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PROFESSOR: OK, let's start.
00:00:25.880 --> 00:00:29.825
So we've been trying to
understand critical points.
00:00:36.570 --> 00:00:41.080
And this refers to the
experimental observation
00:00:41.080 --> 00:00:44.570
that in a number
of systems we can
00:00:44.570 --> 00:00:48.570
be changing some parameters,
such as temperature,
00:00:48.570 --> 00:00:53.610
and you encounter a transition
to some other type of behavior
00:00:53.610 --> 00:00:54.760
at some point.
00:00:54.760 --> 00:00:58.950
So the temperature, let's
say, in this behavior
00:00:58.950 --> 00:01:00.070
is the control parameter.
00:01:00.070 --> 00:01:02.330
And you have to
see, for example,
00:01:02.330 --> 00:01:05.390
this will be normal to
superfluid transition.
00:01:05.390 --> 00:01:08.336
You have one now [INAUDIBLE]
change temperature and going
00:01:08.336 --> 00:01:10.210
through this point.
00:01:10.210 --> 00:01:12.500
For other systems,
such as magnets,
00:01:12.500 --> 00:01:14.730
you actually have two knobs.
00:01:14.730 --> 00:01:18.120
There is also the
magnetic field.
00:01:18.120 --> 00:01:21.360
And there, you have
to turn two knobs
00:01:21.360 --> 00:01:25.630
in order to end at
this critical point,
00:01:25.630 --> 00:01:29.060
also in the case of the liquid
gas system in the pressure
00:01:29.060 --> 00:01:33.553
temperature plane, you
have to tune two things
00:01:33.553 --> 00:01:35.420
to get this point.
00:01:35.420 --> 00:01:38.630
And the interesting thing
was that in the vicinity
00:01:38.630 --> 00:01:43.420
of his point, the singular
parts of various thermodynamic
00:01:43.420 --> 00:01:48.710
quantities are interestingly
independent of the type
00:01:48.710 --> 00:01:50.210
of material.
00:01:50.210 --> 00:01:55.320
So if we, for example, establish
a coordinate at t and h
00:01:55.320 --> 00:01:59.400
describing deviations
from this critical point,
00:01:59.400 --> 00:02:02.880
we have, let's say,
the singular part
00:02:02.880 --> 00:02:06.820
of free energy as a
function of t and h
00:02:06.820 --> 00:02:10.710
has a form like t to
the 2 minus alpha,
00:02:10.710 --> 00:02:15.440
some scaling function
ht to the delta,
00:02:15.440 --> 00:02:18.856
and these exponents, alpha
and delta, other things
00:02:18.856 --> 00:02:21.700
that are universe.
00:02:21.700 --> 00:02:25.080
For example, we
could get from that
00:02:25.080 --> 00:02:28.660
by taking two derivatives with
respect to h, the singularity
00:02:28.660 --> 00:02:33.120
and the divergence of
the susceptibility.
00:02:33.120 --> 00:02:35.776
And we said that the
diverging susceptibility also
00:02:35.776 --> 00:02:39.660
immediately tells you that
there is a correlation then
00:02:39.660 --> 00:02:44.790
that diverges, and
in particular, we
00:02:44.790 --> 00:02:48.410
indicated its divergence
to an exponent u.
00:02:48.410 --> 00:02:53.210
WE could for that also
establish a scaling form
00:02:53.210 --> 00:02:56.140
on how the correlation
then diverges
00:02:56.140 --> 00:03:01.980
on approaching this point
generally in the ht plane.
00:03:01.980 --> 00:03:08.820
So this was the general picture.
00:03:08.820 --> 00:03:14.340
And building on that, we made
one observation last time,
00:03:14.340 --> 00:03:18.370
which is that any point when
you are away from h and t
00:03:18.370 --> 00:03:23.130
equals to 0, you have
a correlation length.
00:03:23.130 --> 00:03:28.760
And then we concluded that if
you are at t and h equals to 0,
00:03:28.760 --> 00:03:30.460
you have a form of
scaling vertices.
00:03:38.560 --> 00:03:45.470
And basically what
that means is that when
00:03:45.470 --> 00:03:48.150
you are at that point,
you look at your system,
00:03:48.150 --> 00:03:52.510
it's a fluctuating system,
and the fluctuations are
00:03:52.510 --> 00:03:56.035
such that you can't
associate a scale with them.
00:03:56.035 --> 00:03:59.400
The scale has already
gone into the correlation
00:03:59.400 --> 00:04:02.180
length that is infinite.
00:04:02.180 --> 00:04:03.990
And we said that
therefore, if I were
00:04:03.990 --> 00:04:06.295
to look at some kind of
a correlation function,
00:04:06.295 --> 00:04:12.270
such as a magnetization
in the case of a magnet,
00:04:12.270 --> 00:04:17.700
that the only way that
it became its separation
00:04:17.700 --> 00:04:23.550
is as a power of a distance.
00:04:23.550 --> 00:04:26.160
And this clearly has
a property that if we
00:04:26.160 --> 00:04:30.990
were to rescale x and
y by a certain amount,
00:04:30.990 --> 00:04:34.090
this correlation function
nearly gets multiplied
00:04:34.090 --> 00:04:38.610
by a factor that is
dependent on this rescale.
00:04:38.610 --> 00:04:41.810
And this is after
we do the averaging,
00:04:41.810 --> 00:04:47.200
so it's a kind of
statistical self-singularity,
00:04:47.200 --> 00:04:52.715
as opposed to some factor such
as Sierpinkski gasket, which
00:04:52.715 --> 00:04:54.840
are identically and
deterministically
00:04:54.840 --> 00:04:58.000
self-similar in that each
piece, if you blow it up,
00:04:58.000 --> 00:05:01.320
looks like the entire thing.
00:05:01.320 --> 00:05:07.900
So what we have in our system
is that if we have, let's say,
00:05:07.900 --> 00:05:14.245
a box which could be
containing our liquid gas
00:05:14.245 --> 00:05:20.400
system at its critical
point, or maybe a magnet
00:05:20.400 --> 00:05:23.980
at its critical point, will
have a statistical field,
00:05:23.980 --> 00:05:26.327
this m of x.
00:05:26.327 --> 00:05:31.120
And it will fluctuate
across the system.
00:05:31.120 --> 00:05:36.490
So maybe this would be a picture
of the density fluctuation.
00:05:36.490 --> 00:05:43.966
What I can do is to take a scan
along some particular axis--
00:05:43.966 --> 00:05:49.860
let's call it x-- and plot
what the fluctuations are
00:05:49.860 --> 00:05:51.600
of this magnetization.
00:05:51.600 --> 00:05:54.660
Let's say m of x.
00:05:54.660 --> 00:05:56.780
Now the average
will be 0, but it
00:05:56.780 --> 00:05:59.700
will have fluctuations
around the average.
00:05:59.700 --> 00:06:01.615
And so maybe it
will look something
00:06:01.615 --> 00:06:12.540
like this-- kind of like
a picture of a mountain,
00:06:12.540 --> 00:06:13.040
for example.
00:06:16.910 --> 00:06:20.140
Now one thing that
we should remember
00:06:20.140 --> 00:06:25.420
is that this object would
be piece of iron or nickle,
00:06:25.420 --> 00:06:29.160
and clearly I don't
really mean that this
00:06:29.160 --> 00:06:33.220
is what is going on at
the scale of a single atom
00:06:33.220 --> 00:06:34.920
or molecule of my substance.
00:06:34.920 --> 00:06:38.930
I had to do some kind
of averaging in order
00:06:38.930 --> 00:06:42.300
to get the statistical field
that I'm presenting here.
00:06:42.300 --> 00:06:45.470
So let's keep in
mind that there is,
00:06:45.470 --> 00:06:54.360
in fact, some implicit
analog of lattice size
00:06:54.360 --> 00:06:57.630
or some implicit
shortest distance,
00:06:57.630 --> 00:07:00.810
shortest wavelength that I
allow for my frustrations.
00:07:06.270 --> 00:07:14.035
Now I can sort of make this
idea of scale invariance
00:07:14.035 --> 00:07:19.910
of a set of pictures, such
as this one, more precise,
00:07:19.910 --> 00:07:26.470
as follows, by going
through a procedure
00:07:26.470 --> 00:07:30.676
that I will call renormalization
that has the following three
00:07:30.676 --> 00:07:31.175
steps.
00:07:38.040 --> 00:07:45.575
So the first step, what I will
do is to coarse-grain further.
00:07:51.490 --> 00:08:10.630
And by this, I mean averaging
m of x over a scale ta.
00:08:10.630 --> 00:08:15.950
So previously, I had done my
averaging of whatever means,
00:08:15.950 --> 00:08:17.360
et cetera.
00:08:17.360 --> 00:08:21.260
We're giving contribution
to the overall magnetization
00:08:21.260 --> 00:08:25.320
over some number,
let's 100 by 100 by 100
00:08:25.320 --> 00:08:30.180
spins and a was my
scaling distance.
00:08:30.180 --> 00:08:31.880
Why should I choose 100?
00:08:31.880 --> 00:08:36.880
Why not choose 200, some factor
of what I had originally?
00:08:36.880 --> 00:08:40.929
So coarse-graining means
increasing this minimum length
00:08:40.929 --> 00:08:43.620
scale from a to ba.
00:08:43.620 --> 00:08:46.850
And then I define a
coarse-grained version
00:08:46.850 --> 00:08:48.690
of my field.
00:08:48.690 --> 00:08:50.760
So previously, I had m of x.
00:08:50.760 --> 00:08:57.630
Now I have m tilda of x, which
is obtained by averaging,
00:08:57.630 --> 00:09:06.400
let's say, over volume around
the point x that I had before.
00:09:06.400 --> 00:09:16.970
And this volume is a box
of dimension ba to the d.
00:09:16.970 --> 00:09:24.220
And then I basically
average over that.
00:09:24.220 --> 00:09:28.920
I guess let's call it
original distance a equals 1,
00:09:28.920 --> 00:09:34.988
so I don't really have to bother
by the dimensionality of y,
00:09:34.988 --> 00:09:35.487
et cetera.
00:09:38.760 --> 00:09:40.240
OK?
00:09:40.240 --> 00:09:42.980
So if I were to apply
that to the picture
00:09:42.980 --> 00:09:45.670
that I have up
there, what do I get?
00:09:45.670 --> 00:09:49.435
I will get an m tilda
as a function of x.
00:09:55.500 --> 00:09:58.770
And essentially,
let's say if I were
00:09:58.770 --> 00:10:04.250
to choose a factor
of b that was like 2,
00:10:04.250 --> 00:10:07.970
I would take the average
of the fluctuations
00:10:07.970 --> 00:10:11.270
that they have over 2 of
those of those intervals.
00:10:11.270 --> 00:10:14.190
And so the picture
that I would get it
00:10:14.190 --> 00:10:17.660
would be kind of a
smoothened out version
00:10:17.660 --> 00:10:20.690
of what I have
before over there.
00:10:23.910 --> 00:10:27.080
I will still have
some fluctuations,
00:10:27.080 --> 00:10:31.480
but kind of ironed out.
00:10:31.480 --> 00:10:35.070
And basically,
essentially, it means
00:10:35.070 --> 00:10:40.640
that if you were to imagine
having taken a photograph,
00:10:40.640 --> 00:10:45.830
previously you had the
pixel size that was 1.
00:10:45.830 --> 00:10:47.830
Now your pixel size is larger.
00:10:47.830 --> 00:10:51.510
It is factor of b.
00:10:51.510 --> 00:10:55.790
So it's this kind of
detuning and averaging
00:10:55.790 --> 00:10:58.170
of the fluctuations
that has gone.
00:10:58.170 --> 00:11:00.378
And so you have here now b.
00:11:08.690 --> 00:11:12.650
Now if I were to give you
a photograph like that
00:11:12.650 --> 00:11:15.430
and a photograph
like this, you would
00:11:15.430 --> 00:11:18.370
say that they are not identical.
00:11:18.370 --> 00:11:22.796
One of them is clearly much
grainier than the other.
00:11:22.796 --> 00:11:27.010
So I say, OK, I can restore
some amount of similarity
00:11:27.010 --> 00:11:30.530
between them by
doing a rescaling.
00:11:34.070 --> 00:11:41.520
So I call a new variable x prime
to be my old variable x divided
00:11:41.520 --> 00:11:44.220
by a factor of b.
00:11:44.220 --> 00:11:47.871
So when I do that
to this picture,
00:11:47.871 --> 00:11:53.710
I will get m tilda as
a function of x prime.
00:11:53.710 --> 00:12:00.120
x prime can go in further
less, because all I do
00:12:00.120 --> 00:12:03.900
is I take this and squeeze
it by a factor of b.
00:12:03.900 --> 00:12:05.840
So I will get a
picture that maybe
00:12:05.840 --> 00:12:08.500
looks something like this.
00:12:19.140 --> 00:12:27.630
Now if I were to look at this
picture and this picture,
00:12:27.630 --> 00:12:30.240
you would also see a difference.
00:12:30.240 --> 00:12:31.920
That is, there is a contrast.
00:12:31.920 --> 00:12:34.890
So here, there would be,
let's say, black and white.
00:12:34.890 --> 00:12:37.340
And as you scan the
picture, you sort of see
00:12:37.340 --> 00:12:39.840
some variation of
black and white.
00:12:39.840 --> 00:12:44.010
If you look at this, you say
the contrast is just too big.
00:12:44.010 --> 00:12:46.890
You have big fluctuations
as you go across
00:12:46.890 --> 00:12:49.750
compared to what
I had over there.
00:12:49.750 --> 00:12:58.620
So there's another step, which
is called renormalize, which
00:12:58.620 --> 00:13:05.590
is that you define
m prime to be m
00:13:05.590 --> 00:13:09.040
by a change of
contrast factor zeta.
00:13:09.040 --> 00:13:13.240
So you take a knob that
corresponds to contrast
00:13:13.240 --> 00:13:21.480
and you reduce it
until you see pictures
00:13:21.480 --> 00:13:29.270
that kind of statistically look
like what you started with.
00:13:29.270 --> 00:13:36.910
So in order to sort of generate
pictures that are self-similar,
00:13:36.910 --> 00:13:38.980
you have this one knob.
00:13:38.980 --> 00:13:45.030
Basically, scaling variance
means the change of size.
00:13:45.030 --> 00:13:48.350
But there is associated
with change of size
00:13:48.350 --> 00:13:51.290
a change of contrast
for whatever variable
00:13:51.290 --> 00:13:52.660
you are looking at.
00:13:52.660 --> 00:13:55.500
It turns out that that
change of contrast
00:13:55.500 --> 00:13:58.140
would eventually map to
one of these exponents
00:13:58.140 --> 00:13:59.320
that we have over there.
00:13:59.320 --> 00:14:00.530
Yes.
00:14:00.530 --> 00:14:03.200
STUDENT: Are you
using m or n tilda?
00:14:03.200 --> 00:14:04.702
PROFESSOR: m tilda, thank you.
00:14:08.080 --> 00:14:14.530
So I guess the green
is m tilda of x prime,
00:14:14.530 --> 00:14:18.900
and the pink is m
prime of x prime.
00:14:30.560 --> 00:14:41.940
So what I have done
mathematically is as follows.
00:14:41.940 --> 00:14:47.230
I have defined an
m prime of x prime,
00:14:47.230 --> 00:14:51.740
which is 1 over zeta,
this contrast factor
00:14:51.740 --> 00:14:56.506
b to the d because of the
averaging over a volume
00:14:56.506 --> 00:15:06.230
that involved b to the d pixels
of the original field centered
00:15:06.230 --> 00:15:09.300
at a location that
was bx prime plus y.
00:15:13.750 --> 00:15:23.030
So in principle, I
can go and generate
00:15:23.030 --> 00:15:27.686
lots and lots of configurations
of my magnetization,
00:15:27.686 --> 00:15:32.480
or lots and lots of pictures
of a system at the liquid gas
00:15:32.480 --> 00:15:35.760
critical point, or
magnetic systems
00:15:35.760 --> 00:15:37.010
at their critical point.
00:15:37.010 --> 00:15:39.920
I can generate lots and
lots of these pictures
00:15:39.920 --> 00:15:42.410
and construct this
transformation.
00:15:45.080 --> 00:15:48.990
And associated with
this transformation
00:15:48.990 --> 00:15:52.520
is a change of
probability, because there
00:15:52.520 --> 00:15:57.200
was some probability--
let's call it P old,
00:15:57.200 --> 00:16:02.980
that was describing my
original configurations m of x.
00:16:02.980 --> 00:16:07.470
Let's forget the vector
notation for the time being.
00:16:07.470 --> 00:16:15.930
Then there will be, after this
transformation, probability
00:16:15.930 --> 00:16:18.212
that describes these
configurations m
00:16:18.212 --> 00:16:18.920
prime of x prime.
00:16:23.230 --> 00:16:26.730
Now you know that
averaging is not something
00:16:26.730 --> 00:16:29.080
that you can reverse.
00:16:29.080 --> 00:16:32.490
So this transformation
going from here and here,
00:16:32.490 --> 00:16:35.010
I cannot go back.
00:16:35.010 --> 00:16:39.270
There are many
configurations over here
00:16:39.270 --> 00:16:41.690
that would correspond
to the same average,
00:16:41.690 --> 00:16:43.810
like up, down or down,
up would give you
00:16:43.810 --> 00:16:45.900
the same average, right?
00:16:45.900 --> 00:16:50.420
So a number of
possibilities here
00:16:50.420 --> 00:16:54.340
have to be summed up to
generate for you this object.
00:16:59.100 --> 00:17:02.540
Now the statement of
self-similarity presumably
00:17:02.540 --> 00:17:08.380
is that this weight is
the same as this weight.
00:17:08.380 --> 00:17:12.170
You can't tell apart that
you generated configurations
00:17:12.170 --> 00:17:14.020
before or after that scaling.
00:17:16.970 --> 00:17:26.310
So this is same
at critical point.
00:17:35.610 --> 00:17:37.880
I've not constructed
either weight,
00:17:37.880 --> 00:17:42.420
so it really doesn't
amount to much.
00:17:42.420 --> 00:17:53.390
But Kadanoff
introduced this concept
00:17:53.390 --> 00:17:59.730
of doing this and thinking of
it as a kind of group operation
00:17:59.730 --> 00:18:03.200
called renormalization
group that I
00:18:03.200 --> 00:18:08.380
describe a little bit better
and evolve the description as we
00:18:08.380 --> 00:18:10.890
go along.
00:18:10.890 --> 00:18:19.610
So if I look at my
original system,
00:18:19.610 --> 00:18:22.560
I said that
self-similarity occurs,
00:18:22.560 --> 00:18:25.720
let's say, exactly at this
point that corresponds to t
00:18:25.720 --> 00:18:26.820
and h equals to 0.
00:18:35.090 --> 00:18:40.200
Now presumably, I
can, in some sense,
00:18:40.200 --> 00:18:45.240
force these things, if I were
to take its log, for example.
00:18:45.240 --> 00:18:49.110
I can construct some
kind of a weight that
00:18:49.110 --> 00:18:52.620
is associated with
m, and this would
00:18:52.620 --> 00:18:59.720
be a new weight that is
associated with m prime.
00:19:05.090 --> 00:19:07.750
Presumably, right at
the critical point,
00:19:07.750 --> 00:19:09.749
these two would be
the same weight,
00:19:09.749 --> 00:19:11.290
and it would be the
same Hamiltonian.
00:19:14.050 --> 00:19:16.980
What happens, if I
do this procedure,
00:19:16.980 --> 00:19:23.460
to a system that is initially
away from the critical point?
00:19:23.460 --> 00:19:28.690
So my initial system is
characterized by deviations t
00:19:28.690 --> 00:19:34.660
and h from this scale
in variant ways, which
00:19:34.660 --> 00:19:38.850
means that over here I
have a correlation length.
00:19:44.834 --> 00:19:46.750
Now I go through all of
these transformations.
00:19:49.460 --> 00:19:53.080
I can do those
transformations also
00:19:53.080 --> 00:19:56.880
for a point that is not
at the critical point.
00:19:56.880 --> 00:19:59.260
But at the end of
the day, I certainly
00:19:59.260 --> 00:20:02.020
will not get back
my original weight,
00:20:02.020 --> 00:20:05.420
because I look at the picture
after this transformation.
00:20:05.420 --> 00:20:09.300
Before the transformation, I
had a long correlation length,
00:20:09.300 --> 00:20:11.910
let's say a mile.
00:20:11.910 --> 00:20:15.640
When I do this transformation,
that correlation length
00:20:15.640 --> 00:20:19.600
is reduced by a factor of b.
00:20:19.600 --> 00:20:26.605
So the new system has deviated
more from the critical point.
00:20:30.080 --> 00:20:33.630
Because the further you go
away from the critical point,
00:20:33.630 --> 00:20:37.540
you have a larger
correlation length.
00:20:37.540 --> 00:20:42.320
So the idea is that right
at the critical point,
00:20:42.320 --> 00:20:45.180
the two weights are the same.
00:20:45.180 --> 00:20:47.350
Deviation from
the critical point
00:20:47.350 --> 00:20:52.430
is described by these
two parameters, t and h.
00:20:52.430 --> 00:20:56.890
And if you do the
renormalization procedure
00:20:56.890 --> 00:20:59.720
on a Hamiltonian
that deviates, you
00:20:59.720 --> 00:21:03.420
will get a Hamiltonian
that more deviates, still
00:21:03.420 --> 00:21:09.420
describable by parameters
t and h that have changed.
00:21:09.420 --> 00:21:16.045
So again, this says
that c was, in fact,
00:21:16.045 --> 00:21:23.380
b times c of t
prime and h prime,
00:21:23.380 --> 00:21:28.070
and t prime and h
prime are further away.
00:21:32.610 --> 00:21:39.870
Now the next thing that Kadanoff
said was, OK, therefore there
00:21:39.870 --> 00:21:43.820
is a transformation
that tells me
00:21:43.820 --> 00:21:48.400
after I do a rescaling
by a factor of b how
00:21:48.400 --> 00:21:55.390
the new t and the new h depend
on the old t and the old h.
00:22:03.060 --> 00:22:06.130
So there is a mapping
in this space.
00:22:06.130 --> 00:22:08.510
So a point that was
here will go over there.
00:22:08.510 --> 00:22:11.240
Maybe a point that is
here will map over there.
00:22:11.240 --> 00:22:14.200
A point that is here
will map over here.
00:22:14.200 --> 00:22:17.550
So there is a mapping
that tells you
00:22:17.550 --> 00:22:23.140
how th get transformed
under this procedure.
00:22:23.140 --> 00:22:29.765
Actually the reason this is
called a renormalization group,
00:22:29.765 --> 00:22:32.260
groups we are really
thinking usually
00:22:32.260 --> 00:22:36.520
in terms of operations
that are invertible.
00:22:36.520 --> 00:22:39.940
This transformation
is not invertible.
00:22:39.940 --> 00:22:41.110
But this is a mapping.
00:22:41.110 --> 00:22:44.090
So potentially this
mapping is invertible.
00:22:44.090 --> 00:22:48.220
You can say that if this
point came from this point
00:22:48.220 --> 00:22:51.781
under inversion, it will go
back to the original point,
00:22:51.781 --> 00:22:52.703
and so forth.
00:22:55.540 --> 00:23:02.740
The next part of the argument
is what did we do over here?
00:23:02.740 --> 00:23:07.230
We got rid of some short
wavelength fluctuations.
00:23:07.230 --> 00:23:10.740
Now one of the things that I
said right at the beginning
00:23:10.740 --> 00:23:14.510
was that as long as you are
getting rid of short scale
00:23:14.510 --> 00:23:18.720
fluctuations, you are summing
over a cube that his 100
00:23:18.720 --> 00:23:20.230
square, 200 cube.
00:23:20.230 --> 00:23:23.350
It doesn't matter,
100 cube, 200 cube--
00:23:23.350 --> 00:23:27.130
you are doing some
analytical function.
00:23:27.130 --> 00:23:30.087
So the transformation
that relates these
00:23:30.087 --> 00:23:34.460
to these, the old to new,
should be analytical,
00:23:34.460 --> 00:23:39.050
and hence you should be able to
write a Taylor series for it.
00:23:39.050 --> 00:23:42.690
So let's try to make a
Taylor series for this.
00:23:42.690 --> 00:23:45.480
Taylor series start
with a constant.
00:23:45.480 --> 00:23:49.150
But we know that the constant
has to be 0 in both cases
00:23:49.150 --> 00:23:52.070
because the starting
point was the point that
00:23:52.070 --> 00:23:56.350
was scale invariant and
was mapping onto itself.
00:23:56.350 --> 00:23:59.120
So the first thing that I can
write down are linear terms.
00:23:59.120 --> 00:24:04.870
So there could be a term
that is proportional to t.
00:24:04.870 --> 00:24:08.262
There could be a term
that is proportional to h.
00:24:08.262 --> 00:24:13.690
There could be a term here
that is proportional to h.
00:24:13.690 --> 00:24:16.550
There could be a term
that is proportional--
00:24:16.550 --> 00:24:18.880
well, let's call this t.
00:24:18.880 --> 00:24:21.810
Let's call this h.
00:24:21.810 --> 00:24:23.420
And then there
will be terms that
00:24:23.420 --> 00:24:26.150
will be order of t
squared and higher.
00:24:32.710 --> 00:24:36.610
So I just did an
analytical expansion,
00:24:36.610 --> 00:24:41.670
justified by this summing over
just finite degrees of freedom
00:24:41.670 --> 00:24:44.990
at short scale.
00:24:44.990 --> 00:24:47.440
Now if I have a structure,
such as the one that I
00:24:47.440 --> 00:24:51.460
have over there, I
also know some things
00:24:51.460 --> 00:24:53.590
on the basis of symmetry.
00:24:53.590 --> 00:24:59.540
Like if I'm on the line that
corresponds to h equals to 0,
00:24:59.540 --> 00:25:02.410
there is no difference
between up and down.
00:25:02.410 --> 00:25:04.910
Under rescaling, I still
don't know the difference
00:25:04.910 --> 00:25:06.620
between up and down.
00:25:06.620 --> 00:25:11.810
So I should not generate
an h if h was originally 0
00:25:11.810 --> 00:25:14.160
just because t deviated from 0.
00:25:14.160 --> 00:25:18.610
So by symmetry, that
has to be absent.
00:25:18.610 --> 00:25:20.740
And similarly, by
symmetry, there
00:25:20.740 --> 00:25:24.790
is no difference between
h positive and h negative.
00:25:24.790 --> 00:25:29.310
As far as t is concerned, h and
minus h should behave the same.
00:25:29.310 --> 00:25:32.530
So this series should start at
order of h squared and not h,
00:25:32.530 --> 00:25:34.330
so that term should be absent.
00:25:38.230 --> 00:25:44.990
So at this level, we have a nice
separation into t prime is at
00:25:44.990 --> 00:25:45.900
and h Prime.
00:25:45.900 --> 00:25:46.470
Is dh.
00:25:49.332 --> 00:25:53.080
Now we know
something more, which
00:25:53.080 --> 00:25:55.930
is that the procedure
that we are doing
00:25:55.930 --> 00:26:00.360
has some kind of a group
character, in that if I,
00:26:00.360 --> 00:26:06.180
let's say, originally
transform by some factor b1,
00:26:06.180 --> 00:26:14.170
change by a factor of 2,
then change by a factor of 3,
00:26:14.170 --> 00:26:18.580
the answer is equivalent to
changing by a factor of 2
00:26:18.580 --> 00:26:20.960
times 3, or 3 times 2.
00:26:20.960 --> 00:26:23.990
Doesn't matter in
which order I do them.
00:26:23.990 --> 00:26:28.010
So also, I would get,
if I were to do b1 first
00:26:28.010 --> 00:26:30.364
and b2 later, it would
be the same thing.
00:26:33.330 --> 00:26:35.590
So what does that imply?
00:26:35.590 --> 00:26:39.930
That if I do two of
these transformation,
00:26:39.930 --> 00:26:47.930
I find that my new t is obtained
in one case by the product,
00:26:47.930 --> 00:26:52.848
in the other case by the
product of the two a's.
00:26:58.116 --> 00:27:02.880
So that's, again, some
kind of a group character.
00:27:02.880 --> 00:27:08.100
And furthermore, if I don't
change the length scale,
00:27:08.100 --> 00:27:10.300
everything should
stay where it is.
00:27:13.500 --> 00:27:16.650
So you glance at
those, and you find
00:27:16.650 --> 00:27:19.390
that there is only
one possibility,
00:27:19.390 --> 00:27:23.350
that a as a function of b
should be b to some power.
00:27:30.090 --> 00:27:35.720
So you know therefore
that at the lowest
00:27:35.720 --> 00:27:40.670
order under rescaling
by a factor of b,
00:27:40.670 --> 00:27:43.670
t prime should be b
to some y-- I called
00:27:43.670 --> 00:27:48.490
it yt-- times t
plus higher orders,
00:27:48.490 --> 00:27:54.770
while h prime is b to some other
power of yh times h plus higher
00:27:54.770 --> 00:27:55.270
orders.
00:28:00.940 --> 00:28:04.976
And you say, OK, fine.
00:28:08.640 --> 00:28:09.730
What's this good for?
00:28:12.300 --> 00:28:18.460
Well, let's take a look
at what we did over there.
00:28:18.460 --> 00:28:22.360
We said that I take some bunch
of initial configurations,
00:28:22.360 --> 00:28:24.450
sum their weights
to get the weight
00:28:24.450 --> 00:28:27.530
of the new configuration.
00:28:27.530 --> 00:28:31.970
What happens if I sum over
all initial configurations?
00:28:31.970 --> 00:28:35.860
Well, if I sum over all
initial configuration,
00:28:35.860 --> 00:28:40.560
I will get the
partition function.
00:28:45.330 --> 00:28:47.880
Now essentially,
all the original
00:28:47.880 --> 00:28:52.630
configurations I
regrouped and put
00:28:52.630 --> 00:29:00.530
into these coarse-grained
configurations that
00:29:00.530 --> 00:29:01.760
are weighted this way.
00:29:04.670 --> 00:29:06.780
So there could be
an overall constant
00:29:06.780 --> 00:29:09.250
that emerges from this.
00:29:09.250 --> 00:29:16.330
But this really implies that
the singular part of log z,
00:29:16.330 --> 00:29:19.790
and presumably this
depends on how far
00:29:19.790 --> 00:29:24.660
away I am from the
critical point,
00:29:24.660 --> 00:29:32.060
is the same as log z that
singular after I do this t
00:29:32.060 --> 00:29:32.957
prime and h prime.
00:29:41.730 --> 00:29:44.073
Now there is one other
issue, which is extensivity.
00:29:46.620 --> 00:29:49.470
Up to signs, factors
of beta, et cetera,
00:29:49.470 --> 00:29:55.520
this is b times an
intensive free energy,
00:29:55.520 --> 00:29:57.260
which is a function of t and h.
00:30:01.880 --> 00:30:07.200
So this is the same as v prime,
because the volume shrunk.
00:30:07.200 --> 00:30:11.610
I took all of my scales and
shrunk it by a factor of v,
00:30:11.610 --> 00:30:15.717
v prime, f of t
prime and h prime.
00:30:21.460 --> 00:30:25.080
So now let's go this way.
00:30:25.080 --> 00:30:30.300
Note that v prime is
the original v divided
00:30:30.300 --> 00:30:35.450
by b to the d scaling factor.
00:30:35.450 --> 00:30:37.570
So you do the
divisions here, and you
00:30:37.570 --> 00:30:43.630
find that f as a
function of t and h
00:30:43.630 --> 00:30:48.736
is the ratio of v prime to
v, which is b to the minus d,
00:30:48.736 --> 00:30:52.530
f as a function of
t prime and h prime.
00:30:52.530 --> 00:30:56.720
But t prime we said to lowest
order is b to the yt t.
00:30:56.720 --> 00:30:59.140
h prime is b to the yh h.
00:31:08.100 --> 00:31:10.520
This is actually the
more correct form
00:31:10.520 --> 00:31:12.250
of writing a
homogeneous function.
00:31:19.950 --> 00:31:23.390
So previously in
last lecture, we
00:31:23.390 --> 00:31:29.350
assumed that the free energy
had a homogeneous form.
00:31:29.350 --> 00:31:32.900
Now subject to these
conditions and assumptions
00:31:32.900 --> 00:31:35.340
of renormalization
group, we have
00:31:35.340 --> 00:31:39.850
concluded that it should
have that homogeneous form.
00:31:39.850 --> 00:31:42.880
Now you say this
homogeneous form does not
00:31:42.880 --> 00:31:44.770
look like the
homogeneous forms that I
00:31:44.770 --> 00:31:48.310
had written for you before.
00:31:48.310 --> 00:31:50.080
I say, OK.
00:31:50.080 --> 00:31:53.350
Presumably this is true
for any factor of b
00:31:53.350 --> 00:31:56.540
that I want to choose.
00:31:56.540 --> 00:32:09.820
Let me choose a b, a rescaling
factor such that v to the yt t
00:32:09.820 --> 00:32:11.940
is of the order of 1.
00:32:11.940 --> 00:32:13.320
Could be 1, could be pi.
00:32:13.320 --> 00:32:14.020
I don't care.
00:32:16.660 --> 00:32:19.050
Which means that
I chose a factor
00:32:19.050 --> 00:32:24.816
of b that will scale with t
as t to the minus 1 over yt.
00:32:28.008 --> 00:32:34.540
I put this b-- this expression
is true for all choices of b.
00:32:34.540 --> 00:32:39.750
If I chose that particular
value, what I get
00:32:39.750 --> 00:32:46.570
is t to the d over
yt, some function.
00:32:46.570 --> 00:32:50.080
First argument has now
become 1 or some constant.
00:32:50.080 --> 00:32:53.820
Really it only depends
on the second argument
00:32:53.820 --> 00:33:00.520
in the combination h and t
to the power of yh over yt.
00:33:06.200 --> 00:33:10.090
So you can see that
this is, in fact,
00:33:10.090 --> 00:33:13.650
the same as the first
line that I have above.
00:33:13.650 --> 00:33:17.730
And I have identified
that 2 minus alpha
00:33:17.730 --> 00:33:22.130
would be related to
this factor of yt, which
00:33:22.130 --> 00:33:27.230
is how you would scale
under renormalization,
00:33:27.230 --> 00:33:30.700
the parameters t and h.
00:33:30.700 --> 00:33:34.935
And the gap exponent is related
to the ratio of yh over yt.
00:33:45.350 --> 00:33:49.657
Similarly, we had that
the correlation length--
00:33:49.657 --> 00:33:50.490
I have a line there.
00:33:50.490 --> 00:33:56.860
Psi of t and h is b psi
of t prime and h prime.
00:33:56.860 --> 00:34:02.706
So I have that psi as
a function of t and h
00:34:02.706 --> 00:34:10.750
is b times psi as a function of
b to the yt t, b to the yh h.
00:34:10.750 --> 00:34:16.520
So that's also correct.
00:34:16.520 --> 00:34:24.389
I can again choose this value
of v, substitute it over there.
00:34:24.389 --> 00:34:25.699
What do I get?
00:34:25.699 --> 00:34:29.239
I get that psi as a
function of t and h
00:34:29.239 --> 00:34:34.159
would be t to the
minus 1 over yt,
00:34:34.159 --> 00:34:40.139
some scaling function-- let's
call it g psi-- of, again, h
00:34:40.139 --> 00:34:41.810
to the power of yh over yt.
00:34:48.969 --> 00:34:56.100
So I have got an answer
that nu should be 1 over yt.
00:34:56.100 --> 00:35:00.800
I can get the scaling form
for the correlation length.
00:35:00.800 --> 00:35:04.170
I identify the divergence
of correlation length
00:35:04.170 --> 00:35:05.680
with inverse of this.
00:35:05.680 --> 00:35:10.140
And by the way, I get, if I
substitute nu as 1 over yt
00:35:10.140 --> 00:35:14.260
here, the Josephson hyperscale
in relation to minus alpha
00:35:14.260 --> 00:35:14.860
equals to b.
00:35:27.540 --> 00:35:31.040
I can go further if I want.
00:35:31.040 --> 00:35:36.440
I can calculate magnetization
as a function of t and h,
00:35:36.440 --> 00:35:38.880
would correspond to
basically the behaviors
00:35:38.880 --> 00:35:46.710
that we identify with exponents
beta or delta as d log z.
00:35:55.074 --> 00:36:01.500
Yeah, let's say f by dh.
00:36:01.500 --> 00:36:05.180
If I take a
derivative over there,
00:36:05.180 --> 00:36:07.000
you can immediately
see that what
00:36:07.000 --> 00:36:13.510
that gives me is b to the
power of yh minus d, and then
00:36:13.510 --> 00:36:16.800
some scaling function which is
the derivative of this scaling
00:36:16.800 --> 00:36:22.510
function b the yt
t, b to the yh h.
00:36:25.030 --> 00:36:29.090
And again, if I
make this choice,
00:36:29.090 --> 00:36:39.980
then this goes over to t to the
power of d minus yh over yt,
00:36:39.980 --> 00:36:44.755
and then some scaling
function of h t to the delta.
00:36:47.790 --> 00:36:50.240
So I can continue with my table.
00:36:50.240 --> 00:36:52.833
And for example,
I will have beta
00:36:52.833 --> 00:36:58.980
to be d minus yh divided by yt.
00:36:58.980 --> 00:37:01.420
I can go and calculate
delta, et cetera.
00:37:06.700 --> 00:37:11.730
Actually I was a
little bit careless
00:37:11.730 --> 00:37:18.770
with this factor
zeta, which presumably
00:37:18.770 --> 00:37:24.270
is implicit in all of these
transformations that I have.
00:37:24.270 --> 00:37:26.480
And I have to do special
things to figure out
00:37:26.480 --> 00:37:30.180
what zeta is so that I will
get self-similarity right
00:37:30.180 --> 00:37:32.780
at the critical point.
00:37:32.780 --> 00:37:36.202
But we can see that
already we have
00:37:36.202 --> 00:37:41.970
the analog of a rescaling for m.
00:37:41.970 --> 00:37:47.530
And so it is easy to sort of
look at those two equations
00:37:47.530 --> 00:37:54.420
and identify that my zeta
should be precisely this one.
00:37:54.420 --> 00:38:01.560
So the zeta is not
independent of the relevance
00:38:01.560 --> 00:38:04.220
of the magnetic field.
00:38:04.220 --> 00:38:09.850
And if you think about it, the
field and the magnetization
00:38:09.850 --> 00:38:14.710
are conjugate variables in the
sense that in the weight here,
00:38:14.710 --> 00:38:20.360
I will have a term that is like
hm-- integrated, of course.
00:38:20.360 --> 00:38:23.150
And so hm integrated,
you can see
00:38:23.150 --> 00:38:27.830
that up to a factor of b
to the d from integration,
00:38:27.830 --> 00:38:31.930
the dimensionality that I assign
to h and the dimensionality
00:38:31.930 --> 00:38:37.700
that I assign to m
should be related.
00:38:37.700 --> 00:38:40.770
And not only for
the magnetization,
00:38:40.770 --> 00:38:45.145
but for any pair of variables
that are so conjugate--
00:38:45.145 --> 00:38:47.970
there's some f, and
there's some x--
00:38:47.970 --> 00:38:51.760
there will be a corresponding
relation between what
00:38:51.760 --> 00:38:58.740
would happen to this x at the
critical point and this factor
00:38:58.740 --> 00:39:01.501
f when I deviate from
the critical point.
00:39:11.730 --> 00:39:17.800
So all of this is
kind of nice, but it's
00:39:17.800 --> 00:39:20.310
a little bit hand waving.
00:39:20.310 --> 00:39:30.450
I essentially traded one set of
assumptions about homogeneity
00:39:30.450 --> 00:39:34.345
and scaling of free
energy correlation length
00:39:34.345 --> 00:39:40.380
to some other set of assumptions
about two parameters moving
00:39:40.380 --> 00:39:45.040
away from a scale
invariant critical point.
00:39:45.040 --> 00:39:47.990
I didn't calculate anything
about what the scale invariant
00:39:47.990 --> 00:39:49.640
probability is.
00:39:49.640 --> 00:39:53.830
I didn't show that, indeed,
two parameters are sufficient,
00:39:53.830 --> 00:39:57.740
that this kind of scaling
takes place, et cetera.
00:39:57.740 --> 00:40:01.750
So we need to be
much more precise
00:40:01.750 --> 00:40:05.740
if we want to do, ultimately,
calculations that give us
00:40:05.740 --> 00:40:09.530
what these numbers
yt and yh are.
00:40:09.530 --> 00:40:13.770
So let's try to put
this hand waving
00:40:13.770 --> 00:40:15.720
on a little bit
more firm setting.
00:40:20.220 --> 00:40:24.600
So let's see how
we should proceed.
00:40:27.620 --> 00:40:37.040
We start with some experimental
system, critical point.
00:40:40.410 --> 00:40:46.420
So I tell you that somebody in
the experiment, the liquid gas
00:40:46.420 --> 00:40:51.000
system, they saw a diverging
correlation length,
00:40:51.000 --> 00:40:53.600
critical opalescence, et cetera.
00:40:53.600 --> 00:40:59.920
So then I associate
with that some kind
00:40:59.920 --> 00:41:02.390
of a statistical field.
00:41:07.080 --> 00:41:11.160
And let's kind of
stick with the notation
00:41:11.160 --> 00:41:12.520
that we have for the magnet.
00:41:12.520 --> 00:41:14.622
Let's call it m of x.
00:41:14.622 --> 00:41:18.800
And in general, this
would be the part
00:41:18.800 --> 00:41:21.900
where one needs to put
in a lot of thinking.
00:41:21.900 --> 00:41:24.830
That is, the experimentalist
comes and tells you
00:41:24.830 --> 00:41:29.260
that I see a system that
undergoes a phase transition.
00:41:29.260 --> 00:41:30.790
There are some
response functions
00:41:30.790 --> 00:41:32.970
that are divergent, et cetera.
00:41:32.970 --> 00:41:36.110
You have to put in some
thinking to think about what
00:41:36.110 --> 00:41:39.600
the appropriate
order parameter is.
00:41:39.600 --> 00:41:47.510
And based on that order
parameter or statistical field,
00:41:47.510 --> 00:42:00.610
you construct the most
general weight consistent
00:42:00.610 --> 00:42:06.840
with symmetries, with
not only asymmetries
00:42:06.840 --> 00:42:12.300
but the kind of assumptions that
we have been putting in play.
00:42:12.300 --> 00:42:18.790
So we put in assumptions
about locality, symmetry.
00:42:22.190 --> 00:42:24.040
Stability is, of
course, paramount.
00:42:24.040 --> 00:42:28.120
But there is a list of things
that you have to think about.
00:42:28.120 --> 00:42:32.770
So once you do
that you say, OK, I
00:42:32.770 --> 00:42:38.602
associate with my
configurations m of x some set
00:42:38.602 --> 00:42:39.310
of probabilities.
00:42:43.650 --> 00:42:46.390
Probabilities are
certainly positive.
00:42:46.390 --> 00:42:50.830
So I can take its log, call its
minus its log to be some kind
00:42:50.830 --> 00:42:56.325
of a weight, beta h, that
governs these m of x's.
00:42:59.440 --> 00:43:03.010
If I say that I'm
obeying locality,
00:43:03.010 --> 00:43:05.820
then I would write the answer,
for example, like this.
00:43:05.820 --> 00:43:07.830
But it doesn't have to be.
00:43:07.830 --> 00:43:11.010
I have to write some
particular example.
00:43:11.010 --> 00:43:13.530
But you may construct
your example
00:43:13.530 --> 00:43:16.060
depending on the
system of interest.
00:43:16.060 --> 00:43:19.750
And let's say we are looking
at something like a superfluid,
00:43:19.750 --> 00:43:23.210
maybe, that we don't even have
the analog of magnetic field,
00:43:23.210 --> 00:43:25.060
and we go and
construct terms that
00:43:25.060 --> 00:43:29.880
are symmetric and made
for a two component m.
00:43:29.880 --> 00:43:32.470
And I will write a
few of these terms
00:43:32.470 --> 00:43:36.590
to emphasize that this is,
in principle, a long list.
00:43:36.590 --> 00:43:40.290
There's coefficient
of m to the sixth.
00:43:40.290 --> 00:43:44.440
We saw that the gradient
terms could start with this k.
00:43:44.440 --> 00:43:47.670
But maybe there's a
higher order gradient,
00:43:47.670 --> 00:43:53.080
and there's essentially
and infinity of terms
00:43:53.080 --> 00:43:56.920
that you can write down that
are consistent with these
00:43:56.920 --> 00:43:59.064
assumptions that you
have made so far.
00:44:03.020 --> 00:44:05.940
So you say, OK.
00:44:05.940 --> 00:44:12.080
Now I take this, and implicit
in all of these calculations
00:44:12.080 --> 00:44:16.470
is, indeed, some kind
of a short scale cutoff.
00:44:16.470 --> 00:44:19.770
To construct the
statistical field,
00:44:19.770 --> 00:44:29.440
I do apply the three steps of
RG-- renormalization group,
00:44:29.440 --> 00:44:31.660
as I described before.
00:44:31.660 --> 00:44:38.130
And this will give me
a new configuration
00:44:38.130 --> 00:44:40.570
for each of the
old configurations
00:44:40.570 --> 00:44:43.205
through the formula that
I gave you over there.
00:44:55.500 --> 00:44:59.740
So in principle, this is just
a transformation from one
00:44:59.740 --> 00:45:03.630
set of variables to a
new set of variables.
00:45:03.630 --> 00:45:05.810
So if I do this
transformation, I
00:45:05.810 --> 00:45:13.206
can calculate the weight
of the new configurations,
00:45:13.206 --> 00:45:14.930
m prime of x prime.
00:45:18.350 --> 00:45:20.830
I can take minus
the log of that.
00:45:23.420 --> 00:45:27.510
And again, up to
some constant, it
00:45:27.510 --> 00:45:30.090
will be the same
as a probability.
00:45:30.090 --> 00:45:32.010
So there could be,
in this procedure,
00:45:32.010 --> 00:45:34.780
some set of constants
that are generated
00:45:34.780 --> 00:45:36.730
that don't depend on m.
00:45:39.800 --> 00:45:41.370
And then there
will be a function
00:45:41.370 --> 00:45:45.700
that depends on m
prime of x prime.
00:45:45.700 --> 00:45:48.220
Now the statement
is that since I
00:45:48.220 --> 00:45:57.090
wrote the most general function
over here, whatever I put here
00:45:57.090 --> 00:46:01.340
will have to have exactly
the same form, because I said
00:46:01.340 --> 00:46:03.420
put anything that
you can think of that
00:46:03.420 --> 00:46:06.120
is consistent with
symmetries over here.
00:46:06.120 --> 00:46:08.240
So you put everything there.
00:46:08.240 --> 00:46:12.120
What I put here should have
exactly the same functional
00:46:12.120 --> 00:46:15.080
form, but with coefficients
that have changed.
00:46:28.550 --> 00:46:32.120
So you basically
prime everything,
00:46:32.120 --> 00:46:36.380
but you have this whole thing.
00:46:36.380 --> 00:46:40.050
Now this may seem like
truly difficult thing.
00:46:40.050 --> 00:46:42.170
But we will actually do this.
00:46:42.170 --> 00:46:46.410
We will carry out this
transformation explicitly
00:46:46.410 --> 00:46:48.450
in particular cases.
00:46:48.450 --> 00:46:51.920
And we will show that this
transformation amounts
00:46:51.920 --> 00:46:56.240
to constructing a rescaling of
each one of these parameters--
00:46:56.240 --> 00:47:05.010
t prime, u prime, v prime, k
prime, l prime, and so forth--
00:47:05.010 --> 00:47:07.987
as functions of
the old parameters.
00:47:24.590 --> 00:47:28.730
So this is, if you
like, a mapping.
00:47:32.360 --> 00:47:41.040
You take some set of
parameters-- t, u, v, k, l,
00:47:41.040 --> 00:47:45.270
blah, blah, blah-- and
you construct a mapping,
00:47:45.270 --> 00:47:51.877
s prime, which is some
function of the original set
00:47:51.877 --> 00:47:52.460
of parameters.
00:47:56.090 --> 00:47:59.840
So this is a huge
dimensional space.
00:47:59.840 --> 00:48:02.960
Any points that you start
on the transformation
00:48:02.960 --> 00:48:05.210
will go to another point.
00:48:05.210 --> 00:48:08.750
But the key is that we
wrote the most general form
00:48:08.750 --> 00:48:13.406
that we could, so we had
to stay within this space.
00:48:22.770 --> 00:48:26.430
So why are you doing this?
00:48:26.430 --> 00:48:33.170
Well, I started by saying that
the key to this whole thing
00:48:33.170 --> 00:48:39.710
is have to having a handle as
to what this self-similar scale
00:48:39.710 --> 00:48:43.210
invariant probability is.
00:48:43.210 --> 00:48:47.470
I can't construct
that just by guessing.
00:48:47.470 --> 00:48:49.750
But I can do what we
usually do, let's say,
00:48:49.750 --> 00:48:53.070
in constructing wave functions
in quantum mechanics that
00:48:53.070 --> 00:48:55.410
have some particular symmetry.
00:48:55.410 --> 00:48:57.880
Maybe you start with
some wave function that
00:48:57.880 --> 00:48:59.830
doesn't have the full
symmetry, and then
00:48:59.830 --> 00:49:02.050
you rotate it and
rotate it again,
00:49:02.050 --> 00:49:03.880
and you average
over all of them,
00:49:03.880 --> 00:49:05.710
and you end up with
some function that
00:49:05.710 --> 00:49:07.830
has the right symmetry.
00:49:07.830 --> 00:49:11.940
So we start with a
weight that I don't
00:49:11.940 --> 00:49:16.930
know whether it has the
property that I want.
00:49:16.930 --> 00:49:19.530
And I apply the action
of the group, which
00:49:19.530 --> 00:49:22.020
is this scaling
variance, to see what
00:49:22.020 --> 00:49:23.780
happens to it under
that transformation.
00:49:26.290 --> 00:49:29.060
But the point that
I am interested,
00:49:29.060 --> 00:49:31.940
or the behavior that
I am interested,
00:49:31.940 --> 00:49:37.030
is where I basically get
the same probability back.
00:49:37.030 --> 00:49:41.250
So I'm very interested
at the point where,
00:49:41.250 --> 00:49:45.770
under the transformation,
I go back to myself.
00:49:45.770 --> 00:49:47.360
And that's called a fixed point.
00:49:52.670 --> 00:49:59.530
So S is a shorthand for this
infinite vector of parameters.
00:49:59.530 --> 00:50:05.200
I want to find the point s
star in this parameter space.
00:50:05.200 --> 00:50:10.051
Actually, let me call
this transformation R
00:50:10.051 --> 00:50:14.990
and indicate that I'm
renormalizing by a scale b,
00:50:14.990 --> 00:50:19.760
such that, when I
renormalize by a scale
00:50:19.760 --> 00:50:23.390
b, my original
set of parameters,
00:50:23.390 --> 00:50:29.440
if I am at this fixed point,
I will end up at that point.
00:50:29.440 --> 00:50:32.790
So clearly, this
is a system that
00:50:32.790 --> 00:50:38.160
has exactly these properties
that I was harping in
00:50:38.160 --> 00:50:39.150
at the beginning.
00:50:39.150 --> 00:50:42.440
This is the point that
is truly scale invariant.
00:50:42.440 --> 00:50:47.000
That's the point that
I want to get at.
00:50:47.000 --> 00:50:49.436
So again, once we have
done this transformation
00:50:49.436 --> 00:50:55.340
in a specific case, we'll figure
out what this fixed point is.
00:50:55.340 --> 00:51:00.680
But for the time being,
let's think a little bit away
00:51:00.680 --> 00:51:11.500
from this and deviate
from fixed point.
00:51:17.460 --> 00:51:22.530
So I start with
an initial point S
00:51:22.530 --> 00:51:30.690
that is, let's write it, S
star plus a little bit away.
00:51:30.690 --> 00:51:32.820
Just like in the
picture that I have
00:51:32.820 --> 00:51:34.450
here, I started
with a fixed point,
00:51:34.450 --> 00:51:36.330
and I said I go
away by an amount
00:51:36.330 --> 00:51:39.360
that I had parameterized
by t and h.
00:51:39.360 --> 00:51:42.700
Now I have essentially
a whole line
00:51:42.700 --> 00:51:46.070
of deviations forming a vector.
00:51:46.070 --> 00:51:55.710
I act with Rb on this, and I
note that if delta S goes to 0,
00:51:55.710 --> 00:51:57.700
then I should go back to S star.
00:52:00.880 --> 00:52:08.460
But if delta S is
small, maybe I can
00:52:08.460 --> 00:52:14.000
look at the delta
S prime, which is
00:52:14.000 --> 00:52:16.800
a linearized version of
these transformations.
00:52:16.800 --> 00:52:19.770
So basically these
transformations
00:52:19.770 --> 00:52:26.140
are highly nonlinear just as
the transformation over here,
00:52:26.140 --> 00:52:29.680
in principle, would have
been highly nonlinear.
00:52:29.680 --> 00:52:34.940
But then I expanded it around
the point t and h equals to 0.
00:52:34.940 --> 00:52:39.870
Similarly, I'm assuming
that this delta S is small,
00:52:39.870 --> 00:52:47.300
and therefore delta S prime
can be related to delta S
00:52:47.300 --> 00:52:52.772
through the action of a matrix
that is a linearized version.
00:52:52.772 --> 00:52:54.770
Let's call it here RL of b.
00:52:54.770 --> 00:53:06.630
So this is a linearized
transformation,
00:53:06.630 --> 00:53:08.544
which means that
it's really a matrix.
00:53:12.260 --> 00:53:16.730
In this particular case, in
principle, I started with a 2
00:53:16.730 --> 00:53:18.150
by 2 matrix.
00:53:18.150 --> 00:53:20.570
The off diagonal
terms were 0, so it
00:53:20.570 --> 00:53:23.360
was only the diagonal
terms that mattered.
00:53:23.360 --> 00:53:26.600
But in general, it
would be a matrix,
00:53:26.600 --> 00:53:28.610
which would be the
square of whatever
00:53:28.610 --> 00:53:31.632
the size of the parameter
space is that I am looking at.
00:53:36.350 --> 00:53:41.590
Now then you have
a matrix, it's good
00:53:41.590 --> 00:53:46.280
always to think in terms of its
eigenvalues and eigendirection.
00:53:46.280 --> 00:53:48.620
In this problem that
I had over here,
00:53:48.620 --> 00:53:51.540
symmetries had already
diagonalized the matrix.
00:53:51.540 --> 00:53:53.730
I didn't have off
diagonal terms.
00:53:53.730 --> 00:53:54.860
But I don't know here.
00:53:54.860 --> 00:53:57.880
It could be all kinds
of off diagonal terms.
00:53:57.880 --> 00:54:08.740
So the properties are
captured by diagonalize, RL,
00:54:08.740 --> 00:54:18.120
which means that I find a set
of vectors in this space--
00:54:18.120 --> 00:54:23.680
let's call them Oi-- such
that under action of this,
00:54:23.680 --> 00:54:29.070
I will get lambda Oi, lambda i.
00:54:29.070 --> 00:54:31.770
Of course, the transformation
depends on the rescaling
00:54:31.770 --> 00:54:34.856
parameter, so there
should be a b here.
00:54:39.620 --> 00:54:42.600
Now of course, you will get
a totally different matrix
00:54:42.600 --> 00:54:45.310
for each b.
00:54:45.310 --> 00:54:48.790
So is it really hopeless that
for each b I have to look
00:54:48.790 --> 00:54:54.700
at a new matrix, new
diagonalization, et cetera?
00:54:54.700 --> 00:54:58.270
Well, exactly this thing
that we had over here
00:54:58.270 --> 00:55:01.230
now comes into
play, because I know
00:55:01.230 --> 00:55:06.980
that if I make a
transformation size b1 followed
00:55:06.980 --> 00:55:11.880
by a transformation size b2,
the answer is a transformation
00:55:11.880 --> 00:55:13.750
size b1, b2.
00:55:13.750 --> 00:55:16.290
And it doesn't matter
in which order I do it.
00:55:20.138 --> 00:55:23.790
AUDIENCE: Can't you just mix
notation, because L used to be
00:55:23.790 --> 00:55:24.337
[INAUDIBLE]?
00:55:24.337 --> 00:55:25.045
PROFESSOR: Sorry.
00:55:46.200 --> 00:55:54.800
So in particular, I see
that these linearized
00:55:54.800 --> 00:55:59.500
matrices commute with each
other for different values of b.
00:55:59.500 --> 00:56:01.590
And again, from your
quantum mechanics,
00:56:01.590 --> 00:56:05.410
you probably know that
if matrices commute then
00:56:05.410 --> 00:56:08.880
they have the same eigenvectors.
00:56:08.880 --> 00:56:13.295
So essentially, I was correct
here in putting no index b
00:56:13.295 --> 00:56:15.660
on these eigenvectors,
because it's
00:56:15.660 --> 00:56:18.750
independent of eigenvector,
whereas the eigenvalues,
00:56:18.750 --> 00:56:21.964
in principle, depend on b.
00:56:21.964 --> 00:56:25.000
And how they depend on
b is also determined
00:56:25.000 --> 00:56:32.500
by this transformation, that is
lambda i of b1, lambda i of b2
00:56:32.500 --> 00:56:35.880
should be the same
thing as lambda b1, b2.
00:56:39.610 --> 00:56:44.490
And of course, lambda
i of 1 should be 1.
00:56:44.490 --> 00:56:48.065
If you don't change scale,
nothing should change.
00:56:48.065 --> 00:56:51.190
And this is exactly the
same set of conditions
00:56:51.190 --> 00:56:54.110
as we have over
here, which means
00:56:54.110 --> 00:56:59.875
that we know that the
eigenvalue's lambda i can
00:56:59.875 --> 00:57:04.460
be written as b to the
power of some set of yi.
00:57:08.330 --> 00:57:11.970
So we just generalized
what we had done before,
00:57:11.970 --> 00:57:15.430
now to this space that
includes many parameters.
00:57:24.940 --> 00:57:28.510
So the story is now
something like this.
00:57:28.510 --> 00:57:32.090
There is this
multi-dimensional space
00:57:32.090 --> 00:57:37.730
with lots and lots
of parameters--
00:57:37.730 --> 00:57:41.710
t, u, v, blah, blah,
blah, many of them.
00:57:41.710 --> 00:57:46.095
And somewhere in this space
of parameters, presumably
00:57:46.095 --> 00:57:49.460
there is a fixed point, S star.
00:57:55.110 --> 00:57:58.810
Now in the vicinity
of that S star,
00:57:58.810 --> 00:58:03.710
I have established that there
are some particular directions
00:58:03.710 --> 00:58:07.015
that I can obtain by
diagonalizing this.
00:58:07.015 --> 00:58:10.090
So let's imagine that
this is one direction,
00:58:10.090 --> 00:58:13.460
this is another direction,
this is a third direction.
00:58:16.630 --> 00:58:24.480
And that if I start
with a beta h-- well,
00:58:24.480 --> 00:58:26.085
actually, let's do this.
00:58:26.085 --> 00:58:34.430
That is, if I start with an S
that is S star plus whatever
00:58:34.430 --> 00:58:38.640
is a projection of
my components are
00:58:38.640 --> 00:58:42.480
along these different
dimensions, so let's call them,
00:58:42.480 --> 00:58:47.480
let's say, ai along
these Oi hat-- just
00:58:47.480 --> 00:58:51.940
make sure we kind of
think of them as vectors--
00:58:51.940 --> 00:58:55.520
that under rescaling,
I will go to S
00:58:55.520 --> 00:59:03.560
prime, which is S star plus
sum over i, ai b to the yi Oi.
00:59:08.240 --> 00:59:12.140
That is, some of
these directions,
00:59:12.140 --> 00:59:15.215
the component will get
stretched if yi is positive.
00:59:15.215 --> 00:59:19.980
It will get diminished
if yi is negative.
00:59:19.980 --> 00:59:25.480
And so now some notation
comes into play.
00:59:51.970 --> 00:59:58.570
If yi is positive, the
corresponding direction
00:59:58.570 --> 00:59:59.804
is called relevant.
01:00:04.890 --> 01:00:15.115
Eigendirection is relevant.
01:00:20.190 --> 01:00:27.390
If yi is negative, the
corresponding eigendirection
01:00:27.390 --> 01:00:28.362
is irrelevant.
01:00:34.270 --> 01:00:41.770
And very occasionally,
we may run into the case
01:00:41.770 --> 01:00:43.940
where yi is 0.
01:00:43.940 --> 01:00:45.590
And there is a terminology.
01:00:45.590 --> 01:00:48.680
The corresponding
eigendirection is marginal.
01:00:51.980 --> 01:00:57.250
And what that means is that I
need to resort to higher order
01:00:57.250 --> 01:01:00.840
terms to see whether it
is attracted or repelled
01:01:00.840 --> 01:01:02.586
by the fixed point.
01:01:02.586 --> 01:01:04.580
So we need higher orders.
01:01:07.180 --> 01:01:11.105
After all, so far I have only
linearized the transformation.
01:01:18.720 --> 01:01:42.967
Now the set of irrelevant
directions to this particular S
01:01:42.967 --> 01:01:51.426
fixed point, S star, defines
basing of attraction of S star.
01:02:06.010 --> 01:02:11.030
So let me go back to the
picture that I have over here
01:02:11.030 --> 01:02:18.640
and be precise and use the arrow
going away as an indication
01:02:18.640 --> 01:02:21.810
that the corresponding
b is positive,
01:02:21.810 --> 01:02:25.330
and I'm forced out
along this direction.
01:02:25.330 --> 01:02:31.640
Let me choose going
in as an indicator
01:02:31.640 --> 01:02:34.920
that the corresponding
y is negative.
01:02:34.920 --> 01:02:38.260
And as I make b
larger and larger,
01:02:38.260 --> 01:02:41.230
I shrink along this axis.
01:02:41.230 --> 01:02:44.190
So in this three
dimensional representation
01:02:44.190 --> 01:02:49.200
that I have over there, I have
one relevant direction and two
01:02:49.200 --> 01:02:51.180
irrelevant directions.
01:02:51.180 --> 01:02:55.730
The two irrelevant directions
will define the plane
01:02:55.730 --> 01:02:58.810
in this three
dimensional space, which
01:02:58.810 --> 01:03:02.070
is the basing of attraction.
01:03:02.070 --> 01:03:09.460
So basically these
two define a surface,
01:03:09.460 --> 01:03:15.400
and presumably any point that
is in this surface in the three
01:03:15.400 --> 01:03:18.910
dimensional picture
under looking
01:03:18.910 --> 01:03:21.960
at larger and larger
things will get
01:03:21.960 --> 01:03:23.940
attracted to the fixed point.
01:03:23.940 --> 01:03:25.760
If you are away
from the surface,
01:03:25.760 --> 01:03:27.820
maybe you will approach
here, and then you
01:03:27.820 --> 01:03:28.830
will be pushed out.
01:03:32.910 --> 01:03:36.680
All right, fine.
01:03:36.680 --> 01:03:40.150
Now let's go and look
at the following.
01:03:40.150 --> 01:03:44.300
We have a formula,
psi of t and h.
01:03:44.300 --> 01:03:47.740
Or quite generally,
psi under rescaling
01:03:47.740 --> 01:03:52.190
is b times the new psi.
01:03:52.190 --> 01:03:56.200
Or the new psi under any one
of these transformation, psi
01:03:56.200 --> 01:04:01.500
prime, is the old psi divided
by the old correlation
01:04:01.500 --> 01:04:03.605
length divided by a factor of b.
01:04:07.400 --> 01:04:11.100
So if I look at
the fixed point--
01:04:11.100 --> 01:04:16.140
so if I ask what is psi
at the fixed point--
01:04:16.140 --> 01:04:22.890
then under the transformation,
I have the same parameters.
01:04:22.890 --> 01:04:24.660
So psi at the fixed
point should be
01:04:24.660 --> 01:04:27.990
the psi of the fixed
point divided by b.
01:04:27.990 --> 01:04:30.870
There are only two
solutions to this.
01:04:30.870 --> 01:04:38.896
Either psi of S star is 0 or
psi of S star is infinite.
01:04:45.530 --> 01:04:48.020
Now we introduce physics.
01:04:48.020 --> 01:04:51.860
Psi being 0 means
that I have units
01:04:51.860 --> 01:04:54.860
that are completely
uncorrelated to each other.
01:04:54.860 --> 01:04:58.010
Each one of them does
whatever it wants.
01:04:58.010 --> 01:05:01.750
So this describes
essentially, let's say,
01:05:01.750 --> 01:05:03.610
a system of infinite
temperature.
01:05:03.610 --> 01:05:06.095
Every degree of freedom
does whatever it wants.
01:05:10.520 --> 01:05:18.310
Well, I should say this
corresponds to disordered
01:05:18.310 --> 01:05:22.390
or ordered phases.
01:05:25.420 --> 01:05:29.070
Because after all,
we said that when
01:05:29.070 --> 01:05:32.210
we go to the
ordered states also,
01:05:32.210 --> 01:05:34.590
there is an overall
magnetization,
01:05:34.590 --> 01:05:38.550
but fluctuations around
the overall magnetization
01:05:38.550 --> 01:05:41.520
have only a finite
correlation length.
01:05:41.520 --> 01:05:44.780
And as you go further and
further into the ordered phase,
01:05:44.780 --> 01:05:47.510
that correlation
length shrinks to 0.
01:05:47.510 --> 01:05:50.400
So there is a similarity
between what goes on
01:05:50.400 --> 01:05:52.730
at very high
temperature and what
01:05:52.730 --> 01:05:55.050
goes on at very
low temperature as
01:05:55.050 --> 01:05:58.340
far as the correlation of
fluctuations is concerned.
01:05:58.340 --> 01:06:00.290
There is, of course,
a long range order
01:06:00.290 --> 01:06:02.880
in one case that is
absent in the other.
01:06:02.880 --> 01:06:05.630
But the correlation
of fluctuations
01:06:05.630 --> 01:06:08.650
in both of those cases
basically becomes
01:06:08.650 --> 01:06:13.920
finite, and under rescaling,
goes all the way to 0.
01:06:13.920 --> 01:06:17.000
And clearly this is
the interesting case,
01:06:17.000 --> 01:06:19.615
where it corresponds
to critical point.
01:06:28.090 --> 01:06:37.150
So we've established that,
once we found this fixed point,
01:06:37.150 --> 01:06:44.180
that those set of parameters
are what can give us the scale
01:06:44.180 --> 01:06:45.960
invariant behavioral
that we want.
01:06:49.580 --> 01:06:56.040
Now this list is
hundreds of parameters.
01:06:56.040 --> 01:07:00.140
So this point corresponds
to a very special point
01:07:00.140 --> 01:07:04.150
in this hundreds
of parameter space.
01:07:04.150 --> 01:07:06.660
So let's say there is one
point somewhere there which
01:07:06.660 --> 01:07:08.980
is the fixed point.
01:07:08.980 --> 01:07:11.170
And then you take
your magnet and you
01:07:11.170 --> 01:07:13.630
change your critical
temperature,
01:07:13.630 --> 01:07:15.710
are we going to hit that point?
01:07:15.710 --> 01:07:16.880
The answer is, no.
01:07:16.880 --> 01:07:20.000
Generically, you are not
going to hit that point.
01:07:20.000 --> 01:07:22.010
But that's no problem.
01:07:22.010 --> 01:07:22.760
Why?
01:07:22.760 --> 01:07:24.395
Because if this
basing of attraction.
01:07:27.340 --> 01:07:44.410
Because for any point on basing
of attraction, I do rescaling,
01:07:44.410 --> 01:07:48.665
and I find that psi
prime is psi over b.
01:07:48.665 --> 01:07:50.140
It becomes smaller.
01:07:50.140 --> 01:07:54.960
So you generically
tend to become smaller.
01:07:54.960 --> 01:07:57.140
But ultimately, you
end up at this point.
01:07:57.140 --> 01:08:01.730
And this point, the
correlation length is infinite.
01:08:01.730 --> 01:08:06.540
So any point on this basing
of attraction, in fact,
01:08:06.540 --> 01:08:09.840
has infinite correlation length.
01:08:09.840 --> 01:08:15.950
So every point on the basis
of psi prime equals to psi,
01:08:15.950 --> 01:08:20.680
and hence psi has
to be infinite.
01:08:25.670 --> 01:08:26.423
Yes.
01:08:26.423 --> 01:08:27.370
AUDIENCE: Question.
01:08:27.370 --> 01:08:31.590
Why should there be
only one fixed point?
01:08:31.590 --> 01:08:32.906
PROFESSOR: There is no reason.
01:08:32.906 --> 01:08:33.899
AUDIENCE: OK.
01:08:33.899 --> 01:08:35.123
So this is just an example?
01:08:35.123 --> 01:08:35.789
PROFESSOR: Yeah.
01:08:35.789 --> 01:08:39.310
So locally, let's say that
we found such a fixed point.
01:08:39.310 --> 01:08:41.819
Maybe globally, there
is hundreds of them.
01:08:41.819 --> 01:08:42.990
I don't know.
01:08:42.990 --> 01:08:45.815
So that will always be
a question in our minds.
01:08:45.815 --> 01:08:50.520
So if I just write down for
you the most general set
01:08:50.520 --> 01:08:54.170
of transformations, who
knows what's happening?
01:08:54.170 --> 01:08:57.890
Ultimately, we have to
be guided by physics.
01:08:57.890 --> 01:09:02.560
We have to say that, if in the
space of all parametrization,
01:09:02.560 --> 01:09:05.300
there are some that have
no physical correspondence,
01:09:05.300 --> 01:09:07.710
we throw them out,
we seek things
01:09:07.710 --> 01:09:09.830
that can be matched to
our physical system.
01:09:15.330 --> 01:09:16.224
Yes?
01:09:16.224 --> 01:09:18.099
AUDIENCE: If there are
multiple fixed points,
01:09:18.099 --> 01:09:21.013
do the planes of the
basing of attraction
01:09:21.013 --> 01:09:22.429
have to be parallel
to each other?
01:09:26.160 --> 01:09:28.750
PROFESSOR: They may have
to have some conditions
01:09:28.750 --> 01:09:31.380
on non-intersecting or whatever.
01:09:31.380 --> 01:09:35.710
These are only linear in the
vicinity of the fixed point.
01:09:35.710 --> 01:09:39.210
So in principle, they could
be highly curved surfaces
01:09:39.210 --> 01:09:44.569
with all kinds of structures
and things that I don't know.
01:09:44.569 --> 01:09:46.004
Yes?
01:09:46.004 --> 01:09:50.260
AUDIENCE: Is there any reason
why you might or might not
01:09:50.260 --> 01:09:54.940
have attracting point
that is actually
01:09:54.940 --> 01:09:57.630
a more complicated
structure, like say,
01:09:57.630 --> 01:10:00.534
a limit cycle or
even a [INAUDIBLE]?
01:10:00.534 --> 01:10:01.200
PROFESSOR: Yeah.
01:10:01.200 --> 01:10:06.950
So again, we are governed
ultimately by physics.
01:10:06.950 --> 01:10:09.230
When I write these
equations, they
01:10:09.230 --> 01:10:13.310
are as general as equations as
the people in dynamical systems
01:10:13.310 --> 01:10:19.270
use that also includes
cycles, chaotic attractors,
01:10:19.270 --> 01:10:21.740
all kinds of strange things.
01:10:21.740 --> 01:10:27.450
And we have to hope that
when we apply this procedure
01:10:27.450 --> 01:10:30.450
to an appropriate
physical system,
01:10:30.450 --> 01:10:33.015
the kind of
equations that we get
01:10:33.015 --> 01:10:37.870
are such that their behavior
is indicative of the physics.
01:10:37.870 --> 01:10:43.960
So there is one case I know
where people sort of found
01:10:43.960 --> 01:10:47.190
chaotic renormalization
group trajectories
01:10:47.190 --> 01:10:49.640
for some kind of a
[INAUDIBLE] system.
01:10:49.640 --> 01:10:53.900
But always, again, this is
a very general procedure.
01:10:53.900 --> 01:10:57.900
We have to limit
mathematics, ultimately,
01:10:57.900 --> 01:11:00.450
by what the physical process is.
01:11:00.450 --> 01:11:03.880
So it's good that you know
that these equations can
01:11:03.880 --> 01:11:06.140
do all kinds of strange things.
01:11:06.140 --> 01:11:09.510
But then we take a
particular physical system,
01:11:09.510 --> 01:11:12.904
we have to beat on them
until they behave properly.
01:11:17.380 --> 01:11:23.270
So let's imagine that we have
a situation, such as this,
01:11:23.270 --> 01:11:27.330
where we have three parameters.
01:11:27.330 --> 01:11:28.880
Two of them are irrelevant.
01:11:28.880 --> 01:11:31.700
One of them is relevant.
01:11:31.700 --> 01:11:35.020
Then presumably, I
take my physical system
01:11:35.020 --> 01:11:37.810
at some temperature
and it would correspond
01:11:37.810 --> 01:11:41.220
to being on some point
in this phase diagram.
01:11:44.290 --> 01:11:47.070
Some color that we don't have.
01:11:47.070 --> 01:11:48.890
Let's say over here.
01:11:48.890 --> 01:11:51.030
And I change the temperature.
01:11:51.030 --> 01:11:56.490
And I will take
some trajectory--
01:11:56.490 --> 01:11:59.440
in this case, three
dimensional space.
01:11:59.440 --> 01:12:04.850
And this is a line in this
three dimensional space.
01:12:04.850 --> 01:12:08.750
And experimentally, I've
been told that if I take,
01:12:08.750 --> 01:12:12.170
let's say, my piece of iron
and I change temperature,
01:12:12.170 --> 01:12:18.270
at some point I go
through a point that
01:12:18.270 --> 01:12:20.300
has infinite correlations.
01:12:20.300 --> 01:12:25.990
So I have to conclude that
my trajectory for iron
01:12:25.990 --> 01:12:31.670
will intersect with
surface at some point.
01:12:31.670 --> 01:12:34.210
And I'll say, OK, I take nickel.
01:12:34.210 --> 01:12:37.430
Nickel would be something else.
01:12:37.430 --> 01:12:39.570
And I change
temperature of nickel,
01:12:39.570 --> 01:12:43.070
and I will be doing something
completely different.
01:12:43.070 --> 01:12:46.300
But that experimentalist
also has a point
01:12:46.300 --> 01:12:48.490
where you have
ferromagnetic transition,
01:12:48.490 --> 01:12:51.210
so it must hit this surface.
01:12:51.210 --> 01:12:54.960
Then you do cobalt, where
some other trajectory
01:12:54.960 --> 01:12:58.590
comes and hits off the surface.
01:12:58.590 --> 01:13:02.510
Now what we now
know is that when
01:13:02.510 --> 01:13:06.760
we rescale the system
sufficiently, all of them
01:13:06.760 --> 01:13:11.030
ultimately are described
at the point where
01:13:11.030 --> 01:13:13.225
they have infinite
correlation length by what
01:13:13.225 --> 01:13:15.810
is going on over here.
01:13:15.810 --> 01:13:18.930
So if I take iron,
nickel, cobalt, clearly
01:13:18.930 --> 01:13:20.570
at the level of
atoms and molecules,
01:13:20.570 --> 01:13:24.550
they are very different
from each other.
01:13:24.550 --> 01:13:28.170
And the difference between
ironness, nickelness,
01:13:28.170 --> 01:13:34.090
cobaltness is really in all of
these irrelevant parameters.
01:13:34.090 --> 01:13:36.810
And as I go and look at
larger and larger scale,
01:13:36.810 --> 01:13:40.170
they all diminish and go away.
01:13:40.170 --> 01:13:42.970
And at large scale,
I see the same thing,
01:13:42.970 --> 01:13:47.610
where all of the individual
details has been washed out.
01:13:47.610 --> 01:13:51.650
So this is able to capture
the idea of universality.
01:13:54.610 --> 01:13:56.860
But there is a very
important caveat
01:13:56.860 --> 01:14:02.970
to this, which is that
the experimental system,
01:14:02.970 --> 01:14:05.920
whether you take iron or
cobalt or some mixture
01:14:05.920 --> 01:14:09.940
of these different elements,
you change one parameter
01:14:09.940 --> 01:14:14.610
temperature, and you always see
a transition from, let's say,
01:14:14.610 --> 01:14:16.720
paramagnetic to
ferromagnetic behavior.
01:14:19.240 --> 01:14:24.640
Now if I have, say, a line
here in three dimensional space
01:14:24.640 --> 01:14:26.770
and I draw another
line that corresponds
01:14:26.770 --> 01:14:30.220
to change in temperature,
I will not intersect it.
01:14:30.220 --> 01:14:36.200
I have to do something very
special to intersect that line.
01:14:36.200 --> 01:14:41.180
So in order that genetically I
have a phase transition-- which
01:14:41.180 --> 01:14:44.960
is what my experimentalist
friends tell me--
01:14:44.960 --> 01:14:51.260
I know that I can only have
one relevant direction,
01:14:51.260 --> 01:14:55.880
because the dimensionality
of the basing of attraction
01:14:55.880 --> 01:15:00.200
is the dimensionality of
the space minus however
01:15:00.200 --> 01:15:02.570
many relevant directions I have.
01:15:02.570 --> 01:15:04.790
And I've been told
by experimentalists
01:15:04.790 --> 01:15:07.290
that they exchange
one parameters,
01:15:07.290 --> 01:15:10.310
and generically they
hit the surface.
01:15:10.310 --> 01:15:12.220
So that's part of the story.
01:15:12.220 --> 01:15:16.490
I better find a theory that,
at the end of the day, when
01:15:16.490 --> 01:15:20.790
I do all of this, I
find a fixed point that
01:15:20.790 --> 01:15:24.320
not only is well-behaved
and is not a limit cycle,
01:15:24.320 --> 01:15:28.470
but also a fixed point
that has one and only one
01:15:28.470 --> 01:15:31.316
relevant direction, if
that's the physical system
01:15:31.316 --> 01:15:34.460
that I'm describing.
01:15:34.460 --> 01:15:37.400
Now of course, maybe that
was for the superfluid, where
01:15:37.400 --> 01:15:39.360
they could only
change temperature,
01:15:39.360 --> 01:15:42.630
and you have a situation where
the magnet comes into play
01:15:42.630 --> 01:15:46.270
and they say, oh, actually we
also have the magnetic field.
01:15:46.270 --> 01:15:50.810
And we really have to go
to the space of zero field.
01:15:50.810 --> 01:15:54.800
And then if I expand
my space of parameters
01:15:54.800 --> 01:15:59.720
here to include terms
that break the symmetry,
01:15:59.720 --> 01:16:02.360
in that generalized
space, I should only
01:16:02.360 --> 01:16:05.870
have two relevant directions.
01:16:05.870 --> 01:16:10.480
So it is kind of strange story,
that all we are doing here
01:16:10.480 --> 01:16:11.650
is mathematics.
01:16:11.650 --> 01:16:13.300
But at the end of
the day, we have
01:16:13.300 --> 01:16:16.925
to get the mathematics to have
very specific properties that
01:16:16.925 --> 01:16:20.250
are dictated by very rough
things about experiments.
01:16:23.770 --> 01:16:28.650
So this was kind of
conceptually rich.
01:16:28.650 --> 01:16:31.770
So I'll let you digest
that for a while.
01:16:31.770 --> 01:16:37.140
And next lecture, we will start
actually doing this procedure
01:16:37.140 --> 01:16:40.060
and finding these kinds
of [INAUDIBLE] relations.