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PROFESSOR: OK, let's start.
00:00:25.438 --> 00:00:31.740
So we have been looking at the
problem of phase transitions
00:00:31.740 --> 00:00:35.995
from the perspective of
a simple system which
00:00:35.995 --> 00:00:37.360
is a piece of magnet.
00:00:40.550 --> 00:00:48.313
And we find that if we change
the temperature of the system,
00:00:48.313 --> 00:00:53.010
there is a critical
temperature, Tc,
00:00:53.010 --> 00:00:56.840
that separates
paramagnetic behavior
00:00:56.840 --> 00:00:59.700
on the high temperature side
and ferromagnetic behavior
00:00:59.700 --> 00:01:03.080
on the low temperature side.
00:01:03.080 --> 00:01:08.680
Clearly in the
vicinity of this point,
00:01:08.680 --> 00:01:11.010
whether you're on one
side or the other,
00:01:11.010 --> 00:01:14.132
the magnetization
is small and we
00:01:14.132 --> 00:01:18.700
are relying on that to
make us a good parameter
00:01:18.700 --> 00:01:21.260
to expand things in.
00:01:21.260 --> 00:01:24.150
The other thing is
that we anticipated
00:01:24.150 --> 00:01:26.520
that over here there
are long wavelength
00:01:26.520 --> 00:01:29.970
fluctuations of the
magnetization field
00:01:29.970 --> 00:01:35.780
and so we did averaging and
defined a statistical field,
00:01:35.780 --> 00:01:39.180
this magnetization as
a function of position.
00:01:42.120 --> 00:01:49.610
Then we said, OK, if I'm
just changing temperature,
00:01:49.610 --> 00:01:54.040
what potential is the
behavior of the probability
00:01:54.040 --> 00:01:57.806
that I will see in my sample
some particular configuration
00:01:57.806 --> 00:01:59.510
of this magnetization field?
00:01:59.510 --> 00:02:03.360
So there is a functional
that governs that.
00:02:03.360 --> 00:02:06.290
And the statement
that we made was
00:02:06.290 --> 00:02:09.220
that whatever this
functional is,
00:02:09.220 --> 00:02:11.800
I can write it as the
exponential of something
00:02:11.800 --> 00:02:12.660
if I want.
00:02:12.660 --> 00:02:15.220
This probability is positive.
00:02:15.220 --> 00:02:21.030
I will assume that it is--
locally there is a probability
00:02:21.030 --> 00:02:24.200
density that I will
integrate across the system.
00:02:24.200 --> 00:02:26.660
Probability density
is a function
00:02:26.660 --> 00:02:31.040
of whatever magnetization
I have around point x.
00:02:31.040 --> 00:02:34.210
And so then when we expanded
this, what did we have?
00:02:34.210 --> 00:02:37.850
We said that the terms that
are consistent with rotational
00:02:37.850 --> 00:02:44.380
symmetry have to be things like
m squared, m to the fourth,
00:02:44.380 --> 00:02:46.440
and so forth.
00:02:46.440 --> 00:02:48.420
In principal, there
is a long series
00:02:48.420 --> 00:02:50.570
but hopefully
since m is small, I
00:02:50.570 --> 00:02:54.620
don't have to include that
many terms in the series.
00:02:54.620 --> 00:02:58.870
And additionally, I can have
an expansion in gradient
00:02:58.870 --> 00:03:01.570
and the lowest order
term in that series
00:03:01.570 --> 00:03:05.700
was the gradient of m squared
and potentially higher
00:03:05.700 --> 00:03:08.280
order terms.
00:03:08.280 --> 00:03:10.990
OK?
00:03:10.990 --> 00:03:14.930
We said that we
could, if we relied
00:03:14.930 --> 00:03:19.460
on looking at the most probable
configuration of this weight,
00:03:19.460 --> 00:03:21.470
make a connection
between what is going
00:03:21.470 --> 00:03:25.000
on here and the
experimental observation.
00:03:25.000 --> 00:03:27.960
And essentially the only
thing that we needed to do
00:03:27.960 --> 00:03:31.090
was to basically
start T from here,
00:03:31.090 --> 00:03:36.620
so T was made to be
proportional to T minus Tc.
00:03:36.620 --> 00:03:40.190
And then we could explain
these kinds of phenomena
00:03:40.190 --> 00:03:41.892
by looking at the
behavior of the most
00:03:41.892 --> 00:03:42.850
probable magnetization.
00:03:45.950 --> 00:03:50.240
Now I kind of said
that we are going
00:03:50.240 --> 00:03:53.680
to have long wavelength
fluctuations.
00:03:53.680 --> 00:03:57.580
There was one case
where we actually
00:03:57.580 --> 00:04:01.220
saw a video of those long
wavelength fluctuations
00:04:01.220 --> 00:04:03.950
and that was for the case
of critical opalescence
00:04:03.950 --> 00:04:09.580
taking place at the liquid gas
mixture at its critical point.
00:04:09.580 --> 00:04:13.850
Can we try to quantify
that a little bit better?
00:04:13.850 --> 00:04:18.634
The answer is yes, we can do so
through scattering experiments.
00:04:23.197 --> 00:04:25.990
And looking at the
sample was an example
00:04:25.990 --> 00:04:27.690
of a scattering
experiment, which
00:04:27.690 --> 00:04:30.320
if you want to do
more quantitatively,
00:04:30.320 --> 00:04:33.600
we can do the following-
we can say that there
00:04:33.600 --> 00:04:41.090
is some incoming field,
electromagnetic wave that
00:04:41.090 --> 00:04:43.500
is impingent on the system.
00:04:43.500 --> 00:04:48.130
It's a pass on
incoming wave vector k.
00:04:48.130 --> 00:04:51.410
It sort of goes
through the sample
00:04:51.410 --> 00:04:54.860
and then when it comes out,
it gets scattered and so what
00:04:54.860 --> 00:04:59.190
I will see is some
k [INAUDIBLE] that
00:04:59.190 --> 00:05:01.960
comes from the other
part of the system.
00:05:01.960 --> 00:05:07.700
In principle, I guess I can
put a probe here and measure
00:05:07.700 --> 00:05:10.120
what is coming out.
00:05:10.120 --> 00:05:15.580
And essentially it will
depend on the angle
00:05:15.580 --> 00:05:19.740
towards which this has rotated.
00:05:19.740 --> 00:05:25.940
If I asked well, how
much has been scattered?
00:05:25.940 --> 00:05:28.720
We'd say, well it's
a complicated problem
00:05:28.720 --> 00:05:30.040
in quantum mechanics.
00:05:30.040 --> 00:05:32.620
Let's say this is a quantum
mechanical procedure-
00:05:32.620 --> 00:05:35.960
you would say that there's
an amplitude that you have
00:05:35.960 --> 00:05:47.270
the scattering that is
proportional to some overlap
00:05:47.270 --> 00:05:50.890
between the kind of state
that you started with,
00:05:50.890 --> 00:05:56.720
what we started is an
incoming wave with k initial,
00:05:56.720 --> 00:06:02.650
presumably there is the
initial state of my sample
00:06:02.650 --> 00:06:06.780
before the wave hits
it, and then I end up
00:06:06.780 --> 00:06:12.170
with the final configuration
which is k f, whatever
00:06:12.170 --> 00:06:17.570
the final version
of my system is.
00:06:17.570 --> 00:06:21.550
Now between these two,
I have to put whatever
00:06:21.550 --> 00:06:26.540
is responsible for
scattering this wave so there
00:06:26.540 --> 00:06:30.450
is in some sense some
overall potential
00:06:30.450 --> 00:06:33.970
that I have to put over here.
00:06:33.970 --> 00:06:36.690
Now let's think about the
case of this thing being,
00:06:36.690 --> 00:06:41.180
say, a mixture of
gas and liquid,
00:06:41.180 --> 00:06:43.460
well what is scattering light?
00:06:43.460 --> 00:06:47.410
Well, it is the individual
atoms that are scattering light
00:06:47.410 --> 00:06:49.290
and there are lots of them.
00:06:49.290 --> 00:06:55.240
So basically I have to sum over
all of the scattering elements
00:06:55.240 --> 00:06:57.420
that I have my system.
00:06:57.420 --> 00:07:02.420
Let's say I have a u for a
scattering element, i that
00:07:02.420 --> 00:07:07.130
is located at position--
maybe bad choice,
00:07:07.130 --> 00:07:10.070
let's call it sum over alpha.
00:07:10.070 --> 00:07:12.150
X alpha is the
position of let's say
00:07:12.150 --> 00:07:15.000
the atom that is scattered here.
00:07:15.000 --> 00:07:16.760
OK?
00:07:16.760 --> 00:07:21.870
So now since I'm dealing
with, say, linear order,
00:07:21.870 --> 00:07:25.330
not multiple scattering,
what I can do
00:07:25.330 --> 00:07:29.230
is I can basically
take this sum outside.
00:07:29.230 --> 00:07:32.070
So this thing is
related to a sum
00:07:32.070 --> 00:07:35.840
over alpha of the
scattering I would have
00:07:35.840 --> 00:07:39.200
for individual elements
that are scattering.
00:07:39.200 --> 00:07:43.350
And then roughly each
individual element
00:07:43.350 --> 00:07:47.900
will scatter an amount
that I will call sigma q.
00:07:47.900 --> 00:07:51.610
If you have elastic
scattering what happens
00:07:51.610 --> 00:07:58.680
is that essentially your
initial k simply gets rotated
00:07:58.680 --> 00:08:01.300
without changing its magnitude.
00:08:01.300 --> 00:08:05.460
So what happens is that
essentially everything
00:08:05.460 --> 00:08:10.480
will end up being a function
of this momentum transfer which
00:08:10.480 --> 00:08:15.970
is proportional
to q k f minus k i
00:08:15.970 --> 00:08:20.750
whose magnitude would be
twice the magnitude of your k
00:08:20.750 --> 00:08:23.740
sine of the half of
the angle if you just
00:08:23.740 --> 00:08:26.410
do the simple
geometry over there.
00:08:26.410 --> 00:08:28.940
So this is for
elastic scattering
00:08:28.940 --> 00:08:31.415
which is what we will
be thinking about.
00:08:37.370 --> 00:08:42.549
Now the amount that each
individual element scatters
00:08:42.549 --> 00:08:44.950
like each atom is
indeed a function
00:08:44.950 --> 00:08:52.310
of your momentum transferred
from the scattering probe.
00:08:52.310 --> 00:08:55.600
But the thing that
you're scattering from
00:08:55.600 --> 00:08:59.110
is something that is
very small like an atom,
00:08:59.110 --> 00:09:05.270
so it turns out that the
resulting q will give
00:09:05.270 --> 00:09:09.610
significant-- well, the
resulting sigma will vary all
00:09:09.610 --> 00:09:12.580
the over scales
where q is related
00:09:12.580 --> 00:09:15.360
to the inverse of
whatever is scattering
00:09:15.360 --> 00:09:18.560
which is something
that is very large.
00:09:18.560 --> 00:09:23.310
So most of this stuff that
is happening at small q, most
00:09:23.310 --> 00:09:25.900
of the variation
that is observed,
00:09:25.900 --> 00:09:28.950
comes from summing
over the contributions
00:09:28.950 --> 00:09:31.170
of the different elements.
00:09:31.170 --> 00:09:36.880
So going to the continuum
limit, this becomes an integral
00:09:36.880 --> 00:09:42.880
across your system of whatever
the density of the thing that
00:09:42.880 --> 00:09:44.760
is scattering is.
00:09:44.760 --> 00:09:47.870
Indeed if I'm thinking about
the light scattering experiment
00:09:47.870 --> 00:09:51.300
that we saw with critical
opalescence, what you would
00:09:51.300 --> 00:09:57.080
be looking at is this
density of liquid
00:09:57.080 --> 00:10:01.250
versus gas, which if I
want to convert to q,
00:10:01.250 --> 00:10:03.280
I have to do a Fourier
transform here.
00:10:05.910 --> 00:10:13.900
And so this is the amplitude of
scattering that I expect and we
00:10:13.900 --> 00:10:17.350
can see that it is
directly probing
00:10:17.350 --> 00:10:20.670
the fluctuations of
the system, Fourier
00:10:20.670 --> 00:10:25.750
transform [INAUDIBLE] number q.
00:10:25.750 --> 00:10:28.680
Eventually of course
this is the amplitude
00:10:28.680 --> 00:10:33.050
what you will be seeing is
the amount that is scattered,
00:10:33.050 --> 00:10:39.580
s of q will be proportional
to this amplitude squared.
00:10:39.580 --> 00:10:46.100
We'll have a part that at
small q is roughly constant,
00:10:46.100 --> 00:10:51.230
so basically at small q I can
regard this as a constant.
00:10:51.230 --> 00:10:54.120
So at small q all
of the variations
00:10:54.120 --> 00:10:56.800
is going to come from
this Roth q squared.
00:10:59.770 --> 00:11:05.540
Of course again thinking about
the case of the liquid gas
00:11:05.540 --> 00:11:11.250
system, where we were seeing the
picture, there were variations,
00:11:11.250 --> 00:11:15.530
so there's essentially
lots and lots
00:11:15.530 --> 00:11:18.820
of these Roth q's depending
on which instant of time
00:11:18.820 --> 00:11:21.090
you're looking at.
00:11:21.090 --> 00:11:25.980
And then it would be useful to
do some kind of a time average
00:11:25.980 --> 00:11:29.920
and hope that the
time average comes
00:11:29.920 --> 00:11:34.880
from the result of a probability
measure such as this.
00:11:34.880 --> 00:11:35.520
OK?
00:11:35.520 --> 00:11:38.330
So that's the procedure
that we'll follow.
00:11:38.330 --> 00:11:43.540
We're going to go slightly
beyond what we did before.
00:11:43.540 --> 00:11:46.610
What we did before was we
started with the probability
00:11:46.610 --> 00:11:48.340
distribution, such
as this that we
00:11:48.340 --> 00:11:54.100
posed on the basis of symmetry
and then calculated singular
00:11:54.100 --> 00:11:57.070
behavior of various
thermodynamic functions
00:11:57.070 --> 00:12:00.890
such as heat capacity,
susceptibility, magnetization,
00:12:00.890 --> 00:12:05.060
et cetera, all of them at
macroscopic quantities.
00:12:05.060 --> 00:12:07.880
But this is a
probability that also
00:12:07.880 --> 00:12:10.850
works at the level
of microscopics.
00:12:10.850 --> 00:12:14.770
It's really a probability as a
function of our configurations
00:12:14.770 --> 00:12:17.470
and the way that
that is probed is
00:12:17.470 --> 00:12:19.830
through scattering experiments.
00:12:19.830 --> 00:12:22.980
So scattering experiments
really probe the Fourier
00:12:22.980 --> 00:12:28.220
transform of this probability
that we have posed over here.
00:12:28.220 --> 00:12:30.590
OK?
00:12:30.590 --> 00:12:36.340
Now again the full probability
that I have written down there
00:12:36.340 --> 00:12:38.120
is rather difficult.
00:12:38.120 --> 00:12:41.490
Let's say this in the case
of the liquid gas system,
00:12:41.490 --> 00:12:43.980
this row would be explicitly
the magnetization.
00:12:43.980 --> 00:12:47.880
It would be the fluctuations
of the magnetization
00:12:47.880 --> 00:12:50.880
around the mean.
00:12:50.880 --> 00:12:53.370
I should note that in
the case of the magnet,
00:12:53.370 --> 00:12:55.920
you can say, well how
do you probe things?
00:12:55.920 --> 00:12:59.870
In that case you need
something, some probe
00:12:59.870 --> 00:13:03.700
that scatters from
magnetization at each point.
00:13:03.700 --> 00:13:08.400
And the appropriate probe for
magnetization is neutrons.
00:13:08.400 --> 00:13:12.320
So you basically hit the
system with a beam of neutrons
00:13:12.320 --> 00:13:15.480
that may be polarizing them
in this particular direction,
00:13:15.480 --> 00:13:18.850
their spins-- and they
hit the spins of whatever
00:13:18.850 --> 00:13:21.630
is in your sample and
they get scattered
00:13:21.630 --> 00:13:23.730
according to this mechanism.
00:13:23.730 --> 00:13:26.610
And what you will
be seeing at small q
00:13:26.610 --> 00:13:32.770
is related to fluctuations
of this magnetization field.
00:13:37.390 --> 00:13:41.755
Think he was looking for that.
00:13:41.755 --> 00:13:52.780
Now I realize-- I'm not
going to run after him.
00:13:52.780 --> 00:13:53.520
OK.
00:13:53.520 --> 00:13:59.350
So teaches them to
leave the room earlier.
00:13:59.350 --> 00:14:00.750
OK.
00:14:00.750 --> 00:14:04.870
So let's see what
we have to-- we can
00:14:04.870 --> 00:14:08.662
do for the case of
calculating this quantity.
00:14:08.662 --> 00:14:11.600
Now I'm not going to calculate
it for the nonlinear form,
00:14:11.600 --> 00:14:13.510
it's rather difficult.
00:14:13.510 --> 00:14:17.780
What I'm going to do is to
sort of expand on the trick
00:14:17.780 --> 00:14:21.210
that we were using
last time which
00:14:21.210 --> 00:14:23.500
led to this other
point which is to look
00:14:23.500 --> 00:14:25.095
at the most probable state.
00:14:32.300 --> 00:14:36.240
So basically we looked
at that function
00:14:36.240 --> 00:14:38.640
where we were calculating
the subtle point
00:14:38.640 --> 00:14:40.310
integration and the
first thing that we
00:14:40.310 --> 00:14:42.610
did was to find
the configuration
00:14:42.610 --> 00:14:47.160
of the magnetization field
that was the most likely.
00:14:47.160 --> 00:14:50.830
And the answer was that
because k is positive,
00:14:50.830 --> 00:14:55.170
your magnetization that
extremizes that probability
00:14:55.170 --> 00:15:00.460
is something that is uniform,
does not depend on x,
00:15:00.460 --> 00:15:03.060
and is pointing all
in one direction.
00:15:03.060 --> 00:15:06.010
Let's call it e hat 1.
00:15:06.010 --> 00:15:08.670
That is indicating
this one thing
00:15:08.670 --> 00:15:12.790
is symmetry breaking in
the zero field limit.
00:15:12.790 --> 00:15:16.230
Of course that spontaneous
symmetry breaking
00:15:16.230 --> 00:15:24.796
only occurs when t is negative
and for t positive m bar is 0.
00:15:24.796 --> 00:15:28.860
For t negative just
minimizing the expression
00:15:28.860 --> 00:15:31.570
t m squared plus
u m to the fourth
00:15:31.570 --> 00:15:36.060
gives you minus t over 4u.
00:15:39.092 --> 00:15:41.380
OK?
00:15:41.380 --> 00:15:45.300
So that's the most
probable configuration.
00:15:45.300 --> 00:15:48.720
What this thing is
probing is fluctuations so
00:15:48.720 --> 00:15:52.960
let's expand around the
most probable configuration.
00:15:52.960 --> 00:15:58.420
So let's say that we-- say that
I have thermally excited and m
00:15:58.420 --> 00:16:05.690
of x which is m bar plus
a little bit that varies
00:16:05.690 --> 00:16:09.120
from each location
to another location,
00:16:09.120 --> 00:16:12.360
like if I'm looking at
this critical opalescence,
00:16:12.360 --> 00:16:14.620
it's the variation
in density from one
00:16:14.620 --> 00:16:17.660
location to another location.
00:16:17.660 --> 00:16:22.120
But that is true as
long as I'm looking
00:16:22.120 --> 00:16:26.770
at the case of something
that has only one component.
00:16:26.770 --> 00:16:29.640
If I have multiple
components, I can also
00:16:29.640 --> 00:16:34.770
have fluctuations in the
remaining m minus 1 directions,
00:16:34.770 --> 00:16:45.620
and this is n of let's say alpha
go from 2 of phi t of e alpha.
00:16:45.620 --> 00:16:51.250
So I have broken the
fluctuations into two types.
00:16:51.250 --> 00:16:56.845
I've said that let's say
if you were n equals to 2
00:16:56.845 --> 00:17:02.440
your m bar would be pointing
in some particular direction.
00:17:02.440 --> 00:17:09.530
And so phi l's correspond
to increasing the length
00:17:09.530 --> 00:17:13.700
or decreasing the length
whereas phi t corresponds
00:17:13.700 --> 00:17:18.550
to going in the
orthogonal direction which
00:17:18.550 --> 00:17:23.069
in general would be n minus
1, different components.
00:17:23.069 --> 00:17:25.000
OK?
00:17:25.000 --> 00:17:28.250
So I want to ask,
what's the probability
00:17:28.250 --> 00:17:31.190
of this set of fluctuations?
00:17:31.190 --> 00:17:34.940
All I need to do and
again this is x-dependent
00:17:34.940 --> 00:17:39.940
is to substitute into my general
expression for the probability
00:17:39.940 --> 00:17:42.480
so for that I need a few things.
00:17:42.480 --> 00:17:48.110
One of the things that I need
is the gradient of m squared.
00:17:48.110 --> 00:17:52.640
The uniform part has no
gradient so I will either
00:17:52.640 --> 00:17:58.760
get the gradient from phi l
squared or I will get gradient
00:17:58.760 --> 00:18:04.410
of the n minus 1 component
vector field phi t
00:18:04.410 --> 00:18:09.480
that I will simply write as
gradient of phi t squared.
00:18:09.480 --> 00:18:14.560
So phi t is an n
minus 1 component.
00:18:14.560 --> 00:18:18.430
I can ask, what is m squared?
00:18:18.430 --> 00:18:22.085
Basically the first term
I need to put over there,
00:18:22.085 --> 00:18:25.120
but m squared, I have to
square this expression.
00:18:25.120 --> 00:18:32.520
I will get m bar squared to m
bar phi l plus phi l squared
00:18:32.520 --> 00:18:35.930
that comes from the
component that is along e 1.
00:18:35.930 --> 00:18:39.505
All the other components
here will add up
00:18:39.505 --> 00:18:46.330
to give me the magnitude of
this transverse field that
00:18:46.330 --> 00:18:51.390
exists in the other
m minus 1 directions.
00:18:51.390 --> 00:18:55.450
The other term that I
need is m to the fourth
00:18:55.450 --> 00:18:59.650
and in particular we saw that
it is absolutely necessary
00:18:59.650 --> 00:19:04.590
if t is negative to include
the m to the fourth term
00:19:04.590 --> 00:19:06.920
because otherwise the
probability that we were
00:19:06.920 --> 00:19:09.740
writing just didn't
make sense and we need
00:19:09.740 --> 00:19:12.590
to write expressions for
probability that are physically
00:19:12.590 --> 00:19:14.510
sensible.
00:19:14.510 --> 00:19:18.700
So I just take the line
above and square it.
00:19:18.700 --> 00:19:23.520
But I want to only keep
terms to quadratic order.
00:19:23.520 --> 00:19:28.670
So to zero order I have m bar
to the fourth to third order
00:19:28.670 --> 00:19:36.680
I have-- to first order I
have 4 m bar cubed phi l
00:19:36.680 --> 00:19:41.840
then there are a bunch of terms
that are order of phi squared.
00:19:41.840 --> 00:19:45.400
Squaring this will give
me 4 m bar squared phi l
00:19:45.400 --> 00:19:49.250
squared, but the dot
product of these two terms
00:19:49.250 --> 00:19:52.940
will also gives me two
m bar squared phi l
00:19:52.940 --> 00:20:01.230
squared for a total of six
m bar squared phi l squared.
00:20:01.230 --> 00:20:05.210
And then the phi t
squared comes simply
00:20:05.210 --> 00:20:08.065
from twice m bar
squared phi t squared.
00:20:12.587 --> 00:20:19.470
And then there's higher order
terms cubic and fourth order m
00:20:19.470 --> 00:20:22.960
phi t and phi l
that I don't write
00:20:22.960 --> 00:20:27.610
assuming that the fluctuations
that I'm looking at our small
00:20:27.610 --> 00:20:30.220
around the most probable state.
00:20:30.220 --> 00:20:32.150
OK?
00:20:32.150 --> 00:20:35.610
So if I stick with
this quadratics
00:20:35.610 --> 00:20:39.370
then the probability
of fluctuations
00:20:39.370 --> 00:20:45.650
across my system characterized
by phi l of x and phi t
00:20:45.650 --> 00:20:52.700
of x is proportional
to exponential
00:20:52.700 --> 00:20:59.975
of minus integral dd x.
00:21:02.850 --> 00:21:08.790
I have an overall factor-- well
I have a factor of K over 2
00:21:08.790 --> 00:21:09.910
for the first term.
00:21:09.910 --> 00:21:15.830
I have a gradient
of phi l squared
00:21:15.830 --> 00:21:24.230
and then you can see that
I have a bunch of them.
00:21:24.230 --> 00:21:30.200
Let's put the K over 2,
let's put the one-half here.
00:21:30.200 --> 00:21:36.380
I have K phi l gradient
of phi l squared.
00:21:36.380 --> 00:21:39.870
I'm going to put everything
that has phi l squared in it.
00:21:39.870 --> 00:21:46.330
I have here t over 2 phi l
squared from t over 2m squared.
00:21:46.330 --> 00:21:50.250
So I have phi l squared.
00:21:50.250 --> 00:21:53.800
I have t over 2.
00:21:53.800 --> 00:21:57.855
Actually I have taken--
I'm going to make mistakes
00:21:57.855 --> 00:22:03.010
unless I put the
one-half over here, too.
00:22:03.010 --> 00:22:09.830
I have another phi l
squared from over here.
00:22:09.830 --> 00:22:14.320
That gets multiplied
by u so I will get here
00:22:14.320 --> 00:22:20.230
plus 12 u m bar squared,
12 rather than 6
00:22:20.230 --> 00:22:21.648
because I divided by 2.
00:22:24.580 --> 00:22:28.660
And then I have a term that is
K over 2 gradient of the vector
00:22:28.660 --> 00:22:29.872
phi t squared.
00:22:32.550 --> 00:22:39.010
And then I have t
over 2 phi t squared
00:22:39.010 --> 00:22:42.980
and then 2 m bar
squared multiplied by u
00:22:42.980 --> 00:22:47.250
becomes 4u m bar squared.
00:22:47.250 --> 00:22:51.246
And there are higher order terms
that I will not write down.
00:22:56.066 --> 00:22:57.012
Yes?
00:22:57.012 --> 00:22:57.512
Did I--
00:22:57.512 --> 00:23:01.380
AUDIENCE: You said
phi l [INAUDIBLE].
00:23:01.380 --> 00:23:02.070
PROFESSOR: Good.
00:23:02.070 --> 00:23:05.780
Yes, so the question
is I immediately
00:23:05.780 --> 00:23:11.330
jumped to second order, so what
happened to the linear term?
00:23:11.330 --> 00:23:14.536
There's a linear term here and
there's a linear term here.
00:23:14.536 --> 00:23:16.400
AUDIENCE: [INAUDIBLE]
00:23:16.400 --> 00:23:18.420
PROFESSOR: Let's
write them down.
00:23:18.420 --> 00:23:21.010
So the coefficient
of phi l would
00:23:21.010 --> 00:23:34.280
be t over 2 m bar plus 4 u
m bar cubed and minimizing
00:23:34.280 --> 00:23:39.080
that expression is setting
this first derivative to zero
00:23:39.080 --> 00:23:43.050
so if you are expanding around
an extreme on the most probable
00:23:43.050 --> 00:23:46.570
state then by
construction you're
00:23:46.570 --> 00:23:50.260
not going to get any terms
that are linear either m phi l
00:23:50.260 --> 00:23:52.822
or m phi t.
00:23:52.822 --> 00:23:55.808
OK?
00:23:55.808 --> 00:23:56.308
Yes?
00:23:56.308 --> 00:23:58.798
AUDIENCE: Is there-- what's
the reason for not including
00:23:58.798 --> 00:24:03.280
a term in our
general probability,
00:24:03.280 --> 00:24:06.525
a term like [INAUDIBLE]
of m squared?
00:24:06.525 --> 00:24:08.750
PROFESSOR: We could.
00:24:08.750 --> 00:24:13.110
He said that
essentially that amount
00:24:13.110 --> 00:24:18.500
over here to an expansion in
powers of the gradient, which
00:24:18.500 --> 00:24:22.270
if I go to Fourier space
this would become q
00:24:22.270 --> 00:24:27.240
squared plus n squared would
be q to the fourth, et cetera.
00:24:27.240 --> 00:24:31.570
If you are looking at more q
or large wavelengths and so
00:24:31.570 --> 00:24:35.410
we are going to focus
on the first few terms.
00:24:35.410 --> 00:24:37.640
But they exist just
as they existed
00:24:37.640 --> 00:24:39.200
for the phonon spectrum.
00:24:39.200 --> 00:24:42.170
We looked at the linear
portion and then realized
00:24:42.170 --> 00:24:44.810
that going away
from q equals to 0
00:24:44.810 --> 00:24:47.000
generates all kinds
of other steps.
00:24:50.740 --> 00:24:51.240
OK?
00:24:51.240 --> 00:24:56.930
I mean, these are very important
things to ask again and again
00:24:56.930 --> 00:25:00.640
to ultimately convince
yourself, because in reality
00:25:00.640 --> 00:25:03.180
that expansion
that I have written
00:25:03.180 --> 00:25:06.380
has an infinity of terms in it.
00:25:06.380 --> 00:25:09.520
You have to always
convince yourself
00:25:09.520 --> 00:25:13.230
that close enough to
the critical point all
00:25:13.230 --> 00:25:16.185
of those other terms
that I don't write down
00:25:16.185 --> 00:25:20.670
are not going to
make any difference.
00:25:20.670 --> 00:25:23.390
OK?
00:25:23.390 --> 00:25:25.000
All right.
00:25:25.000 --> 00:25:27.490
So that's the weight.
00:25:27.490 --> 00:25:31.460
What I'm going to do is
the same thing that we did
00:25:31.460 --> 00:25:33.950
last time for the case
of Goldstone modes,
00:25:33.950 --> 00:25:38.930
et cetera, which is to go
to a Fourier presentation
00:25:38.930 --> 00:25:44.230
so any one of the components, be
the longitudinal or transverse,
00:25:44.230 --> 00:25:52.110
I will write as a sum over q e
to the i q dot, x some Fourier
00:25:52.110 --> 00:25:59.790
component and then
get a square root of v
00:25:59.790 --> 00:26:04.430
just so that the normalization
would look simple.
00:26:04.430 --> 00:26:11.910
And if I substitute for phi
of x in terms of phi of q,
00:26:11.910 --> 00:26:16.525
just as we saw last
time the probability
00:26:16.525 --> 00:26:21.550
will decompose into independent
contribution for each q
00:26:21.550 --> 00:26:24.380
because once you
substitute it here,
00:26:24.380 --> 00:26:30.220
every quadratic term will have
both a sum over x and-- sorry,
00:26:30.220 --> 00:26:32.650
and integral over x
and sums over q and q
00:26:32.650 --> 00:26:37.930
prime, e to the i q plus q prime
x the integration over x forces
00:26:37.930 --> 00:26:42.730
q and q prime to be
the same up to a sign.
00:26:42.730 --> 00:26:46.811
So then we find that the
probability distribution
00:26:46.811 --> 00:26:50.340
as a function of these
Fourier amplitudes
00:26:50.340 --> 00:26:59.140
phi alpha and phi t
decomposes into a product.
00:26:59.140 --> 00:27:03.020
Basically each q mode
is acting independently
00:27:03.020 --> 00:27:05.330
of all the others.
00:27:05.330 --> 00:27:07.570
And also at the
quadratic level we
00:27:07.570 --> 00:27:11.080
see that there is no
crosstalk between transverse
00:27:11.080 --> 00:27:16.690
and longitudinal so we will have
one weight for the transverse,
00:27:16.690 --> 00:27:19.370
one weight for the longitudinal.
00:27:19.370 --> 00:27:22.230
And what's it actually
going to look like?
00:27:22.230 --> 00:27:24.315
Essentially it's going
to look like something
00:27:24.315 --> 00:27:31.776
that is proportional,
let's say, for phi l,
00:27:31.776 --> 00:27:36.190
it is proportional to
phi l of q squared.
00:27:36.190 --> 00:27:42.440
I have here something and
then I have k q squared over 2
00:27:42.440 --> 00:27:46.386
so then a Fourier transform that
I will get k q squared over 2.
00:27:46.386 --> 00:27:52.710
So let's write it in this
fashion, q over 2 q squared
00:27:52.710 --> 00:27:59.400
and then that's something
that's not q dependent.
00:27:59.400 --> 00:28:05.346
And by convention I will
write it as x e l minus 2.
00:28:05.346 --> 00:28:10.070
It has dimensions of
inverse length scale
00:28:10.070 --> 00:28:13.630
because q has dimensions
of inverse length scale
00:28:13.630 --> 00:28:19.840
so I will shortly define
a length so that these two
00:28:19.840 --> 00:28:23.340
terms would have the
same dimensional,
00:28:23.340 --> 00:28:26.080
so c l is defined
in that fashion.
00:28:26.080 --> 00:28:28.956
And similarly I
have an exponential
00:28:28.956 --> 00:28:37.210
that governs k over 2 q
squared plus c t to the minus 2
00:28:37.210 --> 00:28:41.510
phi tilde of t of q squared
and this is a vector
00:28:41.510 --> 00:28:45.152
so there are n minus
1 components there.
00:28:45.152 --> 00:28:46.500
OK?
00:28:46.500 --> 00:28:51.310
And you can see that basically
potentially these two terms,
00:28:51.310 --> 00:28:55.010
c l and c t are different.
00:28:55.010 --> 00:28:59.010
In fact, let's just
write down what they are.
00:28:59.010 --> 00:29:06.340
So this coefficient
K over c l squared
00:29:06.340 --> 00:29:14.645
is defined to be t plus
12 u m bar squared.
00:29:14.645 --> 00:29:15.144
OK?
00:29:18.856 --> 00:29:20.192
Question?
00:29:20.192 --> 00:29:21.500
AUDIENCE: [INAUDIBLE]
00:29:21.500 --> 00:29:23.210
PROFESSOR: OK.
00:29:23.210 --> 00:29:25.170
Now this depends
on whether you are
00:29:25.170 --> 00:29:29.620
for t positive or t
negative-- better be positive
00:29:29.620 --> 00:29:34.140
since I have written it as one
over some positive quantity--
00:29:34.140 --> 00:29:40.850
for t positive, m bar
is 0 so this is just t.
00:29:40.850 --> 00:29:50.490
For t negative, then m bar
squared is minus t over 4 u
00:29:50.490 --> 00:29:56.450
so this becomes minus 3t,
so this becomes minus 2t.
00:29:59.250 --> 00:29:59.750
OK?
00:30:02.910 --> 00:30:12.960
And k over c t squared is
t plus 4u m bar squared.
00:30:12.960 --> 00:30:16.940
It is t for t positive.
00:30:16.940 --> 00:30:20.550
For t negative,
substitute it for m bar
00:30:20.550 --> 00:30:22.304
squared, it will give me 0.
00:30:26.980 --> 00:30:27.480
OK?
00:30:30.130 --> 00:30:33.910
Actually the top one
hopefully you recognize
00:30:33.910 --> 00:30:35.820
or you remember from last time.
00:30:35.820 --> 00:30:39.450
We had exactly this
expression t and minus
00:30:39.450 --> 00:30:44.460
2t when we calculated
the susceptibility.
00:30:44.460 --> 00:30:49.010
This was the inverse
susceptibility.
00:30:49.010 --> 00:30:51.460
In fact, I can be
now more precise
00:30:51.460 --> 00:30:56.570
and call that the inverse of
the longitudinal susceptibility.
00:30:56.570 --> 00:31:01.070
And what we have
here is the inverse
00:31:01.070 --> 00:31:04.780
of the transverse
susceptibility.
00:31:04.780 --> 00:31:06.820
What does that mean?
00:31:06.820 --> 00:31:10.230
Let me remind you what
susceptibility is.
00:31:10.230 --> 00:31:12.560
Susceptibility is
you have a system
00:31:12.560 --> 00:31:15.210
and you put on a
little bit of the field
00:31:15.210 --> 00:31:19.390
and then see how the
magnetization responds.
00:31:19.390 --> 00:31:24.500
If you are in the ordered
phase so that your system is
00:31:24.500 --> 00:31:27.635
spontaneously pointing
in one direction,
00:31:27.635 --> 00:31:30.990
then if you put the
field in this direction,
00:31:30.990 --> 00:31:34.900
you have to climb this
Mexican hat potential
00:31:34.900 --> 00:31:38.680
and you have to pay
a cost to do so.
00:31:38.680 --> 00:31:44.010
Whereas if I put it the field
perpendicular, all that happens
00:31:44.010 --> 00:31:47.090
is that the
magnetization rotates,
00:31:47.090 --> 00:31:51.970
so it can respond without any
cost and that's what this is.
00:31:51.970 --> 00:31:54.280
These are really
the Goldstone modes
00:31:54.280 --> 00:31:58.720
that we were discussing are the
transverse fluctuations that I
00:31:58.720 --> 00:32:00.345
have written before.
00:32:00.345 --> 00:32:03.290
So again, we discussed
last time you
00:32:03.290 --> 00:32:07.060
break a continual symmetry
you will have Goldstone modes
00:32:07.060 --> 00:32:09.830
and these Goldstone
modes are the ones that
00:32:09.830 --> 00:32:15.578
are perpendicular to the average
magnetization, if you like.
00:32:15.578 --> 00:32:16.078
OK?
00:32:22.870 --> 00:32:27.090
So now we have a prediction.
00:32:27.090 --> 00:32:34.370
We say that if I look at
these phi phi fluctuations,
00:32:34.370 --> 00:32:37.616
I can pick a particular
q, let's say,
00:32:37.616 --> 00:32:40.870
and here I have to
put q star in order
00:32:40.870 --> 00:32:45.810
to get something
that is non-zero.
00:32:45.810 --> 00:32:51.240
Actually that's put here q prime
and then pick two different
00:32:51.240 --> 00:32:54.556
and this is alpha and beta.
00:32:54.556 --> 00:32:58.910
Well, if I look at this
average since the weight is
00:32:58.910 --> 00:33:03.020
the product of contributions
from different q's the answer
00:33:03.020 --> 00:33:08.270
will be 0 unless q and
q prime add up to 0.
00:33:11.280 --> 00:33:14.420
And if I'm looking
at the same q,
00:33:14.420 --> 00:33:17.990
I better make sure that I'm
looking at the same component
00:33:17.990 --> 00:33:21.680
because the longitudinal and
transverse component or any
00:33:21.680 --> 00:33:25.410
of the n minus 1 transverse
components among each other
00:33:25.410 --> 00:33:30.090
have completely
independent Gaussian rates.
00:33:30.090 --> 00:33:34.790
If I'm now looking at the same
Gaussian, for the same Gaussian
00:33:34.790 --> 00:33:38.410
I can just immediately
read off it's variance
00:33:38.410 --> 00:33:46.600
which is K q squared plus
whatever the appropriate c is
00:33:46.600 --> 00:33:49.060
for that direction
whether it's c l
00:33:49.060 --> 00:33:53.660
or c t potentially
would make a difference.
00:33:53.660 --> 00:33:54.160
OK?
00:33:57.040 --> 00:33:57.730
Right.
00:33:57.730 --> 00:34:04.360
So now we have a prediction
for our experimentals.
00:34:04.360 --> 00:34:09.280
I said that these guys
can go and measure
00:34:09.280 --> 00:34:13.090
the scattering as a function
of angle at small angle
00:34:13.090 --> 00:34:16.750
and they can fit how
much is scattered
00:34:16.750 --> 00:34:23.320
as you go as a function of angle
and fit it as a function of q.
00:34:23.320 --> 00:34:28.600
So we predict that if
they look at something
00:34:28.600 --> 00:34:33.150
that's say like phi l
squared, if you're thinking
00:34:33.150 --> 00:34:35.870
about the liquid gas
system that's really
00:34:35.870 --> 00:34:39.269
the only thing that
you have because there
00:34:39.269 --> 00:34:43.530
is no transverse component if
you have a Scalar variable.
00:34:43.530 --> 00:34:51.320
We claim that if you go and look
at those critical opalescent
00:34:51.320 --> 00:34:57.270
pictures that we saw,
and do it more precisely
00:34:57.270 --> 00:35:02.145
and see what happens as a
function of the scattered wave
00:35:02.145 --> 00:35:05.860
number q, that you
will get a shape that
00:35:05.860 --> 00:35:10.920
is 1 over q squared
plus c to the minus 2.
00:35:10.920 --> 00:35:19.040
This kind of shape that
is called a Lorentzian
00:35:19.040 --> 00:35:23.210
is indeed what you commonly
see for all kinds of scattering
00:35:23.210 --> 00:35:25.420
line shapes.
00:35:25.420 --> 00:35:26.280
OK?
00:35:26.280 --> 00:35:30.230
So we have a prediction.
00:35:30.230 --> 00:35:32.440
Of course, the
reason that it works
00:35:32.440 --> 00:35:36.160
is because in principle we
know that this series will have
00:35:36.160 --> 00:35:37.825
higher order terms
as we discussed
00:35:37.825 --> 00:35:41.150
like q to the fourth, q
to the sixth, et cetera,
00:35:41.150 --> 00:35:44.960
but they fall way down
here where you're not
00:35:44.960 --> 00:35:48.910
going to be seeing
all that much anyway.
00:35:48.910 --> 00:35:54.440
Now the place where
this curve turns around
00:35:54.440 --> 00:36:01.860
from being something that
is dominated by 1 over K c
00:36:01.860 --> 00:36:09.410
to the minus 2 to something
that falls off as 1
00:36:09.410 --> 00:36:14.010
over K q squared or
maybe even faster,
00:36:14.010 --> 00:36:20.160
the borderline is this inverse
length scale that we indicated,
00:36:20.160 --> 00:36:21.464
c l minus 1.
00:36:25.550 --> 00:36:31.480
So now what happens if I go
closer to the phase transition
00:36:31.480 --> 00:36:33.160
point?
00:36:33.160 --> 00:36:36.930
As I go closer to the
phase transition point,
00:36:36.930 --> 00:36:44.850
t goes to zero, this c
inverse goes towards zero.
00:36:44.850 --> 00:36:50.110
So if this is for some
temperature above Tc
00:36:50.110 --> 00:36:54.040
and I go to some lower
temperature, then what happens
00:36:54.040 --> 00:37:05.460
is that the curve will start
higher and then [INAUDIBLE].
00:37:08.998 --> 00:37:11.960
Yeah?
00:37:11.960 --> 00:37:14.900
Actually it doesn't
cross the other curve
00:37:14.900 --> 00:37:17.210
which is what I wrote down.
00:37:17.210 --> 00:37:24.550
Just because it starts
higher it can bend and go
00:37:24.550 --> 00:37:29.090
and join this curve
at a further point.
00:37:29.090 --> 00:37:34.970
Eventually when you go through
exactly the critical point then
00:37:34.970 --> 00:37:40.010
you get the union of all of
these curves, which is a 1
00:37:40.010 --> 00:37:43.650
over q squared type of curve.
00:37:43.650 --> 00:37:49.660
So right at the point
where t equals to 0
00:37:49.660 --> 00:37:54.270
the prediction is that
the Lorentzian shape,
00:37:54.270 --> 00:37:58.870
the coefficient that appears
in front of q squared
00:37:58.870 --> 00:38:02.640
vanishes you will see
one over q squared.
00:38:02.640 --> 00:38:04.666
OK?
00:38:04.666 --> 00:38:08.060
Now the results of experiments
in reality that are very
00:38:08.060 --> 00:38:12.780
happily fitted, the
Lorentzian, when you're away
00:38:12.780 --> 00:38:16.000
from the critical
point they claim
00:38:16.000 --> 00:38:18.570
that when you are exactly
at the critical point,
00:38:18.570 --> 00:38:21.320
it's not quite 1 over q squared.
00:38:21.320 --> 00:38:23.572
Seems to be slightly different.
00:38:26.940 --> 00:38:30.670
At Tc, the scattering
appears to be
00:38:30.670 --> 00:38:37.760
more similar to 1 over q
to 2 minus a small amount.
00:38:37.760 --> 00:38:41.740
That's where another
critical exponent theta
00:38:41.740 --> 00:38:46.240
is introduced so that's another
thing that ultimately you
00:38:46.240 --> 00:38:49.766
have to try to figure
out and understand.
00:38:49.766 --> 00:38:52.580
OK?
00:38:52.580 --> 00:38:59.410
Of course I drew the curves
for the longitudinal component.
00:38:59.410 --> 00:39:02.410
If I look at the curves for
the transverse components,
00:39:02.410 --> 00:39:08.000
and again, by appropriate choice
of spin polarized neutrons
00:39:08.000 --> 00:39:11.190
you can decompose
different components
00:39:11.190 --> 00:39:14.180
of scattering from
the magnetization
00:39:14.180 --> 00:39:17.160
field of a piece of
iron, for example.
00:39:17.160 --> 00:39:19.350
If you are above
Tc, there is really
00:39:19.350 --> 00:39:23.410
no difference between
longitudinal and transverse
00:39:23.410 --> 00:39:26.500
because there is no
direction that is selected.
00:39:26.500 --> 00:39:30.450
And you can see that the forms
that you will get above Tc
00:39:30.450 --> 00:39:32.990
would be exactly the same.
00:39:32.990 --> 00:39:37.020
When you go below Tc, that's
where the difference appears
00:39:37.020 --> 00:39:40.670
because the length
scale that would appear
00:39:40.670 --> 00:39:44.380
for the longitudinal
parameters would be finite
00:39:44.380 --> 00:39:48.100
and it corresponds
to having to push
00:39:48.100 --> 00:39:53.840
the magnetization above this
bottom of the Mexican hat
00:39:53.840 --> 00:39:59.060
potential whereas there is no
cost in the other direction.
00:39:59.060 --> 00:40:03.510
So if you can probe in
the fluctuations that
00:40:03.510 --> 00:40:08.040
would correspond to
these Goldstone modes,
00:40:08.040 --> 00:40:13.282
you would see the 1 over q
squared type of behavior.
00:40:13.282 --> 00:40:15.580
OK?
00:40:15.580 --> 00:40:21.410
So the story that we were
talking about last time around
00:40:21.410 --> 00:40:23.920
about the Goldstone
modes and they're
00:40:23.920 --> 00:40:29.620
fluctuating a lot because of
their low cost also certainly
00:40:29.620 --> 00:40:31.600
remains in this case.
00:40:31.600 --> 00:40:34.630
We have now explicitly
separated out
00:40:34.630 --> 00:40:37.620
the longitudinal
fluctuations that
00:40:37.620 --> 00:40:41.150
are finite because
they are controlled
00:40:41.150 --> 00:40:46.300
by this stiffness of going up
the bottom of the potential
00:40:46.300 --> 00:40:49.380
whereas there is no
stiffness associated
00:40:49.380 --> 00:40:55.430
with the transverse ones.
00:40:55.430 --> 00:40:57.900
OK?
00:40:57.900 --> 00:40:59.420
All right, so good.
00:40:59.420 --> 00:41:03.120
So we've talked about
some of the things that
00:41:03.120 --> 00:41:07.806
are experimentally observed.
00:41:07.806 --> 00:41:08.610
Any questions?
00:41:14.210 --> 00:41:21.530
Now we looked at things here in
the Fourier space corresponding
00:41:21.530 --> 00:41:25.350
to this momentum transfer in
the scattering experiment,
00:41:25.350 --> 00:41:30.030
but we can also ask about what
is happening in physical space.
00:41:30.030 --> 00:41:33.270
That is if I have a
fluctuation at one point,
00:41:33.270 --> 00:41:36.690
how much does the influence
of that fluctuation
00:41:36.690 --> 00:41:38.960
propagate in space?
00:41:38.960 --> 00:41:46.070
So for that I need to calculate
things like phi-- let's say,
00:41:46.070 --> 00:41:49.270
do we need to put an
invert, why not?--
00:41:49.270 --> 00:41:54.580
phi l of x phi l of x prime.
00:41:54.580 --> 00:41:59.293
Let's say we want to
calculate this quantity.
00:41:59.293 --> 00:41:59.792
OK?
00:42:02.580 --> 00:42:05.400
Now I can certainly
decompose phi l
00:42:05.400 --> 00:42:09.970
of x in terms of these
Fourier components.
00:42:09.970 --> 00:42:11.510
And so what do I get?
00:42:11.510 --> 00:42:18.490
I will get a sum over q-- maybe
I should write it in one case
00:42:18.490 --> 00:42:25.350
explicitly, q and q
prime e to i cube of x e
00:42:25.350 --> 00:42:30.130
to the i q prime the x prime.
00:42:30.130 --> 00:42:33.770
Two factors of root
3 giving me the V
00:42:33.770 --> 00:42:42.300
and then I have phi l
of q phi l of q prime
00:42:42.300 --> 00:42:48.484
and the expectation
value will go over here.
00:42:48.484 --> 00:42:53.110
Now we said that the
different q's and q primes
00:42:53.110 --> 00:42:55.230
are uncorrelated.
00:42:55.230 --> 00:43:00.130
So here I immediately
will have a delta function
00:43:00.130 --> 00:43:06.530
mq plus q prime but if I'm
looking at the same q and q
00:43:06.530 --> 00:43:14.150
prime, I have this factor
of 1 over K q squared
00:43:14.150 --> 00:43:20.940
plus c to the minus
2-- c l to the minus 2.
00:43:20.940 --> 00:43:22.830
OK?
00:43:22.830 --> 00:43:31.580
So then the whole thing becomes
due to the delta function
00:43:31.580 --> 00:43:46.410
the sum over 1 q of 1
over V each with i q
00:43:46.410 --> 00:43:52.800
dot x minus x prime because
q prime was 2 minus q.
00:43:52.800 --> 00:44:00.598
And then I have K 2 squared
plus c l to the minus.
00:44:00.598 --> 00:44:03.560
OK?
00:44:03.560 --> 00:44:07.350
So then I go to the continuum
limit of a large size,
00:44:07.350 --> 00:44:10.400
the sum over q gets
replaced with integral
00:44:10.400 --> 00:44:16.790
over q d times the
density of state.
00:44:16.790 --> 00:44:19.620
The V's disappear,
I will have a factor
00:44:19.620 --> 00:44:30.228
of 2 pi to the d and what I have
is the Fourier transform of k q
00:44:30.228 --> 00:44:34.220
squared plus c to the
minus-- l to the minus 2.
00:44:38.220 --> 00:44:48.870
And I will write this
as minus 1 over K
00:44:48.870 --> 00:44:54.470
a function I that
depends on d dimension
00:44:54.470 --> 00:44:58.410
and clearly depends
on the separation
00:44:58.410 --> 00:45:04.960
x minus x prime at the
correlation length c l.
00:45:04.960 --> 00:45:10.490
Why I said correlation length
shortly becomes apparent.
00:45:10.490 --> 00:45:16.825
So I introduce a function I
d which depends on x and c
00:45:16.825 --> 00:45:27.900
to be minus integral over q 2 pi
to the d Fourier transform of q
00:45:27.900 --> 00:45:30.730
squared plus c to the minus 2.
00:45:33.310 --> 00:45:37.270
If that c was not there
that's the integral that we
00:45:37.270 --> 00:45:39.890
did last time and
it was the Coulomb
00:45:39.890 --> 00:45:42.790
potential in d dimension.
00:45:42.790 --> 00:45:46.970
So this presumably
is related to that.
00:45:46.970 --> 00:45:51.720
And we can use the same
trick that we employed
00:45:51.720 --> 00:45:56.230
to make it explicit
last time around.
00:45:56.230 --> 00:46:03.040
We can take the Laplacian of
this potential i d and what
00:46:03.040 --> 00:46:06.910
happens is that I will
bring two factors of i
00:46:06.910 --> 00:46:10.370
q squared so the
minus goes away.
00:46:10.370 --> 00:46:15.210
I will have the integral
d d cubed 2 pi to the d.
00:46:15.210 --> 00:46:19.310
I will have a q
squared, denominator
00:46:19.310 --> 00:46:22.660
is q squared plus
c to the minus 2.
00:46:22.660 --> 00:46:28.420
I add and subtract the c to
the minus 2 to the numerator
00:46:28.420 --> 00:46:30.652
and I have to Fourier transform.
00:46:34.620 --> 00:46:37.600
OK?
00:46:37.600 --> 00:46:44.370
The first part if I divide by
the denominator is simply 1.
00:46:44.370 --> 00:46:46.370
Integral of Fourier
transform of 1
00:46:46.370 --> 00:46:48.038
will give me a delta function.
00:46:52.920 --> 00:46:58.470
And then what I have is minus
c squared, the same integral
00:46:58.470 --> 00:47:03.576
that I used to define i d.
00:47:03.576 --> 00:47:12.960
So this becomes plus i d
of x divided by c squared.
00:47:12.960 --> 00:47:15.430
OK?
00:47:15.430 --> 00:47:21.010
So whereas in the
absence of c you have
00:47:21.010 --> 00:47:27.934
the potential do to a
charge, the presence of c
00:47:27.934 --> 00:47:32.080
adds this additional
term that corresponds
00:47:32.080 --> 00:47:34.390
to some kind of a damping.
00:47:34.390 --> 00:47:37.560
So this equation
you probably have
00:47:37.560 --> 00:47:42.120
seen in the context of
screened Coulomb interaction
00:47:42.120 --> 00:47:44.700
and giving rise to the
[INAUDIBLE] potential
00:47:44.700 --> 00:47:46.780
in three dimensions.
00:47:46.780 --> 00:47:50.240
We would like to look
at it in d dimension
00:47:50.240 --> 00:47:54.180
so that we know what the
behavior is in general.
00:47:54.180 --> 00:47:54.680
OK?
00:48:01.400 --> 00:48:11.660
So again what I'm looking
at is the potential
00:48:11.660 --> 00:48:16.420
that is due to a
charge at the origin
00:48:16.420 --> 00:48:20.620
so this idea of x
in principle only
00:48:20.620 --> 00:48:26.850
depends on the magnitude of x
and not on the direction of it.
00:48:26.850 --> 00:48:30.870
It is something that has
general spherical symmetry
00:48:30.870 --> 00:48:31.825
in d dimensions.
00:48:34.580 --> 00:48:44.660
So I use that fact
of spherical symmetry
00:48:44.660 --> 00:48:48.600
to write down what the
expression for the Laplacian
00:48:48.600 --> 00:48:49.980
is.
00:48:49.980 --> 00:48:52.360
OK?
00:48:52.360 --> 00:48:54.740
We can again use Gauss's
law if you've forgotten
00:48:54.740 --> 00:48:58.580
or whatever but in the
presence of spherical symmetry
00:48:58.580 --> 00:49:08.000
the general expression for
Laplacian in d dimension
00:49:08.000 --> 00:49:11.444
is this.
00:49:11.444 --> 00:49:12.430
OK?
00:49:12.430 --> 00:49:16.330
So if this d was
equal to 1 it would
00:49:16.330 --> 00:49:18.820
be a simple second derivative.
00:49:18.820 --> 00:49:23.350
And in higher dimensions you
would have additional factors
00:49:23.350 --> 00:49:28.300
if you basically apply Gauss's
law to shares around here.
00:49:28.300 --> 00:49:31.340
You can very easily
convince yourself of that.
00:49:31.340 --> 00:49:33.730
This is some kind
of an aerial term
00:49:33.730 --> 00:49:37.390
that comes in d
minus 1 dimension.
00:49:37.390 --> 00:49:43.980
And then I can write this as
either the second derivative
00:49:43.980 --> 00:49:47.845
if d by the x acts on
this, the x to the minus 1
00:49:47.845 --> 00:49:51.830
disappears or if it
acts on this one,
00:49:51.830 --> 00:49:54.730
it will gives me d
minus 1 x to the d
00:49:54.730 --> 00:50:02.130
minus 2, x to the d minus 1
gives me an x, the i by d x.
00:50:02.130 --> 00:50:05.460
So the equation
that I have to solve
00:50:05.460 --> 00:50:13.129
is this object equals to i over
c squared plus a delta function
00:50:13.129 --> 00:50:13.629
[INAUDIBLE].
00:50:18.398 --> 00:50:18.898
OK?
00:50:24.180 --> 00:50:29.010
Now if you vary one
dimension you wouldn't
00:50:29.010 --> 00:50:33.310
have this term at all and you
would have-- except that x
00:50:33.310 --> 00:50:36.880
equals to 0, the second
derivative proportional
00:50:36.880 --> 00:50:40.040
to the function
divided by c squared.
00:50:40.040 --> 00:50:41.690
So you will
immediately write a way
00:50:41.690 --> 00:50:48.060
x equals to 0 that the answer
is e to the minus x over c.
00:50:48.060 --> 00:50:50.290
OK?
00:50:50.290 --> 00:50:51.880
Actually proportional
because you
00:50:51.880 --> 00:50:54.380
have to fit out with the
amplitude, et cetera.
00:50:57.630 --> 00:51:03.650
Now in higher
dimensions what happens
00:51:03.650 --> 00:51:06.590
is that this solution
gets modified,
00:51:06.590 --> 00:51:11.430
falls off with some
additional x to the p
00:51:11.430 --> 00:51:13.920
but we have to be somewhat
careful with this.
00:51:13.920 --> 00:51:19.210
So let's look at this a
little bit more closely.
00:51:19.210 --> 00:51:24.810
If I were to substitute this
ansatz into this expression,
00:51:24.810 --> 00:51:26.980
what would happen?
00:51:26.980 --> 00:51:30.360
What I need to do is to take
the first and the second
00:51:30.360 --> 00:51:30.860
derivative.
00:51:35.800 --> 00:51:39.880
Now if I take the
first derivative,
00:51:39.880 --> 00:51:42.940
the derivative either
acts on this factor,
00:51:42.940 --> 00:51:46.490
gives me a factor of
minus 1 over c and then
00:51:46.490 --> 00:51:50.720
the exponential back,
so I can get the i back.
00:51:50.720 --> 00:51:53.100
If I had an exponential
I take a derivative,
00:51:53.100 --> 00:51:56.870
I will get just minus
1 over psi exponential.
00:51:56.870 --> 00:52:02.690
If I act on x to the
minus b I will get minus p
00:52:02.690 --> 00:52:06.870
x to the p minus 1, which is
different from the original
00:52:06.870 --> 00:52:10.565
solution by a
factor of p over x.
00:52:10.565 --> 00:52:13.547
OK?
00:52:13.547 --> 00:52:22.600
If I now take two derivatives I
can take the second derivative
00:52:22.600 --> 00:52:26.380
on I itself and then d
I by d x will give me I
00:52:26.380 --> 00:52:29.420
back with this factor.
00:52:29.420 --> 00:52:38.390
So I will get 1 over c squared
plus 2 P c x plus P squared
00:52:38.390 --> 00:52:44.115
over x squared with
I but that's not
00:52:44.115 --> 00:52:48.125
the whole story because the
derivative can also leave I
00:52:48.125 --> 00:52:52.420
aside and act on P over
x, which if it does so,
00:52:52.420 --> 00:52:55.500
it will get P over
x squared so that
00:52:55.500 --> 00:52:58.960
will be an additional term here.
00:52:58.960 --> 00:53:02.340
So that's the second derivative.
00:53:02.340 --> 00:53:09.120
So now what I have done
is I have evaluated
00:53:09.120 --> 00:53:12.870
with this ansatz the
terms that should appear
00:53:12.870 --> 00:53:16.390
in that equation of a from
x equals to 0, so let's
00:53:16.390 --> 00:53:17.740
substitute it.
00:53:17.740 --> 00:53:22.010
Everything now I have is
proportional to I so I just
00:53:22.010 --> 00:53:23.690
forget about the I.
00:53:23.690 --> 00:53:26.450
I have the second
derivative 1 over c
00:53:26.450 --> 00:53:31.960
squared plus 2 P
divided by x c plus p p
00:53:31.960 --> 00:53:37.130
plus 1 divided by x squared.
00:53:37.130 --> 00:53:40.300
And then I have d minus
1, the first derivative,
00:53:40.300 --> 00:53:50.720
so I have minus d minus 1 over
c minus d minus 1 p over x.
00:53:50.720 --> 00:53:53.160
Both of these terms get
an additional factor
00:53:53.160 --> 00:53:58.650
of x because of here so I
will get x c and x squared
00:53:58.650 --> 00:54:03.050
and what I have on the right
hand side from the origin
00:54:03.050 --> 00:54:04.710
is I over c squared.
00:54:04.710 --> 00:54:07.575
Divide by the I, I
have 1 over c squared.
00:54:12.430 --> 00:54:16.930
Now if I'm moving away
from x I can organize
00:54:16.930 --> 00:54:20.080
things in powers of 1 over x.
00:54:20.080 --> 00:54:23.110
The most important
term is the constant
00:54:23.110 --> 00:54:27.740
and clearly you can see
that I chose the decay
00:54:27.740 --> 00:54:31.920
constant of the
exponential correctly as
00:54:31.920 --> 00:54:36.590
evidenced by the
absence or removal of 1
00:54:36.590 --> 00:54:41.210
over the constant
term on the two sides.
00:54:41.210 --> 00:54:45.040
But now I have two terms,
two types of terms.
00:54:45.040 --> 00:54:50.490
Terms that are proportional
to x squared and terms
00:54:50.490 --> 00:54:54.650
that are proportional
to x psi and there's
00:54:54.650 --> 00:55:01.860
no way that I can simultaneously
satisfy both of these.
00:55:01.860 --> 00:55:07.170
So the assumption that the
solution of this equation
00:55:07.170 --> 00:55:10.606
is a single exponential
divided by a power law
00:55:10.606 --> 00:55:13.950
is in fact not correct.
00:55:13.950 --> 00:55:17.910
But it can be correct
in two regions.
00:55:17.910 --> 00:55:26.980
So for x that is much
less than psi then
00:55:26.980 --> 00:55:35.130
the more important term is
the 1 over x squared part.
00:55:35.130 --> 00:55:38.540
For x but going towards
0 the 1 over x squared
00:55:38.540 --> 00:55:41.560
is more important than 1 over x.
00:55:41.560 --> 00:55:44.560
OK so then what I do is
I will match these two
00:55:44.560 --> 00:55:49.300
terms and those two terms that
are 1 over x squared tell me
00:55:49.300 --> 00:55:59.435
that P p plus 1
should be P d minus 1.
00:55:59.435 --> 00:56:01.770
OK?
00:56:01.770 --> 00:56:06.340
And that immediately getting
rid of the p's tells me
00:56:06.340 --> 00:56:13.485
that the P in this
regime is d minus 2.
00:56:13.485 --> 00:56:14.762
OK?
00:56:14.762 --> 00:56:18.320
Now the d minus 2
you recall is what
00:56:18.320 --> 00:56:21.723
we had for the
Coulomb potential.
00:56:21.723 --> 00:56:22.670
Right?
00:56:22.670 --> 00:56:27.080
So basically at
short distances you
00:56:27.080 --> 00:56:31.560
are still not screened
by this additional term.
00:56:31.560 --> 00:56:36.490
You don't see its effect
and you get essentially
00:56:36.490 --> 00:56:45.930
the standard Coulomb potential
whereas if you are away
00:56:45.930 --> 00:56:50.520
what you get is that you
have to match the terms that
00:56:50.520 --> 00:56:53.900
are proportional to x c because
they're more important than 1
00:56:53.900 --> 00:56:55.300
over x squared.
00:56:55.300 --> 00:56:59.482
And there you get that
2 P should be d minus 1
00:56:59.482 --> 00:57:02.990
or P should be d minus 1 over 2.
00:57:06.448 --> 00:57:09.420
OK?
00:57:09.420 --> 00:57:15.800
So let's just plot that
function over here.
00:57:15.800 --> 00:57:23.260
So if I plot this function as
a function of the separation x
00:57:23.260 --> 00:57:26.630
and it only depends
on the magnitude,
00:57:26.630 --> 00:57:29.790
in fact, what I
should plot is minus i
00:57:29.790 --> 00:57:32.170
d because it's
the minus i d that
00:57:32.170 --> 00:57:37.900
depends on the
fluctuations once I divide
00:57:37.900 --> 00:57:42.250
by K. I find that
it has two regimes.
00:57:42.250 --> 00:57:47.530
Let's say above two dimensions
you have one regime that
00:57:47.530 --> 00:57:51.040
is a simple Coulomb
type of potential
00:57:51.040 --> 00:57:52.950
and the Coulomb
potential last time
00:57:52.950 --> 00:57:55.260
actually we normalized properly.
00:57:55.260 --> 00:58:03.600
We saw that it is x to the
2 minus d S d d minus 2.
00:58:08.550 --> 00:58:11.160
The e to x into the
minus x over c I
00:58:11.160 --> 00:58:12.910
can in fact ignore
in this regime
00:58:12.910 --> 00:58:15.870
because I'm at distance x
that is much less than c
00:58:15.870 --> 00:58:20.010
so the exponential term
has not kicked in yet
00:58:20.010 --> 00:58:25.950
whereas I go at large distances
and the exponential term does
00:58:25.950 --> 00:58:27.360
kick in.
00:58:27.360 --> 00:58:32.230
So the overall behavior is
e to the minus x over c.
00:58:32.230 --> 00:58:36.330
That's the most dominant
behavior that you have.
00:58:36.330 --> 00:58:38.570
On top of that, we
have a power log
00:58:38.570 --> 00:58:42.857
which is x to the power
of d minus 1 over 2.
00:58:45.779 --> 00:58:53.350
Now those of you who know
what the screened Coulomb
00:58:53.350 --> 00:58:58.090
potential is know that the
screened Coulomb potential
00:58:58.090 --> 00:59:02.320
in three dimensions is the 1
over r, the Coulomb potential,
00:59:02.320 --> 00:59:05.070
and you put an exponential
on top of that.
00:59:05.070 --> 00:59:07.030
There is no difference
in the powers
00:59:07.030 --> 00:59:09.270
that you have whether
or not you are
00:59:09.270 --> 00:59:12.575
smaller than this
correlation length or larger.
00:59:12.575 --> 00:59:16.570
You can check here, if
I put d equals to 3,
00:59:16.570 --> 00:59:21.350
this becomes a 1 over x and
this becomes a 1 over x.
00:59:21.350 --> 00:59:24.020
So it's just an accident
of three dimensions
00:59:24.020 --> 00:59:26.220
that the screened
Coulomb potential
00:59:26.220 --> 00:59:29.470
is the 1 over r with
an exponential on top.
00:59:29.470 --> 00:59:35.160
In general dimensions you
have different powers.
00:59:35.160 --> 00:59:37.270
But having different
powers also means
00:59:37.270 --> 00:59:41.230
that somehow the
amplitude that goes over
00:59:41.230 --> 00:59:47.130
here has to carry dimensions so
that it can be matched to what
00:59:47.130 --> 00:59:50.910
we have here at
this distance of C.
00:59:50.910 --> 00:59:54.570
And so if I try to match
those terms, roughly
00:59:54.570 --> 00:59:58.000
when you're at order of c,
what I would do is I would put
00:59:58.000 --> 01:00:04.576
s d d minus 2 and c to
the 3 minus d over 2.
01:00:04.576 --> 01:00:09.610
And now you can check
that the two expressions
01:00:09.610 --> 01:00:12.840
will have the right dimension
and will match roughly
01:00:12.840 --> 01:00:16.310
at order of c.
01:00:16.310 --> 01:00:18.340
OK?
01:00:18.340 --> 01:00:29.170
So essentially what it says is
that if I ask in my system what
01:00:29.170 --> 01:00:31.810
is the nature of
these fluctuations,
01:00:31.810 --> 01:00:35.180
how correlated they
are, they would
01:00:35.180 --> 01:00:40.830
know to be more or less the same
although falling off as if you
01:00:40.830 --> 01:00:42.700
were at the critical
point because we said
01:00:42.700 --> 01:00:46.160
that the critical point or
when you have Goldstone modes,
01:00:46.160 --> 01:00:49.920
you have just this term.
01:00:49.920 --> 01:00:52.380
But then they know that
you are not exactly
01:00:52.380 --> 01:00:56.640
sitting at the
critical point and then
01:00:56.640 --> 01:00:59.850
they are no longer correlated.
01:00:59.850 --> 01:01:03.400
So basically there
is this length scale
01:01:03.400 --> 01:01:05.060
that we also saw
when we were looking
01:01:05.060 --> 01:01:07.720
at these critical
opalescence and we
01:01:07.720 --> 01:01:11.380
were seeing things that
were moving together.
01:01:11.380 --> 01:01:14.780
That length scale where
things are moving together
01:01:14.780 --> 01:01:19.240
is this parameter c that
we have defined over here.
01:01:19.240 --> 01:01:27.420
So what we have is-- where
do we want to put it?
01:01:27.420 --> 01:01:28.330
Let's put it here.
01:01:45.610 --> 01:01:55.800
A correlation length
which measures
01:01:55.800 --> 01:02:01.330
the extent to which things
are fluctuating together,
01:02:01.330 --> 01:02:03.920
although when I'm saying
fluctuating together,
01:02:03.920 --> 01:02:07.990
they are still correlations
that are falling off
01:02:07.990 --> 01:02:10.630
but they're not falling
off exponentially.
01:02:10.630 --> 01:02:12.800
They start to fall
off exponentially
01:02:12.800 --> 01:02:16.830
when you are beyond
this length scale c.
01:02:16.830 --> 01:02:20.670
And we have the formula for c.
01:02:20.670 --> 01:02:23.210
So what we find
is that if I were
01:02:23.210 --> 01:02:29.450
to invert that, for
example, what I find for c l
01:02:29.450 --> 01:02:44.530
as a function of t is that it is
simply square root of k over t
01:02:44.530 --> 01:02:48.320
when I am on the
t positive side.
01:02:48.320 --> 01:02:50.890
When I go to the
t negative side,
01:02:50.890 --> 01:03:02.296
it just becomes square
root of K minus 2 t.
01:03:02.296 --> 01:03:02.796
OK?
01:03:05.670 --> 01:03:14.480
So this correlation length
I indicated we could state
01:03:14.480 --> 01:03:17.980
has behavior close
to a transition,
01:03:17.980 --> 01:03:20.310
there's a divergence.
01:03:20.310 --> 01:03:25.190
We can parametrize those
divergences through something
01:03:25.190 --> 01:03:32.510
like t minus Tc to exponent
u, potentially different
01:03:32.510 --> 01:03:35.530
on the two sides
of the transition.
01:03:35.530 --> 01:03:38.630
But this t is simply
proportional to the real t
01:03:38.630 --> 01:03:43.520
minus Tc so we
conclude that u plus is
01:03:43.520 --> 01:03:47.190
the same as u minus we've
just indicated by u,
01:03:47.190 --> 01:03:48.310
should be one-half.
01:03:52.130 --> 01:03:56.630
The amplitudes themselves
depend on all kinds of things.
01:03:56.630 --> 01:03:58.840
We don't know much about them.
01:03:58.840 --> 01:04:01.850
But we can see that
the amplitude ratio B
01:04:01.850 --> 01:04:09.220
plus over B minus, if I
were to divide those two
01:04:09.220 --> 01:04:13.130
the ratio of those
two is universal,
01:04:13.130 --> 01:04:15.940
it gives me a factor
of square root of 2.
01:04:21.590 --> 01:04:23.760
OK?
01:04:23.760 --> 01:04:29.810
If I were to plot
c t, for example,
01:04:29.810 --> 01:04:36.340
on the high temperature side c t
and c l are of course the same.
01:04:36.340 --> 01:04:40.040
On the low temperature side, we
said that the Goldstone modes
01:04:40.040 --> 01:04:42.170
have these long
range correlations.
01:04:42.170 --> 01:04:46.260
They fall off or grow according
to the Coulomb potential
01:04:46.260 --> 01:04:49.280
but there is no length
scale so in some sense
01:04:49.280 --> 01:04:52.119
the correlation length
for the transverse modes
01:04:52.119 --> 01:04:52.910
is always infinity.
01:04:58.530 --> 01:04:59.030
OK.
01:05:13.430 --> 01:05:17.550
Now actually in the
second lecture what I said
01:05:17.550 --> 01:05:21.800
was that the fact
that the response
01:05:21.800 --> 01:05:25.660
function such as
susceptibility diverges
01:05:25.660 --> 01:05:29.260
immediately tells you
that there have to be long
01:05:29.260 --> 01:05:31.470
range correlations,
so we had predicted
01:05:31.470 --> 01:05:35.320
before that c has to diverge.
01:05:35.320 --> 01:05:40.530
But we were not sufficiently
precise about the way
01:05:40.530 --> 01:05:44.471
that it does, so
let's try to do that.
01:05:44.471 --> 01:05:44.970
Let's see.
01:05:44.970 --> 01:05:50.850
A relationship more
precisely with susceptibility
01:05:50.850 --> 01:05:55.900
and these correlation
lengths, so what we said more
01:05:55.900 --> 01:06:00.560
generally was that the
susceptibilities up
01:06:00.560 --> 01:06:03.540
to various factors of data,
et cetera, that are not
01:06:03.540 --> 01:06:11.330
that important are related to
the integrated magnetization
01:06:11.330 --> 01:06:15.130
to magnetization
connected correlation.
01:06:15.130 --> 01:06:17.505
So basically, what
I have to do is
01:06:17.505 --> 01:06:22.102
to look at m minus
its average at x,
01:06:22.102 --> 01:06:25.400
m minus its average at
some other point, which
01:06:25.400 --> 01:06:27.370
means that what I'm
really looking at
01:06:27.370 --> 01:06:29.387
is the phi phi averages.
01:06:34.229 --> 01:06:34.728
OK?
01:06:37.870 --> 01:06:41.850
Now what we have
shown right now is
01:06:41.850 --> 01:06:47.236
that these averages
are significant.
01:06:50.380 --> 01:06:54.480
These phi phi correlations
are significant
01:06:54.480 --> 01:06:59.720
only over a distance that
is this correlation length
01:06:59.720 --> 01:07:02.010
and then they die off.
01:07:02.010 --> 01:07:06.450
So we could basically as
far as scaling and things
01:07:06.450 --> 01:07:11.520
like that is concerned
terminate this integration at c.
01:07:11.520 --> 01:07:18.350
And that when we are looking at
distances that are below that,
01:07:18.350 --> 01:07:20.560
you don't see the effect
of the exponential,
01:07:20.560 --> 01:07:24.170
you just see the
Coulomb power law
01:07:24.170 --> 01:07:28.060
so you would see
here fluctuations
01:07:28.060 --> 01:07:35.850
that decay as x
to the 2 minus d.
01:07:35.850 --> 01:07:37.850
Right?
01:07:37.850 --> 01:07:43.880
So essentially what you're doing
is integrating x to the 2 minus
01:07:43.880 --> 01:07:46.790
d in d dimension of space.
01:07:46.790 --> 01:07:51.590
So you can see that
immediately gets related
01:07:51.590 --> 01:07:55.050
to the square of the
correlation length.
01:07:55.050 --> 01:07:59.910
X to the minus d n d d
x, the d part vanishes,
01:07:59.910 --> 01:08:02.970
there's a 2 that remains
and gives you c squared.
01:08:02.970 --> 01:08:05.790
If you like you can write
it in spherical coordinates,
01:08:05.790 --> 01:08:08.620
et cetera, but dimensions
have to work out
01:08:08.620 --> 01:08:10.640
to be something like this.
01:08:10.640 --> 01:08:12.240
So now we can-- yes?
01:08:12.240 --> 01:08:15.350
AUDIENCE: Just to clarify, when
you sat phi of x and phi of 0,
01:08:15.350 --> 01:08:19.740
are those both longitudinal
or both transverse?
01:08:19.740 --> 01:08:25.920
PROFESSOR: I wasn't precise
so if I, thinking about chi l,
01:08:25.920 --> 01:08:28.399
these will be both longitudinal.
01:08:28.399 --> 01:08:29.016
OK?
01:08:29.016 --> 01:08:31.859
And then we have
this expression.
01:08:31.859 --> 01:08:37.649
If I'm talking about chi t
and I'm above the transition
01:08:37.649 --> 01:08:40.340
temperature, there's no problem.
01:08:40.340 --> 01:08:42.460
If I'm below the
transition temperature,
01:08:42.460 --> 01:08:44.500
I can use the same
thing but have
01:08:44.500 --> 01:08:47.420
to set c to infinity
so I have to integrate
01:08:47.420 --> 01:08:49.047
all the way to infinity.
01:08:52.719 --> 01:08:54.076
OK?
01:08:54.076 --> 01:08:59.220
But now you can see that the
divergence of susceptibility
01:08:59.220 --> 01:09:03.760
is very much related to the
divergence of correlations,
01:09:03.760 --> 01:09:07.319
in some sense very
precisely in that
01:09:07.319 --> 01:09:13.180
if this goes like t
to the minus gamma
01:09:13.180 --> 01:09:19.630
and the correlation length
diverges as t to the minus u,
01:09:19.630 --> 01:09:22.076
then gamma should be 2 nu.
01:09:22.076 --> 01:09:24.859
And indeed, our ne is one-half.
01:09:24.859 --> 01:09:28.700
We had seen previously
that gamma was 2 nu.
01:09:28.700 --> 01:09:33.359
Secondly that the amplitude
ratio for susceptibility
01:09:33.359 --> 01:09:36.029
should be the square
of the amplitude ratio
01:09:36.029 --> 01:09:38.880
for the correlation
length and again this
01:09:38.880 --> 01:09:40.990
is something that
we have seen before.
01:09:40.990 --> 01:09:43.920
The amplitude ratio
for susceptibility
01:09:43.920 --> 01:09:46.288
was the square root of 2.
01:09:49.090 --> 01:09:50.840
Now it turns out
that all of this
01:09:50.840 --> 01:09:55.300
is a gain within this
[INAUDIBLE] point approximation
01:09:55.300 --> 01:09:58.720
looking at the most
probable state, et cetera.
01:09:58.720 --> 01:10:07.220
Because what we find in reality
is that at the critical point,
01:10:07.220 --> 01:10:11.390
the correlations don't decay
simply according to the Coulomb
01:10:11.390 --> 01:10:16.560
law but there is
this additional eta
01:10:16.560 --> 01:10:21.820
which is the same eta
that we had over here.
01:10:21.820 --> 01:10:23.100
OK?
01:10:23.100 --> 01:10:27.360
And that because of that
eta, here what you would have
01:10:27.360 --> 01:10:32.380
is 2 minus eta and you
would get an example
01:10:32.380 --> 01:10:36.440
of a number of things that
we will see a lot later on.
01:10:36.440 --> 01:10:40.530
That is there even if you don't
know what the exponents are,
01:10:40.530 --> 01:10:42.650
you know that there
are relationships
01:10:42.650 --> 01:10:44.020
among the exponents.
01:10:44.020 --> 01:10:47.585
This is an example of an
exponent identity called
01:10:47.585 --> 01:10:49.550
a Fisher exponent
and there are several
01:10:49.550 --> 01:10:51.236
of these exponents identities.
01:10:54.117 --> 01:10:54.617
OK?
01:10:58.490 --> 01:11:05.010
But that also brings
us to the following-
01:11:05.010 --> 01:11:11.550
that we did all of this
work and we came up
01:11:11.550 --> 01:11:23.990
with answers for the singular
behaviors at critical points
01:11:23.990 --> 01:11:27.650
and why they are universal.
01:11:27.650 --> 01:11:31.410
And actually as far as the
thermodynamic quantities
01:11:31.410 --> 01:11:34.850
were concerned, all
we ended up doing
01:11:34.850 --> 01:11:38.990
was to write some expression
that was analytical
01:11:38.990 --> 01:11:40.880
and then find its minimum.
01:11:40.880 --> 01:11:46.200
And we found that the minimum
of an analytical expression
01:11:46.200 --> 01:11:50.530
always has the same type
of singularities, which
01:11:50.530 --> 01:11:53.530
we can characterize
by these exponents.
01:11:53.530 --> 01:11:56.600
So maybe it's now a
good time to check
01:11:56.600 --> 01:12:00.090
how these match
with the experiment.
01:12:00.090 --> 01:12:11.500
So let's look at the various
types of phase transition,
01:12:11.500 --> 01:12:16.540
an example of the material
that undergoes that phase
01:12:16.540 --> 01:12:24.830
transition, and what the
exponents alpha, beta, gamma,
01:12:24.830 --> 01:12:28.680
and u are that are
experimentally obtained.
01:12:37.974 --> 01:12:40.920
AUDIENCE: What is this again?
01:12:40.920 --> 01:12:43.660
PROFESSOR: The material
that undergoes a transition,
01:12:43.660 --> 01:12:51.950
so for example when we are
talking about the ferromagnet
01:12:51.950 --> 01:12:58.070
to paramagnet transition,
you could look at material
01:12:58.070 --> 01:13:05.870
such as iron or
nickel and if we ask
01:13:05.870 --> 01:13:09.620
in the context of
this systematics
01:13:09.620 --> 01:13:12.620
that we were developing for
the Landau-Ginzburg what they
01:13:12.620 --> 01:13:17.930
correspond to, they are things
that have three components
01:13:17.930 --> 01:13:22.620
or fields so they
correspond to n equals to 3.
01:13:22.620 --> 01:13:27.370
Of course everything that I will
be talking to in this column
01:13:27.370 --> 01:13:29.800
will correspond to
3-dimensional systems.
01:13:29.800 --> 01:13:32.930
Later we'll talk also about
2-dimensional and other
01:13:32.930 --> 01:13:39.060
systems, but let's stick with
real 3-dimensional world.
01:13:39.060 --> 01:13:41.360
So that would be one set.
01:13:41.360 --> 01:13:43.543
We will look at super fluidity.
01:13:47.170 --> 01:13:53.130
Let's say in helium, which
we discussed last semester,
01:13:53.130 --> 01:13:55.500
that corresponds
to n equals to 2.
01:13:58.030 --> 01:14:04.310
We will talk about various
examples of liquid gas
01:14:04.310 --> 01:14:17.020
transition which correspond to
a scalar density difference.
01:14:17.020 --> 01:14:23.415
And this could be anything
from say carbon dioxide, neon,
01:14:23.415 --> 01:14:26.570
argon, whatever gas we like.
01:14:26.570 --> 01:14:41.310
And also talk about
superconductors
01:14:41.310 --> 01:14:43.840
which to all
intents and purposes
01:14:43.840 --> 01:14:49.680
should have the same type of
symmetries as super fluids.
01:14:49.680 --> 01:14:53.440
An example of a quantum system
should be n equals to 2,
01:14:53.440 --> 01:14:57.692
gained lots of different cases
such as aluminum, copper,
01:14:57.692 --> 01:14:58.192
whatever.
01:15:01.110 --> 01:15:05.280
So what do we find
for the exponent?
01:15:05.280 --> 01:15:07.270
Actually for
ferromagnetic system
01:15:07.270 --> 01:15:09.510
the heat capacity
does not diverge.
01:15:09.510 --> 01:15:14.300
It has a discontinuous
derivative at the transition
01:15:14.300 --> 01:15:18.340
and kind of goes in a manner
that if you take its derivative
01:15:18.340 --> 01:15:21.470
then the derivative appears
to be singular and corresponds
01:15:21.470 --> 01:15:22.440
to an alpha.
01:15:22.440 --> 01:15:28.935
If you try to fit it to
it's slightly negative.
01:15:28.935 --> 01:15:34.170
The superfluid has this famous
lambda shape for its heat
01:15:34.170 --> 01:15:39.012
capacity and a lambda
shape is very well fitted
01:15:39.012 --> 01:15:42.446
to a logarithm type of function.
01:15:42.446 --> 01:15:46.590
The logarithm is the
limit of a power law
01:15:46.590 --> 01:15:50.400
as the exponent goes to 0 so we
can more or less indicate that
01:15:50.400 --> 01:15:55.460
by an alpha of 0 or really
it's a divergent log.
01:15:58.560 --> 01:16:02.060
These objects, the
liquid gas transition
01:16:02.060 --> 01:16:06.210
does have weakly
divergent heat capacity
01:16:06.210 --> 01:16:11.110
so the alpha is around 0.1.
01:16:11.110 --> 01:16:14.970
The values of betas are
all less than one-half,
01:16:14.970 --> 01:16:21.350
for ferromagnet system
is of the order of 0.4.
01:16:21.350 --> 01:16:29.380
It is almost one-third, slightly
less for superfluid helium
01:16:29.380 --> 01:16:34.610
and less for the
liquid gas system.
01:16:34.610 --> 01:16:39.990
Gammas something like 1.4.
01:16:39.990 --> 01:16:42.040
We don't have a
gamma for superfluid,
01:16:42.040 --> 01:16:44.730
you can't put a magnetic
field on the superfluid.
01:16:44.730 --> 01:16:48.300
There's nothing that is
conjugate to the quantum phase.
01:16:48.300 --> 01:16:51.680
Here it is more like 1.3.
01:16:51.680 --> 01:16:59.498
Mu is 0.-- it's
not-- [INAUDIBLE].
01:17:06.758 --> 01:17:08.210
OK.
01:17:08.210 --> 01:17:18.081
So what I have here it is more
like 1.3, 1.24, 0.7, 0.67,
01:17:18.081 --> 01:17:20.962
0.63.
01:17:20.962 --> 01:17:21.462
OK?
01:17:24.360 --> 01:17:30.260
Now they are different from
the predictions that we had.
01:17:30.260 --> 01:17:36.520
Predictions that we had where
alpha was 0 discontinuous.
01:17:36.520 --> 01:17:38.970
Beta goes to one-half.
01:17:38.970 --> 01:17:43.920
Gamma was 1, mu
equals to one-half.
01:17:43.920 --> 01:17:49.420
And actually these
predictions that we just made
01:17:49.420 --> 01:17:52.120
happen to match extremely
well with all kinds
01:17:52.120 --> 01:17:55.164
of super conducting
systems that you look at.
01:17:58.120 --> 01:18:00.870
So again it is
important to state
01:18:00.870 --> 01:18:04.405
that within a particular
class like liquid gas
01:18:04.405 --> 01:18:07.430
you can do a lot of
different systems.
01:18:07.430 --> 01:18:10.090
We saw that curve in
the second lecture.
01:18:10.090 --> 01:18:13.470
They all correspond to
this same set of exponents,
01:18:13.470 --> 01:18:18.110
singularly for a different
magnet and so forth.
01:18:18.110 --> 01:18:21.150
So there is something
that is universal
01:18:21.150 --> 01:18:24.950
but our Landau-Ginzburg
approach with this looking
01:18:24.950 --> 01:18:27.660
at the most probable state
and fluctuations around
01:18:27.660 --> 01:18:31.120
it has not captured
it for most cases
01:18:31.120 --> 01:18:33.070
but for some reason
has captured it
01:18:33.070 --> 01:18:35.760
for the case of superconductors.
01:18:35.760 --> 01:18:40.410
So we have that puzzle and
starting from next lecture
01:18:40.410 --> 01:18:43.160
we'll start to unravel that.