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PROFESSOR: OK, let's start.
00:00:25.770 --> 00:00:31.060
So recapping what
we have been doing,
00:00:31.060 --> 00:00:34.800
we said that many systems that
undergo phase transition-- so
00:00:34.800 --> 00:00:38.830
there's some material that
undergoes phase transition-- we
00:00:38.830 --> 00:00:41.130
could look at it
and characterize it
00:00:41.130 --> 00:00:43.900
through a statistical field.
00:00:43.900 --> 00:00:49.290
But my analogy in the case
of magnetization of magnet
00:00:49.290 --> 00:00:53.890
will be noted by m that
varies as a function
00:00:53.890 --> 00:00:56.350
of the position on the sample.
00:00:56.350 --> 00:00:59.210
And it's a vector.
00:00:59.210 --> 00:01:01.950
And this vector
has n components.
00:01:01.950 --> 00:01:04.879
And we said that basically
we could distinguish
00:01:04.879 --> 00:01:09.470
different types of systems by
the number of components of n.
00:01:09.470 --> 00:01:13.600
And for the case of
things like liquid gas,
00:01:13.600 --> 00:01:17.060
we had a scale or
density difference,
00:01:17.060 --> 00:01:18.980
which is one component.
00:01:18.980 --> 00:01:27.590
For the case of superfluid,
we had the phase
00:01:27.590 --> 00:01:31.670
of a quantum mechanical
wave function, which
00:01:31.670 --> 00:01:36.450
had, therefore, two components
when we included the magnitude.
00:01:36.450 --> 00:01:45.860
And for the case of, say,
[INAUDIBLE] ferromagnet,
00:01:45.860 --> 00:01:47.589
we had n equals 3.
00:01:52.080 --> 00:01:57.380
We said that basically all of
these systems in the vicinity
00:01:57.380 --> 00:02:02.180
of the transition point
where the field n of x
00:02:02.180 --> 00:02:05.970
is presumably fluctuating
around a small quantity
00:02:05.970 --> 00:02:08.259
and the correlation
lengths are large,
00:02:08.259 --> 00:02:12.640
we could describe in
terms of weight that
00:02:12.640 --> 00:02:15.570
was constructed on
the basis of symmetry
00:02:15.570 --> 00:02:20.240
and a form of
locality which allowed
00:02:20.240 --> 00:02:24.970
us to express the
weight in powers of m
00:02:24.970 --> 00:02:33.760
squared integrated in the
vicinity of some point x.
00:02:33.760 --> 00:02:38.040
Then the connection between
the different clients
00:02:38.040 --> 00:02:42.514
was captured through terms
that involves gradients of m.
00:02:46.600 --> 00:02:51.850
And higher order derivatives
are also possible.
00:02:51.850 --> 00:02:56.250
And that easy back to deviate
from the symmetry axis,
00:02:56.250 --> 00:02:58.590
we could add a term that is h.m.
00:03:04.350 --> 00:03:07.860
So that was this
statistical weight
00:03:07.860 --> 00:03:12.370
that we assigned to
configurations of this field.
00:03:12.370 --> 00:03:15.810
Now we said that when
you do measurements
00:03:15.810 --> 00:03:18.770
of these kinds of
systems, for example,
00:03:18.770 --> 00:03:21.325
you will see singularities
in heat capacity.
00:03:28.730 --> 00:03:33.580
And those in the vicinity
of the phase transitions
00:03:33.580 --> 00:03:35.950
were characterized
by an exponent alpha.
00:03:44.730 --> 00:03:50.250
Now the value of alpha, you can
go and look at various system,
00:03:50.250 --> 00:03:54.000
you find for liquid gas
systems many different versions
00:03:54.000 --> 00:03:57.820
of carbon dioxide, et
cetera, and other systems
00:03:57.820 --> 00:04:00.600
that you would
correspond to n equals 1,
00:04:00.600 --> 00:04:03.560
like binary mixtures
such as the one that
00:04:03.560 --> 00:04:07.440
is in the first
problem set, correspond
00:04:07.440 --> 00:04:17.430
to a value of alpha divergence
that is roughly around 0.11.
00:04:17.430 --> 00:04:21.250
For the case of
superfluid, we saw
00:04:21.250 --> 00:04:24.220
curves that described
this lambda point.
00:04:24.220 --> 00:04:27.620
There is, again, a
divergence, or the divergence
00:04:27.620 --> 00:04:29.010
is weaker than the [INAUDIBLE].
00:04:29.010 --> 00:04:34.370
It is approximately a
logarithmic divergence.
00:04:34.370 --> 00:04:37.520
Whereas for ferromagnets,
there is a cost singularity.
00:04:37.520 --> 00:04:39.420
There's no divergence.
00:04:39.420 --> 00:04:42.400
And the singularity
can be expressed
00:04:42.400 --> 00:04:45.610
in terms of a negative alpha.
00:04:45.610 --> 00:04:47.920
So there are these
classes, depending
00:04:47.920 --> 00:04:53.860
on the value of this parameter
n, which are all the same.
00:04:53.860 --> 00:04:58.300
And in our case, they are all
described by this same field
00:04:58.300 --> 00:05:04.350
theory, with different number of
components of this quantity n.
00:05:04.350 --> 00:05:09.140
So we asked whether or not
we could get that result.
00:05:09.140 --> 00:05:15.600
So what we did was we said, OK,
let's calculate the partition
00:05:15.600 --> 00:05:18.560
function that corresponds
to this system
00:05:18.560 --> 00:05:22.260
by integrating over all
configurations of this field.
00:05:25.340 --> 00:05:29.300
And this is actually
just the singular part,
00:05:29.300 --> 00:05:32.400
because in the process
of going from whatever
00:05:32.400 --> 00:05:36.850
microscopic variables we have
to these variables that describe
00:05:36.850 --> 00:05:40.320
the statistical field,
we have to integrate
00:05:40.320 --> 00:05:43.090
over many microscopic
configurations.
00:05:43.090 --> 00:05:46.510
So there could be a
non-singular part that emerges.
00:05:46.510 --> 00:05:50.960
But the singularities are due to
the appearance of magnetization
00:05:50.960 --> 00:05:52.480
spontaneously.
00:05:52.480 --> 00:05:56.660
So they should be reflected
in calculating the partition
00:05:56.660 --> 00:05:58.010
function of this component.
00:06:01.370 --> 00:06:05.830
Now what we did
was, then, to say,
00:06:05.830 --> 00:06:08.490
OK, this is difficult thing.
00:06:08.490 --> 00:06:18.240
What I am going to do is do
a subtle point approximation,
00:06:18.240 --> 00:06:22.415
which really amounted to
finding the most probable state.
00:06:31.280 --> 00:06:39.280
And that most probable
state corresponded to the m
00:06:39.280 --> 00:06:47.700
being uniform across the system,
value m bar, that potentially
00:06:47.700 --> 00:06:55.640
would be directed along
the magnetic field.
00:06:55.640 --> 00:06:58.650
But there's only the limit
that magnetic field goes to 0,
00:06:58.650 --> 00:07:02.780
spontaneously select
some kind of a direction.
00:07:02.780 --> 00:07:07.210
But of course, this
m bar would be 0
00:07:07.210 --> 00:07:10.320
if you are above the
transition, which
00:07:10.320 --> 00:07:13.180
in this most
probable state occurs
00:07:13.180 --> 00:07:17.255
for t's that are
positive at h equals 0.
00:07:20.340 --> 00:07:26.000
While for t negative,
minimizing tm squared plus um
00:07:26.000 --> 00:07:30.920
to the fourth gave us a value of
square root of minus t over 4.
00:07:39.370 --> 00:07:52.490
Then our z singular in the
subtle point approximation
00:07:52.490 --> 00:08:00.240
evaluated as a function
of t for h equals 0
00:08:00.240 --> 00:08:05.990
is simply related to the value
of this most probable state
00:08:05.990 --> 00:08:08.700
at this particular point.
00:08:08.700 --> 00:08:14.480
And we found that the answer
was exponential of minus because
00:08:14.480 --> 00:08:17.450
of the integration over space.
00:08:17.450 --> 00:08:18.695
But everything is uniform.
00:08:18.695 --> 00:08:21.740
It will be
proportional to volume.
00:08:21.740 --> 00:08:24.870
And then multiplied
by a function
00:08:24.870 --> 00:08:30.140
that was either 0, if you
were looking at t positive.
00:08:30.140 --> 00:08:33.679
Whereas for t negative,
substituting that value
00:08:33.679 --> 00:08:38.790
of h bar, gave us minus
t squared over 16.
00:08:45.476 --> 00:08:45.975
Yeah.
00:08:55.640 --> 00:08:58.820
So essentially, it's a function.
00:08:58.820 --> 00:09:04.860
There is really no magnetization
above the critical point.
00:09:04.860 --> 00:09:06.690
And you get 0.
00:09:06.690 --> 00:09:09.740
Below the critical
point, what you
00:09:09.740 --> 00:09:13.550
have is this quadratic
behavior in t.
00:09:13.550 --> 00:09:17.280
So if I were to take two
derivatives of it, which
00:09:17.280 --> 00:09:19.950
would give me something that
is proportional to the heat
00:09:19.950 --> 00:09:27.450
capacity-- so from here I would
get a heat capacity evaluated
00:09:27.450 --> 00:09:33.570
in the subtle point method
as a function of t for h
00:09:33.570 --> 00:09:39.545
equals 0, which would be
either 0 or 1 over 8u,
00:09:39.545 --> 00:09:42.440
if I'm taking this derivative.
00:09:42.440 --> 00:09:48.440
So the prediction is that
you have an alpha which is 0
00:09:48.440 --> 00:09:52.520
because there is no
power law dependence.
00:09:52.520 --> 00:09:56.795
And what it really
reflects is that there
00:09:56.795 --> 00:10:00.750
is a discontinuity
in heat capacity.
00:10:00.750 --> 00:10:02.870
So none of the
examples that I showed
00:10:02.870 --> 00:10:07.670
you aboved-- the liquid gas,
the superfluid, ferromagnet--
00:10:07.670 --> 00:10:09.820
have a discontinuous
heat capacity.
00:10:09.820 --> 00:10:12.560
So this does not seem to work.
00:10:12.560 --> 00:10:15.320
On the other hand,
this discontinuity
00:10:15.320 --> 00:10:21.323
is observed for
superconductor transitions.
00:10:29.080 --> 00:10:32.070
So that's the state
of the affairs.
00:10:32.070 --> 00:10:36.250
What we have to understand
now is, first of all,
00:10:36.250 --> 00:10:39.060
why doesn't it work in general?
00:10:39.060 --> 00:10:43.620
Secondly, why does it
work for superconductors?
00:10:43.620 --> 00:10:46.978
So that's the task for today.
00:10:46.978 --> 00:10:49.740
All right?
00:10:49.740 --> 00:10:53.780
So the one thing
that is certainly
00:10:53.780 --> 00:10:57.220
a glaring approximation
is to replace
00:10:57.220 --> 00:11:01.550
this integration over all
configuration by just the one
00:11:01.550 --> 00:11:03.900
most probable state.
00:11:03.900 --> 00:11:07.190
But we did precisely
that when we
00:11:07.190 --> 00:11:11.870
were talking about the subtle
point method of integration
00:11:11.870 --> 00:11:17.110
in the previous class in 8 333.
00:11:17.110 --> 00:11:22.770
So let's examine why it
was legitimate to do so
00:11:22.770 --> 00:11:24.690
at that point.
00:11:24.690 --> 00:11:29.020
So there we are evaluating
essentially an integration
00:11:29.020 --> 00:11:31.360
that involved one variable.
00:11:31.360 --> 00:11:33.360
Let's call it m.
00:11:33.360 --> 00:11:42.410
And we had a large number that
was appearing in the exponent.
00:11:42.410 --> 00:11:53.120
And we had some function
that we were looking at,
00:11:53.120 --> 00:11:54.870
depending on the
variable of integration.
00:11:57.640 --> 00:12:00.490
Now the most probable
value of this
00:12:00.490 --> 00:12:04.500
occurs for some
particular m bar.
00:12:04.500 --> 00:12:09.520
And what we can do,
without essentially
00:12:09.520 --> 00:12:13.980
doing any approximation
at this point,
00:12:13.980 --> 00:12:19.130
is to make a Taylor
expansion of the function
00:12:19.130 --> 00:12:21.130
around its maximum.
00:12:21.130 --> 00:12:30.520
So the function I can write as
psi evaluated at this extremum.
00:12:30.520 --> 00:12:33.180
But since I am looking
at an extremum,
00:12:33.180 --> 00:12:35.630
if I make a Taylor
expansion, the term
00:12:35.630 --> 00:12:38.650
that is proportional to the
first derivative is absent.
00:12:38.650 --> 00:12:41.910
I'm expanding
around an extremum.
00:12:41.910 --> 00:12:52.020
The term that is proportional to
the second derivative evaluated
00:12:52.020 --> 00:12:57.600
at m bar will go with
m minus m bar squared.
00:12:57.600 --> 00:13:00.370
And in principle, there are
higher and higher order terms
00:13:00.370 --> 00:13:01.710
I can put in this expansion.
00:13:05.610 --> 00:13:10.410
Now the value at the
most probable position,
00:13:10.410 --> 00:13:13.020
which is the subtle point
value, is a constant.
00:13:13.020 --> 00:13:16.730
I can put it outside.
00:13:16.730 --> 00:13:20.410
So essentially,
terminating here is exactly
00:13:20.410 --> 00:13:24.470
like what I was doing
over there, more or less.
00:13:24.470 --> 00:13:28.140
But then I have
fluctuations around
00:13:28.140 --> 00:13:30.540
this most probably state.
00:13:30.540 --> 00:13:33.840
So I can do the
integration, let's say,
00:13:33.840 --> 00:13:38.100
in the variable delta n.
00:13:38.100 --> 00:13:44.176
I have the
differential of delta m
00:13:44.176 --> 00:13:51.070
into the minus 1/2
psi double m bar
00:13:51.070 --> 00:13:57.060
m minus delta m bar
delta m squared.
00:13:57.060 --> 00:13:59.680
And then I have
higher order terms.
00:13:59.680 --> 00:14:02.790
And principle, those
higher order terms I
00:14:02.790 --> 00:14:05.330
can start expanding over here.
00:14:14.600 --> 00:14:17.510
And I forgot the very
important factor,
00:14:17.510 --> 00:14:21.530
which is that this whole
thing is proportional to n.
00:14:21.530 --> 00:14:25.200
And indeed, all of
these terms over here
00:14:25.200 --> 00:14:29.500
will also be proportional to n.
00:14:29.500 --> 00:14:30.870
OK?
00:14:30.870 --> 00:14:33.650
But the first term
in the series is just
00:14:33.650 --> 00:14:36.080
the Gaussian integration.
00:14:36.080 --> 00:14:43.680
And so I know that the leading
correction to the subtle point
00:14:43.680 --> 00:14:52.520
comes from this factor of root 2
pi n psi double prime of m bar.
00:14:52.520 --> 00:14:58.370
And then, in principle, there
will be higher order terms.
00:14:58.370 --> 00:15:03.120
And if you keep track of
how many factors of delta m
00:15:03.120 --> 00:15:05.160
allowed-- delta m
cubed is certainly not
00:15:05.160 --> 00:15:07.890
allowed because of
the evenness of what
00:15:07.890 --> 00:15:09.690
I'm integrating against.
00:15:09.690 --> 00:15:13.040
So the next order term will
be delta m to the fourth.
00:15:13.040 --> 00:15:15.770
Evaluated against
this Gaussian, it
00:15:15.770 --> 00:15:19.330
will give you something that
is order of 1 over n squared.
00:15:19.330 --> 00:15:22.420
Multiplied by n, you will get
corrections of the order of 1
00:15:22.420 --> 00:15:25.680
over n.
00:15:25.680 --> 00:15:29.770
So very systematically,
we could see
00:15:29.770 --> 00:15:33.900
that if I called the result
of this integration i,
00:15:33.900 --> 00:15:40.440
that log of i has a
term that is dominated
00:15:40.440 --> 00:15:45.580
by the most probable
value of the integrant.
00:15:45.580 --> 00:15:52.260
And then there are corrections,
such as this factor of log n
00:15:52.260 --> 00:15:57.170
psi double prime
m bar over 2 pi,
00:15:57.170 --> 00:16:02.190
and lower order corrections
of order of 1 over n.
00:16:02.190 --> 00:16:09.400
Basically all of these
terms in the limit of n
00:16:09.400 --> 00:16:13.300
being much larger than
1, you can ignore.
00:16:13.300 --> 00:16:16.340
And essentially, this term
will dominate everything.
00:16:20.840 --> 00:16:24.660
So what we did over there
kind of looks the same.
00:16:24.660 --> 00:16:29.700
So let's repeat that for
our functional integral.
00:16:29.700 --> 00:16:31.970
So I have z.
00:16:31.970 --> 00:16:35.250
Actually it is the singular
part of the partition
00:16:35.250 --> 00:16:40.790
function, which is obtained
by integrating over
00:16:40.790 --> 00:16:44.560
all functions m of x.
00:16:44.560 --> 00:16:47.210
And for the time being,
let's just focus on the h
00:16:47.210 --> 00:16:49.400
equals 0 part.
00:16:49.400 --> 00:16:53.450
So I have exponential
of minus integral
00:16:53.450 --> 00:17:00.420
over x, t over 2m
square, um to the fourth,
00:17:00.420 --> 00:17:05.595
k over 2 gradient of m
squared, and so forth.
00:17:12.130 --> 00:17:18.700
And repeat what
we did over there.
00:17:18.700 --> 00:17:22.530
So what we need over here
was to basically pick
00:17:22.530 --> 00:17:27.819
the most probable state
and then expand around
00:17:27.819 --> 00:17:30.280
the most probable state.
00:17:30.280 --> 00:17:37.300
So going beyond just picking the
contribution of most probable
00:17:37.300 --> 00:17:41.670
state involves including
these fluctuations.
00:17:41.670 --> 00:17:50.810
So let me write my m of x
to be essentially m bar,
00:17:50.810 --> 00:17:53.130
but allowing a little
bit of fluctuation.
00:17:53.130 --> 00:17:57.080
And we saw that we could
divide the fluctuations
00:17:57.080 --> 00:17:59.100
into a longitudinal part.
00:17:59.100 --> 00:18:02.290
Let's call it e1 hat.
00:18:02.290 --> 00:18:09.610
And the transfers part, which
is an n minus 1 component
00:18:09.610 --> 00:18:13.570
vector in the n minus
1 transfers directions.
00:18:17.280 --> 00:18:21.320
And then I substitute
this over here.
00:18:21.320 --> 00:18:24.200
So what do I get?
00:18:24.200 --> 00:18:26.920
Just like here, I
can pull out the term
00:18:26.920 --> 00:18:29.160
that corresponds to
the subtle point.
00:18:29.160 --> 00:18:31.770
In fact, I had
calculated it up there.
00:18:31.770 --> 00:18:37.646
So I have exponential
of minus v,
00:18:37.646 --> 00:18:40.520
the value of this thing
at the subtle point.
00:18:48.400 --> 00:18:52.960
And then I have essentially
replaced the variable m
00:18:52.960 --> 00:18:56.040
with the integration
over fluctuations.
00:18:56.040 --> 00:19:02.890
So now I have to integrate over
the longitudinal fluctuations
00:19:02.890 --> 00:19:05.260
and the transfers fluctuations.
00:19:08.590 --> 00:19:15.720
And what I need to do is
to expand this quantity up
00:19:15.720 --> 00:19:18.130
to second order.
00:19:18.130 --> 00:19:20.310
But that's exactly
what we did last time,
00:19:20.310 --> 00:19:24.880
where you were looking at how
the system was scattering.
00:19:24.880 --> 00:19:31.510
So we can rely on the
result from last time
00:19:31.510 --> 00:19:34.290
for what the quadratic part is.
00:19:34.290 --> 00:19:39.130
So we saw that the answer
could be written as minus k
00:19:39.130 --> 00:19:43.942
over 2, integral ddx.
00:19:43.942 --> 00:19:49.270
Well, actually, let's
keep it this way.
00:19:49.270 --> 00:20:02.790
We have cl to the minus
2 plus phi l squared
00:20:02.790 --> 00:20:11.560
plus gradient of phi l squared.
00:20:11.560 --> 00:20:14.700
So this is what I did.
00:20:14.700 --> 00:20:21.300
What we had to do was to
replace this function.
00:20:21.300 --> 00:20:24.035
The only part that has a
contribution from variation
00:20:24.035 --> 00:20:29.260
in space, and hence contributes
to gradient, comes from phi.
00:20:29.260 --> 00:20:32.830
So from here, we will
get a k over 2 gradient
00:20:32.830 --> 00:20:33.790
of phi l squared.
00:20:37.090 --> 00:20:41.930
Then there is a
contribution from t,
00:20:41.930 --> 00:20:44.350
and one that comes
from expanding m
00:20:44.350 --> 00:20:47.720
to the fourth to
quadratic folder, that
00:20:47.720 --> 00:20:51.190
are proportional
to phi l squared.
00:20:51.190 --> 00:20:53.430
And the coefficient
of both of them
00:20:53.430 --> 00:20:57.190
we combine to write
as cl to the minus 2.
00:20:57.190 --> 00:21:01.070
And if I go back to
what we had last time,
00:21:01.070 --> 00:21:09.510
our result was that
k over cl squared
00:21:09.510 --> 00:21:17.120
was either t, if I
was for t positive,
00:21:17.120 --> 00:21:20.055
or minus 2t if I
was for t negative.
00:21:22.850 --> 00:21:30.050
Whereas, when I expanded
the transfers component,
00:21:30.050 --> 00:21:35.750
what I got above
tc, for t positive
00:21:35.750 --> 00:21:38.255
there is no difference between
longitudinal transfers,
00:21:38.255 --> 00:21:40.380
so I had the same result.
00:21:40.380 --> 00:21:45.330
Below, there was no cost
for these Goldstone modes,
00:21:45.330 --> 00:21:48.860
and the answer was 0.
00:21:48.860 --> 00:21:51.740
But essentially, I have a
similar expression, then,
00:21:51.740 --> 00:22:05.050
to write for the
transfers component.
00:22:11.020 --> 00:22:16.130
So this part amounts
to essentially
00:22:16.130 --> 00:22:18.711
what I have over here.
00:22:18.711 --> 00:22:24.880
And in principle, I can put
a whole bunch of other things
00:22:24.880 --> 00:22:30.470
that would correspond to higher
order fluctuations, effects
00:22:30.470 --> 00:22:33.710
beyond the quadratic.
00:22:33.710 --> 00:22:37.420
But again, our
anticipation is that, just
00:22:37.420 --> 00:22:41.200
like what is happening
here, the leading correction
00:22:41.200 --> 00:22:47.340
to the subtle point will already
come from the quadratic part.
00:22:47.340 --> 00:22:51.080
So let's evaluate that.
00:22:51.080 --> 00:22:53.460
So let's continue.
00:22:53.460 --> 00:22:59.155
So this is exponential of
the subtle point phi energy.
00:23:05.460 --> 00:23:11.370
And then I have to do all of
these integrations over phi
00:23:11.370 --> 00:23:12.560
l and phi q.
00:23:18.210 --> 00:23:23.350
Now what I can do, and I already
did this also last time around,
00:23:23.350 --> 00:23:30.120
is we introduced an
expansion of phi.
00:23:30.120 --> 00:23:34.120
We said each phi
of x I can write
00:23:34.120 --> 00:23:37.510
as a sum over
Fourier components--
00:23:37.510 --> 00:23:44.790
e to the iq.x phi tilda
of q, and with a root
00:23:44.790 --> 00:23:48.200
phi for normalization
convenience.
00:23:48.200 --> 00:23:53.560
So I can certainly replace
both phi l and phi t,
00:23:53.560 --> 00:23:58.610
just as I did last time, in
terms of Fourier component.
00:23:58.610 --> 00:24:03.660
And then the integration over
all configurations of phi
00:24:03.660 --> 00:24:05.720
is equivalent to
integrating over
00:24:05.720 --> 00:24:10.760
all configurations
of the phi tilda
00:24:10.760 --> 00:24:12.513
of q's, sll the
Fourier amplitudes.
00:24:16.320 --> 00:24:19.280
But the advantage is that
when we look at the Fourier
00:24:19.280 --> 00:24:23.680
amplitudes, the
different q's are
00:24:23.680 --> 00:24:26.650
completely independent
of each other.
00:24:26.650 --> 00:24:30.190
So this integration
over here that
00:24:30.190 --> 00:24:33.430
was not the
one-dimensional integration
00:24:33.430 --> 00:24:37.880
becomes a product of
one-dimensional integrations
00:24:37.880 --> 00:24:42.410
when we go to the Fourier
component representation.
00:24:42.410 --> 00:24:45.360
So now I have to
integrate for each q.
00:24:45.360 --> 00:24:49.190
I have either phi
l of q, or I have
00:24:49.190 --> 00:24:52.620
the n minus 1
component phi p of q.
00:24:52.620 --> 00:24:58.190
So these are whole bunch
of one dimensional Gaussian
00:24:58.190 --> 00:24:59.710
integrations.
00:24:59.710 --> 00:25:03.720
Because when I look at
what these rates are doing,
00:25:03.720 --> 00:25:11.800
I get e to the minus k over 2,
q squared plus cl to the minus 2
00:25:11.800 --> 00:25:18.060
phi l of q squared for
the longitudinal mode,
00:25:18.060 --> 00:25:22.620
and a very similar
factor k over 2 q
00:25:22.620 --> 00:25:27.930
squared plus ct to the
minus 2, phi t of q
00:25:27.930 --> 00:25:30.460
squared for the
transfers vectors.
00:25:33.352 --> 00:25:35.830
I have a whole bunch of
these different things.
00:25:39.080 --> 00:25:43.870
Now we can, again, follow
like what we had before.
00:25:43.870 --> 00:25:51.200
The leading behavior is minus v
t m bar squared over 2, u m bar
00:25:51.200 --> 00:25:53.510
to the fourth.
00:25:53.510 --> 00:26:00.150
And then I have a product
of Gaussian integrations.
00:26:00.150 --> 00:26:04.860
For each one of these
longitudinal modes,
00:26:04.860 --> 00:26:08.090
just like here, I
will get a factor
00:26:08.090 --> 00:26:18.140
of 2 pi divided by
k q squared plus cl
00:26:18.140 --> 00:26:23.040
to the minus 2 square root.
00:26:23.040 --> 00:26:26.950
And for each one of the
transfers components,
00:26:26.950 --> 00:26:34.170
I will get 2 pi k q 2
plus ct to the minus 2.
00:26:34.170 --> 00:26:37.940
And there are n
minus 1 of these.
00:26:37.940 --> 00:26:39.460
So I will get that factor.
00:26:42.250 --> 00:26:45.370
And then presume,
again, there will
00:26:45.370 --> 00:26:50.110
be corrections due
to higher orders that
00:26:50.110 --> 00:26:52.200
will be multiplying
the whole thing.
00:27:00.220 --> 00:27:04.880
So the quantity that
we are interested
00:27:04.880 --> 00:27:11.510
is, in fact, something
like phi energy.
00:27:11.510 --> 00:27:13.710
So we take log of z.
00:27:13.710 --> 00:27:16.700
Let's look at the singular part.
00:27:16.700 --> 00:27:18.930
Let's divide it by
volume, because we
00:27:18.930 --> 00:27:21.580
expect this to be an
extensive quantity,
00:27:21.580 --> 00:27:25.156
just like this other result
was proportional to n.
00:27:25.156 --> 00:27:28.010
And let's put a minus
sign-- typically
00:27:28.010 --> 00:27:30.850
you have to change
sign in any case--
00:27:30.850 --> 00:27:36.890
so that the leading term
then becomes this tm squared
00:27:36.890 --> 00:27:42.700
plus um to the fourth,
which, let me remind you,
00:27:42.700 --> 00:27:46.250
is-- actually,
let's just write it.
00:27:46.250 --> 00:27:50.140
tm bar squared over 2 plus
u m bar to the fourth.
00:27:52.940 --> 00:28:00.122
And then when I take the
log, this product over q
00:28:00.122 --> 00:28:04.162
will go to a sum over q.
00:28:07.858 --> 00:28:13.040
And the sum over q in
the continuum limit,
00:28:13.040 --> 00:28:19.926
I will replace by v integral
over q divided by 3 pi
00:28:19.926 --> 00:28:20.708
to the d.
00:28:23.640 --> 00:28:27.150
So then the next
step of the process,
00:28:27.150 --> 00:28:29.540
I will have a sum
over q which I replace
00:28:29.540 --> 00:28:31.830
with v times the integration.
00:28:31.830 --> 00:28:37.230
But the volume will go
away, and what I'm left with
00:28:37.230 --> 00:28:39.230
is the integration.
00:28:39.230 --> 00:28:46.530
So I have the integral
vdq 2 pi to the d.
00:28:46.530 --> 00:28:50.030
And I have the log of whatever
is appearing over here.
00:28:53.810 --> 00:28:59.930
So what I have
there is log of k q
00:28:59.930 --> 00:29:06.900
squared plus k cl to
the minus 2 with 1/2.
00:29:06.900 --> 00:29:07.960
Why the 1/2?
00:29:07.960 --> 00:29:10.120
Because it's the square root.
00:29:10.120 --> 00:29:12.750
I take it to the
exponential because it
00:29:12.750 --> 00:29:15.200
becomes one half of the log.
00:29:15.200 --> 00:29:17.160
In fact, it is in
the denominator.
00:29:17.160 --> 00:29:18.440
So there's a minus sign.
00:29:18.440 --> 00:29:22.690
And the minus sign cancels
the minus sign out here.
00:29:22.690 --> 00:29:25.580
And then the next term from
the transfers component,
00:29:25.580 --> 00:29:31.310
I will get n minus 1
over 2, integral dbq 2 pi
00:29:31.310 --> 00:29:39.650
to the d log of kq squared
plus kct to the minus 2.
00:29:39.650 --> 00:29:43.470
And presumably, if I go ahead
with higher and higher order
00:29:43.470 --> 00:29:45.258
corrections, there
will be other things.
00:29:45.258 --> 00:29:45.758
Yes, Carter.
00:29:45.758 --> 00:29:47.252
AUDIENCE: So [INAUDIBLE].
00:29:52.740 --> 00:29:53.980
PROFESSOR: No.
00:29:53.980 --> 00:29:56.860
It's just, like
the subtle point,
00:29:56.860 --> 00:30:00.790
I'm trying to calculate
a systematic expansion
00:30:00.790 --> 00:30:02.420
around the subtle point.
00:30:02.420 --> 00:30:07.390
So I've calculated so far
the lowest order term,
00:30:07.390 --> 00:30:11.120
although I haven't explicitly
told you what its behavior is.
00:30:11.120 --> 00:30:14.790
Once I'm satisfied with what
kind of connection that this,
00:30:14.790 --> 00:30:18.900
I need to go beyond and include
higher and higher order terms,
00:30:18.900 --> 00:30:22.510
and maybe show you that they
are explicitly unimportant,
00:30:22.510 --> 00:30:24.580
like they are in the
ordinary subtle point,
00:30:24.580 --> 00:30:25.920
or that they are important.
00:30:25.920 --> 00:30:28.160
At this stage, we are agnostic.
00:30:28.160 --> 00:30:29.341
We don't say anything.
00:30:37.060 --> 00:30:39.330
One thing to note--
of course, there
00:30:39.330 --> 00:30:42.310
are all of these
factors of 2 pi.
00:30:42.310 --> 00:30:46.900
Now if you go to a
mathematician and show them
00:30:46.900 --> 00:30:51.570
a functional integral, they
say it's an undefined quantity.
00:30:51.570 --> 00:30:54.190
And part of the reason
for undefined quantity
00:30:54.190 --> 00:30:57.440
is, well, how many factors
of 2 pi do you have?
00:30:57.440 --> 00:31:01.270
And what are the limits
of this integration?
00:31:01.270 --> 00:31:03.970
So from the perspective
of mathematics,
00:31:03.970 --> 00:31:05.680
a functional
integral is something
00:31:05.680 --> 00:31:08.610
that is very sick
and ill-behaved.
00:31:08.610 --> 00:31:11.190
In our case, there
is no problem,
00:31:11.190 --> 00:31:14.110
because we know that our
field, although I wrote it
00:31:14.110 --> 00:31:16.830
as a continuous
function, it is really
00:31:16.830 --> 00:31:20.100
a continuous function that
has a limited set of Fourier
00:31:20.100 --> 00:31:21.390
components.
00:31:21.390 --> 00:31:24.430
This product over
q will not extend
00:31:24.430 --> 00:31:26.452
to arbitrary short wavelength.
00:31:26.452 --> 00:31:27.910
There's a characteristic
wavelength
00:31:27.910 --> 00:31:31.370
which is the scale over which
I did the coarse graining,
00:31:31.370 --> 00:31:33.710
and I don't have
anything beyond that.
00:31:33.710 --> 00:31:36.110
So these are finite
number of Fourier modes
00:31:36.110 --> 00:31:37.440
that I'm integrating here.
00:31:37.440 --> 00:31:41.080
There is a finite number of 2,
2 pi, et cetera, that one has.
00:31:45.780 --> 00:31:47.310
All right.
00:31:47.310 --> 00:31:50.320
So fine, so this
is the behavior.
00:31:50.320 --> 00:31:55.640
Again, I have looked only
as a function of t setting h
00:31:55.640 --> 00:31:58.230
equals to 0.
00:31:58.230 --> 00:32:01.300
I didn't include
the effect of h.
00:32:01.300 --> 00:32:07.110
And let's explicitly look at
what these things are for t
00:32:07.110 --> 00:32:12.850
positive that I will write
above, and t negative
00:32:12.850 --> 00:32:14.402
that I will write below.
00:32:14.402 --> 00:32:17.810
We saw that above, this is 0.
00:32:17.810 --> 00:32:20.510
Below, it is minus
t squared over 16u.
00:32:24.020 --> 00:32:29.950
That this quantity kcl to
the minus 2, it is t above
00:32:29.950 --> 00:32:33.660
and it is minus 2t below.
00:32:33.660 --> 00:32:39.820
This quantity kc to the minus t
squared is t above and 0 below.
00:32:43.650 --> 00:32:46.710
Why do I bother to write that?
00:32:46.710 --> 00:32:52.250
Because I want to go and address
this question of heat capacity.
00:32:52.250 --> 00:32:57.370
And we said that heat
capacity is ultimately
00:32:57.370 --> 00:33:04.620
related to taking two
derivatives of this log c
00:33:04.620 --> 00:33:08.850
singular with respect
to temperature and beta
00:33:08.850 --> 00:33:09.740
and all of that.
00:33:09.740 --> 00:33:11.740
But let's write it
as a proportionality.
00:33:11.740 --> 00:33:12.850
It goes like this.
00:33:15.716 --> 00:33:16.215
Yes?
00:33:19.470 --> 00:33:20.900
Yes?
00:33:20.900 --> 00:33:23.850
AUDIENCE: So the third
line on the top board,
00:33:23.850 --> 00:33:30.020
you have under this continuous
product over all elements of q.
00:33:30.020 --> 00:33:31.510
PROFESSOR: So this
product over q
00:33:31.510 --> 00:33:33.818
goes all the way to the
end of the line, yes.
00:33:33.818 --> 00:33:34.442
AUDIENCE: Yeah.
00:33:34.442 --> 00:33:38.387
So can [INAUDIBLE]
be in the exponents,
00:33:38.387 --> 00:33:39.470
or are they still outside?
00:33:42.630 --> 00:33:44.593
PROFESSOR: What infinitesimals?
00:33:44.593 --> 00:33:46.485
AUDIENCE: d phi l and d phi t.
00:33:49.330 --> 00:33:51.140
PROFESSOR: OK, so
what I have left out,
00:33:51.140 --> 00:33:53.285
and you're quite
right, is the integral.
00:33:56.310 --> 00:34:02.730
So for each q, I have
to do n integrations
00:34:02.730 --> 00:34:06.335
over this variable and
this n minus 1 component.
00:34:06.335 --> 00:34:08.670
So I forgot the integral
sign, so that's correct.
00:34:18.530 --> 00:34:20.100
All right.
00:34:20.100 --> 00:34:21.280
So what do we have?
00:34:24.040 --> 00:34:29.400
So for t positive, if I take two
derivative of this with respect
00:34:29.400 --> 00:34:35.410
to t-- and actually there is
a minus sign involved here,
00:34:35.410 --> 00:34:35.909
sorry.
00:34:38.719 --> 00:34:42.350
Above the transition,
I will get 0.
00:34:42.350 --> 00:34:46.020
Below the transition, I
will get this 1 over 8u.
00:34:46.020 --> 00:34:49.710
So this is the discontinuity
that I had calculated before.
00:34:52.850 --> 00:34:56.400
Now above the transition,
I have to take
00:34:56.400 --> 00:35:03.490
a derivative of log of tkq
squared plus t with respect
00:35:03.490 --> 00:35:04.780
to t.
00:35:04.780 --> 00:35:07.960
Taking the derivative
of log will give me
00:35:07.960 --> 00:35:11.010
1 over its argument.
00:35:11.010 --> 00:35:14.820
Taking the second
derivative will give me
00:35:14.820 --> 00:35:16.740
the argument squared.
00:35:16.740 --> 00:35:20.310
Because of the minus sign, I
forget about the minus sign.
00:35:20.310 --> 00:35:23.310
So two derivatives of this
object with respect to t
00:35:23.310 --> 00:35:26.430
will bring down a factor of
kq squared plus t squared.
00:35:29.030 --> 00:35:31.800
And I have to
integrate that over q.
00:35:36.760 --> 00:35:43.540
And there is one from here, and
there's n minus 1 from here.
00:35:43.540 --> 00:35:47.070
So there is a total
of n over 2 of that.
00:35:54.070 --> 00:35:57.810
Below the transition, I
have to take a derivative,
00:35:57.810 --> 00:36:02.090
except that plus thing is
replaced with minus 2t.
00:36:02.090 --> 00:36:04.200
So every time I
take a derivative,
00:36:04.200 --> 00:36:07.380
I will get an
additional factor of 2.
00:36:07.380 --> 00:36:14.765
So rather than 1/2, I will
end up with 2 integral over q
00:36:14.765 --> 00:36:21.950
2 pi to the d 1 over kq
squared minus 2t squared
00:36:21.950 --> 00:36:24.360
from the longitudinal part.
00:36:24.360 --> 00:36:26.860
And the transfers part
has no t dependence,
00:36:26.860 --> 00:36:28.137
so it doesn't contribute.
00:36:35.300 --> 00:36:41.090
So the entire
thing, you can see,
00:36:41.090 --> 00:36:45.740
is what I had calculated
at the subtle point.
00:36:45.740 --> 00:36:51.990
And to this order in expansions
around the subtle point, which
00:36:51.990 --> 00:36:57.190
corresponds, essentially,
only to the quadratic part,
00:36:57.190 --> 00:36:58.430
I have found a correction.
00:37:01.440 --> 00:37:06.070
And generically, we see
that these corrections
00:37:06.070 --> 00:37:15.512
are proportional to an
integral over q 2 pi to the d.
00:37:15.512 --> 00:37:17.690
I can actually
pull out one factor
00:37:17.690 --> 00:37:24.220
of k squared outside so that the
integral more looks more nice
00:37:24.220 --> 00:37:29.430
with some characteristic
lengths scale, which is either
00:37:29.430 --> 00:37:32.000
coming from t or from minus 2t.
00:37:32.000 --> 00:37:34.160
So I can write it as cl squared.
00:37:39.640 --> 00:37:47.120
So in order to understand how
important these corrections
00:37:47.120 --> 00:37:50.350
are-- and here the corrections
were under control.
00:37:50.350 --> 00:37:54.920
So really, I'm asking question,
are these corrections small
00:37:54.920 --> 00:37:57.270
compared to what I
started in the same sense
00:37:57.270 --> 00:38:00.080
that log n is small
compared to n?
00:38:00.080 --> 00:38:02.060
Well, what I need to
do is to understand
00:38:02.060 --> 00:38:03.920
how this integral behaves.
00:38:03.920 --> 00:38:06.650
There is no factor
of log n versus n,
00:38:06.650 --> 00:38:08.650
because you can see
both of those terms
00:38:08.650 --> 00:38:10.921
have the factor of volume.
00:38:10.921 --> 00:38:14.770
So the issue is not that you
have something like square,
00:38:14.770 --> 00:38:17.950
log of the volume that will
give you small quantity.
00:38:17.950 --> 00:38:20.080
You have to hope
that for some reason
00:38:20.080 --> 00:38:25.890
or other this whole integral
here is not important.
00:38:25.890 --> 00:38:32.520
So if I look at the integrand--
well, I can do one more thing.
00:38:32.520 --> 00:38:37.260
I can note that it behaves
as 1 over k squared.
00:38:37.260 --> 00:38:38.880
There is a combination
that you will
00:38:38.880 --> 00:38:43.230
see appearing many, many
times in this course.
00:38:43.230 --> 00:38:45.380
Because this is
spherically symmetric,
00:38:45.380 --> 00:38:49.030
I can write it as
some solid angle q
00:38:49.030 --> 00:38:51.340
to the d minus 1 with q.
00:38:51.340 --> 00:38:54.310
And so the whole thing is
proportional to the ratio
00:38:54.310 --> 00:38:59.430
of solid angle divided by 2 pi
to the d, which will occur so
00:38:59.430 --> 00:39:03.150
many times in this course that
we will give it a name k sub d.
00:39:05.760 --> 00:39:09.230
And then the eventual
integral is simply
00:39:09.230 --> 00:39:16.350
an integral over one variable
q, q to the d minus 1.
00:39:16.350 --> 00:39:21.370
And then I have q squared
plus c to the minus 2 squared.
00:39:26.910 --> 00:39:27.540
Yes?
00:39:27.540 --> 00:39:30.040
AUDIENCE: Should that 1
over kd be 1 over k squared?
00:39:34.840 --> 00:39:37.855
PROFESSOR: There is a q
over k-- Yeah, that's right.
00:39:37.855 --> 00:39:39.090
I already had it.
00:39:39.090 --> 00:39:40.410
Yes, 1 over k squared.
00:39:40.410 --> 00:39:41.701
And then there's 1 over k.
00:39:41.701 --> 00:39:42.200
Thank you.
00:39:48.260 --> 00:39:52.770
So how does this
integrand look like,
00:39:52.770 --> 00:39:56.120
the thing that I
have to integrate?
00:39:56.120 --> 00:39:58.250
As a function of q,
I have to integrate
00:39:58.250 --> 00:40:01.680
a function that at
least that small q has
00:40:01.680 --> 00:40:06.320
no problem of singularity,
divergence, et cetera.
00:40:06.320 --> 00:40:10.390
It is simply q to
the minus something
00:40:10.390 --> 00:40:12.280
with the coefficient
that is like c.
00:40:15.510 --> 00:40:21.131
At large distances, however,
let's say three dimensions,
00:40:21.131 --> 00:40:27.906
it would fall off as q to the
power of d minus 1 minus 4.
00:40:27.906 --> 00:40:30.960
At large q, I can ignore
whatever is from here
00:40:30.960 --> 00:40:34.720
and just look at
the powers of q.
00:40:34.720 --> 00:40:38.020
But then if I'm at
sufficiently large dimension,
00:40:38.020 --> 00:40:42.030
the function will keep growing.
00:40:42.030 --> 00:40:45.100
So basically, depending
on which dimensions
00:40:45.100 --> 00:40:49.040
you are, and the borderline
dimension is clearly for,
00:40:49.040 --> 00:40:51.190
it's an integration
that you can either
00:40:51.190 --> 00:40:55.130
perform without any difficulty
going all the way to infinity
00:40:55.130 --> 00:41:00.010
in q, or you have to worry
about the upper column.
00:41:00.010 --> 00:41:01.450
OK?
00:41:01.450 --> 00:41:07.600
So if you are in
dimensions greater than 4,
00:41:07.600 --> 00:41:16.070
what you find is that this cf
in dimensions that are larger
00:41:16.070 --> 00:41:22.640
than 4, as you go to
larger and larger q,
00:41:22.640 --> 00:41:24.770
you are integrating
something that
00:41:24.770 --> 00:41:27.550
is getting bigger and bigger.
00:41:27.550 --> 00:41:30.690
And you have to worry
about that being infinity.
00:41:30.690 --> 00:41:33.110
Except, as I told you,
we don't have any worries
00:41:33.110 --> 00:41:36.690
about infinity,
because our q has
00:41:36.690 --> 00:41:41.160
to be cut off by the inverse of
the character wavelength, which
00:41:41.160 --> 00:41:44.460
is the length scale over which
I am doing the coarse grain.
00:41:44.460 --> 00:41:48.020
So let's call that cut
off lambda, presumably
00:41:48.020 --> 00:41:53.510
this inverse of some kind
of lattice-like spacing.
00:41:53.510 --> 00:41:55.060
It's not the lattice spacing.
00:41:55.060 --> 00:41:58.580
It's the coarse graining scale.
00:41:58.580 --> 00:42:02.010
So if I'm doing this,
then this integral,
00:42:02.010 --> 00:42:05.500
I can really forget about
what's happening here.
00:42:05.500 --> 00:42:06.930
Most of the integral
contribution
00:42:06.930 --> 00:42:10.350
will come from the large
lambda, and so the answer
00:42:10.350 --> 00:42:13.600
will be proportional
to 1 over k squared
00:42:13.600 --> 00:42:16.093
and whatever this
other cut off is
00:42:16.093 --> 00:42:20.336
raised to the
power of t minus 4.
00:42:20.336 --> 00:42:21.920
It's proportion.
00:42:21.920 --> 00:42:24.670
I don't care about constants
of proportionality, et cetera.
00:42:27.700 --> 00:42:32.430
However, if I am at dimensions
that is 3 less than 4,
00:42:32.430 --> 00:42:36.670
any dimension less than 4, I can
as well say the upper cut off
00:42:36.670 --> 00:42:40.440
go all the way to infinity,
because the contribution that I
00:42:40.440 --> 00:42:44.180
get by replacing
lambda to infinity
00:42:44.180 --> 00:42:47.150
is going to be very small.
00:42:47.150 --> 00:42:50.790
So then it becomes like
a definite integral.
00:42:50.790 --> 00:42:53.700
And it becomes more
like a definite integral
00:42:53.700 --> 00:42:57.370
if I scale q by c inverse.
00:42:57.370 --> 00:43:02.920
And then what you have to do
is you have 1 over k squared.
00:43:02.920 --> 00:43:06.410
You have c inverse to
the power of t minus 4
00:43:06.410 --> 00:43:09.980
or c to the power of
4 minus t, and then
00:43:09.980 --> 00:43:13.700
some definite integral,
which is 0 to infinity dx,
00:43:13.700 --> 00:43:18.000
x to the d minus 1 divided by x
squared plus 1 to the squared.
00:43:18.000 --> 00:43:20.340
I don't really care
what the number is.
00:43:20.340 --> 00:43:23.606
It's just some number that
goes in this proportionality.
00:43:23.606 --> 00:43:27.504
So this is what happens
for d less than 4.
00:43:32.690 --> 00:43:38.397
So let's see what
all of this means.
00:43:38.397 --> 00:43:41.180
So we are trying to understand
the behavior of the heat
00:43:41.180 --> 00:43:49.270
capacity of the system as a
function of this parameter t.
00:43:49.270 --> 00:43:55.820
And actually, only the
part that corresponds
00:43:55.820 --> 00:43:57.466
to integrating the
magnetization field.
00:43:57.466 --> 00:44:00.230
As I said, there's
phonon contributions,
00:44:00.230 --> 00:44:02.700
all kinds of other
phonon contributions
00:44:02.700 --> 00:44:05.020
that give you some
kind of a background.
00:44:05.020 --> 00:44:08.870
Let's subtract that background
and see what we have.
00:44:08.870 --> 00:44:14.410
So what we have is that
from the subtle point part,
00:44:14.410 --> 00:44:17.130
we get this continuity.
00:44:17.130 --> 00:44:19.100
So let's draw the
subtle point part.
00:44:19.100 --> 00:44:22.816
So the subtle point
part is-- oops.
00:44:22.816 --> 00:44:25.140
Wrong direction.
00:44:25.140 --> 00:44:31.550
Above 0, it's 0.
00:44:31.550 --> 00:44:36.370
Below 0, it jumps to 1 over 8u.
00:44:36.370 --> 00:44:40.330
So it's a behavior such as this.
00:44:40.330 --> 00:44:44.120
So this part is the c
of the subtle point.
00:44:50.770 --> 00:44:54.390
But to that, I have
to add a correction.
00:44:54.390 --> 00:44:56.200
So let's look at the correction.
00:44:56.200 --> 00:44:58.390
First of all, if I'm
looking at the correction
00:44:58.390 --> 00:45:03.790
above four dimensions,
whether I'm above or below,
00:45:03.790 --> 00:45:08.100
I have to add one
of these quantities.
00:45:08.100 --> 00:45:11.700
These quantities don't have
any explicit dependence
00:45:11.700 --> 00:45:14.150
on t itself.
00:45:14.150 --> 00:45:17.510
So what happens is that if
I add that, presumably there
00:45:17.510 --> 00:45:21.505
is a correction that I will
get from below and a correction
00:45:21.505 --> 00:45:23.680
that I will get from above.
00:45:23.680 --> 00:45:30.760
So this is cf for d
that is larger than 4.
00:45:30.760 --> 00:45:36.000
So what it certainly does is
when I add this part to what
00:45:36.000 --> 00:45:42.450
I had before, I will change the
magnitude of the discontinuity.
00:45:42.450 --> 00:45:43.430
But so what?
00:45:43.430 --> 00:45:47.220
The discontinuity itself was not
something that was important,
00:45:47.220 --> 00:45:50.030
because u was not
a universal number.
00:45:50.030 --> 00:45:55.260
So there was some singularity
before, some singularity above.
00:45:55.260 --> 00:46:00.230
We see that the corrections
for dimensions greater than 4
00:46:00.230 --> 00:46:03.620
do not change the qualitative
statement that the heat
00:46:03.620 --> 00:46:07.800
capacity should have
a discontinuity.
00:46:07.800 --> 00:46:12.580
But if I go to
dimensions less than 4
00:46:12.580 --> 00:46:16.780
and I realize that my c goes
like the square root of t--
00:46:16.780 --> 00:46:22.060
there is the formulas for c over
there, or t to the minus 1/2--
00:46:22.060 --> 00:46:26.245
we find that this quantity
is proportional to t
00:46:26.245 --> 00:46:29.715
to the minus 4 minus d over 2.
00:46:32.900 --> 00:46:38.850
So below four
dimensions, what we get
00:46:38.850 --> 00:46:44.310
is that the correction
that we calculated
00:46:44.310 --> 00:46:45.435
is actually divergent.
00:46:48.270 --> 00:46:53.570
So this is cf for d less than 4.
00:46:53.570 --> 00:46:57.360
There is a divergence as t
goes to 0 that, let's say,
00:46:57.360 --> 00:47:00.030
if you're sitting
three dimensions
00:47:00.030 --> 00:47:02.160
would be an exponent
t to the minus 1/2.
00:47:07.140 --> 00:47:09.850
So you started with a
subtle point prediction
00:47:09.850 --> 00:47:14.220
that the heat capacity
should be discontinuous.
00:47:14.220 --> 00:47:16.640
You add the analog
of these corrections
00:47:16.640 --> 00:47:19.340
to the subtle point
calculation, and you
00:47:19.340 --> 00:47:24.620
find that the correction is
much, much more important
00:47:24.620 --> 00:47:26.245
than the original discontinuity.
00:47:26.245 --> 00:47:30.530
It completely changes
your conclusions.
00:47:30.530 --> 00:47:33.860
So once we go beyond
this approximation
00:47:33.860 --> 00:47:36.840
that we did over here,
the subtle point,
00:47:36.840 --> 00:47:40.240
and the difference between
our problematic and the one
00:47:40.240 --> 00:47:44.705
that we did in 8 333 is that
we don't have one variable
00:47:44.705 --> 00:47:46.510
that we are integrating.
00:47:46.510 --> 00:47:48.630
We are integrating
over fluctuations
00:47:48.630 --> 00:47:51.170
over the entirety of the system.
00:47:51.170 --> 00:47:53.330
And we see that
these fluctuations
00:47:53.330 --> 00:47:57.580
over the entirety of the system
are so severe, at least close
00:47:57.580 --> 00:48:00.230
to the transition point,
that they completely
00:48:00.230 --> 00:48:03.670
invalidate the results that
you had from the subtle point.
00:48:03.670 --> 00:48:04.340
Yes?
00:48:04.340 --> 00:48:08.780
AUDIENCE: So obviously you have
some high order [INAUDIBLE].
00:48:08.780 --> 00:48:11.140
And here you're basically
completing [INAUDIBLE].
00:48:11.140 --> 00:48:12.540
PROFESSOR: Exactly.
00:48:12.540 --> 00:48:16.070
AUDIENCE: Is there
an easy way to argue
00:48:16.070 --> 00:48:19.140
that for b greater than
4 there is no divergence
00:48:19.140 --> 00:48:22.850
lurking in the
higher order terms?
00:48:22.850 --> 00:48:25.080
PROFESSOR: Actually,
the answer is no.
00:48:25.080 --> 00:48:29.750
If I look at this integral
that I have over here,
00:48:29.750 --> 00:48:31.610
it depends on t.
00:48:31.610 --> 00:48:34.760
If I take sufficiently
high derivatives of it,
00:48:34.760 --> 00:48:37.520
I will encounter a singularity.
00:48:37.520 --> 00:48:41.810
So indeed, what I
have focused here
00:48:41.810 --> 00:48:43.690
is at the level of
the heat capacity.
00:48:43.690 --> 00:48:46.410
But if I were to look at the
fifth derivative of the phi
00:48:46.410 --> 00:48:48.585
energy, I will
see singularities.
00:48:48.585 --> 00:48:50.793
AUDIENCE: No, I'm talking
about the second derivative
00:48:50.793 --> 00:48:52.342
for higher order terms.
00:48:56.590 --> 00:48:59.160
PROFESSOR: These higher
order terms, the phis?
00:48:59.160 --> 00:49:00.090
OK, all right.
00:49:00.090 --> 00:49:02.100
So that was my next one.
00:49:02.100 --> 00:49:09.040
So you may be tempted to say,
OK, I found the divergence.
00:49:09.040 --> 00:49:14.160
Let's say that the heat capacity
diverges with exponent of 1/2.
00:49:14.160 --> 00:49:15.710
And no.
00:49:15.710 --> 00:49:19.470
The only thing that it says
is that your starting point
00:49:19.470 --> 00:49:21.710
was wrong.
00:49:21.710 --> 00:49:23.930
Any conclusion that
you want to make
00:49:23.930 --> 00:49:29.120
based on what we are
doing here is wrong.
00:49:29.120 --> 00:49:33.050
There is no point in my
going beyond and calculating
00:49:33.050 --> 00:49:35.040
the higher order term,
because I already
00:49:35.040 --> 00:49:38.430
see that the lowest order
correction is invalidating
00:49:38.430 --> 00:49:39.190
my result.
00:49:39.190 --> 00:49:42.200
AUDIENCE: So you [INAUDIBLE]
conclude that mean field theory
00:49:42.200 --> 00:49:44.340
is good for bigger than 4.
00:49:53.870 --> 00:49:55.960
PROFESSOR: From what
I have told you,
00:49:55.960 --> 00:50:00.000
I've shown you that the
discontinuity in the heat
00:50:00.000 --> 00:50:04.030
capacity is maintained.
00:50:04.030 --> 00:50:07.870
It is true that if I look at
sufficiently high derivatives,
00:50:07.870 --> 00:50:11.760
I may encounter some
difficulty in justifying
00:50:11.760 --> 00:50:18.880
why d greater than 4 or less
that 4 is making a difference.
00:50:18.880 --> 00:50:23.590
But certainly, as we will
build on what we know later
00:50:23.590 --> 00:50:27.420
on in the course, I will
be able to convince you
00:50:27.420 --> 00:50:29.220
that the mean field
theory is certainly
00:50:29.220 --> 00:50:32.790
valid in dimensions
greater than 4.
00:50:32.790 --> 00:50:39.270
But right now, I guess the only
thing that we can say for sure
00:50:39.270 --> 00:50:43.880
is that the subtle point method
cannot be applied when you are
00:50:43.880 --> 00:50:46.955
dealing with a field that is
varying all over the space.
00:50:50.490 --> 00:50:56.980
So we have this situation.
00:50:56.980 --> 00:51:00.730
On the other hand,
you say, well,
00:51:00.730 --> 00:51:02.522
if it is so bad,
why does it work
00:51:02.522 --> 00:51:03.980
for the case of
the superconductor?
00:51:06.940 --> 00:51:11.290
So let's see if we can
try to understand that.
00:51:11.290 --> 00:51:15.320
Again, sticking with the
language of the heat capacity,
00:51:15.320 --> 00:51:21.140
we see that if I am, let's
say, sitting in some dimensions
00:51:21.140 --> 00:51:26.760
below 4, to the
lowest order I will
00:51:26.760 --> 00:51:31.390
predict that there is a
discontinuity in the singular
00:51:31.390 --> 00:51:40.898
part and that the fluctuations
lead to a correction
00:51:40.898 --> 00:51:42.235
where it should be divergent.
00:51:47.024 --> 00:51:49.130
Now it is
mathematically correct.
00:51:49.130 --> 00:51:52.540
But let's see how
you would go and see
00:51:52.540 --> 00:51:53.940
that in the experiments.
00:51:53.940 --> 00:51:57.890
So presumably in the experiment,
in the analog of your t going
00:51:57.890 --> 00:52:03.560
to 0 is that you have a
t that passes through tc.
00:52:03.560 --> 00:52:06.370
And what you are doing
in the experiment
00:52:06.370 --> 00:52:09.140
is that you are
making measurements,
00:52:09.140 --> 00:52:11.789
let's say, at this point, at
this point, at this point,
00:52:11.789 --> 00:52:13.247
and then you are
going all the way.
00:52:15.770 --> 00:52:18.380
Now we can see that
there could potentially
00:52:18.380 --> 00:52:23.600
be a difference, depending on
the amplitude of this term.
00:52:23.600 --> 00:52:27.490
If it is like that, and I can
resolve things at this scale
00:52:27.490 --> 00:52:29.940
that I have indicated
here, there's no problem.
00:52:29.940 --> 00:52:32.650
I should see the divergence.
00:52:32.650 --> 00:52:37.020
But suppose the amplitude
is much, much smaller
00:52:37.020 --> 00:52:41.110
and it is something that
is looking like this,
00:52:41.110 --> 00:52:43.320
and you are taking
measurements that
00:52:43.320 --> 00:52:47.240
correspond to, essentially,
intervals such as this,
00:52:47.240 --> 00:52:50.420
then you really
integrate across this.
00:52:50.420 --> 00:52:53.740
You don't see the peak.
00:52:53.740 --> 00:52:55.820
You don't sufficient resolution.
00:52:55.820 --> 00:53:00.200
It's kind of searching for a
delta function more or less.
00:53:00.200 --> 00:53:03.860
And so whether or not
you are in one situation
00:53:03.860 --> 00:53:07.360
or another situation
could tell you
00:53:07.360 --> 00:53:11.820
about the result of
experimental observation.
00:53:11.820 --> 00:53:16.250
So how do I find out
something about that?
00:53:16.250 --> 00:53:21.410
Well, I want the
amplitude of this
00:53:21.410 --> 00:53:24.950
to be at least as large
as the discontinuity
00:53:24.950 --> 00:53:27.330
for me to be able to state it.
00:53:27.330 --> 00:53:31.540
That is, I want
to have a c that I
00:53:31.540 --> 00:53:35.910
have from the subtle point,
which is a discontinuity that
00:53:35.910 --> 00:53:38.900
is of the order of one
over 8u, so there's
00:53:38.900 --> 00:53:41.900
at a discontinuity
heat capacity.
00:53:41.900 --> 00:53:48.015
This discontinuity should be
of the order of this quantity 1
00:53:48.015 --> 00:53:54.830
over k squared c to
the power of 4 minus t.
00:53:54.830 --> 00:54:00.240
But now it becomes
kind of non-universal
00:54:00.240 --> 00:54:05.870
because I really want to compare
things, compare amplitudes.
00:54:05.870 --> 00:54:12.330
I know that my c is predicted
from the subtle point
00:54:12.330 --> 00:54:18.570
to go like t to the minus
1/2, where t is kind
00:54:18.570 --> 00:54:20.856
a rescaled version
of temperature.
00:54:20.856 --> 00:54:25.370
So t is, let's say,
tc minus t over tc.
00:54:25.370 --> 00:54:28.530
It is something that
is dimensionless.
00:54:28.530 --> 00:54:30.600
And so all of the
dimensions should
00:54:30.600 --> 00:54:33.940
be carried by some kind
of a prefactor here,
00:54:33.940 --> 00:54:36.610
that is some kind
of a landscape.
00:54:36.610 --> 00:54:40.260
So the correlation,
then, is a length scale.
00:54:40.260 --> 00:54:43.060
There is some prefactor
that is also a length scale,
00:54:43.060 --> 00:54:45.460
and then this
reduced temperature
00:54:45.460 --> 00:54:49.960
that controls the
functional divergence.
00:54:49.960 --> 00:54:55.730
Actually, I can read off what
this c0 should depend on.
00:54:55.730 --> 00:55:03.240
You can see that c0 should
scale like k square root of k.
00:55:09.050 --> 00:55:12.750
So then you can see
that this object
00:55:12.750 --> 00:55:16.070
k scales like c0 squared.
00:55:16.070 --> 00:55:20.570
So this scales like 1
over c0 to fourth power.
00:55:20.570 --> 00:55:24.910
And this scales like c0
to the power of 4 minus t.
00:55:24.910 --> 00:55:28.090
And then I have this
reduced temperature
00:55:28.090 --> 00:55:31.766
to the power of
d minus 4 over 2.
00:55:37.200 --> 00:55:43.080
So you can see that for these
things to be compatible,
00:55:43.080 --> 00:55:48.610
I should reduce my
t to a value such
00:55:48.610 --> 00:55:53.400
that this divergence
compensates for the combination
00:55:53.400 --> 00:56:01.220
c0 to the d delta csp,
should be of the order
00:56:01.220 --> 00:56:05.210
of some minimal value of t.
00:56:05.210 --> 00:56:08.280
Let's call it tc.
00:56:08.280 --> 00:56:15.930
Actually, let's call it tg to
the power of d minus 4 over 2.
00:56:20.440 --> 00:56:34.960
Or tg is of the
order of delta csp c0
00:56:34.960 --> 00:56:38.250
to the d, the whole
thing to the power of 2
00:56:38.250 --> 00:56:41.369
divided by d minus 4.
00:56:47.357 --> 00:56:50.040
Let me wrote that
slightly better.
00:56:50.040 --> 00:56:57.660
So tg goes off the order
of delta cp, delta csp
00:56:57.660 --> 00:57:03.510
to the power of
minus 2 4 minus t,
00:57:03.510 --> 00:57:07.290
since we are going to be
looking at dimensions such as 3,
00:57:07.290 --> 00:57:13.240
and then c0 to the
power of minus 2
00:57:13.240 --> 00:57:17.752
divided by 2d
divided by 4 minus d.
00:57:23.470 --> 00:57:28.700
So we can see that the
resolution that you need,
00:57:28.700 --> 00:57:32.720
how close you have to go
to the critical point,
00:57:32.720 --> 00:57:35.900
very much depends
on this quantity c0.
00:57:35.900 --> 00:57:38.870
It does depend
also on delta csp.
00:57:38.870 --> 00:57:43.230
But we can argue that that is
a less important contribution.
00:57:43.230 --> 00:57:49.770
Let's focus, for the time being,
on the dependence on this c0.
00:57:49.770 --> 00:57:54.060
So c0 presumably
has something to do
00:57:54.060 --> 00:57:57.260
with the physics of the system
that you are looking at.
00:57:57.260 --> 00:58:01.390
So then we are leaving the realm
of things that were universal.
00:58:01.390 --> 00:58:05.900
And we have to think about the
system under consideration.
00:58:05.900 --> 00:58:09.390
And we have to identify
a length scale associated
00:58:09.390 --> 00:58:12.770
with the system that
is under consideration.
00:58:12.770 --> 00:58:20.000
Now if I think about
something like liquid gas,
00:58:20.000 --> 00:58:25.980
well, one kind of length scale
that immediately comes to mind
00:58:25.980 --> 00:58:31.880
is the length scale over which
the particles are interacting.
00:58:31.880 --> 00:58:35.290
Also I can look at the
kind of phase diagrams
00:58:35.290 --> 00:58:40.990
that we were looking get, and
there was some critical volume
00:58:40.990 --> 00:58:45.770
where this transition
from one type of isotherm
00:58:45.770 --> 00:58:48.500
to another type of
isotherm occurs,
00:58:48.500 --> 00:58:54.700
I can ask that critical volume
how many angstroms it is.
00:58:54.700 --> 00:58:57.080
But again, everything
here, we have
00:58:57.080 --> 00:59:00.410
to try to be as
dimensionless as possible.
00:59:00.410 --> 00:59:04.350
So let's say this critical
volume corresponds
00:59:04.350 --> 00:59:06.830
to how many particles.
00:59:06.830 --> 00:59:08.840
And let's take the
cube root of that
00:59:08.840 --> 00:59:11.980
and convert it to a
length scale over which
00:59:11.980 --> 00:59:15.820
these number of particles are
confined in three dimensions.
00:59:15.820 --> 00:59:18.950
And what we find is,
for liquid gas systems,
00:59:18.950 --> 00:59:24.980
that number c0 that you get
in units of atomic spacing
00:59:24.980 --> 00:59:30.490
is of the order of 1
to 10 atomic spacings.
00:59:37.224 --> 00:59:38.186
Yes.
00:59:38.186 --> 00:59:43.010
AUDIENCE: Scale on which atoms
interact with each other?
00:59:43.010 --> 00:59:44.800
PROFESSOR: Well, it could be.
00:59:44.800 --> 00:59:48.660
But for the case of, say,
particles in this room,
00:59:48.660 --> 00:59:54.290
the range of interaction is not
that different than the size
00:59:54.290 --> 00:59:56.180
of the particles
coming together.
00:59:56.180 --> 00:59:58.510
It's maybe a few times that.
00:59:58.510 --> 01:00:02.060
So that's basically a few times
of [INAUDIBLE] saying here.
01:00:02.060 --> 01:00:04.170
And I'm not going
to argue whether it
01:00:04.170 --> 01:00:06.493
is twice that or 10 times that.
01:00:06.493 --> 01:00:07.742
It really makes no difference.
01:00:10.340 --> 01:00:11.870
The thing is that
when I'm looking
01:00:11.870 --> 01:00:19.040
about the problem of
superconductivity,
01:00:19.040 --> 01:00:23.130
this is the only place
where we introduce
01:00:23.130 --> 01:00:25.050
a little bit of physics.
01:00:25.050 --> 01:00:28.410
When one is looking at
something like aluminum
01:00:28.410 --> 01:00:32.400
that goes into being
a superconductor,
01:00:32.400 --> 01:00:38.320
it is an ordering of
bosons in the same sense
01:00:38.320 --> 01:00:41.050
that we have for liquid helium.
01:00:41.050 --> 01:00:43.640
But the difference is
that what is ordering
01:00:43.640 --> 01:00:45.960
in superconductivity
is not bosons,
01:00:45.960 --> 01:00:48.780
but it is fermions or electrons.
01:00:48.780 --> 01:00:51.480
And electrons have
Coulomb repulsion.
01:00:51.480 --> 01:00:53.570
So what has to
happen is that there
01:00:53.570 --> 01:00:56.850
is some mechanism,
phonons or whatever, that
01:00:56.850 --> 01:01:01.960
gives an effective attraction
between electrons and pairs
01:01:01.960 --> 01:01:04.960
them together into
a Cooper pair.
01:01:04.960 --> 01:01:08.060
The characteristic
size of a Cooper pair,
01:01:08.060 --> 01:01:12.820
because of the
repulsion that you
01:01:12.820 --> 01:01:17.350
have between electrons, rather
than being 1 to 10, say,
01:01:17.350 --> 01:01:28.350
angstroms, is c0 is suddenly of
the order of 1,000 angstroms.
01:01:28.350 --> 01:01:32.520
Now note that if you
are in three dimensions,
01:01:32.520 --> 01:01:35.110
this is something that is
raised to the sixth power.
01:01:38.400 --> 01:01:44.140
So if I think of this after
dividing by an atomic size
01:01:44.140 --> 01:01:46.860
or whatever, to a number that
is of the order of, let's say,
01:01:46.860 --> 01:01:51.510
100 or even 1,000 and I
raise it to the sixth power,
01:01:51.510 --> 01:01:53.670
you can see that the
kind of resolution
01:01:53.670 --> 01:01:58.100
that you need when you raise
something large to a huge power
01:01:58.100 --> 01:02:03.200
corresponds to t that is of the
order of 10 to the minus 12,
01:02:03.200 --> 01:02:05.490
10 to the minus 15, et cetera.
01:02:05.490 --> 01:02:07.090
And that's just
not the resolution
01:02:07.090 --> 01:02:08.640
that you have in experiment.
01:02:08.640 --> 01:02:12.880
So basically experiment
will go over this
01:02:12.880 --> 01:02:14.760
without really seeing it.
01:02:14.760 --> 01:02:19.800
Essentially the units
are so big that you
01:02:19.800 --> 01:02:25.210
don't have that many of them
to fluctuate across the system.
01:02:25.210 --> 01:02:28.760
The effect of fluctuations
is much diminished
01:02:28.760 --> 01:02:32.850
compared to superfluid helium
or compared to liquid gas,
01:02:32.850 --> 01:02:35.110
where over the
size of the system,
01:02:35.110 --> 01:02:39.860
you have many, many fluctuations
that can take place.
01:02:39.860 --> 01:02:43.170
This condition,
whether or not you're
01:02:43.170 --> 01:02:47.320
going to be able to see
the effects of fluctuations
01:02:47.320 --> 01:02:49.910
and something that is
[INAUDIBLE] field like,
01:02:49.910 --> 01:02:52.441
I'll call it t sub g, because
it's called a Ginzburg
01:02:52.441 --> 01:02:52.940
criterion.
01:03:06.440 --> 01:03:12.150
So this basically
answers the questions
01:03:12.150 --> 01:03:15.520
that we had over here.
01:03:15.520 --> 01:03:17.770
For all of our
phase transitions,
01:03:17.770 --> 01:03:21.110
we constructed the
Landau-Ginzburg theory,
01:03:21.110 --> 01:03:23.380
and we evaluated
its consequences
01:03:23.380 --> 01:03:25.250
for phase transition,
such as divergence
01:03:25.250 --> 01:03:29.120
of heat capacity using
the subtle point method.
01:03:29.120 --> 01:03:31.360
We saw that the results
worked extremely well
01:03:31.360 --> 01:03:36.000
for superconductors, but
not for anything else.
01:03:36.000 --> 01:03:39.890
And the answer to that is
that for superconductors,
01:03:39.890 --> 01:03:42.530
fluctuations are
not so important.
01:03:42.530 --> 01:03:45.090
And the most probable
state gives you
01:03:45.090 --> 01:03:47.410
a good idea of
what is happening.
01:03:47.410 --> 01:03:49.990
Whereas for super
helium, for liquid gas,
01:03:49.990 --> 01:03:53.320
et cetera, fluctuations
are very important,
01:03:53.320 --> 01:03:56.550
and the starting point that is
the subtle point, most probable
01:03:56.550 --> 01:04:00.695
state, is simply
not good enough.
01:04:00.695 --> 01:04:01.665
Yes.
01:04:01.665 --> 01:04:05.545
AUDIENCE: So when
you were giving us
01:04:05.545 --> 01:04:09.920
the system of different phase
transitions [INAUDIBLE],
01:04:09.920 --> 01:04:12.096
you only talked about
the critical exponents,
01:04:12.096 --> 01:04:18.370
because, for instance, there is
a discontinuity of [INAUDIBLE]
01:04:18.370 --> 01:04:21.040
heat capacity for all
phase transitions.
01:04:21.040 --> 01:04:24.372
But it's often masked with
fixed singularity, right?
01:04:29.040 --> 01:04:31.340
PROFESSOR: Once you
have a divergence,
01:04:31.340 --> 01:04:36.064
I don't know how you would be
talking about a singularity.
01:04:36.064 --> 01:04:40.297
AUDIENCE: If you roughly measure
the heat capacity further away
01:04:40.297 --> 01:04:42.920
from singularity,
wouldn't it kind of
01:04:42.920 --> 01:04:47.300
converges left and right
of two different values?
01:04:47.300 --> 01:04:48.010
PROFESSOR: OK.
01:04:48.010 --> 01:04:54.070
So if I draw a random
function that has divergence,
01:04:54.070 --> 01:04:57.390
the chances are very, very
good that, if I go a little bit
01:04:57.390 --> 01:04:59.200
further, the two
of them will not
01:04:59.200 --> 01:05:00.990
be exactly at the same height.
01:05:00.990 --> 01:05:02.970
There will be an asymmetry.
01:05:02.970 --> 01:05:06.690
So are you talking about
the asymmetry in amplitudes?
01:05:06.690 --> 01:05:10.230
Because I know the
amplitudes are not symmetric.
01:05:10.230 --> 01:05:14.250
If I go very, very far away,
then all kinds of other things
01:05:14.250 --> 01:05:15.110
come in to play.
01:05:15.110 --> 01:05:18.350
There's the phonon, heat
capacity, et cetera.
01:05:18.350 --> 01:05:21.640
So the statement
that you make, I
01:05:21.640 --> 01:05:24.390
have never heard
before, in fact.
01:05:24.390 --> 01:05:26.290
But I'm trying to
see whether or not
01:05:26.290 --> 01:05:28.150
it's even mathematically
conceivable.
01:05:31.366 --> 01:05:36.390
AUDIENCE: Another
question with this series
01:05:36.390 --> 01:05:38.580
you just wrote out with
[INAUDIBLE] singularity,
01:05:38.580 --> 01:05:44.111
doesn't it give you that
exponent for the singularity?
01:05:44.111 --> 01:05:44.694
PROFESSOR: No.
01:05:44.694 --> 01:05:47.630
AUDIENCE: It's a
[INAUDIBLE] number.
01:05:47.630 --> 01:05:50.850
PROFESSOR: It is 1/2, yes.
01:05:50.850 --> 01:05:52.870
So there is a theory.
01:05:52.870 --> 01:05:54.920
There is a mathematical
theory that
01:05:54.920 --> 01:05:58.270
has this 1/2
exponent divergence.
01:05:58.270 --> 01:05:59.880
What is that theory?
01:05:59.880 --> 01:06:06.550
It's a theory that is cut
off at the Gaussian level.
01:06:06.550 --> 01:06:11.770
So if we had some system
for which we were sure
01:06:11.770 --> 01:06:16.110
that when we write our
statistical field theory,
01:06:16.110 --> 01:06:19.980
I can terminate at the
level of Gaussian terms,
01:06:19.980 --> 01:06:22.170
m squared, gradient of
m squared, et cetera.
01:06:22.170 --> 01:06:24.300
If such a theory
existed, it would
01:06:24.300 --> 01:06:27.140
have exactly this divergence.
01:06:27.140 --> 01:06:31.305
But I don't see any reason
for eliminating all those--
01:06:31.305 --> 01:06:33.694
AUDIENCE: So we still
have not found the reason
01:06:33.694 --> 01:06:36.210
why the actual experimental
exponents are--
01:06:36.210 --> 01:06:38.072
PROFESSOR: No, we
have not found.
01:06:38.072 --> 01:06:39.940
Yes.
01:06:39.940 --> 01:06:43.265
AUDIENCE: So how do we
interpret the larger
01:06:43.265 --> 01:06:46.800
signal of superconductity?
01:06:46.800 --> 01:06:50.260
Does that mean the correlation
actually is longer?
01:06:50.260 --> 01:06:51.310
PROFESSOR: Yes, yes.
01:06:51.310 --> 01:06:54.340
AUDIENCE: But then
why are we saying
01:06:54.340 --> 01:06:57.690
that the fluctuation
there is not so important?
01:06:57.690 --> 01:07:00.566
We have longer correlation,
then usually that
01:07:00.566 --> 01:07:02.440
means we have bigger
fluctuation [INAUDIBLE].
01:07:26.950 --> 01:07:30.710
PROFESSOR: OK, so let's
see if we can unpack that.
01:07:30.710 --> 01:07:44.570
So our correlation length is
some c0 t to the minus 1/2.
01:07:44.570 --> 01:07:49.520
And indeed, what
that says is that
01:07:49.520 --> 01:07:54.130
at some particular same
value of how far I am away
01:07:54.130 --> 01:08:00.860
from the critical point, the
correlations are longer ranged.
01:08:00.860 --> 01:08:05.906
If I go and look at the
amplitudes of the fluctuations
01:08:05.906 --> 01:08:21.649
that I have, then I am
closer as a function of q
01:08:21.649 --> 01:08:25.460
to a situation such as this.
01:08:25.460 --> 01:08:33.330
So c0 is large c0.
01:08:33.330 --> 01:08:34.979
Inverse would be smaller.
01:08:34.979 --> 01:08:38.270
So that's correct.
01:08:38.270 --> 01:08:43.620
And then in real space,
what it would mean
01:08:43.620 --> 01:08:49.439
is that if I look at my
system, what I would have
01:08:49.439 --> 01:08:55.930
is that there are parts that
are of the order of c0 t
01:08:55.930 --> 01:09:00.689
to the minus 1/2 that
are doing the same thing.
01:09:14.496 --> 01:09:16.260
Let me understand your question.
01:09:16.260 --> 01:09:19.920
So it is true that
the superconductor
01:09:19.920 --> 01:09:26.319
you have more correlations,
and what that means is
01:09:26.319 --> 01:09:34.350
that the number of independent
modes that you have that
01:09:34.350 --> 01:09:39.770
can contribute and
fluctuate is less.
01:09:39.770 --> 01:09:46.290
And what we will
see ultimately is
01:09:46.290 --> 01:09:52.470
that the reason for all of
these exponents being different
01:09:52.470 --> 01:09:56.810
from what we have
in superconductivity
01:09:56.810 --> 01:10:02.280
is that there is essentially
a much more broader range
01:10:02.280 --> 01:10:04.700
of the influence
that is contributing
01:10:04.700 --> 01:10:07.740
to the whole thing.
01:10:07.740 --> 01:10:12.670
So I'm not sure if I'm
answering your question.
01:10:12.670 --> 01:10:15.790
Let's go back and think
about your question.
01:10:15.790 --> 01:10:20.620
So basically for superconductor,
certainly everything
01:10:20.620 --> 01:10:23.740
that we said, including
being able to express it
01:10:23.740 --> 01:10:26.440
in terms of this
statistical field theory,
01:10:26.440 --> 01:10:29.830
having large correlation lengths
close to the critical point,
01:10:29.830 --> 01:10:32.480
all of that is correct.
01:10:32.480 --> 01:10:34.800
The only thing
that is not correct
01:10:34.800 --> 01:10:43.060
is that the diversity of
fluctuations here is less.
01:10:43.060 --> 01:10:47.130
And this lack of diversity
of fluctuations compared
01:10:47.130 --> 01:10:51.620
to something like
liquid gas gives you
01:10:51.620 --> 01:10:53.745
more subtle point,
like exponents.
01:10:56.760 --> 01:11:05.720
AUDIENCE: So you mean the limit
for my integrand with respect
01:11:05.720 --> 01:11:07.291
to q is smaller?
01:11:07.291 --> 01:11:07.790
[INAUDIBLE]
01:11:10.940 --> 01:11:14.500
So the q space I'm
integrating is smaller.
01:11:14.500 --> 01:11:16.210
PROFESSOR: Yes.
01:11:16.210 --> 01:11:21.875
AUDIENCE: But if I calculate
fluctuation function,
01:11:21.875 --> 01:11:22.716
something?
01:11:22.716 --> 01:11:23.340
PROFESSOR: Yes.
01:11:23.340 --> 01:11:28.060
So this is what I was trying
to calculate here, yes.
01:11:28.060 --> 01:11:30.505
AUDIENCE: Then it
should be larger than--
01:11:30.505 --> 01:11:31.130
PROFESSOR: Yes.
01:11:31.130 --> 01:11:36.100
But it is larger for a
smaller limit of q's.
01:11:36.100 --> 01:11:37.860
So I guess what
you are saying is
01:11:37.860 --> 01:11:39.960
that if I look at
the superconductor,
01:11:39.960 --> 01:11:42.030
I will see something like this.
01:11:42.030 --> 01:11:47.010
If I look at the liquid gas, I
will see something like this.
01:11:47.010 --> 01:11:50.560
AUDIENCE: And [INAUDIBLE]
just intuitively interpret
01:11:50.560 --> 01:11:53.280
what's the behavior of the
heat capacity from this--
01:11:57.665 --> 01:11:59.560
PROFESSOR: [INAUDIBLE],
because if you
01:11:59.560 --> 01:12:03.160
look at something like this, and
particular its t dependence--
01:12:03.160 --> 01:12:05.720
after all, everything
that we are interested
01:12:05.720 --> 01:12:10.880
is how things change as
a function of t minus tc.
01:12:10.880 --> 01:12:13.900
So presumably, when we do
that for superconductor,
01:12:13.900 --> 01:12:16.770
if you do some kind of
a scattering experiment,
01:12:16.770 --> 01:12:19.410
you will see some peak
like this emerging,
01:12:19.410 --> 01:12:21.890
but the peak never
expanding as much
01:12:21.890 --> 01:12:24.090
as it would do for these things.
01:12:24.090 --> 01:12:27.130
You should be able,
based on that,
01:12:27.130 --> 01:12:30.350
to deduce that the range
of wavelengths that
01:12:30.350 --> 01:12:32.680
are fluctuating in
the superconductor
01:12:32.680 --> 01:12:35.860
is less compared to
the liquid gas system.
01:12:35.860 --> 01:12:42.730
And so there is not much range
in the diversity of length
01:12:42.730 --> 01:12:46.000
scales that are contributing
to the fluctuations
01:12:46.000 --> 01:12:46.945
in a superconductor.
01:12:51.130 --> 01:12:54.460
AUDIENCE: So that explains why
we have only very narrow peak
01:12:54.460 --> 01:12:55.430
in a cp?
01:12:59.410 --> 01:13:00.140
PROFESSOR: Yes.
01:13:00.140 --> 01:13:02.950
You have to go
very close in order
01:13:02.950 --> 01:13:05.360
to expand the range
of wavelengths.
01:13:08.024 --> 01:13:11.170
But then you go a little
bit one side or the other,
01:13:11.170 --> 01:13:13.200
and then you are
passed that range
01:13:13.200 --> 01:13:14.920
that you can see very
large wavelengths.
01:13:17.500 --> 01:13:18.230
Yes.
01:13:18.230 --> 01:13:20.420
AUDIENCE: So in our subtle
point, the approximation
01:13:20.420 --> 01:13:23.610
we found our maximum
when we looked
01:13:23.610 --> 01:13:25.855
at the second derivative,
if we had considered more
01:13:25.855 --> 01:13:28.640
derivatives, would we have
captured those exponents?
01:13:32.080 --> 01:13:34.970
PROFESSOR: So if we
think about things,
01:13:34.970 --> 01:13:40.230
and mathematical consistence,
here we have a parameter.
01:13:40.230 --> 01:13:45.360
And we can explicitly calculate
higher and higher order terms
01:13:45.360 --> 01:13:49.630
and how they are smaller and
become more and more small
01:13:49.630 --> 01:13:53.140
as the parameter becomes
larger and larger.
01:13:53.140 --> 01:13:58.140
Now what we have here is
the following situation.
01:13:58.140 --> 01:14:02.140
If I presumably stick at
some value that is away
01:14:02.140 --> 01:14:09.611
from the critical point, let's
say t of 10 to the minus 1,
01:14:09.611 --> 01:14:12.260
at that point I
calculate subtle point.
01:14:12.260 --> 01:14:13.940
And then I calculate
fluctuations
01:14:13.940 --> 01:14:17.140
around subtle point, and
I add more and more term,
01:14:17.140 --> 01:14:19.070
eventually, I think,
I will converge
01:14:19.070 --> 01:14:21.970
to some value for
the heat capacity.
01:14:21.970 --> 01:14:26.010
The problem is that I don't
want to stick to one value of t.
01:14:26.010 --> 01:14:29.990
I want to see what's the
singularity as I approach 0.
01:14:32.530 --> 01:14:37.150
Now we can see that
the problem here
01:14:37.150 --> 01:14:43.610
is that this correction
gives a functional form that
01:14:43.610 --> 01:14:45.750
is divergent.
01:14:45.750 --> 01:14:54.130
And then I would say that if I
go to from t of minus 1 to 10
01:14:54.130 --> 01:15:00.870
to the minus 3, then I'm less
sure about the first correction
01:15:00.870 --> 01:15:03.680
and maybe I will do many,
many more corrections,
01:15:03.680 --> 01:15:05.880
and then I would
get something else.
01:15:05.880 --> 01:15:09.590
And presumably, the closer
I get to t equals to 0,
01:15:09.590 --> 01:15:13.130
I have to go further and
further down in the series.
01:15:13.130 --> 01:15:15.513
And so that becomes
essentially useless.
01:15:20.250 --> 01:15:23.140
Now we will actually do
later on another version
01:15:23.140 --> 01:15:27.020
of this problem, where
we say the following.
01:15:27.020 --> 01:15:31.030
What I did for you here
was calculating essentially
01:15:31.030 --> 01:15:33.220
Gaussian integrals.
01:15:33.220 --> 01:15:36.390
And I know how to do
Gaussian integrals.
01:15:36.390 --> 01:15:40.165
And for Gaussian theory,
this result is exact.
01:15:40.165 --> 01:15:43.610
I will get alpha equals to 1/2.
01:15:43.610 --> 01:15:47.550
Maybe what I can do, instead
of doing subtle points
01:15:47.550 --> 01:15:49.690
approximation,
approach the problem
01:15:49.690 --> 01:15:52.130
completely in a
different fashion.
01:15:52.130 --> 01:15:54.590
I will start with
the Gaussian part,
01:15:54.590 --> 01:15:59.800
and then I do a perturbation
in all of these nonlinearities.
01:15:59.800 --> 01:16:01.190
That's another approach.
01:16:01.190 --> 01:16:06.190
You can say, OK, I know
the problem for u equals 0,
01:16:06.190 --> 01:16:10.570
and so let's say I got this
result for u equals to 0.
01:16:10.570 --> 01:16:13.170
And I want to calculate
what the correction will
01:16:13.170 --> 01:16:17.850
be in proportion to u,
u squared, et cetera.
01:16:17.850 --> 01:16:21.960
But what we find is that
we start expanding in u
01:16:21.960 --> 01:16:24.180
and calculate the
first correction.
01:16:24.180 --> 01:16:25.860
And the first
correction, you'll find,
01:16:25.860 --> 01:16:36.980
is proportional to uc to the
power of t minus 4 over 2.
01:16:36.980 --> 01:16:40.210
So exactly the same
problem over here
01:16:40.210 --> 01:16:43.490
reappears when we try to
do preservation theory.
01:16:43.490 --> 01:16:47.600
You think you are preserving
around a small quantity,
01:16:47.600 --> 01:16:51.140
but as you go to
t equals to 0, you
01:16:51.140 --> 01:16:55.810
find that the coefficient of the
first term in the preservation
01:16:55.810 --> 01:17:00.010
theory actually blows up.
01:17:00.010 --> 01:17:03.920
So we will try a
number of these methods
01:17:03.920 --> 01:17:10.690
to try to extract the right
answer out of this expression.
01:17:10.690 --> 01:17:13.160
This expression is,
in fact, correct.
01:17:13.160 --> 01:17:15.385
The difficulty is mathematical.
01:17:15.385 --> 01:17:19.830
We don't know how to deal
with this kind of integration.
01:17:19.830 --> 01:17:26.520
And I was just listening to the
story of Oppenheimer and Pauli.
01:17:26.520 --> 01:17:28.540
And Oppenheimer,
when he was young,
01:17:28.540 --> 01:17:33.530
goes to-- actually, not
Pauli but [INAUDIBLE].
01:17:33.530 --> 01:17:36.270
And he says, I am
working on some problem,
01:17:36.270 --> 01:17:38.285
and I'm not having any progress.
01:17:38.285 --> 01:17:41.120
He says is the problem,
the difficulty,
01:17:41.120 --> 01:17:43.560
mathematical or physical?
01:17:43.560 --> 01:17:46.360
And Oppenheimer is
flustered because he
01:17:46.360 --> 01:17:48.590
didn't know the answer.
01:17:48.590 --> 01:17:52.520
So here, we know the
problem is mathematical,
01:17:52.520 --> 01:17:56.570
because the physics is
entirely captured here.
01:17:56.570 --> 01:17:58.790
We haven't done anything.
01:17:58.790 --> 01:18:01.790
Now the question,
however, is whether
01:18:01.790 --> 01:18:05.120
the mathematical
problem will be resolved
01:18:05.120 --> 01:18:08.820
by mathematical insights
or physics insights.
01:18:08.820 --> 01:18:11.500
And the interesting thing is
that, in a number of cases
01:18:11.500 --> 01:18:14.940
where the problem originates
from physics, eventually
01:18:14.940 --> 01:18:19.040
the mathematical solution
is provided also by physics.
01:18:19.040 --> 01:18:23.450
So ultimately, people develop
this idea of a normalization
01:18:23.450 --> 01:18:28.190
group that I will be developing
for you in future lectures,
01:18:28.190 --> 01:18:32.590
which is how to solve this
mathematical problem, which we
01:18:32.590 --> 01:18:34.820
have addressed from
this perspective.
01:18:34.820 --> 01:18:39.330
We will try to approach from
the perturbative perspective.
01:18:39.330 --> 01:18:43.045
And it just doesn't
work until we introduce
01:18:43.045 --> 01:18:46.470
a more physical way
of looking at it.