
So for example, if I
take two derivatives
01:13:42.940 --> 01:13:49.600
to get the heat
capacity, what I would
01:13:49.600 --> 01:13:56.920
plot as a function of T at
Tc should have no signatures.
01:13:56.920 --> 01:14:01.040
So basically what you
would see because if this
01:14:01.040 --> 01:14:02.730
is that the curve
just continues.
01:14:02.730 --> 01:14:06.465
There is no signature of
a transition at this heat
01:14:06.465 --> 01:14:08.170
capacity.
01:14:08.170 --> 01:14:10.070
And indeed, people
later on, they
01:14:10.070 --> 01:14:12.870
did numerical
simulations, et cetera.
01:14:12.870 --> 01:14:15.430
What they find is that
the heat capacity actually
01:14:15.430 --> 01:14:20.240
has kind of smooth peak a
little bit later than Tc, which
01:14:20.240 --> 01:14:23.580
is the location where there's
lots and lots of vortex
01:14:23.580 --> 01:14:25.060
unbinding going, gone.
01:14:25.060 --> 01:14:28.576
But at Tc itself, there is no
signature of a singularity.
01:14:36.520 --> 01:14:43.480
As far as I know, there's
no experimental case
01:14:43.480 --> 01:14:45.810
with this correlation
length as we observed.
01:14:55.950 --> 01:15:06.500
So the lesson that we can take
from this particular system
01:15:06.500 --> 01:15:13.300
is that two dimensional
system are kind of potentially
01:15:13.300 --> 01:15:16.710
interesting and different.
01:15:16.710 --> 01:15:20.000
We had this
Mermin-Wagner Theorem
01:15:20.000 --> 01:15:21.690
that we mentioned in
the beginning that
01:15:21.690 --> 01:15:24.010
said that there should
be no true long range
01:15:24.010 --> 01:15:25.990
order than two dimensions.
01:15:25.990 --> 01:15:27.780
That is still true.
01:15:27.780 --> 01:15:30.856
But despite that, there
could be phase transitions
01:15:30.856 --> 01:15:36.240
with quite observable
consequences.
01:15:36.240 --> 01:15:42.670
Add a particular type of
transition in two dimension
01:15:42.670 --> 01:15:47.440
that we will pursue next
lecture-- so I'll give you
01:15:47.440 --> 01:15:51.428
a preview-- is that of melting.
01:15:56.670 --> 01:16:01.210
So the prototype of a phase
transition you may think of
01:16:01.210 --> 01:16:05.130
is either liquid gas
or a liquid solid.
01:16:05.130 --> 01:16:09.640
And you can say, well, you
have studied phase transitions
01:16:09.640 --> 01:16:11.340
to such a degree.
01:16:11.340 --> 01:16:16.470
Why not go back and talk
about the melting transition,
01:16:16.470 --> 01:16:18.800
for example?
01:16:18.800 --> 01:16:24.320
The reason is that the straight
melting transition is typically
01:16:24.320 --> 01:16:25.620
first order.
01:16:25.620 --> 01:16:28.670
And we've seen that universality
and all of those things
01:16:28.670 --> 01:16:33.290
emerge when you have a
diverging correlation length.
01:16:33.290 --> 01:16:38.130
So you want to
have a place where
01:16:38.130 --> 01:16:40.420
there is a possible potential
for continuous phase
01:16:40.420 --> 01:16:41.530
transition.
01:16:41.530 --> 01:16:46.140
And it turns out that melting
in two dimensions provides that.
01:16:46.140 --> 01:16:48.730
So in two dimensions,
you could have
01:16:48.730 --> 01:16:53.570
a bunch of points that
could, for example,
01:16:53.570 --> 01:16:58.270
in a minimum energy
configuration at T close to 0
01:16:58.270 --> 01:17:00.560
form a triangular lattice.
01:17:03.360 --> 01:17:05.540
Now when you go to
finite temperature--
01:17:05.540 --> 01:17:08.750
as we discussed, again, at
the very first lecture--
01:17:08.750 --> 01:17:13.990
you will start to have
distortions around this.
01:17:13.990 --> 01:17:15.920
We can describe
these distortions
01:17:15.920 --> 01:17:19.630
to effect of u of x and y.
01:17:19.630 --> 01:17:23.380
And then go to that
appropriate continuum limit
01:17:23.380 --> 01:17:27.920
that describes the
elasticity of these things.
01:17:27.920 --> 01:17:30.180
And it is going
to look very much
01:17:30.180 --> 01:17:32.220
like that gradient
of theta squared term
01:17:32.220 --> 01:17:35.970
that we wrote at the beginning,
except that since this u is
01:17:35.970 --> 01:17:39.780
a vector, as we saw, even
for an isotropic material,
01:17:39.780 --> 01:17:41.970
you will have the
potential for having
01:17:41.970 --> 01:17:45.180
multiple elastic constants.
01:17:45.180 --> 01:17:49.110
But modeled on that, the
conclusion that you would have
01:17:49.110 --> 01:17:53.430
is that as long
as it is OK for me
01:17:53.430 --> 01:17:58.800
to make an expansion that
is like the elastic theory--
01:17:58.800 --> 01:18:03.170
some kind of a gradient
of u expansion--
01:18:03.170 --> 01:18:08.170
the conclusion would be
that the correlations in u
01:18:08.170 --> 01:18:11.880
will grow logarithmically
as a function of size.
01:18:11.880 --> 01:18:16.670
And you will not have
too long range of order,
01:18:16.670 --> 01:18:19.640
but you will have some kind
of a power of behavior,
01:18:19.640 --> 01:18:23.730
such as the one that we
have indicated over there.
01:18:23.730 --> 01:18:26.290
On the other hand, when you
go to very high temperature,
01:18:26.290 --> 01:18:28.612
presumably this
whole thing melts.
01:18:28.612 --> 01:18:31.290
There is no reason
to have correlations
01:18:31.290 --> 01:18:35.520
beyond a few atoms
that are close to you.
01:18:35.520 --> 01:18:37.500
Add so typically at
high temperature,
01:18:37.500 --> 01:18:41.140
correlations will
decay exponentially.
01:18:41.140 --> 01:18:43.280
So this is low
temperature expansion
01:18:43.280 --> 01:18:47.200
is elastic theory expansion
that we have written down
01:18:47.200 --> 01:18:50.990
has to break down
also in this case.
01:18:50.990 --> 01:18:55.220
And a particular mechanism for
its breakdown in two dimensions
01:18:55.220 --> 01:18:58.590
is to create these
topological defects, which
01:18:58.590 --> 01:19:01.890
in the case of solid will
correspond to these location
01:19:01.890 --> 01:19:04.670
lines that, for
example, correspond
01:19:04.670 --> 01:19:10.040
to adding an addition row of
particles here terminating
01:19:10.040 --> 01:19:11.990
at some point.
01:19:11.990 --> 01:19:15.620
And we can go through exactly
the same kind of story
01:19:15.620 --> 01:19:21.570
as we had before and conclude
that these dislocations,
01:19:21.570 --> 01:19:24.370
because of the competition
between their energy
01:19:24.370 --> 01:19:27.950
cost soaring logarithmically
and their entropy gain growing
01:19:27.950 --> 01:19:31.150
logarithmically,
we need to unbind
01:19:31.150 --> 01:19:33.740
at the critical temperature.
01:19:33.740 --> 01:19:38.870
And so that provides a
mechanism for describing
01:19:38.870 --> 01:19:41.875
the melting of
two-dimensional materials
01:19:41.875 --> 01:19:45.690
in a language that is
very similar to this,
01:19:45.690 --> 01:19:51.380
except for the complications
that have to do with this being
01:19:51.380 --> 01:19:55.110
a vector rather than
the scale of quantity.
01:19:55.110 --> 01:19:59.650
And so what we find is that
these topological charges
01:19:59.650 --> 01:20:02.920
are different from
minus 2 plus 2 pi.
01:20:02.920 --> 01:20:06.930
The interactions between
them is a particular version
01:20:06.930 --> 01:20:12.900
of the Coulomb interaction, but
that many of the other results
01:20:12.900 --> 01:20:14.135
go through.
01:20:14.135 --> 01:20:22.820
And we will get an idea of what
happens when the solid melts
01:20:22.820 --> 01:20:26.520
because of the unbinding
of these locations.
01:20:26.520 --> 01:20:28.150
But there is a
puzzle that we will
01:20:28.150 --> 01:20:31.530
find if we're not
melting to a liquid,
01:20:31.530 --> 01:20:34.090
but into something which is
more like a liquid crystal.
01:20:34.090 --> 01:20:37.670
So [INAUDIBLE] did they
discover also something
01:20:37.670 --> 01:20:39.675
about liquid crystals.