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PROFESSOR: OK, let's start.
00:00:25.410 --> 00:00:32.580
So last time we started thinking
about phase transitions.
00:00:32.580 --> 00:00:35.310
We said that a simple
example obtained
00:00:35.310 --> 00:00:42.260
by taking a piece of
magnet such as iron, nickel
00:00:42.260 --> 00:00:44.605
and seeing what happens as
a function of temperature.
00:00:47.800 --> 00:00:53.556
And there's a phase transition
between a paramagnet
00:00:53.556 --> 00:00:57.240
at high temperature
and a ferromagnet
00:00:57.240 --> 00:01:00.440
at low temperatures.
00:01:00.440 --> 00:01:05.880
This transition takes place
at a characteristic Tc.
00:01:05.880 --> 00:01:12.490
And a nice way to describe what
was happening thermodynamically
00:01:12.490 --> 00:01:17.490
was to also look at the space
that included a magnetic field.
00:01:17.490 --> 00:01:20.370
And then it was clear
that various thermodynamic
00:01:20.370 --> 00:01:27.650
properties had a singularity,
had a discontinuity at the line
00:01:27.650 --> 00:01:32.330
h equals to 0 for all T
less than or equal to Tc.
00:01:36.120 --> 00:01:42.440
Then we saw that if we looked at
characteristic isotherms going
00:01:42.440 --> 00:01:47.450
from high temperatures
going to low temperatures
00:01:47.450 --> 00:01:50.950
where there was a
discontinuity, we more or less
00:01:50.950 --> 00:01:54.950
had to conclude by
continuity that the one that
00:01:54.950 --> 00:01:59.750
goes along Tc, the magnetization
as a function of field,
00:01:59.750 --> 00:02:06.830
has to come and hug the
axis at 90 degree angle
00:02:06.830 --> 00:02:10.840
corresponding to having
infinite susceptibility.
00:02:10.840 --> 00:02:14.700
We also said that once you
have infinite susceptibility,
00:02:14.700 --> 00:02:16.560
you can pretty much
conclude that you
00:02:16.560 --> 00:02:21.350
are going to have long-range
correlations across the sample.
00:02:21.350 --> 00:02:23.400
So that if you make
a fluctuation here,
00:02:23.400 --> 00:02:28.440
the influence is going to be
felt at large distances away.
00:02:28.440 --> 00:02:29.590
OK?
00:02:29.590 --> 00:02:33.040
So given with that
piece of knowledge,
00:02:33.040 --> 00:02:36.190
we said what we can
do is to basically do
00:02:36.190 --> 00:02:37.490
some kind of averaging.
00:02:37.490 --> 00:02:40.290
I can take pieces of the sample.
00:02:40.290 --> 00:02:44.620
And at each piece,
I can find locally
00:02:44.620 --> 00:02:49.930
what the magnetization
is and have this field
00:02:49.930 --> 00:02:53.820
m as a function of x that
varies from one point
00:02:53.820 --> 00:02:55.560
to another point.
00:02:55.560 --> 00:03:02.840
Presumably, close to this point
either below or above, this m
00:03:02.840 --> 00:03:07.150
fluctuates across the
sample over large distances
00:03:07.150 --> 00:03:10.380
and is typically small.
00:03:10.380 --> 00:03:13.230
So then we use those
pieces of information
00:03:13.230 --> 00:03:15.210
to proceed as follows.
00:03:15.210 --> 00:03:18.360
We said all thermodynamic
properties of the system
00:03:18.360 --> 00:03:22.870
can in principle be obtained by
looking at a kind of partition
00:03:22.870 --> 00:03:25.750
function or Gibbs partition
function that depends
00:03:25.750 --> 00:03:31.250
on temperature, let's say,
which is obtained by tracing
00:03:31.250 --> 00:03:37.980
the Hamiltonian that governs all
microscopic degrees of freedom
00:03:37.980 --> 00:03:39.300
that describe this system.
00:03:39.300 --> 00:03:45.650
The electrons, their spins,
nuclei, all kinds of things.
00:03:45.650 --> 00:03:49.750
Now naturally, this I cannot do.
00:03:49.750 --> 00:03:56.980
But I can focus on this
magnetization field close to Tc
00:03:56.980 --> 00:03:59.835
and say that each
configuration of magnetization
00:03:59.835 --> 00:04:02.440
has some kind of a weight.
00:04:02.440 --> 00:04:04.960
And in principle,
what I can do is
00:04:04.960 --> 00:04:08.890
I can subdivide
all configurations
00:04:08.890 --> 00:04:11.680
of microscopic
degrees of freedom
00:04:11.680 --> 00:04:17.680
that are consistent with a
particular macroscopic weight.
00:04:17.680 --> 00:04:20.190
Macroscopic field m of x.
00:04:20.190 --> 00:04:26.230
And hence, in principle
compute what that weight is.
00:04:30.500 --> 00:04:34.030
The analog of tracing over
all degrees of freedom
00:04:34.030 --> 00:04:36.545
would now become
integrating over
00:04:36.545 --> 00:04:40.140
all configurations
of this magnetization
00:04:40.140 --> 00:04:43.700
that I indicate
through this symbol
00:04:43.700 --> 00:04:46.250
of functional integration
over all configurations.
00:04:49.520 --> 00:04:52.920
Now, clearly I can
no more obtain this
00:04:52.920 --> 00:04:55.930
than I can do the
original trace.
00:04:55.930 --> 00:04:57.650
So what did we do?
00:04:57.650 --> 00:05:02.260
We said I can guess what
this is going to look like.
00:05:02.260 --> 00:05:09.670
Because in the
absence of the field,
00:05:09.670 --> 00:05:12.280
it's a function that
has rotational symmetry.
00:05:14.960 --> 00:05:21.680
So what I can do is I can write
the log of that probability
00:05:21.680 --> 00:05:22.910
as something.
00:05:22.910 --> 00:05:25.100
So far I haven't done anything.
00:05:25.100 --> 00:05:28.130
I made the assumption
that I can write it
00:05:28.130 --> 00:05:31.790
as an integral over
space of some kind
00:05:31.790 --> 00:05:34.710
of a density at each point.
00:05:34.710 --> 00:05:39.460
So that was this kind of
quasi-locality assumption.
00:05:39.460 --> 00:05:44.270
And then I would
write anything that
00:05:44.270 --> 00:05:46.400
comes to my mind
that is consistent
00:05:46.400 --> 00:05:49.080
with rotational symmetry.
00:05:49.080 --> 00:05:52.590
Now, since m is small in
the vicinity of this point,
00:05:52.590 --> 00:05:55.550
it makes sense that I
should make something
00:05:55.550 --> 00:05:58.010
like a Taylor expansion.
00:05:58.010 --> 00:06:01.770
So the Taylor expansion will
start not with a linear term,
00:06:01.770 --> 00:06:05.260
which violates rotational
symmetry, but something
00:06:05.260 --> 00:06:08.090
that is quadratic.
00:06:08.090 --> 00:06:11.330
And I can add any
even power such as m
00:06:11.330 --> 00:06:17.440
to the fourth and
higher order terms.
00:06:17.440 --> 00:06:22.680
And I can add all kinds of
gradients that are consistent,
00:06:22.680 --> 00:06:25.680
again, with rotational symmetry.
00:06:25.680 --> 00:06:29.720
And the first of those terms
is gradient of m squared.
00:06:29.720 --> 00:06:30.780
And there are many more.
00:06:39.210 --> 00:06:42.960
Of course, there can be
an overall constant term.
00:06:42.960 --> 00:06:45.220
Maybe I will write it out front.
00:06:45.220 --> 00:06:52.030
Insert a Z regular, which means
that you-- in the process,
00:06:52.030 --> 00:06:55.190
you have all kinds of
other degrees of freedom
00:06:55.190 --> 00:06:58.340
that are not reflected
in the magnetization.
00:06:58.340 --> 00:07:00.450
You're going to have
phonon degrees of freedom.
00:07:00.450 --> 00:07:03.030
So there will be a
contribution from the phonons
00:07:03.030 --> 00:07:05.480
to the partition
function of the system.
00:07:05.480 --> 00:07:08.240
We are not interested
in any of those things.
00:07:08.240 --> 00:07:12.930
We are interested in what
becomes singular over here.
00:07:12.930 --> 00:07:15.850
And the reason I write
that as Z regular
00:07:15.850 --> 00:07:19.590
is because it's presumably some
benign function of temperature
00:07:19.590 --> 00:07:22.040
as I pass through this point.
00:07:22.040 --> 00:07:25.150
So it is, indeed, a
function of temperature.
00:07:25.150 --> 00:07:28.840
It is also worth
emphasizing that not only is
00:07:28.840 --> 00:07:33.990
this parameter that appears
outside representing
00:07:33.990 --> 00:07:35.670
all kinds of other
degrees of freedom
00:07:35.670 --> 00:07:39.100
a function of temperature,
that these phenomenological
00:07:39.100 --> 00:07:41.540
parameters that
I introduced here
00:07:41.540 --> 00:07:43.310
are also functions
of temperature.
00:07:48.120 --> 00:07:53.840
Because the microscopic
weight, the true Hamiltonian,
00:07:53.840 --> 00:07:58.280
is the one that is
scaled by 1 over kt.
00:07:58.280 --> 00:08:00.980
Just because of
analogy, I sometimes
00:08:00.980 --> 00:08:06.280
call this combination of what is
happening here beta H or minus
00:08:06.280 --> 00:08:09.360
beta H as appearing
in the exponent.
00:08:09.360 --> 00:08:15.130
But that, by no means, indicates
that the coefficient here
00:08:15.130 --> 00:08:18.040
are scaling inversely
with temperature
00:08:18.040 --> 00:08:21.070
like the true
microscopic coordinates.
00:08:21.070 --> 00:08:26.250
Because in order to do
this coarse graining,
00:08:26.250 --> 00:08:29.860
I have to integrate over a lot
of different configurations.
00:08:29.860 --> 00:08:32.180
So there is energy
associated with this.
00:08:32.180 --> 00:08:34.659
There is entropy
associated with this.
00:08:34.659 --> 00:08:37.179
There is all kinds
of complicated things
00:08:37.179 --> 00:08:40.340
that go into this--
these parameters.
00:08:40.340 --> 00:08:43.500
So that's important to remember.
00:08:43.500 --> 00:08:46.860
Finally, if I slightly
go away from here just
00:08:46.860 --> 00:08:51.350
to explore the vicinity of
having a finite magnetic field,
00:08:51.350 --> 00:08:53.930
then I can work in
the ensemble where
00:08:53.930 --> 00:08:55.880
I have added the field here.
00:08:55.880 --> 00:08:58.060
And the weight here
will be modified
00:08:58.060 --> 00:09:01.402
by an amount that is h dot m.
00:09:06.330 --> 00:09:17.650
So what is occurring here is
this Landau-Ginzburg model
00:09:17.650 --> 00:09:22.400
that I right now
introduced in the context
00:09:22.400 --> 00:09:25.760
of magnetic systems.
00:09:25.760 --> 00:09:28.020
But very shortly,
I will introduce
00:09:28.020 --> 00:09:31.220
this in the context
of super-fluidity.
00:09:31.220 --> 00:09:33.590
I can go back to
the original example
00:09:33.590 --> 00:09:36.540
that we started with the
liquid gas phenomena,
00:09:36.540 --> 00:09:39.780
replace m with some kind
of a density difference,
00:09:39.780 --> 00:09:42.930
and then it would indicate
a phase transition
00:09:42.930 --> 00:09:44.820
of the liquid gas system.
00:09:44.820 --> 00:09:47.390
So it is supposed
to be very general,
00:09:47.390 --> 00:09:52.050
applicable to a lot of things
because it is constructed
00:09:52.050 --> 00:09:57.890
on the basis of nothing other
than symmetry principles.
00:09:57.890 --> 00:10:01.870
But at the cost of having
no knowledge of how
00:10:01.870 --> 00:10:04.970
these phenomenological
parameters depend
00:10:04.970 --> 00:10:08.210
on the microscopics of the
system-- on temperature.
00:10:08.210 --> 00:10:11.710
Clearly, for example, even
in the context of magnet,
00:10:11.710 --> 00:10:15.330
I would construct the same
theory for iron, nickel,
00:10:15.330 --> 00:10:16.770
cobalt, et cetera.
00:10:16.770 --> 00:10:18.350
But presumably
these coefficients
00:10:18.350 --> 00:10:22.770
will be very different from one
system as opposed to the other.
00:10:25.400 --> 00:10:31.030
All right, so that's supposed to
be, as far as we are concerned,
00:10:31.030 --> 00:10:33.380
sufficient to figure
out what is going
00:10:33.380 --> 00:10:37.380
on in the vicinity
of this transition.
00:10:37.380 --> 00:10:40.500
Maybe I should emphasize
one more thing,
00:10:40.500 --> 00:10:44.730
which is that I said that
after all what we are trying
00:10:44.730 --> 00:10:48.905
to figure out is the nature of
singularities in free energy,
00:10:48.905 --> 00:10:52.110
in phase diagrams, et cetera.
00:10:52.110 --> 00:10:54.720
Yet, when I wrote
this, I insisted
00:10:54.720 --> 00:10:58.420
on making an
analytical expansion.
00:10:58.420 --> 00:11:01.740
And the reason making an
analytical expansion here
00:11:01.740 --> 00:11:06.920
is justified is because
to get this expansion,
00:11:06.920 --> 00:11:09.060
I summed over
degrees of freedom--
00:11:09.060 --> 00:11:12.760
I averaged degrees of freedom
over some finite piece
00:11:12.760 --> 00:11:14.300
of my system.
00:11:14.300 --> 00:11:18.240
Maybe I took a 100 by
100 by 100 Angstrom cubed
00:11:18.240 --> 00:11:22.240
block of material and averaged
the magnetization of spins
00:11:22.240 --> 00:11:25.010
in that area, et cetera.
00:11:25.010 --> 00:11:29.660
And the idea that we also
encountered last semester
00:11:29.660 --> 00:11:31.680
is that as long
as you are dealing
00:11:31.680 --> 00:11:35.610
with a finite system,
all of the manipulations
00:11:35.610 --> 00:11:40.040
that you are doing involve
analytic functions such as e
00:11:40.040 --> 00:11:45.390
to the minus beta H applied to
a finite number of integrations.
00:11:45.390 --> 00:11:49.490
And you simply cannot get
a singularity out of such
00:11:49.490 --> 00:11:50.770
a process.
00:11:50.770 --> 00:11:54.930
So the averaging process that
goes in this coarse graining
00:11:54.930 --> 00:11:57.870
must give you
analytical functions.
00:11:57.870 --> 00:12:00.560
That's not the origin
of the singularity.
00:12:00.560 --> 00:12:02.750
The origin of the
singularity must
00:12:02.750 --> 00:12:08.180
come from taking the size
of the system, the volume
00:12:08.180 --> 00:12:12.230
of this piece of iron,
essentially to infinity.
00:12:12.230 --> 00:12:16.060
That's what gives
us the singularity.
00:12:16.060 --> 00:12:24.070
Also, because of that I expect
that at the end of the day,
00:12:24.070 --> 00:12:28.650
this Z of T and h
that I calculate
00:12:28.650 --> 00:12:31.400
will be something
that is proportional
00:12:31.400 --> 00:12:35.000
to volume or something
like that in the exponent.
00:12:35.000 --> 00:12:36.780
That is, I will have
something like e
00:12:36.780 --> 00:12:43.640
to the-- from this component
minus V some beta f regular.
00:12:47.380 --> 00:12:52.170
And actually, let me just
forget about the beta
00:12:52.170 --> 00:12:54.910
and just write it as some
other function regular.
00:12:54.910 --> 00:12:57.360
It's an extensive quantity.
00:12:57.360 --> 00:13:02.410
And the result of
this integration
00:13:02.410 --> 00:13:05.020
of all of the field
configurations
00:13:05.020 --> 00:13:08.890
should give me
another contribution
00:13:08.890 --> 00:13:12.470
that is proportional to volume.
00:13:12.470 --> 00:13:18.060
And hopefully, all of the
singularities that I expect
00:13:18.060 --> 00:13:19.950
will come from this piece.
00:13:23.800 --> 00:13:28.310
And I am going to expect those
singularities to arise only
00:13:28.310 --> 00:13:30.340
in the limit where
V becomes large.
00:13:33.390 --> 00:13:37.590
Now, last semester
we saw a trick
00:13:37.590 --> 00:13:41.500
that allowed us
to do integrations
00:13:41.500 --> 00:13:45.110
when the final
answer was extensive.
00:13:45.110 --> 00:13:47.200
It was the saddle point result.
00:13:47.200 --> 00:13:52.140
Basically, we said that as we
span the integration range,
00:13:52.140 --> 00:13:56.710
there is a part that corresponds
to extremizing whatever you are
00:13:56.710 --> 00:14:01.010
integrating that gives
you overwhelmingly
00:14:01.010 --> 00:14:05.450
larger weight than any other
part of the integration.
00:14:05.450 --> 00:14:11.090
So let's try to,
without justification--
00:14:11.090 --> 00:14:14.150
and we'll correct this--
apply that same principle
00:14:14.150 --> 00:14:25.850
of a saddle point approximation
to the functional integration
00:14:25.850 --> 00:14:28.260
that we are doing over here.
00:14:28.260 --> 00:14:34.910
That is, rather than
integrating over all functions,
00:14:34.910 --> 00:14:38.100
let's find the extremum.
00:14:38.100 --> 00:14:41.270
Where is this
function maximized,
00:14:41.270 --> 00:14:42.770
the integrand is maximized?
00:14:45.430 --> 00:14:50.660
So what I need to do is to
find maximum of exponent.
00:14:54.580 --> 00:14:57.040
So what is happening over here.
00:15:00.700 --> 00:15:05.270
I am going to make one
further statement here
00:15:05.270 --> 00:15:10.150
before I do that, which is
that this term, the thing that
00:15:10.150 --> 00:15:13.370
is proportional
to gradient of m,
00:15:13.370 --> 00:15:16.810
has something to do
with the way spins
00:15:16.810 --> 00:15:19.360
know about their neighborhood.
00:15:19.360 --> 00:15:23.040
And if I am looking at
something that is ferromagnet,
00:15:23.040 --> 00:15:27.160
the tendency is for neighbors
to be in the same direction.
00:15:27.160 --> 00:15:29.940
That's how
ferromagnetism emerges.
00:15:29.940 --> 00:15:36.640
And that corresponds to
having a K that is positive.
00:15:39.710 --> 00:15:41.890
So that helps me
a lot in finding
00:15:41.890 --> 00:15:44.860
the extremum of the
function because it
00:15:44.860 --> 00:15:48.000
means that the extremum
will occur when m of x
00:15:48.000 --> 00:15:50.530
is uniform across the system.
00:15:50.530 --> 00:15:55.260
Any variations in m of x will
give you a cost from that
00:15:55.260 --> 00:15:58.410
and will reduce the probability.
00:15:58.410 --> 00:16:04.200
So with K positive, I know
that the optimal solution
00:16:04.200 --> 00:16:08.020
is going to correspond
to a uniform value
00:16:08.020 --> 00:16:10.200
across the system.
00:16:10.200 --> 00:16:11.610
What is this uniform value?
00:16:11.610 --> 00:16:15.120
If I put a uniform
value, first of all,
00:16:15.120 --> 00:16:18.080
the integration will
simply give me the volume.
00:16:18.080 --> 00:16:23.240
That's good because that's
what I expect over here.
00:16:23.240 --> 00:16:26.180
So at the end of the
day, what do I get?
00:16:26.180 --> 00:16:38.540
I will get that F singular is,
in fact, the minimum of psi
00:16:38.540 --> 00:16:46.760
of m where psi of m is obtained
by simply evaluating what
00:16:46.760 --> 00:16:51.290
is in the integrand at
a uniform value of m.
00:16:51.290 --> 00:16:54.855
t over 2 m squared
mu m to the fourth.
00:16:54.855 --> 00:16:59.510
Potentially higher orders,
but no K. The K disappeared.
00:16:59.510 --> 00:17:00.968
Minus h m.
00:17:04.849 --> 00:17:14.329
Actually, I kind of expect
also that the uniform solution,
00:17:14.329 --> 00:17:17.690
if I put a magnetic field, will
point along the magnetic field.
00:17:17.690 --> 00:17:20.770
If there's an up field,
the uniform solution
00:17:20.770 --> 00:17:21.980
will be along the field.
00:17:21.980 --> 00:17:23.650
If it's 0, then I don't know.
00:17:23.650 --> 00:17:26.250
If it is down field, it
will be the opposite.
00:17:26.250 --> 00:17:30.620
So it makes sense that
if I have a field,
00:17:30.620 --> 00:17:32.780
the uniform magnetization
should point
00:17:32.780 --> 00:17:37.330
along it, which means that over
here this dot product is really
00:17:37.330 --> 00:17:38.750
just h times m.
00:17:41.850 --> 00:17:47.280
So I reduced the complexity
of the problem subject
00:17:47.280 --> 00:17:49.260
to this saddle
point approximation
00:17:49.260 --> 00:17:54.170
to just minimizing
some function.
00:17:54.170 --> 00:17:56.370
Now, it is important
to note, since I
00:17:56.370 --> 00:18:00.520
was talking about analyticities,
and singularities, and things
00:18:00.520 --> 00:18:05.450
like that, that whereas the
function that is appearing here
00:18:05.450 --> 00:18:11.190
as psi of m as we discussed
is completely analytical,
00:18:11.190 --> 00:18:15.000
the operation of finding
the minimum of a function
00:18:15.000 --> 00:18:17.990
is something that
introduces non-analyticity.
00:18:17.990 --> 00:18:21.920
So that's how we are going
to get that structure that we
00:18:21.920 --> 00:18:24.240
have over there.
00:18:24.240 --> 00:18:28.240
Let's explicitly
show how that occurs.
00:18:28.240 --> 00:18:33.110
So I'm going to show you
the shape of this function
00:18:33.110 --> 00:18:37.260
in the space that is
spanned by the parameters
00:18:37.260 --> 00:18:42.170
that I have here,
which are t and h.
00:18:42.170 --> 00:18:43.910
We will come to u later on.
00:18:43.910 --> 00:18:48.540
But for the time being, since I
want to show things in a plane,
00:18:48.540 --> 00:18:51.170
let's stick with t and h.
00:18:57.080 --> 00:19:03.830
So first of all, let's look at
the regime where t is positive.
00:19:03.830 --> 00:19:04.880
So this is 0.
00:19:04.880 --> 00:19:08.610
To the right is t is positive.
00:19:08.610 --> 00:19:14.280
Right on the axis
when h equals to 0,
00:19:14.280 --> 00:19:18.530
the function is m squared plus
m to the fourth, et cetera.
00:19:18.530 --> 00:19:21.620
The coefficient of m
squared is positive.
00:19:21.620 --> 00:19:24.260
So basically, right on
this axis the function
00:19:24.260 --> 00:19:25.666
is like a parabola.
00:19:25.666 --> 00:19:32.280
So what I am plotting here are
different forms of psi of m.
00:19:32.280 --> 00:19:35.120
So this is m squared
is a parabola.
00:19:35.120 --> 00:19:37.387
If I go further
out, m to the fourth
00:19:37.387 --> 00:19:39.880
is something to worry about.
00:19:39.880 --> 00:19:44.430
But if I'm really looking
at case where the emerging
00:19:44.430 --> 00:19:47.915
magnetization is
small, I can stick just
00:19:47.915 --> 00:19:49.891
with the lowest order
term in the expansion.
00:19:49.891 --> 00:19:50.390
Yes?
00:19:50.390 --> 00:19:52.785
AUDIENCE: Are we
plotting h or psi?
00:19:56.138 --> 00:19:59.970
The axis label was h.
00:19:59.970 --> 00:20:01.000
PROFESSOR: OK.
00:20:01.000 --> 00:20:05.651
So there is a
two-parameter plane t h.
00:20:05.651 --> 00:20:10.600
At each plant in this parameter
plane, I can plot what psi of m
00:20:10.600 --> 00:20:11.590
looks like.
00:20:11.590 --> 00:20:15.700
Unfortunately, I can't
bring it out of the plane,
00:20:15.700 --> 00:20:18.580
so I am going to put it
in the two dimensions.
00:20:18.580 --> 00:20:21.250
So this is why I wrote
psi of m with green
00:20:21.250 --> 00:20:23.760
to distinguish it with
the h that is in white.
00:20:27.960 --> 00:20:30.680
So this is supposed to
be right on the axis.
00:20:30.680 --> 00:20:36.060
If I go up on the axis, what
would psi of m look like?
00:20:36.060 --> 00:20:39.970
So now I have a term
that is positive h m.
00:20:39.970 --> 00:20:45.510
So it starts
linearly to go down,
00:20:45.510 --> 00:20:49.230
and then the m squared
term takes over.
00:20:49.230 --> 00:20:51.985
So now the minimum is over here.
00:20:51.985 --> 00:20:54.280
Whereas, right on
the axis-- again,
00:20:54.280 --> 00:20:57.750
by symmetry-- the
minimum was at 0.
00:20:57.750 --> 00:21:01.820
If I go and look at what
happens at the other side
00:21:01.820 --> 00:21:04.660
when h is negative,
then the slope
00:21:04.660 --> 00:21:06.180
is in the opposite direction.
00:21:06.180 --> 00:21:08.980
So the function kind
of looks like this.
00:21:08.980 --> 00:21:12.670
And the minimum has
shifted over here.
00:21:12.670 --> 00:21:19.060
So if I were to plot
now what the minimum m
00:21:19.060 --> 00:21:25.422
bar is as a function
of the field h,
00:21:25.422 --> 00:21:32.830
as I scan from some positive t--
from h positive to h negative,
00:21:32.830 --> 00:21:37.020
what we have is that
this sequence then
00:21:37.020 --> 00:21:41.250
corresponds to a curve
in which, essentially, m
00:21:41.250 --> 00:21:43.583
is proportional to h.
00:21:43.583 --> 00:21:47.010
Something like this.
00:21:47.010 --> 00:21:49.236
So this is for t
that is positive.
00:21:53.430 --> 00:21:55.910
Now, what happens
if I try to draw
00:21:55.910 --> 00:22:00.320
these same curves for
t that is negative?
00:22:00.320 --> 00:22:03.550
Let's stick first again to
the analog of this curve,
00:22:03.550 --> 00:22:07.390
but now for h equal to 0.
00:22:07.390 --> 00:22:11.200
So for h equals to
0, in plotting psi
00:22:11.200 --> 00:22:15.320
if m, whereas previously
the parabola had
00:22:15.320 --> 00:22:18.590
the positive coefficient,
now the parabola
00:22:18.590 --> 00:22:20.070
has a negative coefficient.
00:22:20.070 --> 00:22:23.750
So it kind of looks like this.
00:22:23.750 --> 00:22:27.210
Now, one of the things that I
know for sure in my system--
00:22:27.210 --> 00:22:32.040
again, mathematics does not tell
us that, physics tells us that.
00:22:32.040 --> 00:22:34.670
That I have a
piece of iron and I
00:22:34.670 --> 00:22:37.730
know that the typical
configurations of magnetization
00:22:37.730 --> 00:22:40.810
are some small number.
00:22:40.810 --> 00:22:45.130
So if the mathematics says that
I have a function such as this,
00:22:45.130 --> 00:22:48.680
it is wrong because it would
say that the extremum would
00:22:48.680 --> 00:22:51.920
go towards infinity.
00:22:51.920 --> 00:22:55.070
But that's where the
u-term comes into play.
00:22:55.070 --> 00:22:59.250
So I need now to have a
u-term with a positive sign
00:22:59.250 --> 00:23:04.520
to ensure that the function
does not have an extremum that
00:23:04.520 --> 00:23:06.470
goes to minus plus infinity.
00:23:06.470 --> 00:23:08.920
Physics tells us
that the function
00:23:08.920 --> 00:23:14.950
that I get if t is negative
should have a positive u.
00:23:14.950 --> 00:23:16.875
Now, what happens if
we take this curve
00:23:16.875 --> 00:23:19.750
and go to h positive?
00:23:19.750 --> 00:23:22.460
Well, now again, like
we did over here,
00:23:22.460 --> 00:23:25.480
I start to shift it
in one direction.
00:23:25.480 --> 00:23:31.510
In particular, what I find is
that the minimum to the left
00:23:31.510 --> 00:23:38.860
goes up and the minimum
to the right goes down.
00:23:38.860 --> 00:23:41.572
What did I do?
00:23:41.572 --> 00:23:42.991
Drew this poorly.
00:23:46.780 --> 00:23:48.285
I start to go like this.
00:23:57.678 --> 00:24:02.320
So now there is a well-defined
minimum over here.
00:24:02.320 --> 00:24:15.750
If I go to negative fields,
I will get the opposite
00:24:15.750 --> 00:24:16.800
Where I have this.
00:24:20.840 --> 00:24:25.560
So now if I follow a
path that corresponds
00:24:25.560 --> 00:24:34.990
to these t negative
structures, what do I get?
00:24:34.990 --> 00:24:41.010
What I get is that the extremum
is at some positive value.
00:24:41.010 --> 00:24:44.680
So this is for t
that is negative.
00:24:44.680 --> 00:24:47.440
As I scan h from
positive values,
00:24:47.440 --> 00:24:48.850
I am tracking this minimum.
00:24:48.850 --> 00:24:52.550
But that minimum ends
up over here at 0.
00:24:52.550 --> 00:24:56.380
So basically, I come to here.
00:24:56.380 --> 00:24:59.320
Whereas, if I come
from the negative side,
00:24:59.320 --> 00:25:02.650
I am basically following
the opposite curve.
00:25:02.650 --> 00:25:05.779
And then I have a
discontinuity exactly at the h
00:25:05.779 --> 00:25:06.570
equals [INAUDIBLE].
00:25:10.130 --> 00:25:16.490
Again, by continuity if I
am exactly at t equals to 0,
00:25:16.490 --> 00:25:20.790
the function is kind of--
rather than the parabola
00:25:20.790 --> 00:25:24.450
is m to the fourth, and
a little bit of thought
00:25:24.450 --> 00:25:28.510
convinces you that the shape
of the curve kind of looks
00:25:28.510 --> 00:25:29.620
like this.
00:25:29.620 --> 00:25:34.492
We will quantify what that shape
is shortly for t equals to 0.
00:25:37.170 --> 00:25:39.960
So what did we do?
00:25:39.960 --> 00:25:46.880
We were able to reproduce
exactly the structure
00:25:46.880 --> 00:25:50.230
of the isotherms
that we are getting
00:25:50.230 --> 00:25:55.600
for the case of the ferromagnet
from this simple theory.
00:25:58.800 --> 00:26:03.710
In order to just match exactly
this with what is going on
00:26:03.710 --> 00:26:07.810
with iron or nickel,
what do I have to do?
00:26:07.810 --> 00:26:12.070
I just have to ensure
that this parameter t goes
00:26:12.070 --> 00:26:16.340
to 0 at Tc of whatever
the material is.
00:26:16.340 --> 00:26:20.310
Now, remember that I
said that my t is really
00:26:20.310 --> 00:26:23.460
a function of temperature.
00:26:23.460 --> 00:26:26.360
So I can certainly
make an expansion of it
00:26:26.360 --> 00:26:27.410
around any point.
00:26:27.410 --> 00:26:30.840
Let's see the Tc
of the material.
00:26:30.840 --> 00:26:35.710
So what I require is that the
first term in that expansion
00:26:35.710 --> 00:26:37.270
has to be 0.
00:26:37.270 --> 00:26:41.480
Then I will have something
that is linear in T, and then
00:26:41.480 --> 00:26:48.630
quadratic in T minus Tc
squared and so forth.
00:26:48.630 --> 00:26:51.840
So there is one condition
that I have to impose,
00:26:51.840 --> 00:26:54.560
that that function
of t-- which again,
00:26:54.560 --> 00:26:56.910
by all of the arguments
that I mentioned,
00:26:56.910 --> 00:27:00.370
completely has to be an analytic
function of temperature.
00:27:00.370 --> 00:27:03.580
Hence, expandable
in a Taylor series.
00:27:03.580 --> 00:27:07.630
The 0 to order term in that
Taylor series has to be 0.
00:27:07.630 --> 00:27:08.900
What else?
00:27:08.900 --> 00:27:11.260
That I can expand
any other function.
00:27:11.260 --> 00:27:17.580
U of t is u0 plus u1 T
minus Tc and so forth.
00:27:17.580 --> 00:27:26.980
K of t is k0 plus k1 T
minus Tc and so forth.
00:27:26.980 --> 00:27:29.860
And I don't really care about
any of these coefficients.
00:27:29.860 --> 00:27:33.670
The only things that
I know are the signs.
00:27:33.670 --> 00:27:37.870
a has to be positive
because the high temperature
00:27:37.870 --> 00:27:40.980
side corresponds to paramagnet.
00:27:40.980 --> 00:27:46.200
u0 has to be positive because
I require the stability
00:27:46.200 --> 00:27:48.560
types of things
that I mentioned.
00:27:48.560 --> 00:27:51.000
And k0 has to be
positive because I
00:27:51.000 --> 00:27:55.780
want to have this kind of
ferromagnetic behavior.
00:27:55.780 --> 00:28:00.650
Apart from that, I
really don't know much.
00:28:00.650 --> 00:28:11.950
So the Landau-Ginzburg
Hamiltonian with one condition
00:28:11.950 --> 00:28:16.370
reproduces the
phenomenology of the magnet
00:28:16.370 --> 00:28:19.240
and all the other
phase transitions,
00:28:19.240 --> 00:28:22.550
which as one of you-- I think
it was David-- was pointing out
00:28:22.550 --> 00:28:31.930
to me, is very much like dealing
with a branch cut singularity
00:28:31.930 --> 00:28:35.380
in the mathematical
sense along this axis
00:28:35.380 --> 00:28:37.670
terminating at that point.
00:28:37.670 --> 00:28:41.300
So this branch cut
singularity is a consequence
00:28:41.300 --> 00:28:45.310
of this minimization
procedure of a purely
00:28:45.310 --> 00:28:47.630
analytical function.
00:28:47.630 --> 00:28:51.580
And the statement of
universality at this level
00:28:51.580 --> 00:28:55.909
is that pick any
analytical function
00:28:55.909 --> 00:28:56.950
and do this minimization.
00:28:56.950 --> 00:29:02.970
You will always get the
same mathematical branch cut
00:29:02.970 --> 00:29:07.115
that we will now explain
in terms of the exponents.
00:29:11.240 --> 00:29:17.030
So we said that experimentally
these phase transitions,
00:29:17.030 --> 00:29:19.230
their universality
was characterized
00:29:19.230 --> 00:29:22.580
by looking at singularities
of various quantities
00:29:22.580 --> 00:29:29.140
and looking at the exponent,
the functional forms.
00:29:29.140 --> 00:29:31.616
So let's first look
at the magnetization.
00:29:38.460 --> 00:29:43.760
So if I do the extremization
of this, what do I do?
00:29:43.760 --> 00:29:48.210
I have to have d psi by dm
equals to 0. d psi by dm
00:29:48.210 --> 00:29:54.350
is tm bar plus 4u m
bar cubed minus h.
00:29:59.360 --> 00:30:07.700
For h equals to 0 along
the symmetry axis,
00:30:07.700 --> 00:30:09.270
then I have two solutions.
00:30:09.270 --> 00:30:14.350
Because that I can write
as t plus 4u m bar squared
00:30:14.350 --> 00:30:17.160
times m bar equals to 0.
00:30:17.160 --> 00:30:20.260
If I state at that
equation, I immediately
00:30:20.260 --> 00:30:26.240
see that the possible
solutions are m bar is 0.
00:30:26.240 --> 00:30:28.800
And that's really
the only solution
00:30:28.800 --> 00:30:32.420
that I have for any
t that is positive.
00:30:35.500 --> 00:30:40.670
While for t negative, in
addition to the solution at 0,
00:30:40.670 --> 00:30:43.400
which is clearly unphysical
because it corresponds
00:30:43.400 --> 00:30:47.520
to a maximum and not a minimum,
I have solutions at minus
00:30:47.520 --> 00:30:51.930
plus square root
of minus t over 4u.
00:30:56.570 --> 00:31:01.180
And so if I were to
plot this magnetization
00:31:01.180 --> 00:31:11.800
as a function of
t, essentially I
00:31:11.800 --> 00:31:18.220
have a kind of coexistence curve
because I have nothing above
00:31:18.220 --> 00:31:20.760
and below I have a
square root singularity.
00:31:24.450 --> 00:31:32.250
So this corresponds to the
exponent beta being 1/2.
00:31:32.250 --> 00:31:38.960
So that's the prediction
[? from each. ?]
00:31:38.960 --> 00:31:42.440
What about the shape of this
green curve, the isotherm
00:31:42.440 --> 00:31:45.090
that you have at t equals to Tc?
00:31:45.090 --> 00:31:47.400
Well, t equals to
Tc, in our language,
00:31:47.400 --> 00:31:51.110
corresponds to
small t equals to 0,
00:31:51.110 --> 00:31:56.830
which means that I have to look
at the equation 4u m bar cubed
00:31:56.830 --> 00:32:04.270
equal to h or m bar
is proportional to--
00:32:04.270 --> 00:32:06.060
well, let's write it.
00:32:06.060 --> 00:32:09.520
h over 4u to the power of 1/3.
00:32:12.210 --> 00:32:16.570
So this green curve that
comes with infinite slope
00:32:16.570 --> 00:32:20.450
corresponds to a
1/3 singularity.
00:32:20.450 --> 00:32:25.150
The exponent delta was defined
to be the inverse of this,
00:32:25.150 --> 00:32:29.160
so this corresponds to having
an exponent delta that is 3.
00:32:38.800 --> 00:32:45.770
We had behavior of
susceptibility characterized
00:32:45.770 --> 00:32:47.123
by another set of exponent.
00:32:54.990 --> 00:32:57.930
Now, the susceptibility,
quite generally,
00:32:57.930 --> 00:33:01.730
is the response of
the magnetization
00:33:01.730 --> 00:33:04.670
if you change the field.
00:33:04.670 --> 00:33:10.490
And typically, we were
interested in the limit where
00:33:10.490 --> 00:33:14.460
we measured a response if
you are just at h equals to 0
00:33:14.460 --> 00:33:17.070
and then you put
a little bit more.
00:33:21.700 --> 00:33:24.830
The equation that we have
that relates h and m bar
00:33:24.830 --> 00:33:29.995
is simply that h equals to
tm bar plus 4u m bar cubed.
00:33:32.840 --> 00:33:37.060
Rather than taking
dm by dh, let me
00:33:37.060 --> 00:33:42.020
evaluate its inverse,
which is dh by dm.
00:33:42.020 --> 00:33:48.350
dh by dm is t plus
12u m bar squared.
00:33:51.460 --> 00:33:55.610
And so this inverse
susceptibility,
00:33:55.610 --> 00:34:01.140
if I am for t positive,
m bar is also 0.
00:34:01.140 --> 00:34:05.130
So the inverse
susceptibility is t.
00:34:05.130 --> 00:34:14.219
If I am for t negative, m bar
squared is minus t over 4u.
00:34:14.219 --> 00:34:17.530
I put a minus t over 4u here.
00:34:17.530 --> 00:34:19.820
And it becomes t minus 3t.
00:34:19.820 --> 00:34:23.120
So it becomes minus 2t.
00:34:23.120 --> 00:34:25.080
Again, nicely positive.
00:34:25.080 --> 00:34:27.889
Response functions
have to be positive.
00:34:27.889 --> 00:34:32.179
If I were to plot this
susceptibility, therefore
00:34:32.179 --> 00:34:35.400
as a function of
temperature, or t
00:34:35.400 --> 00:34:40.320
that is related to T
minus Tc, what do I get?
00:34:40.320 --> 00:34:48.520
I will get a divergence that is
inverse of 1 over T minus Tc--
00:34:48.520 --> 00:34:50.100
on both sides.
00:34:50.100 --> 00:34:55.260
So basically, I get
something like this.
00:34:55.260 --> 00:34:59.390
And we said that the divergence
of the susceptibility we
00:34:59.390 --> 00:35:02.870
characterize by exponent gamma.
00:35:02.870 --> 00:35:05.750
And as I had promised,
we explicitly
00:35:05.750 --> 00:35:08.330
see that, in this
case, the gammas
00:35:08.330 --> 00:35:12.300
on both sides of the transition
are the same and equal
00:35:12.300 --> 00:35:13.060
to unity.
00:35:13.060 --> 00:35:17.050
The inverse vanishes linearly,
so the susceptibility
00:35:17.050 --> 00:35:21.300
diverges with unit exponent.
00:35:21.300 --> 00:35:23.440
But actually, we
are making-- we can
00:35:23.440 --> 00:35:27.070
make an additional
statement here
00:35:27.070 --> 00:35:29.170
that experimentalists
can go and check
00:35:29.170 --> 00:35:33.760
that I hadn't told you before.
00:35:33.760 --> 00:35:37.580
Now, you see in all
of these other cases
00:35:37.580 --> 00:35:40.670
the only thing that I can
say is universal side is
00:35:40.670 --> 00:35:41.990
the functional form.
00:35:41.990 --> 00:35:44.760
This is where the
exponent beta came from.
00:35:44.760 --> 00:35:47.130
But the amplitude of
what is happening there,
00:35:47.130 --> 00:35:50.270
or the amplitude of
what is happening here,
00:35:50.270 --> 00:35:53.440
these are things that depend
on you and all of these things
00:35:53.440 --> 00:35:56.250
that I have no idea about.
00:35:56.250 --> 00:35:59.780
Similarly here,
because I don't know
00:35:59.780 --> 00:36:05.610
what the relationship between
t minus Tc and the parameter t
00:36:05.610 --> 00:36:10.420
is, it involves this number
a that I don't know of.
00:36:10.420 --> 00:36:13.630
But one thing that I notice
is that the ratio of these two
00:36:13.630 --> 00:36:16.690
things is a pure number.
00:36:16.690 --> 00:36:19.730
So I say, OK, what you
have is if you measure
00:36:19.730 --> 00:36:23.530
the susceptibility on the
two sides of the transition,
00:36:23.530 --> 00:36:27.050
you will see amplitudes.
00:36:27.050 --> 00:36:31.560
And then T minus
Tc absolute value
00:36:31.560 --> 00:36:34.540
to minus gamma plus
or gamma minus.
00:36:34.540 --> 00:36:37.180
I have told you that
the gammas are the same.
00:36:37.180 --> 00:36:39.730
I don't know what
the amplitudes are,
00:36:39.730 --> 00:36:43.090
but I can tell for sure that
the ratio of amplitudes--
00:36:43.090 --> 00:36:46.230
if this is the theory
that describes things--
00:36:46.230 --> 00:36:49.190
is a pure number of 2.
00:36:49.190 --> 00:36:53.670
So that's another thing
that you can go and say
00:36:53.670 --> 00:36:55.380
the experimentalists can check.
00:36:55.380 --> 00:36:57.560
They can check the
divergence, and then
00:36:57.560 --> 00:37:01.100
see that the amplitude
ratio is a universal object.
00:37:04.350 --> 00:37:08.640
OK, there's one other response
function that I had mentioned.
00:37:08.640 --> 00:37:11.420
There was the
exponent alpha that
00:37:11.420 --> 00:37:12.733
came from the heat capacity.
00:37:16.870 --> 00:37:19.316
So how do I calculate
heat capacity?
00:37:28.260 --> 00:37:31.600
So the heat capacity, which
is a function of temperature,
00:37:31.600 --> 00:37:37.130
let's focus only in the
case where h equals to 0,
00:37:37.130 --> 00:37:41.155
is obtained by taking a
temperature derivative
00:37:41.155 --> 00:37:43.725
of the internal
energy of the system.
00:37:46.500 --> 00:37:48.820
Now, the energy,
on the other hand,
00:37:48.820 --> 00:37:55.441
is obtained by taking a d by
d beta of log of the partition
00:37:55.441 --> 00:37:55.940
function.
00:38:02.190 --> 00:38:09.580
Now, I have all of my answers in
terms of these parameters t, u,
00:38:09.580 --> 00:38:11.300
et cetera.
00:38:11.300 --> 00:38:13.440
But I know that to
lowest order, there
00:38:13.440 --> 00:38:17.730
is a linear relationship
between small t
00:38:17.730 --> 00:38:20.440
and the real
temperature that I have
00:38:20.440 --> 00:38:25.440
to put into these expressions.
00:38:25.440 --> 00:38:30.850
So in particular, I
can do the following.
00:38:30.850 --> 00:38:34.615
I can say that
something like d by dT d
00:38:34.615 --> 00:38:40.760
by d beta, where
beta is 1 over kt,
00:38:40.760 --> 00:38:46.930
is something
approximately when you
00:38:46.930 --> 00:38:53.090
look at the linear regime that
is close to Tc of the order--
00:38:53.090 --> 00:38:58.400
1 over beta is going
to be kb t squared.
00:38:58.400 --> 00:39:03.220
And then I would have d by
dT, and then another d by dT.
00:39:06.250 --> 00:39:08.840
I have to evaluate
all of these things
00:39:08.840 --> 00:39:12.480
eventually in the vicinity
of the critical point.
00:39:12.480 --> 00:39:17.620
Everything else is going
to be a correction.
00:39:17.620 --> 00:39:20.740
So to lowest order,
I will do that.
00:39:20.740 --> 00:39:28.420
And then I note that this
is related-- derivatives
00:39:28.420 --> 00:39:32.280
between temperature
and small t are
00:39:32.280 --> 00:39:35.570
related through a factor of a.
00:39:35.570 --> 00:39:38.637
So I do this and I
put a squared up here.
00:39:41.910 --> 00:39:42.590
Doesn't matter.
00:39:42.590 --> 00:39:49.710
The only reason I do that is
because through the process
00:39:49.710 --> 00:39:55.220
that we have described here, I
get an idea for what log of Z
00:39:55.220 --> 00:39:55.925
is.
00:39:55.925 --> 00:40:02.780
So in particular,
log of Z has a part
00:40:02.780 --> 00:40:09.080
that comes from all of the
regular degrees of freedom
00:40:09.080 --> 00:40:15.066
and a part that comes from
this additional minimization
00:40:15.066 --> 00:40:16.120
that we are doing.
00:40:16.120 --> 00:40:23.910
So we have a minus V times the
minimum of the function, which
00:40:23.910 --> 00:40:30.370
is t over 2 m bar squared
plus u m bar to the fourth
00:40:30.370 --> 00:40:31.790
when evaluated at h equals to 0.
00:40:35.690 --> 00:40:40.810
So this is log Z, which
is some regular function
00:40:40.810 --> 00:40:44.130
of temperature, and hence t.
00:40:44.130 --> 00:40:48.560
Why is this part singular?
00:40:48.560 --> 00:40:53.720
Because for t that
is positive, m is 0.
00:40:53.720 --> 00:40:56.865
So this is going to give me
0 contribution for t that
00:40:56.865 --> 00:40:59.010
is positive.
00:40:59.010 --> 00:41:02.840
Whereas, for t
that is negative, I
00:41:02.840 --> 00:41:05.130
have to substitute
the value of m bar
00:41:05.130 --> 00:41:10.770
squared as I found above,
which is minus t over 4u.
00:41:10.770 --> 00:41:18.720
So I will get t over 2
times minus t over 4u.
00:41:18.720 --> 00:41:24.770
So that's going to give me
minus t squared over 8u.
00:41:24.770 --> 00:41:28.290
Here I will have-- once I
substitute that formula,
00:41:28.290 --> 00:41:30.630
plus t squared over 16u.
00:41:30.630 --> 00:41:33.975
The overall thing would be
minus t squared over 16u.
00:41:37.910 --> 00:41:42.100
So I have to take two
derivatives of this function.
00:41:42.100 --> 00:41:44.100
You can see that the
function will give me
00:41:44.100 --> 00:41:50.876
0 above for t positive and it
will give me a constant for t
00:41:50.876 --> 00:41:51.770
negative.
00:41:51.770 --> 00:42:00.180
So if I were to plot two
derivatives of this function
00:42:00.180 --> 00:42:04.890
as T or T minus Tc is varied.
00:42:04.890 --> 00:42:06.900
Well, there is a
background part that
00:42:06.900 --> 00:42:09.030
comes from all the other
degrees of freedom.
00:42:09.030 --> 00:42:11.960
So there is basically some kind
of behavior you would have had
00:42:11.960 --> 00:42:13.790
normally.
00:42:13.790 --> 00:42:18.100
What we find is that
above t positive,
00:42:18.100 --> 00:42:21.820
that normal behavior is the
only thing that you have.
00:42:21.820 --> 00:42:25.510
And when you are below, taking
two derivatives of this,
00:42:25.510 --> 00:42:27.630
something is added to this.
00:42:27.630 --> 00:42:33.590
So basically, the prediction
is that the heat capacity
00:42:33.590 --> 00:42:36.220
of the system as a
function of temperature
00:42:36.220 --> 00:42:38.046
should have a jump
discontinuity.
00:42:40.850 --> 00:42:43.910
Now, I said that in
a number of cases
00:42:43.910 --> 00:42:48.040
we see that the heat
capacity actually diverges
00:42:48.040 --> 00:42:51.370
and we introduce
then exponent alpha
00:42:51.370 --> 00:42:53.750
to parametrize that divergence.
00:42:53.750 --> 00:42:56.740
Since we don't have
this divergence,
00:42:56.740 --> 00:43:00.350
people have resorted
to indicating this
00:43:00.350 --> 00:43:02.590
with alpha equals to 0.
00:43:02.590 --> 00:43:06.390
But since alpha equals
to 0 is ambiguous,
00:43:06.390 --> 00:43:11.630
putting a discontinuity
in addition
00:43:11.630 --> 00:43:14.084
to be precise about
what is happening.
00:43:18.130 --> 00:43:24.140
So the predictions of
the saddle point method
00:43:24.140 --> 00:43:26.970
applied to this field
theory are the set
00:43:26.970 --> 00:43:29.550
of exponents and
functional forms
00:43:29.550 --> 00:43:33.612
beta equals to 1/2, gamma
as being 1, et cetera.
00:43:33.612 --> 00:43:36.940
A nice set of predictions.
00:43:36.940 --> 00:43:42.200
And of course, the
test is, do they agree?
00:43:42.200 --> 00:43:46.360
It turns out that there is
one and only one case where
00:43:46.360 --> 00:43:49.310
you do the experiments and you
get these precise exponents.
00:43:49.310 --> 00:43:52.320
And that's something
like a superconductor.
00:43:52.320 --> 00:43:55.230
And the picture that
I showed you last time
00:43:55.230 --> 00:43:58.300
for the gas, et
cetera, corresponds
00:43:58.300 --> 00:44:02.630
to totally different
set of exponents.
00:44:02.630 --> 00:44:07.680
So at this point, we have to
face one of two alternatives.
00:44:07.680 --> 00:44:11.480
One, the starting
point is wrong.
00:44:11.480 --> 00:44:14.640
We put everything we could
think of in the starting point.
00:44:14.640 --> 00:44:17.930
Maybe we forget something,
but it seems OK.
00:44:17.930 --> 00:44:21.270
The other is, maybe we didn't
do the analysis right when
00:44:21.270 --> 00:44:24.120
we did the saddle
point approximation.
00:44:24.120 --> 00:44:26.540
And we'll gradually
build the case
00:44:26.540 --> 00:44:31.010
that that is, indeed, the
case and that we should treat
00:44:31.010 --> 00:44:35.270
the problem in a
slightly better fashion.
00:44:35.270 --> 00:44:37.861
Any questions?
00:44:37.861 --> 00:44:38.360
OK.
00:44:38.360 --> 00:44:39.932
Yes?
00:44:39.932 --> 00:44:42.840
AUDIENCE: Just to remind me,
the saddle point approximation
00:44:42.840 --> 00:44:47.466
was saying m was continuous,
a continuous number
00:44:47.466 --> 00:44:49.610
across the substrate?
00:44:49.610 --> 00:44:51.400
PROFESSOR: The saddle
point approximation
00:44:51.400 --> 00:44:55.890
is to evaluate the functional
integral, which corresponds
00:44:55.890 --> 00:44:58.980
to looking at all
configurations,
00:44:58.980 --> 00:45:03.460
replacing that integration
with the value of the integrand
00:45:03.460 --> 00:45:06.560
at the point that
is most probable.
00:45:06.560 --> 00:45:11.200
In this case, the most probable
point was the uniform case.
00:45:11.200 --> 00:45:14.490
But maybe in some other case,
the most probable configuration
00:45:14.490 --> 00:45:15.640
would be something else.
00:45:15.640 --> 00:45:20.640
The saddle point is to replace
the entire functional integral
00:45:20.640 --> 00:45:24.730
with just one value
of the integral.
00:45:24.730 --> 00:45:25.883
Yes?
00:45:25.883 --> 00:45:30.240
AUDIENCE: So your second claims
that analysis is probably wrong
00:45:30.240 --> 00:45:30.740
somewhere.
00:45:30.740 --> 00:45:33.650
It is most likely
when we are trying
00:45:33.650 --> 00:45:38.320
to compute the energy of the
system from all of field m
00:45:38.320 --> 00:45:41.296
and we just assumed
something incorrectly,
00:45:41.296 --> 00:45:45.380
and that's why we get
incorrect exponents?
00:45:45.380 --> 00:45:46.110
PROFESSOR: No.
00:45:46.110 --> 00:45:51.500
My claim is that up to the place
that I say Landau-Ginzburg,
00:45:51.500 --> 00:45:54.920
I have been extremely general.
00:45:54.920 --> 00:45:57.370
It may be that I
missed something,
00:45:57.370 --> 00:46:00.310
but I will convince you
that that's not the case.
00:46:00.310 --> 00:46:04.080
Then the line below that says
saddle point approximation.
00:46:04.080 --> 00:46:07.738
My claim is that that's
where the error came.
00:46:07.738 --> 00:46:12.112
AUDIENCE: Also, we can do
a similar kind of analysis
00:46:12.112 --> 00:46:15.060
for liquid gas transition
in critical [INAUDIBLE]?
00:46:15.060 --> 00:46:15.945
PROFESSOR: Yes.
00:46:15.945 --> 00:46:18.020
AUDIENCE: And then,
how would it be
00:46:18.020 --> 00:46:21.436
reasonable to assume
uniform density?
00:46:21.436 --> 00:46:25.680
Because I guess the
whole point of behavior--
00:46:25.680 --> 00:46:30.010
PROFESSOR: We did exactly
that approximation in 8.3.3.3.
00:46:30.010 --> 00:46:34.990
I wrote down some theory for
the liquid gas transition
00:46:34.990 --> 00:46:37.776
out of which came the
van der Waals equation.
00:46:37.776 --> 00:46:38.490
AUDIENCE: Yes.
00:46:38.490 --> 00:46:40.110
PROFESSOR: And the
assumption for that
00:46:40.110 --> 00:46:43.790
was that the density in the
grand canonical ensemble,
00:46:43.790 --> 00:46:47.040
in the grand canonical
ensemble was uniform.
00:46:47.040 --> 00:46:48.970
So that you either
got the density
00:46:48.970 --> 00:46:52.180
for the liquid or the
density for the gas.
00:46:52.180 --> 00:46:56.040
But I made there the saddle
point approximation also.
00:46:56.040 --> 00:46:59.409
I assumed that there was a
uniform density that was--
00:46:59.409 --> 00:46:59.950
AUDIENCE: OK.
00:46:59.950 --> 00:47:02.225
So was just likely to
be the point where--
00:47:02.225 --> 00:47:02.850
PROFESSOR: Yes.
00:47:02.850 --> 00:47:04.070
AUDIENCE: --something breaks up.
00:47:04.070 --> 00:47:04.695
PROFESSOR: Yes.
00:47:09.990 --> 00:47:16.250
So before doing that,
let me point out
00:47:16.250 --> 00:47:20.480
to something interesting
that happened.
00:47:20.480 --> 00:47:35.040
And it's just a
matter of terminology.
00:47:35.040 --> 00:47:41.890
Note that we constructed the
Landau-Ginzburg Hamiltonian
00:47:41.890 --> 00:47:46.400
for h equals to 0
on the basis that we
00:47:46.400 --> 00:47:49.200
should have rotational symmetry.
00:47:49.200 --> 00:47:53.870
Nonetheless, even for h
equals to 0, what we find
00:47:53.870 --> 00:47:57.670
are solutions where the
magnetization is pointing
00:47:57.670 --> 00:48:00.090
in one direction or the other.
00:48:00.090 --> 00:48:04.400
So it is possible to
have the state that
00:48:04.400 --> 00:48:08.030
emerges as a result of a
weight that has some symmetry
00:48:08.030 --> 00:48:09.990
to not have that symmetry.
00:48:09.990 --> 00:48:13.880
So the symmetry is
spontaneously broken
00:48:13.880 --> 00:48:19.400
and the direction in
space is selected.
00:48:19.400 --> 00:48:22.480
Now, of course,
what that means is
00:48:22.480 --> 00:48:25.990
that if you apply the
rotation operation to one
00:48:25.990 --> 00:48:29.000
of these ground
states, then you will
00:48:29.000 --> 00:48:31.650
generate another equally
good ground state.
00:48:31.650 --> 00:48:35.090
You can take everything that
is pointed along the z-axis
00:48:35.090 --> 00:48:37.460
and make them all
point along the x-axis.
00:48:37.460 --> 00:48:40.410
That's an equally
good ground state.
00:48:40.410 --> 00:48:44.890
So essentially, you have a
manifold of possible states.
00:48:44.890 --> 00:48:48.330
And making a change
from one state deforming
00:48:48.330 --> 00:48:53.670
to another ground state does
not cost you any energy.
00:48:53.670 --> 00:49:04.980
So one consequence of that
is that slow deformations
00:49:04.980 --> 00:49:07.965
should cost little energy.
00:49:12.040 --> 00:49:14.180
What do I mean by that?
00:49:14.180 --> 00:49:17.250
So let's imagine that
I start with a state
00:49:17.250 --> 00:49:24.360
where after I minimize, I find
that all of my magnetizations
00:49:24.360 --> 00:49:27.940
are pointing up.
00:49:27.940 --> 00:49:35.860
Now, as I said, I could rotate
everybody into this direction
00:49:35.860 --> 00:49:40.360
and the formation of my
state would cost no energy.
00:49:43.070 --> 00:49:45.940
That's a uniform deformation.
00:49:45.940 --> 00:49:50.180
What if I took a deformation
that is very slow?
00:49:50.180 --> 00:49:59.320
So I gradually rotate from
one to the other state.
00:49:59.320 --> 00:50:04.050
Then in the limit where the
wavelength of this deformation
00:50:04.050 --> 00:50:07.680
becomes of the order of
the size of your system,
00:50:07.680 --> 00:50:10.470
you should have no energy cost.
00:50:10.470 --> 00:50:13.740
And it kind of makes sense
that in the limit where
00:50:13.740 --> 00:50:20.090
you have long wavelengths, you
should have little energy cost.
00:50:20.090 --> 00:50:31.400
So you should have slow
wavelength, no energy
00:50:31.400 --> 00:50:42.120
distortions or modes,
called Goldstone modes.
00:50:50.260 --> 00:51:06.780
But you can only have this for
a broken continuous symmetry
00:51:06.780 --> 00:51:09.370
such as what I have
depicted there,
00:51:09.370 --> 00:51:12.670
where all orientations
are equally likely.
00:51:12.670 --> 00:51:15.010
But if I had the
liquid gas system,
00:51:15.010 --> 00:51:19.330
the density was either above
average or below average.
00:51:19.330 --> 00:51:22.320
If I had uniaxial
magnet, the spin
00:51:22.320 --> 00:51:25.240
would be either
pointing up or down.
00:51:25.240 --> 00:51:29.830
Then I can't deform slowly
from one to the other.
00:51:29.830 --> 00:51:33.330
So for discrete symmetries,
you don't have these modes.
00:51:33.330 --> 00:51:36.481
For continuous symmetries,
you have these modes.
00:51:39.430 --> 00:51:44.560
And actually, we've already
seen one set of those modes.
00:51:44.560 --> 00:51:46.910
These were the phonons.
00:51:46.910 --> 00:51:49.490
When in the first lecture
I was constructing
00:51:49.490 --> 00:51:53.390
this theory of
elasticity, I said
00:51:53.390 --> 00:51:58.910
if we take the whole deformation
and move it uniformity,
00:51:58.910 --> 00:52:00.750
there is no cost.
00:52:00.750 --> 00:52:03.750
And then we were
able, based on that,
00:52:03.750 --> 00:52:07.700
to conclude that long wavelength
phonons have little cost.
00:52:07.700 --> 00:52:10.760
And we wrote their dispersion,
relation, et cetera.
00:52:10.760 --> 00:52:14.480
So phonons are an example
of Goldstone modes.
00:52:14.480 --> 00:52:19.730
These kinds of rotations of
spins in a magnet-- magnons
00:52:19.730 --> 00:52:22.930
are another example
of these modes.
00:52:22.930 --> 00:52:25.730
But something else that we said,
therefore in the first lecture,
00:52:25.730 --> 00:52:29.100
is something that we should
start to think about.
00:52:29.100 --> 00:52:34.610
Which is that we said that
because these modes exist
00:52:34.610 --> 00:52:39.345
and they have so little
energy cost, if I
00:52:39.345 --> 00:52:44.930
am at some finite temperature,
I will be able to excite them.
00:52:44.930 --> 00:52:50.010
So I know for sure that if
I'm at finite temperature,
00:52:50.010 --> 00:52:52.950
there are at least
these fluctuations that
00:52:52.950 --> 00:52:54.960
are going on in my system.
00:52:54.960 --> 00:52:57.000
And maybe in lieu
of that, I should
00:52:57.000 --> 00:53:02.280
be wary of assuming that only
the state where everything
00:53:02.280 --> 00:53:04.474
is uniform is the thing
that is contributing.
00:53:04.474 --> 00:53:05.640
What about the fluctuations?
00:53:09.410 --> 00:53:13.760
So let's think about
these fluctuations that
00:53:13.760 --> 00:53:18.590
are easiest and most
easily generated
00:53:18.590 --> 00:53:22.770
and look at their thermal
excitations and consequences
00:53:22.770 --> 00:53:25.210
for the phase in
phase transition.
00:53:25.210 --> 00:53:28.920
And let's do that in the
context of superfluid.
00:53:35.130 --> 00:53:39.590
So we saw the problem
of super fluidity
00:53:39.590 --> 00:53:42.840
towards the end of 8.3.3.3.
00:53:42.840 --> 00:53:47.540
You had helium that
was an ordinary liquid.
00:53:47.540 --> 00:53:51.480
We cooled it below 2.8
degrees and suddenly it
00:53:51.480 --> 00:53:56.420
became a new form of matter
that has this ability
00:53:56.420 --> 00:53:59.380
to flow through
capillaries, et cetera.
00:53:59.380 --> 00:54:01.390
And we pointed
out that there was
00:54:01.390 --> 00:54:04.720
some kind of quite
likely quantum origin
00:54:04.720 --> 00:54:06.900
to that because of
the similarities
00:54:06.900 --> 00:54:10.930
that it showed to
Bose-Einstein condensation.
00:54:10.930 --> 00:54:13.620
And basically, it
was in this context
00:54:13.620 --> 00:54:20.970
that Landau introduced
something like this The theory
00:54:20.970 --> 00:54:25.720
that we write down
where he chose as order
00:54:25.720 --> 00:54:33.810
parameter as the analog of the
m of x that we have over there,
00:54:33.810 --> 00:54:38.060
a complex function psi of x.
00:54:38.060 --> 00:54:42.050
And very roughly,
you can regard this--
00:54:42.050 --> 00:54:48.650
and again, this is very
rough-- as overlap of wave
00:54:48.650 --> 00:54:58.030
function with the
ground state at position
00:54:58.030 --> 00:55:00.790
x in some coarse-grained sense.
00:55:03.490 --> 00:55:06.670
Now, anything
quantum mechanical we
00:55:06.670 --> 00:55:09.460
saw has an amplitude
and a phase.
00:55:09.460 --> 00:55:14.030
So this is actually a
number plus a phase.
00:55:14.030 --> 00:55:18.050
Or if you like, it has a real
part and an imaginary part.
00:55:18.050 --> 00:55:23.780
And there is no way that we
know anything about the phase.
00:55:23.780 --> 00:55:26.480
The phase is not
an unobservable.
00:55:26.480 --> 00:55:30.620
So the probability that
when we scan the system
00:55:30.620 --> 00:55:35.210
we have identified some psi of
x that the probability should
00:55:35.210 --> 00:55:37.700
depend on the phase
is meaningless.
00:55:37.700 --> 00:55:39.500
It's not an observable.
00:55:39.500 --> 00:55:42.310
So this functional
should only depend
00:55:42.310 --> 00:55:45.920
on things like
absolute value of psi.
00:55:45.920 --> 00:55:51.080
If, like Landau, we assume
that it is a local form,
00:55:51.080 --> 00:55:54.100
then the kinds of
terms that we can write
00:55:54.100 --> 00:55:58.750
are absolute value of psi
squared, absolute value of psi
00:55:58.750 --> 00:56:01.140
to the fourth power.
00:56:01.140 --> 00:56:07.000
And the tendency for the order
to expand across the system you
00:56:07.000 --> 00:56:11.640
would put through a term such
as gradient of psi squared.
00:56:15.320 --> 00:56:18.340
Now, for the case
of the superfluid,
00:56:18.340 --> 00:56:21.750
there is no physical field
that corresponds to the h.
00:56:21.750 --> 00:56:25.300
That just you don't
have that field.
00:56:25.300 --> 00:56:29.386
You can convince yourself
that if you write psi
00:56:29.386 --> 00:56:36.700
to be psi 1 plus psi 2, i psi
2-- real and imaginary part.
00:56:36.700 --> 00:56:41.060
And put that in this formula,
that corresponds exactly
00:56:41.060 --> 00:56:44.270
to the theory that we
wrote over there as long
00:56:44.270 --> 00:56:47.530
as we choose a
two-component magnetization.
00:56:47.530 --> 00:56:49.031
So these two theories
are identical.
00:56:56.110 --> 00:57:05.690
Now, if I look at this
system for t negative
00:57:05.690 --> 00:57:12.430
and try to find a
minimum of the functional
00:57:12.430 --> 00:57:17.540
that I have over
there, then the shape
00:57:17.540 --> 00:57:23.970
of the function functional
psi-- poor choice of notation.
00:57:23.970 --> 00:57:27.650
Psi both being the wave
function as well as the function
00:57:27.650 --> 00:57:28.950
that I have to extremize.
00:57:28.950 --> 00:57:30.920
But let's stick with it.
00:57:30.920 --> 00:57:36.220
It has a minimum that
goes along a circuit.
00:57:36.220 --> 00:57:40.110
So basically, take
this picture that we
00:57:40.110 --> 00:57:43.940
have over here that corresponds
to essentially one direction
00:57:43.940 --> 00:57:46.230
and rotate it.
00:57:46.230 --> 00:57:49.400
And what you will get
is what is sometimes
00:57:49.400 --> 00:57:54.340
called the wine bottle type
of shape, or the Mexican hat
00:57:54.340 --> 00:57:56.910
potential, or whatever.
00:57:56.910 --> 00:57:59.100
But essentially,
it means that there
00:57:59.100 --> 00:58:03.770
is a ring of possible
ground states.
00:58:03.770 --> 00:58:10.170
So the minimum
occurs for psi of x
00:58:10.170 --> 00:58:14.220
having some particular
magnitude which corresponds
00:58:14.220 --> 00:58:17.610
to the location of
this ring-- how far
00:58:17.610 --> 00:58:19.400
away it is from the center.
00:58:19.400 --> 00:58:22.610
And that will be given by
the square root of t formula
00:58:22.610 --> 00:58:24.020
that we have up there.
00:58:24.020 --> 00:58:32.900
But then there is a
phase that is something
00:58:32.900 --> 00:58:33.750
that you don't know.
00:58:39.190 --> 00:58:42.040
Now, let's ask the question.
00:58:42.040 --> 00:58:45.880
Suppose I allow
this phase to vary
00:58:45.880 --> 00:58:50.420
from one part of the sample
to another part of the sample.
00:58:50.420 --> 00:58:55.180
So that's the analog
of this slow distortion
00:58:55.180 --> 00:58:57.750
that I was making up there.
00:58:57.750 --> 00:59:01.160
So essentially, as I go
from one part of the sample
00:59:01.160 --> 00:59:04.140
to another part of
the sample, I slowly
00:59:04.140 --> 00:59:07.570
move around this bottom
of this with potential.
00:59:10.320 --> 00:59:15.780
And I ask, what is the cost
of this distortion that I
00:59:15.780 --> 00:59:18.060
impose on the system?
00:59:18.060 --> 00:59:28.700
If I calculate beta H for psi
bar e the i theta x, what I get
00:59:28.700 --> 00:59:33.940
is whatever I have
put over here,
00:59:33.940 --> 00:59:40.700
such as this function
which minimizes the-- which
00:59:40.700 --> 00:59:43.810
is the location of the minimum.
00:59:43.810 --> 00:59:49.950
But because of the variation
in theta that I have allowed,
00:59:49.950 --> 00:59:51.850
there is a cost.
00:59:51.850 --> 00:59:57.370
So let's write that as beta
H0 plus this additional cost.
00:59:57.370 --> 01:00:00.390
The additional cost
comes from this term.
01:00:00.390 --> 01:00:04.540
If I simply put psi bar in
to the i theta over there,
01:00:04.540 --> 01:00:09.780
I will get an
integral d dx k psi
01:00:09.780 --> 01:00:14.420
bar squared over 2
gradient of theta squared.
01:00:21.450 --> 01:00:27.000
So there is an
additional energy cost.
01:00:27.000 --> 01:00:29.700
This is very similar
to the energy cost
01:00:29.700 --> 01:00:31.170
that we had for phonons.
01:00:31.170 --> 01:00:32.870
Because if I Fourier
transform, you
01:00:32.870 --> 01:00:35.640
can see that I get
a k squared just
01:00:35.640 --> 01:00:38.070
like we got for the
case of phonons.
01:00:38.070 --> 01:00:40.770
And just like for
the case of phonons,
01:00:40.770 --> 01:00:43.470
I expect that at some
finite temperature,
01:00:43.470 --> 01:00:46.850
these kinds of modes
are thermally excited.
01:00:46.850 --> 01:00:51.220
So in reality, I expect that if
I'm at some finite temperature,
01:00:51.220 --> 01:00:54.695
this phase will fluctuate
across my system.
01:00:57.820 --> 01:01:04.290
And maybe I should note
that whereas I'm here
01:01:04.290 --> 01:01:08.040
thinking in terms of
thermal fluctuations,
01:01:08.040 --> 01:01:11.330
by appropriate boundary
conditions one can establish
01:01:11.330 --> 01:01:15.200
a gradient of theta that is
uniform across the system.
01:01:15.200 --> 01:01:18.770
And that actually corresponds
to a superfluid flow.
01:01:18.770 --> 01:01:22.700
So the case of a
superfluid flow can
01:01:22.700 --> 01:01:25.920
be regarded by having
a gradient of theta
01:01:25.920 --> 01:01:28.700
being proportional to
velocity, and then this
01:01:28.700 --> 01:01:31.070
is something like
the kinetic energy.
01:01:31.070 --> 01:01:33.460
But that's a different story.
01:01:33.460 --> 01:01:35.370
Let's just stick with
the fact that this
01:01:35.370 --> 01:01:39.180
is the cost of making
these distortions.
01:01:39.180 --> 01:01:43.460
And I want to know, what's
the probability of having
01:01:43.460 --> 01:01:45.410
one of these distorted shapes?
01:01:45.410 --> 01:01:46.232
Yes.
01:01:46.232 --> 01:01:47.216
AUDIENCE: Question.
01:01:47.216 --> 01:01:49.600
When you're introducing the
psi as another parameter
01:01:49.600 --> 01:01:54.590
and you call it overlap of
[INAUDIBLE] what is boundaries
01:01:54.590 --> 01:01:58.070
and what values this sort
of parameter can take?
01:01:58.070 --> 01:02:02.090
So I basically wonder if we
have this Mexican hat-shaped
01:02:02.090 --> 01:02:08.140
potential with minima on
the ring of radius at 1,
01:02:08.140 --> 01:02:12.380
can the value of potential-- of
the other parameter principally
01:02:12.380 --> 01:02:13.540
be further than that?
01:02:16.417 --> 01:02:17.000
PROFESSOR: OK.
01:02:17.000 --> 01:02:21.940
So again, here we are trying
to phenomenologically explain
01:02:21.940 --> 01:02:27.220
an observation that there is a
transition between a case where
01:02:27.220 --> 01:02:30.960
there is no super
fluidity and right when
01:02:30.960 --> 01:02:33.830
a certain small amount
of super fluidity
01:02:33.830 --> 01:02:36.130
has been established
in the system.
01:02:36.130 --> 01:02:38.720
The question that you asked
over here is legitimate,
01:02:38.720 --> 01:02:40.750
but you could have asked
it also for the case
01:02:40.750 --> 01:02:42.870
of the magnetization.
01:02:42.870 --> 01:02:46.450
So you could have
said, why not to have
01:02:46.450 --> 01:02:49.870
that potential with the
minimum somewhere else?
01:02:49.870 --> 01:02:52.630
But that does not
explain the phenomena
01:02:52.630 --> 01:02:54.320
that we are trying to explain.
01:02:54.320 --> 01:02:56.880
The phenomena that we
are trying to explain
01:02:56.880 --> 01:03:01.280
is the observation that
I go from having nothing
01:03:01.280 --> 01:03:03.840
to having a little
bit of something.
01:03:03.840 --> 01:03:06.070
And I choose the
mathematical form
01:03:06.070 --> 01:03:09.961
that is capable of
describing that.
01:03:09.961 --> 01:03:11.460
AUDIENCE: My question
basically was,
01:03:11.460 --> 01:03:13.668
when we were talking about
magnetization, if you take
01:03:13.668 --> 01:03:16.030
a piece of metal,
you can magnetize it
01:03:16.030 --> 01:03:20.057
from 0 to a pretty large value.
01:03:20.057 --> 01:03:20.640
PROFESSOR: No.
01:03:20.640 --> 01:03:23.577
AUDIENCE: If we are interested
in something not too large--
01:03:23.577 --> 01:03:24.160
PROFESSOR: No.
01:03:24.160 --> 01:03:25.660
AUDIENCE: And as I
say, [INAUDIBLE].
01:03:25.660 --> 01:03:27.750
PROFESSOR: What is your scale?
01:03:27.750 --> 01:03:28.805
What is very large?
01:03:31.440 --> 01:03:34.662
For a magnet, there is a
maximum magnetization--
01:03:34.662 --> 01:03:38.415
AUDIENCE: In this case, I mean
that spontaneous magnetization
01:03:38.415 --> 01:03:43.930
for a magnet would be
lower than saturation, no?
01:03:43.930 --> 01:03:45.180
PROFESSOR: What is saturation?
01:03:45.180 --> 01:03:47.487
AUDIENCE: When all spins
are same direction.
01:03:47.487 --> 01:03:48.070
PROFESSOR: OK.
01:03:48.070 --> 01:03:52.200
So you have a microscopic
picture in mind.
01:03:52.200 --> 01:03:54.350
Now, the place
that we are is far
01:03:54.350 --> 01:03:56.110
from that saturation
magnetization.
01:03:56.110 --> 01:03:59.320
Similarly, in this
case, presumably
01:03:59.320 --> 01:04:04.380
if I go to 0 temperature,
there is some uniformity.
01:04:04.380 --> 01:04:08.642
And if I call this an overlap,
the maximum of it will be 1.
01:04:08.642 --> 01:04:09.266
AUDIENCE: Yeah.
01:04:09.266 --> 01:04:11.120
But basically, I just
don't understand--
01:04:11.120 --> 01:04:14.150
what is overlap
of wave function?
01:04:14.150 --> 01:04:15.650
PROFESSOR: Well,
that's why I didn't
01:04:15.650 --> 01:04:17.190
want to go into that detail.
01:04:17.190 --> 01:04:19.830
But basically, the overlap
of two wave functions
01:04:19.830 --> 01:04:24.040
would be the psi
1 star psi 2 of x.
01:04:24.040 --> 01:04:26.650
And if you are thinking
about the ground state,
01:04:26.650 --> 01:04:29.460
let's say that I have
normalized this function
01:04:29.460 --> 01:04:32.940
to have a maximum of 1.
01:04:32.940 --> 01:04:35.776
The point is that
what that maximum is,
01:04:35.776 --> 01:04:41.680
is folded into all of these
parameters-- a, u, et cetera--
01:04:41.680 --> 01:04:45.912
and is pretty irrelevant to
the nature of the transition.
01:04:45.912 --> 01:04:46.453
AUDIENCE: OK.
01:04:50.150 --> 01:04:52.000
PROFESSOR: OK?
01:04:52.000 --> 01:04:54.960
So the probability of a
particular configuration
01:04:54.960 --> 01:05:00.030
of theta across the system
is given by this formula.
01:05:00.030 --> 01:05:03.440
I can unpack that a
little bit better just
01:05:03.440 --> 01:05:08.040
like we did for
the case of phonons
01:05:08.040 --> 01:05:12.000
by writing theta in
terms of Fourier modes.
01:05:12.000 --> 01:05:18.600
e to the i q dot x theta of
q, which for the time being,
01:05:18.600 --> 01:05:22.650
I assume I have discretized
appropriate values of q.
01:05:22.650 --> 01:05:28.460
I choose this normalization
root V in this context.
01:05:28.460 --> 01:05:32.070
If I substitute
that over here, what
01:05:32.070 --> 01:05:36.920
I find is that beta H as a
function of the collection
01:05:36.920 --> 01:05:41.700
of theta q's is,
again, some beta H0,
01:05:41.700 --> 01:05:44.230
which is not important.
01:05:44.230 --> 01:05:46.440
I put gradient of theta.
01:05:46.440 --> 01:05:48.810
Once I do gradient
of theta, you can
01:05:48.810 --> 01:05:51.200
see that I get a factor of iq.
01:05:51.200 --> 01:05:53.640
I will have two of
them, so the answer
01:05:53.640 --> 01:05:58.510
is going to be
proportional to q squared.
01:05:58.510 --> 01:06:02.260
Let's call this combination
k psi bar squared k bar
01:06:02.260 --> 01:06:05.350
so that I don't have to
write it again and again.
01:06:05.350 --> 01:06:07.690
k bar over 2.
01:06:07.690 --> 01:06:12.760
And then theta of q squared.
01:06:12.760 --> 01:06:14.890
OK?
01:06:14.890 --> 01:06:20.800
So the probability of some
particular combination
01:06:20.800 --> 01:06:28.050
of these Fourier amplitudes
is proportional to exponential
01:06:28.050 --> 01:06:29.480
of this.
01:06:29.480 --> 01:06:34.820
And therefore, a
product of independent
01:06:34.820 --> 01:06:36.670
Gaussian-distributed quantities.
01:06:46.037 --> 01:06:47.023
q squared.
01:06:51.480 --> 01:06:53.349
Yes.
01:06:53.349 --> 01:06:57.870
AUDIENCE: When you plug in
your summation of your-- well,
01:06:57.870 --> 01:07:00.785
your Fourier series
into the gradient
01:07:00.785 --> 01:07:03.740
and then you square
that, why don't you
01:07:03.740 --> 01:07:08.390
get interactions between the
different Fourier amplitudes?
01:07:08.390 --> 01:07:10.180
PROFESSOR: OK, let's
do it explicitly.
01:07:10.180 --> 01:07:21.150
I have integral d dx gradient
of theta gradient of theta.
01:07:21.150 --> 01:07:28.220
Gradient of theta
is iq sum-- OK.
01:07:28.220 --> 01:07:39.110
Is i sum over q q e to the
i q dot x theta tilde of q.
01:07:39.110 --> 01:07:40.900
And I have to repeat that twice.
01:07:40.900 --> 01:07:47.160
So I have i sum over q prime
q prime into the i q prime dot
01:07:47.160 --> 01:07:51.330
x theta tilde of q prime.
01:07:51.330 --> 01:07:55.150
So basically, this
went into that.
01:07:55.150 --> 01:07:58.330
This went into that.
01:07:58.330 --> 01:08:00.720
OK?
01:08:00.720 --> 01:08:03.360
So I have a sum over
q, sum over q prime,
01:08:03.360 --> 01:08:05.330
and an integral over x.
01:08:05.330 --> 01:08:13.590
What is the integral over x of
e to the i q plus q prime dot x?
01:08:13.590 --> 01:08:20.450
It is 0, unless q and
q prime add up to 0.
01:08:20.450 --> 01:08:22.240
This is delta function.
01:08:22.240 --> 01:08:23.930
So you put it there.
01:08:23.930 --> 01:08:25.729
Only one sum survives.
01:08:25.729 --> 01:08:28.960
Actually, I had introduced
the normalizations that
01:08:28.960 --> 01:08:33.450
were root V. The normalizations
get rid of this factor of V.
01:08:33.450 --> 01:08:36.180
That's why I had normalized
it with the root V.
01:08:36.180 --> 01:08:39.620
And I will get one
factor of q remaining.
01:08:39.620 --> 01:08:45.500
Since q prime is minus q iq
iq prime becomes q squared.
01:08:45.500 --> 01:08:49.020
And then I have theta
tilde of q theta tilde
01:08:49.020 --> 01:08:51.410
of minus q, which gives me this.
01:09:00.720 --> 01:09:04.140
So each mode is independently
distributed according
01:09:04.140 --> 01:09:07.085
to a Gaussian, which
immediately tells me
01:09:07.085 --> 01:09:11.696
the average of theta of
q tilde is, of course, 0.
01:09:11.696 --> 01:09:16.300
Let's be careful and put the
tildes all over the place.
01:09:16.300 --> 01:09:22.250
While the average of
theta tilde of q squared
01:09:22.250 --> 01:09:26.240
is 1 over k bar q squared.
01:09:26.240 --> 01:09:32.870
Again, all that says is that as
you go to long wavelength modes
01:09:32.870 --> 01:09:36.240
and q goes to 0,
the fluctuations
01:09:36.240 --> 01:09:41.340
become larger because the
energy cost is smaller.
01:09:41.340 --> 01:09:44.490
OK, so that's understandable.
01:09:44.490 --> 01:09:47.209
But now, let's look at what
is happening in real space.
01:09:57.360 --> 01:10:02.990
I pick two points,
x and x prime.
01:10:02.990 --> 01:10:06.340
And I ask, how do
the fluctuations
01:10:06.340 --> 01:10:09.750
vary from one point
to another point?
01:10:09.750 --> 01:10:12.605
So I'm interested in theta x.
01:10:15.540 --> 01:10:29.150
Let's say minus theta.
01:10:29.150 --> 01:10:32.400
Let's calculate the
following first,
01:10:32.400 --> 01:10:35.740
theta of x, theta of x prime
just because the algebra
01:10:35.740 --> 01:10:36.710
is slightly easier.
01:10:39.500 --> 01:10:44.380
Now, theta of x I can write
in terms of theta tilde of q.
01:10:44.380 --> 01:10:53.236
So this becomes sum over q
q prime e to the iq dot x
01:10:53.236 --> 01:10:56.765
e to the iq prime dot x prime.
01:10:56.765 --> 01:10:58.706
There is a factor
of 1 over V that
01:10:58.706 --> 01:11:02.240
comes from the
normalization I chose.
01:11:02.240 --> 01:11:05.350
And then the average
of theta tilde
01:11:05.350 --> 01:11:10.970
of q theta tilde of q prime.
01:11:10.970 --> 01:11:15.180
But we just established
that the different modes
01:11:15.180 --> 01:11:18.100
are independent of each other.
01:11:18.100 --> 01:11:21.490
So basically, this gives
me a delta function
01:11:21.490 --> 01:11:26.060
that forces q and q
prime to add up to 0.
01:11:26.060 --> 01:11:30.450
If they do add up to 0,
the expectation that I get
01:11:30.450 --> 01:11:32.690
is 1 over k bar q squared.
01:11:35.200 --> 01:11:42.470
And so what I find is
that this becomes related
01:11:42.470 --> 01:11:49.420
to a sum over q 1 over
V e to the iq dot x
01:11:49.420 --> 01:11:54.480
minus x prime divided
by k bar q squared.
01:11:57.360 --> 01:12:00.960
If I go to the continuum
limit where the sum over
01:12:00.960 --> 01:12:05.290
q I replace with
an integral over q,
01:12:05.290 --> 01:12:10.570
then I have to introduce
the density of states.
01:12:10.570 --> 01:12:15.080
And so then I find that
theta of x theta of x
01:12:15.080 --> 01:12:25.265
prime is 1 over k
bar integral d dq
01:12:25.265 --> 01:12:32.145
2 pi to the d the Fourier
transform of 1 over q squared.
01:12:37.340 --> 01:12:40.895
Now, the Fourier transform
of 1 over q squared
01:12:40.895 --> 01:12:45.310
is something that
appears all over physics.
01:12:45.310 --> 01:12:48.180
So let's give it a name.
01:12:48.180 --> 01:12:54.730
So we're going to call the
integral d dq 2 pi to the d
01:12:54.730 --> 01:13:01.730
e to the iq dot x
divided by q squared.
01:13:01.730 --> 01:13:04.900
And let's put a minus
sign in front of that.
01:13:04.900 --> 01:13:08.370
And I'll give it the name
the Coulomb potential
01:13:08.370 --> 01:13:09.720
in d dimensional space.
01:13:14.810 --> 01:13:19.770
And for those of you
who haven't seen this,
01:13:19.770 --> 01:13:22.520
the reason this is
the Coulomb potential
01:13:22.520 --> 01:13:24.880
is because if I
take two derivatives
01:13:24.880 --> 01:13:31.470
and construct the Laplacian
of that function--
01:13:31.470 --> 01:13:34.140
so I take two derivative
with respect to x here,
01:13:34.140 --> 01:13:38.650
the two derivatives will
go inside the integral.
01:13:38.650 --> 01:13:43.810
And what they do
is they bring down
01:13:43.810 --> 01:13:48.680
two factors of iq
divided by q squared.
01:13:51.420 --> 01:13:54.330
The minus sign disappears,
q squared over q squared
01:13:54.330 --> 01:13:55.470
goes to 1.
01:13:55.470 --> 01:14:00.290
Fourier transform of e to the iq
x is simply the delta function.
01:14:04.990 --> 01:14:12.070
So this Cd of x is the potential
that would emerge from a unit
01:14:12.070 --> 01:14:16.100
charge at the origin
at a distance x.
01:14:21.070 --> 01:14:25.890
So again, for those who have
forgotten this or not seen it,
01:14:25.890 --> 01:14:31.210
let's calculate it explicitly
using Gauss' theorem.
01:14:31.210 --> 01:14:34.310
The potential due
to a unit charge
01:14:34.310 --> 01:14:37.820
is going to be spherically
symmetric, so it's only
01:14:37.820 --> 01:14:40.180
a function of the
magnitude of x.
01:14:40.180 --> 01:14:44.750
It doesn't depend
on the orientation.
01:14:44.750 --> 01:14:51.033
And Gauss' law states that
the integral of Laplacian
01:14:51.033 --> 01:14:59.230
over volume is the same as
the integral of the analog
01:14:59.230 --> 01:15:03.590
of the electric field, which is
the gradient over the surface.
01:15:03.590 --> 01:15:07.930
So I have the surface
integral of gradient.
01:15:14.540 --> 01:15:17.440
Now, for the case that
we are dealing with,
01:15:17.440 --> 01:15:21.930
the left-hand side Laplacian
is a delta function.
01:15:21.930 --> 01:15:26.650
So when we integrate that over
the sphere, I simply get 1.
01:15:26.650 --> 01:15:28.640
So this gives me 1.
01:15:28.640 --> 01:15:32.865
What do I get on
the right-hand side?
01:15:32.865 --> 01:15:34.854
It's just like the flux
of the electric field
01:15:34.854 --> 01:15:35.895
that you have calculated.
01:15:35.895 --> 01:15:41.910
It is the magnitude
of the electric field
01:15:41.910 --> 01:15:45.930
times the surface area.
01:15:45.930 --> 01:15:49.350
And I am doing this
generally in d dimensions.
01:15:49.350 --> 01:15:51.810
So the surface
area in d dimension
01:15:51.810 --> 01:15:55.280
grows like x to the d minus 1.
01:15:55.280 --> 01:16:00.300
And then there is a factor
such as 2 pi, 4 pi, et cetera,
01:16:00.300 --> 01:16:02.220
which is the solid
angle that you
01:16:02.220 --> 01:16:05.160
would have to put
in d dimensions.
01:16:05.160 --> 01:16:10.790
And to remind you, the
solid angle in d dimensions
01:16:10.790 --> 01:16:17.029
is 2 pi to the d over 2 d
over 2 minus 1 factorial.
01:16:45.170 --> 01:16:48.143
So the magnitude of
this derivative dC
01:16:48.143 --> 01:16:54.900
by dx following
from that is simply
01:16:54.900 --> 01:17:00.000
1 divided by x to the d
minus 1, or x to the 1
01:17:00.000 --> 01:17:03.100
minus d divided by Sd.
01:17:08.990 --> 01:17:12.310
So this is generalizing
how you would calculate
01:17:12.310 --> 01:17:14.830
Gauss' law in three dimensions.
01:17:14.830 --> 01:17:19.090
So now I just integrate that and
I find that the d dimensional
01:17:19.090 --> 01:17:24.450
Coulomb potential is
x to the 2 minus d
01:17:24.450 --> 01:17:29.720
divided by 2 minus d Sd.
01:17:29.720 --> 01:17:32.592
And of course, there could be
some constant of integration.
01:17:37.100 --> 01:17:42.670
So it reproduces the familiar 1
over x law in three dimensions.
01:17:42.670 --> 01:17:45.680
But the thing that
is important to note
01:17:45.680 --> 01:17:49.590
is how much this
Coulomb potential
01:17:49.590 --> 01:17:52.980
depends on dimensions.
01:17:52.980 --> 01:17:58.090
It determines these angle--
angle fluctuation correlations.
01:17:58.090 --> 01:18:00.300
And typically, here
in this context
01:18:00.300 --> 01:18:03.580
we want to know something
about large distances.
01:18:03.580 --> 01:18:06.793
If I make a fluctuation
here, how far away
01:18:06.793 --> 01:18:10.110
is the influence of
that fluctuation felt?
01:18:10.110 --> 01:18:14.460
So I would be interested
in the limit of this
01:18:14.460 --> 01:18:19.723
when the separation is
large-- goes to infinity.
01:18:19.723 --> 01:18:22.370
And we can see that
the answer very much
01:18:22.370 --> 01:18:24.630
depends on the dimensions.
01:18:24.630 --> 01:18:30.140
So we find that for d
that is greater than 2,
01:18:30.140 --> 01:18:33.170
like the Coulomb potential
in three dimensions,
01:18:33.170 --> 01:18:36.150
you basically go to a constant.
01:18:36.150 --> 01:18:40.920
While in d less than
2, it is something
01:18:40.920 --> 01:18:43.325
that grows as a
function of distance
01:18:43.325 --> 01:18:49.550
as x to the 2 minus
d 2 minus d Sd.
01:18:49.550 --> 01:18:53.450
And actually, write at
the borderline dimension
01:18:53.450 --> 01:18:58.150
of d equals to 2, it also
grows at large distances.
01:18:58.150 --> 01:19:00.370
If you do the
integration correctly,
01:19:00.370 --> 01:19:05.853
you will find that it is 1 over
2 pi log x over some distance
01:19:05.853 --> 01:19:06.839
or [INAUDIBLE].
01:19:14.250 --> 01:19:17.240
Now, what you are really
interested-- actually,
01:19:17.240 --> 01:19:21.790
this thing that I wrote down
is not particularly meaningful.
01:19:21.790 --> 01:19:23.410
The thing that you
are interested in
01:19:23.410 --> 01:19:26.040
is what I had
originally written,
01:19:26.040 --> 01:19:30.890
which is that if I look at the
angle that I have at x and then
01:19:30.890 --> 01:19:33.440
I go far away-- because
the angel itself is not
01:19:33.440 --> 01:19:38.030
an observable, but
angle differences are.
01:19:38.030 --> 01:19:41.150
So the average of this
quantity will be 0.
01:19:41.150 --> 01:19:43.130
But there will be
some variance to it,
01:19:43.130 --> 01:19:44.530
so I can look at this quantity.
01:19:47.810 --> 01:19:51.420
And that quantity--
I can expand this--
01:19:51.420 --> 01:19:56.820
is twice the average of theta
at some particular location.
01:19:56.820 --> 01:19:59.740
Presumably, it doesn't matter
which location I look at.
01:19:59.740 --> 01:20:03.990
So it's the variance locally
that you have in the angles.
01:20:03.990 --> 01:20:08.940
And then minus twice theta
of x theta of x prime,
01:20:08.940 --> 01:20:12.535
which is the quantity that
I calculated for you above.
01:20:15.150 --> 01:20:20.290
So all I need to do is to take
the Coulomb potential that I
01:20:20.290 --> 01:20:24.140
calculated, multiply
it by a factor of 2,
01:20:24.140 --> 01:20:28.030
divide by a factor of
k bar that I basically
01:20:28.030 --> 01:20:30.750
indicated as part
of the definition.
01:20:30.750 --> 01:20:36.870
So this object is going
to be 2 x to the 2
01:20:36.870 --> 01:20:43.620
minus d divided by
k bar 2 minus d Sd.
01:20:43.620 --> 01:20:47.070
And actually, the reason
I do this is because now I
01:20:47.070 --> 01:20:51.310
can indicate the overall
constant as follows.
01:20:54.170 --> 01:20:59.490
Remember that all of our
statistical field theories
01:20:59.490 --> 01:21:02.250
are obtained by averaging.
01:21:02.250 --> 01:21:06.240
And I shouldn't believe
any of these formulas
01:21:06.240 --> 01:21:09.260
when I look at very
short wavelengths.
01:21:09.260 --> 01:21:12.770
So I shouldn't really
believe any answer
01:21:12.770 --> 01:21:16.910
that I got from those formulas
when the points x prime and x
01:21:16.910 --> 01:21:19.680
come too close to each other.
01:21:19.680 --> 01:21:24.330
So there is something of the
order of a lattice spacing,
01:21:24.330 --> 01:21:28.340
averaging distance, et
cetera, that I'll call a.
01:21:28.340 --> 01:21:33.140
And by the time I get to a,
I expect that my fluctuations
01:21:33.140 --> 01:21:35.340
vanish because that's
the distance over which
01:21:35.340 --> 01:21:37.320
I'm doing the average.
01:21:37.320 --> 01:21:40.870
So I manage to get rid of
whatever this constant is
01:21:40.870 --> 01:21:46.140
by substituting [INAUDIBLE] with
the scale over which I expect
01:21:46.140 --> 01:21:49.370
my theory to cease to be valid.
01:21:49.370 --> 01:21:52.490
But again, what I
find is that if I
01:21:52.490 --> 01:21:59.130
look at the fluctuation between
two points at large distances,
01:21:59.130 --> 01:22:02.540
if I am in dimensions
greater than 2,
01:22:02.540 --> 01:22:04.680
this thing eventually
goes to a constant.
01:22:08.460 --> 01:22:11.330
Which means that if I'm
in three dimensions,
01:22:11.330 --> 01:22:16.080
the fluctuation in phase between
one place and another place
01:22:16.080 --> 01:22:20.520
are not necessarily small or
large because I don't know what
01:22:20.520 --> 01:22:22.180
the magnitude of
this constant is,
01:22:22.180 --> 01:22:23.800
but they are not
getting bigger as I
01:22:23.800 --> 01:22:26.440
go further and further along.
01:22:26.440 --> 01:22:30.690
Whereas, no matter what I
do in d that is less than
01:22:30.690 --> 01:22:38.600
or equal to 2, this thing
at large distances blows up.
01:22:38.600 --> 01:22:41.340
So I thought that I
had a system where
01:22:41.340 --> 01:22:47.030
I had broken spontaneous
symmetry and all of my spins,
01:22:47.030 --> 01:22:51.470
all of my phases were
pointing in one direction.
01:22:51.470 --> 01:22:55.910
But I see that when I put
these fluctuations, no matter
01:22:55.910 --> 01:23:00.500
how small I make the amplitude,
the amplitude doesn't matter.
01:23:00.500 --> 01:23:04.820
If I go to far enough
distances, fluctuations
01:23:04.820 --> 01:23:08.020
will tell me that I don't know
what the phase is from here
01:23:08.020 --> 01:23:12.090
to here because it has gone
over many multiples of 2 pi,
01:23:12.090 --> 01:23:15.740
so that it has become divergent.
01:23:15.740 --> 01:23:23.350
So what that really means is
that because of fluctuations,
01:23:23.350 --> 01:23:26.740
you cannot have
long-range order.
01:23:26.740 --> 01:23:38.370
So destroy continuous
long-range order
01:23:38.370 --> 01:23:43.390
in dimensions that are
less than or equal to 2.
01:23:43.390 --> 01:23:45.230
This is called the
Mermin-Wagner theorem.
01:23:51.390 --> 01:23:55.490
So you shouldn't have any,
for example, super fluidity,
01:23:55.490 --> 01:23:59.840
magnetization, anything in two
dimensions of one dimension.
01:23:59.840 --> 01:24:04.110
If you go to long
enough, you will
01:24:04.110 --> 01:24:07.430
see that fluctuations
have destroyed your order.
01:24:07.430 --> 01:24:12.590
So we can see already how
important fluctuations are.
01:24:12.590 --> 01:24:15.700
This d equal to 2 is called
the lower critical dimension.
01:24:21.210 --> 01:24:27.760
It is this phenomena of
symmetry breaking, ordering,
01:24:27.760 --> 01:24:29.750
phase transition,
et cetera, that we
01:24:29.750 --> 01:24:34.010
are discussing for
continuous systems--
01:24:34.010 --> 01:24:36.580
for continuous symmetry
breaking can only
01:24:36.580 --> 01:24:40.810
exist in three dimensions
but not in two dimensions.
01:24:40.810 --> 01:24:43.780
We'll see that for
discrete symmetries,
01:24:43.780 --> 01:24:47.090
you can have ordering in
two dimensions, but not one
01:24:47.090 --> 01:24:48.240
dimensions.
01:24:48.240 --> 01:24:51.990
So there, the lower
critical dimension is 1.
01:24:51.990 --> 01:24:52.720
Yes.
01:24:52.720 --> 01:24:54.382
AUDIENCE: And does
this hold for any n?
01:24:54.382 --> 01:24:55.590
This example you were doing--
01:24:55.590 --> 01:24:55.950
PROFESSOR: Yes.
01:24:55.950 --> 01:24:57.575
AUDIENCE: --it just
has two components.
01:24:57.575 --> 01:25:00.960
PROFESSOR: Any n. n equals
to 2, 3, 4, anything.
01:25:00.960 --> 01:25:03.800
We'll see later on, towards
the end of the course,
01:25:03.800 --> 01:25:07.210
that there is a slight proviso
for the case of n equals to 2,
01:25:07.210 --> 01:25:09.746
but that we'll leave for later.