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PROFESSOR: So today, I'd
like to wrap together
00:00:25.490 --> 00:00:29.390
and summarize everything that
we have been doing the last 10,
00:00:29.390 --> 00:00:31.520
12 lectures.
00:00:31.520 --> 00:00:37.320
So the idea started by saying
that you take something
00:00:37.320 --> 00:00:41.450
like a magnet, you
change its temperature.
00:00:41.450 --> 00:00:44.080
You go from one phase
that is paramagnet
00:00:44.080 --> 00:00:49.182
at some critical temperature
Tc to some other phase that
00:00:49.182 --> 00:00:50.430
is a ferromagnet.
00:00:50.430 --> 00:00:52.570
Naturally, the
direction of magnetism
00:00:52.570 --> 00:00:57.520
depends whether you put on a
magnetic field and went to 0.
00:00:57.520 --> 00:01:00.239
So there's a lot of
these transitions
00:01:00.239 --> 00:01:01.280
that involve ferromagnet.
00:01:05.530 --> 00:01:09.570
There were a set of other
transitions that involved,
00:01:09.570 --> 00:01:11.823
for example, superfluids
or superconductivity.
00:01:15.530 --> 00:01:19.820
And the most common
example of phase
00:01:19.820 --> 00:01:23.990
transition being
liquid gas, which
00:01:23.990 --> 00:01:26.390
has a coexistence line
that also terminates
00:01:26.390 --> 00:01:28.160
at the critical point.
00:01:28.160 --> 00:01:30.820
So you have to turn
it around a little bit
00:01:30.820 --> 00:01:36.200
to get a coexistence
line like this.
00:01:36.200 --> 00:01:37.260
Fine.
00:01:37.260 --> 00:01:38.870
So there are phase transitions.
00:01:38.870 --> 00:01:44.160
The interesting thing was that
when people did successively
00:01:44.160 --> 00:01:46.120
better and better
experiments, they
00:01:46.120 --> 00:01:48.810
found that the singularities
in the vicinity
00:01:48.810 --> 00:01:54.910
of these critical
points are universal.
00:01:54.910 --> 00:01:58.150
That is, it doesn't matter
whether you have iron, nickel,
00:01:58.150 --> 00:02:01.410
or some other thing that is
undergoing ferromagnetism.
00:02:01.410 --> 00:02:05.690
You can characterize the
divergence of the heat capacity
00:02:05.690 --> 00:02:12.180
to an exponent alpha, which
ferromagnets is minus 0.12.
00:02:12.180 --> 00:02:17.770
For superfluids, we said
that people even take things
00:02:17.770 --> 00:02:21.670
on satellites to calculate
this exponent to much
00:02:21.670 --> 00:02:24.240
higher accuracy than
I have indicated.
00:02:24.240 --> 00:02:27.710
For the case of liquid gas,
there is a true divergence
00:02:27.710 --> 00:02:33.790
and the exponent is around 0.11.
00:02:33.790 --> 00:02:35.810
There are a whole set
of other exponents
00:02:35.810 --> 00:02:37.505
that I also mentioned.
00:02:37.505 --> 00:02:45.050
There is the exponent beta for
how the magnetization vanishes.
00:02:45.050 --> 00:02:52.106
And the values here
were 0.37, 0.35, 0.33.
00:02:52.106 --> 00:02:58.130
There is the divergence of
the susceptibility gamma that
00:02:58.130 --> 00:03:01.240
is characterized
through exponents
00:03:01.240 --> 00:03:10.950
that are 1.39, 1.32, 1.24.
00:03:10.950 --> 00:03:14.900
And there is a divergence
of the correlation length
00:03:14.900 --> 00:03:26.050
characterized by exponents
[INAUDIBLE] 0.71, 0.67, 0.63.
00:03:26.050 --> 00:03:32.400
So there is this
table of pure numbers
00:03:32.400 --> 00:03:36.590
that don't depend on the
property of the material
00:03:36.590 --> 00:03:38.250
that you are looking at.
00:03:38.250 --> 00:03:45.240
And the fact that you don't
have this material dependence
00:03:45.240 --> 00:03:49.800
suggests that these pure
numbers are some characteristics
00:03:49.800 --> 00:03:52.900
of the collective
behavior that gives rise
00:03:52.900 --> 00:03:56.730
to what's happening at
this critical point.
00:03:56.730 --> 00:04:00.380
And we should be able to
devise some kind of a theory
00:04:00.380 --> 00:04:02.600
to understand
that, maybe extract
00:04:02.600 --> 00:04:07.970
these nice, pure numbers,
which are certainly embedded
00:04:07.970 --> 00:04:12.420
in the physics of the problem
that we are looking at.
00:04:12.420 --> 00:04:19.220
So the first idea that
we explored conceptually
00:04:19.220 --> 00:04:24.760
due to lambda probably,
probably others,
00:04:24.760 --> 00:04:28.205
is that we should construct
the statistical field.
00:04:32.426 --> 00:04:37.530
That is, what is
happening is irrespective
00:04:37.530 --> 00:04:41.070
of whether we are dealing with
nickel or iron, et cetera.
00:04:41.070 --> 00:04:43.690
So the properties of
the microscopic elements
00:04:43.690 --> 00:04:45.560
should disappear
and we should be
00:04:45.560 --> 00:04:50.280
focusing on the quantity that is
undergoing a phase transition.
00:04:50.280 --> 00:04:55.130
And that quantity, we said, is
some kind of a magnetization.
00:04:55.130 --> 00:04:59.330
And what distinguishes
the different systems is
00:04:59.330 --> 00:05:01.340
that for ferromagnet,
it's certainly
00:05:01.340 --> 00:05:03.160
a three-component system.
00:05:03.160 --> 00:05:07.190
But in general, for superfluid
we would have a phase.
00:05:07.190 --> 00:05:09.135
And that's a
two-component object,
00:05:09.135 --> 00:05:12.550
so we introduced
this parameter n
00:05:12.550 --> 00:05:17.300
that characterized the symmetry
of the order parameter.
00:05:17.300 --> 00:05:20.490
And in the same way,
we said let's look
00:05:20.490 --> 00:05:24.150
at things that are
embedded in space.
00:05:24.150 --> 00:05:27.410
That is, in general,
d dimensional.
00:05:27.410 --> 00:05:34.430
So our specification of
the statistical field
00:05:34.430 --> 00:05:37.830
was on the basis
of these things.
00:05:37.830 --> 00:05:42.930
And the idea of lambda was
to construct a probability
00:05:42.930 --> 00:05:47.380
for this field,
configurations across space.
00:05:47.380 --> 00:05:52.530
Once we had that, we could
calculate a partition function,
00:05:52.530 --> 00:05:55.735
let's say, by integrating
over all configurations
00:05:55.735 --> 00:05:59.860
of this field of some
kind of a weight.
00:06:05.800 --> 00:06:10.540
And we constructed the weight.
00:06:10.540 --> 00:06:18.460
We wrote something as beta
H was an integral d dx.
00:06:18.460 --> 00:06:21.800
And then we put a whole
bunch of things in here.
00:06:21.800 --> 00:06:26.630
We said we could have something
like t over 2m squared.
00:06:26.630 --> 00:06:30.530
We had gradient of m squared.
00:06:30.530 --> 00:06:33.860
Potentially, we could
have higher derivatives
00:06:33.860 --> 00:06:37.850
staying at the
order of m squared.
00:06:37.850 --> 00:06:39.630
And so this list
of things that I
00:06:39.630 --> 00:06:45.090
could put that are all order of
m squared is quite extensive.
00:06:45.090 --> 00:06:48.710
Then I could have things
that are fourth order,
00:06:48.710 --> 00:06:51.940
like u m to the fourth.
00:06:51.940 --> 00:06:56.250
And we saw that when we
performed this renormalization
00:06:56.250 --> 00:07:01.520
group last time around, that a
term that we typically had not
00:07:01.520 --> 00:07:04.050
paid attention to was generated.
00:07:04.050 --> 00:07:06.430
Something that was,
again, order of m
00:07:06.430 --> 00:07:11.440
to the fourth, but had
a structure maybe like m
00:07:11.440 --> 00:07:14.870
squared gradient of m
squared, or some other form
00:07:14.870 --> 00:07:18.040
of the two-derivative operator.
00:07:18.040 --> 00:07:20.190
This is OK.
00:07:20.190 --> 00:07:21.880
There could be other
types of things
00:07:21.880 --> 00:07:24.490
that have four m's in it.
00:07:24.490 --> 00:07:29.350
And you could have something
that is m to the sixth, m
00:07:29.350 --> 00:07:30.500
to the eighth, et cetera.
00:07:34.060 --> 00:07:42.650
So the idea of lambda was to
include all kinds of terms
00:07:42.650 --> 00:07:44.760
that you can put.
00:07:44.760 --> 00:07:49.200
But actually, we have
already constrained terms.
00:07:49.200 --> 00:07:53.830
So the idea of
lambda is all terms
00:07:53.830 --> 00:07:59.220
consistent with some
constraints that you put.
00:07:59.220 --> 00:08:02.690
What are the
constraint that we put?
00:08:02.690 --> 00:08:10.480
We put locality in that we wrote
this as an integral over x.
00:08:10.480 --> 00:08:14.040
We considered symmetry.
00:08:14.040 --> 00:08:17.770
So if I am at the 0
field limit, I only
00:08:17.770 --> 00:08:21.670
have terms that are proportional
to m squared, rotationally
00:08:21.670 --> 00:08:22.170
symmetric.
00:08:28.490 --> 00:08:33.200
And there is something
else that is implicit,
00:08:33.200 --> 00:08:34.330
which is analyticity.
00:08:40.890 --> 00:08:42.369
What do I mean?
00:08:42.369 --> 00:08:44.910
I mean that there is,
in some sense here,
00:08:44.910 --> 00:08:49.210
a space of parameters
composed of all
00:08:49.210 --> 00:08:58.150
of these coefficients--
t, k, u, v.
00:08:58.150 --> 00:09:00.520
And these are
supposed to represent
00:09:00.520 --> 00:09:05.390
what is happening to my
system that I have in mind
00:09:05.390 --> 00:09:08.730
as I change the temperature.
00:09:08.730 --> 00:09:13.650
And so in principle, if I
do some averaging procedure
00:09:13.650 --> 00:09:18.200
and arrive at this description,
all of these parameters
00:09:18.200 --> 00:09:21.100
presumably will be
functions of temperature.
00:09:25.260 --> 00:09:29.190
And the statement
is that the process
00:09:29.190 --> 00:09:31.780
of coarse gaining the
degrees of freedom
00:09:31.780 --> 00:09:35.380
and averaging to arrive
at this description
00:09:35.380 --> 00:09:38.210
and the corresponding
parameters involves
00:09:38.210 --> 00:09:40.570
finite number of
degrees of freedom.
00:09:40.570 --> 00:09:43.360
And adding and
integrating finite numbers
00:09:43.360 --> 00:09:48.410
of degrees of freedom can only
lead to analytical functions.
00:09:48.410 --> 00:09:51.980
So the statement here is
that this set of parameters
00:09:51.980 --> 00:09:53.555
are analytical functions.
00:09:56.630 --> 00:10:03.210
So given this
construction by lambda,
00:10:03.210 --> 00:10:07.300
we should be able to figure
out if this is correct,
00:10:07.300 --> 00:10:10.760
what is happening and where
these numbers come from.
00:10:10.760 --> 00:10:13.450
So what did we attempt?
00:10:13.450 --> 00:10:15.400
The first thing
that we attempted
00:10:15.400 --> 00:10:16.585
was to do saddle point.
00:10:19.910 --> 00:10:23.570
And we saw that
doing so just looking
00:10:23.570 --> 00:10:27.850
at the most probable state
fails because fluctuations
00:10:27.850 --> 00:10:30.750
were important.
00:10:30.750 --> 00:10:35.760
We tried to break
the Hamiltonian
00:10:35.760 --> 00:10:39.740
into a part that was
quadratic and Gaussian.
00:10:39.740 --> 00:10:46.930
And we could calculate
everything about it,
00:10:46.930 --> 00:10:53.820
and then treating everybody
else as a perturbation.
00:10:53.820 --> 00:10:57.030
And when we
attempted to do that,
00:10:57.030 --> 00:10:59.550
we found that
perturbation theory
00:10:59.550 --> 00:11:00.870
failed below four dimensions.
00:11:10.660 --> 00:11:15.570
So at this stage, it
was kind of an impasse
00:11:15.570 --> 00:11:19.310
in that as far as
physics is concerned,
00:11:19.310 --> 00:11:22.620
we feel that this thing
captures all of the properties
00:11:22.620 --> 00:11:24.750
that you need in
order to somehow
00:11:24.750 --> 00:11:27.520
be able to explain
those phenomena.
00:11:27.520 --> 00:11:31.000
Yet, we don't have
the mathematical power
00:11:31.000 --> 00:11:36.950
to carry out the integrations
that are implicit in this.
00:11:36.950 --> 00:11:42.060
So then the idea was, can
we go around it somehow?
00:11:42.060 --> 00:11:46.450
And so the next set of
things that we introduced
00:11:46.450 --> 00:11:50.920
were basically
versions of scaling.
00:11:50.920 --> 00:11:55.025
And quite a few
statistical physicists
00:11:55.025 --> 00:11:56.840
were involved with that.
00:11:56.840 --> 00:12:03.950
Names such as [INAUDIBLE],
Fisher, and number of others.
00:12:03.950 --> 00:12:10.510
And the idea is that
if we also consider,
00:12:10.510 --> 00:12:17.610
let's say, introducing a
magnetic field direction here
00:12:17.610 --> 00:12:21.670
and look at the singularities
in the plane that involves
00:12:21.670 --> 00:12:25.640
those two that is necessary
to also characterize
00:12:25.640 --> 00:12:29.300
some of these other
things such as gamma,
00:12:29.300 --> 00:12:34.480
then you have a singular
part for the free energy that
00:12:34.480 --> 00:12:38.340
is a function of how
far you go away from Tc.
00:12:38.340 --> 00:12:42.820
So this t now stands
for T minus Tc.
00:12:42.820 --> 00:12:48.070
And how far you go along the
direction that breaks symmetry.
00:12:48.070 --> 00:12:51.920
And the statement was
that all of the results
00:12:51.920 --> 00:12:59.170
were consistent with the form
that depended on really two
00:12:59.170 --> 00:13:04.020
exponents that could
be bonded together
00:13:04.020 --> 00:13:05.650
into the behavior
of the singular
00:13:05.650 --> 00:13:08.630
part of the free
energy or the singular
00:13:08.630 --> 00:13:11.110
part of the
correlation function.
00:13:11.110 --> 00:13:17.520
And essentially, this
approach immediately
00:13:17.520 --> 00:13:20.550
leads to exponent identities.
00:13:23.570 --> 00:13:26.510
And these exponent identities
we can go back and look
00:13:26.510 --> 00:13:31.890
at the table of numbers
that we have up there.
00:13:31.890 --> 00:13:36.035
And we see that they
are correct and valid.
00:13:38.940 --> 00:13:44.060
But well, what are the
two primary exponents?
00:13:44.060 --> 00:13:46.820
How can we obtain them?
00:13:46.820 --> 00:13:51.830
Well, going and looking at this
scaling behavior a little bit
00:13:51.830 --> 00:14:01.840
further, one could trace back to
some kind of a self-similarity
00:14:01.840 --> 00:14:05.890
that should exist right
at the critical point.
00:14:05.890 --> 00:14:08.280
That is, the
correlation functions,
00:14:08.280 --> 00:14:11.010
et cetera at the
critical point should
00:14:11.010 --> 00:14:14.150
have this kind of
scaling variance.
00:14:14.150 --> 00:14:19.700
And then the question
is, can we somehow manage
00:14:19.700 --> 00:14:22.470
to use that property,
that looking at things
00:14:22.470 --> 00:14:25.210
at different scales at
the critical points,
00:14:25.210 --> 00:14:28.470
should give you the
same thing to divine
00:14:28.470 --> 00:14:30.850
what these exponents are.
00:14:30.850 --> 00:14:34.230
So the next stage
in this progression
00:14:34.230 --> 00:14:45.678
was the work of Kadanoff in
introducing the idea of RG.
00:14:45.678 --> 00:14:52.130
And the idea of RG was to
basically average things
00:14:52.130 --> 00:14:53.976
further.
00:14:53.976 --> 00:14:56.910
Here, implicit in
the calculation
00:14:56.910 --> 00:15:02.440
that we had was some kind of a
short distance [INAUDIBLE] a.
00:15:02.440 --> 00:15:07.070
And if we average
between a and b a,
00:15:07.070 --> 00:15:12.390
then presumably these parameters
mu would change to something
00:15:12.390 --> 00:15:15.520
else-- mu prime--
that correspond
00:15:15.520 --> 00:15:18.970
to rescaling by a factor of b.
00:15:18.970 --> 00:15:22.410
And these mu primes
would be a function
00:15:22.410 --> 00:15:27.770
of the original set
of parameters mu.
00:15:27.770 --> 00:15:32.870
And then Kadanoff's idea was
that the scaling variant points
00:15:32.870 --> 00:15:37.400
would correspond to the points
where you have no changed.
00:15:44.250 --> 00:15:51.840
And that if you then
deviated from that point,
00:15:51.840 --> 00:15:54.175
you would have some
characteristic scale
00:15:54.175 --> 00:15:55.830
in the problem.
00:15:55.830 --> 00:15:58.930
And you could capture
what was happening
00:15:58.930 --> 00:16:07.370
by looking at essentially how
the changes, delta mu prime,
00:16:07.370 --> 00:16:10.980
were related to the
changes delta mu.
00:16:10.980 --> 00:16:15.270
So essentially, linearizing
these relationships.
00:16:15.270 --> 00:16:17.753
So there would be some kind of
a linearized transformation.
00:16:20.380 --> 00:16:26.990
And then the eigenvalues
of this transformation
00:16:26.990 --> 00:16:31.230
would determine how many
relevant quantities you should
00:16:31.230 --> 00:16:33.580
have.
00:16:33.580 --> 00:16:36.350
Now, the physics, the entire
physics of the process,
00:16:36.350 --> 00:16:38.610
then comes into play here.
00:16:38.610 --> 00:16:43.130
That the experiments
tell us that you can,
00:16:43.130 --> 00:16:48.050
let's say, take superfluid and
it has this phase transition.
00:16:48.050 --> 00:16:50.510
You change the pressure of
it, it still has that phase
00:16:50.510 --> 00:16:52.570
transition-- slightly
different temperature,
00:16:52.570 --> 00:16:54.650
but it's the same
phase transition.
00:16:54.650 --> 00:16:56.710
You can add some
impurities to it
00:16:56.710 --> 00:16:58.650
as you did in one
of the problems.
00:16:58.650 --> 00:17:01.920
You still have the
phase transition.
00:17:01.920 --> 00:17:06.430
So basically, the existence
of a phase transition
00:17:06.430 --> 00:17:10.930
as a function of one parameter
that is temperature-like
00:17:10.930 --> 00:17:13.270
is pretty robust.
00:17:13.270 --> 00:17:17.099
And if we think about that in
the language of fixed point,
00:17:17.099 --> 00:17:20.730
it meant that along
the symmetry direction,
00:17:20.730 --> 00:17:25.020
there should be only
one relevant eigenvalue.
00:17:25.020 --> 00:17:29.550
So this construction
that Kadanoff proposed
00:17:29.550 --> 00:17:34.200
is nice and fine, but
one has to demonstrate
00:17:34.200 --> 00:17:40.700
that, indeed, this infinite
number of parameters
00:17:40.700 --> 00:17:43.100
can be boiled down
to a fixed point.
00:17:43.100 --> 00:17:46.200
And that fixed point has
only one relevant direction.
00:17:49.230 --> 00:17:54.050
So the next step
in this progression
00:17:54.050 --> 00:18:01.570
was Wilson who did perturbative
version of this procedure.
00:18:08.690 --> 00:18:17.650
So the idea was that we can
certainly solve beta H0.
00:18:17.650 --> 00:18:23.020
And beta H0 is really a bunch
of Gaussian independent modes
00:18:23.020 --> 00:18:26.690
as long as we look at
things in Fourier space.
00:18:26.690 --> 00:18:29.460
So in Fourier space,
we have a bunch
00:18:29.460 --> 00:18:33.630
of modes that exist over
some [INAUDIBLE] zone.
00:18:33.630 --> 00:18:39.910
And as long as we are looking
at some set of wavelengths
00:18:39.910 --> 00:18:43.360
and no fluctuation shorter
than that wavelength has been
00:18:43.360 --> 00:18:46.390
allowed, there is
a maximum to here.
00:18:46.390 --> 00:18:51.680
And the procedure of
averaging and increasing
00:18:51.680 --> 00:18:54.490
this minimum wavelength
then corresponds
00:18:54.490 --> 00:19:01.100
to integrating out modes that
are sitting outside lambda
00:19:01.100 --> 00:19:06.870
over v and keeping modes
that we call m tilde that
00:19:06.870 --> 00:19:10.333
live within 0 to lambda over p.
00:19:13.300 --> 00:19:20.250
And if we integrate
these modes--
00:19:20.250 --> 00:19:24.570
so if I rewrite this integration
as an integration over Fourier
00:19:24.570 --> 00:19:28.300
modes, do this
decomposition, et cetera--
00:19:28.300 --> 00:19:34.660
what I find is that
my-- after I integrate.
00:19:34.660 --> 00:19:36.805
So step 1, I do a
coarse graining.
00:19:41.210 --> 00:19:45.510
I find a Hamiltonian
that governs
00:19:45.510 --> 00:19:46.890
the coarse-grained modes.
00:19:49.580 --> 00:19:53.120
Well, if I integrate
out the sigma modes
00:19:53.120 --> 00:19:55.850
and treat them as
Gaussians, then there
00:19:55.850 --> 00:19:58.640
will be a contribution to
the free energy trivially
00:19:58.640 --> 00:20:03.360
from those modes proportional
to volume, presumably.
00:20:03.360 --> 00:20:05.710
Since these modes
and these modes
00:20:05.710 --> 00:20:09.290
don't couple at the Gaussian
level, at the Gaussian level
00:20:09.290 --> 00:20:18.370
we also have beta H0 acting on
these modes that I have kept.
00:20:18.370 --> 00:20:22.390
And the hard part is, of
course, the interaction
00:20:22.390 --> 00:20:25.390
between these
types of modes that
00:20:25.390 --> 00:20:29.650
are governed by all of
these non-linearities
00:20:29.650 --> 00:20:31.400
that we have over here.
00:20:31.400 --> 00:20:36.660
And we kind of can formally
write that as minus log of e
00:20:36.660 --> 00:20:42.440
to the minus u that depends
on m tilde and sigma
00:20:42.440 --> 00:20:45.470
when I integrate out
in the Gaussian weight
00:20:45.470 --> 00:20:47.760
the modes that are
the sigma parameters.
00:20:52.740 --> 00:20:55.490
So that's formally correct.
00:20:55.490 --> 00:20:59.120
This is some complicated
function of m tilde
00:20:59.120 --> 00:21:02.690
after I get rid of
the sigma variables.
00:21:02.690 --> 00:21:07.370
But presumably, if I
were to expand and write
00:21:07.370 --> 00:21:11.570
this in powers of m tilde
and powers of gradient,
00:21:11.570 --> 00:21:15.830
it will reproduce back
the original series.
00:21:15.830 --> 00:21:19.830
Because I said that the original
series includes everything
00:21:19.830 --> 00:21:22.550
that could possibly
be generated.
00:21:22.550 --> 00:21:25.650
This is presumably, after
I do all of these things,
00:21:25.650 --> 00:21:28.500
still consistent with
symmetries and so will
00:21:28.500 --> 00:21:31.641
generate those kinds of terms.
00:21:31.641 --> 00:21:33.930
But of course to
evaluate it, then we
00:21:33.930 --> 00:21:36.190
have to do perturbation.
00:21:36.190 --> 00:21:39.870
And so we can start expanding
this in powers of u.
00:21:39.870 --> 00:21:43.770
And the first term
would be u, assuming
00:21:43.770 --> 00:21:45.540
that u is a small quantity.
00:21:45.540 --> 00:21:55.942
Then minus 1/2 u squared minus
u average squared and so forth.
00:22:04.130 --> 00:22:09.150
And of course, this RG
has two other steps.
00:22:09.150 --> 00:22:13.020
After I have
performed this step,
00:22:13.020 --> 00:22:20.610
I have to do rescaling,
which in Fourier space
00:22:20.610 --> 00:22:25.040
means I blow up q.
00:22:25.040 --> 00:22:32.690
So q I will replace
with b inverse q prime.
00:22:32.690 --> 00:22:40.980
And renormalize, which
meant that in Fourier space
00:22:40.980 --> 00:22:45.265
I replace m with z m prime.
00:22:49.750 --> 00:22:53.660
So after I do these
procedures, what do I find?
00:22:53.660 --> 00:22:56.720
I find that to
whatever order I go,
00:22:56.720 --> 00:23:01.120
I start with some
original Hamiltonian
00:23:01.120 --> 00:23:04.840
that includes all terms
consistent with symmetries.
00:23:04.840 --> 00:23:10.490
And I generate a new
log of probability.
00:23:10.490 --> 00:23:11.830
It's not really a Hamiltonian.
00:23:11.830 --> 00:23:14.205
It's the kind of
effective free energy.
00:23:14.205 --> 00:23:16.790
It's the log of the probability
of these configurations.
00:23:20.010 --> 00:23:24.450
So now I should be able to
read this transformation of how
00:23:24.450 --> 00:23:29.110
I go from mu to mu prime.
00:23:29.110 --> 00:23:35.660
So let's go through
this list and do it.
00:23:35.660 --> 00:23:42.320
So t prime is something
that in Fourier space
00:23:42.320 --> 00:23:45.260
went with one
integration over q.
00:23:45.260 --> 00:23:47.295
So I got b to the minus d.
00:23:47.295 --> 00:23:50.760
There's two factors
of m, so it's
00:23:50.760 --> 00:23:54.290
a z square type of contribution.
00:23:54.290 --> 00:24:02.270
And at the 0 order
from here, I have my t.
00:24:02.270 --> 00:24:05.880
And then when I did the
average, from the average
00:24:05.880 --> 00:24:11.220
of u I got a contribution
that was proportional to u.
00:24:11.220 --> 00:24:13.610
There was a degeneracy of 4.
00:24:13.610 --> 00:24:17.560
There were two kinds of diagrams
that were contributing to it,
00:24:17.560 --> 00:24:22.545
and then I had the integral
from lambda over b to lambda,
00:24:22.545 --> 00:24:29.740
d dk 2 pi to the d 1 over
t plus t plus k k squared.
00:24:29.740 --> 00:24:33.260
And just to remind you
the kind of diagrams
00:24:33.260 --> 00:24:36.220
that were contributing
to this, one of them
00:24:36.220 --> 00:24:40.030
was something like
this and the other one
00:24:40.030 --> 00:24:44.420
was something like this.
00:24:44.420 --> 00:24:48.020
This one that had a closed
loop gave me the factor of n
00:24:48.020 --> 00:24:50.810
and the other gave me
what was eventually
00:24:50.810 --> 00:24:54.350
the eighth that I
have observed here.
00:24:54.350 --> 00:25:00.040
So this is what we
found at order of u.
00:25:00.040 --> 00:25:03.540
We went on and
calculated the u squared.
00:25:03.540 --> 00:25:08.880
And the u squared will give
me another contribution.
00:25:08.880 --> 00:25:12.910
There is a coefficient out here
that also similarly involves
00:25:12.910 --> 00:25:14.330
integrals.
00:25:14.330 --> 00:25:17.000
The integrals will depend on t.
00:25:17.000 --> 00:25:18.770
They will depend on k.
00:25:18.770 --> 00:25:21.790
They will depend on this
lambda that I'm integrating.
00:25:21.790 --> 00:25:24.430
They will depend
on t, et cetera.
00:25:24.430 --> 00:25:27.910
So there is some function here.
00:25:27.910 --> 00:25:35.180
And we argued last time that
we don't need to evaluate it,
00:25:35.180 --> 00:25:36.305
but let's write it down.
00:25:36.305 --> 00:25:39.710
And let's make sure that
it doesn't contribute.
00:25:39.710 --> 00:25:44.890
But this is only looking
at the effect of this u.
00:25:44.890 --> 00:25:48.330
And I know that I have
all of these other terms.
00:25:48.330 --> 00:25:51.150
So presumably, I
will get something
00:25:51.150 --> 00:25:57.800
that will be of the order
of, let's say, v squared uv.
00:25:57.800 --> 00:26:01.850
I will certainly get something
that is of the order of u6.
00:26:01.850 --> 00:26:05.950
If I think of u6 as something
that has six legs associated
00:26:05.950 --> 00:26:08.370
with it, I can
certainly join two
00:26:08.370 --> 00:26:10.790
of these legs, two of these
legs and have something
00:26:10.790 --> 00:26:16.670
that is two legs leftover, just
as I did getting from the u4 m
00:26:16.670 --> 00:26:17.920
to the 4 2m squared.
00:26:17.920 --> 00:26:19.620
I certainly can do that.
00:26:19.620 --> 00:26:24.380
So there is all kinds of higher
order terms here in principle
00:26:24.380 --> 00:26:26.352
that we have to keep track of.
00:26:29.870 --> 00:26:34.780
Now, the next term in
the series is the k.
00:26:34.780 --> 00:26:38.120
Compared to the t-terms, it
had two additional factor
00:26:38.120 --> 00:26:39.000
of gradients.
00:26:39.000 --> 00:26:43.030
When we do everything, it turns
out that it will be minus 2
00:26:43.030 --> 00:26:47.530
because of the two gradients
that became q squared.
00:26:47.530 --> 00:26:51.480
It's still second order
in m, so I would get this.
00:26:51.480 --> 00:26:55.200
And then I start with k.
00:26:55.200 --> 00:26:58.240
Now, the interesting
and important thing
00:26:58.240 --> 00:27:02.920
is that when we do the
calculation at order of u,
00:27:02.920 --> 00:27:07.170
we don't get any
correction to k.
00:27:07.170 --> 00:27:10.470
The only diagrams that
could have contributed to k
00:27:10.470 --> 00:27:14.480
were diagrams of this variety.
00:27:14.480 --> 00:27:16.690
But diagrams of
this variety, we saw
00:27:16.690 --> 00:27:20.040
that when I performed those
integrals the result just
00:27:20.040 --> 00:27:21.480
doesn't depend on q.
00:27:21.480 --> 00:27:24.010
It only corrected
the constant term
00:27:24.010 --> 00:27:26.990
that is proportional
to t squared.
00:27:26.990 --> 00:27:30.960
But that structure
will not preserve.
00:27:30.960 --> 00:27:34.510
If I go to order
of u squared, there
00:27:34.510 --> 00:27:37.760
will be some kind
of a correction that
00:27:37.760 --> 00:27:40.020
is order of u squared.
00:27:40.020 --> 00:27:44.410
And we kind of had in the
table in the table of 6
00:27:44.410 --> 00:27:47.260
by 6 things that I
had a diagram that
00:27:47.260 --> 00:27:49.650
gives contribution such as this.
00:27:49.650 --> 00:27:51.810
That, in fact, look
something like this.
00:27:59.500 --> 00:28:02.420
So basically, this is
a four-point vertex,
00:28:02.420 --> 00:28:04.170
a four-point vertex.
00:28:04.170 --> 00:28:05.060
I joined them.
00:28:05.060 --> 00:28:07.650
I make these kinds
of calculation.
00:28:07.650 --> 00:28:10.560
Now, once you do
that, you'll find
00:28:10.560 --> 00:28:12.710
that the difference
between this diagram
00:28:12.710 --> 00:28:17.840
and let's say that diagram
is that the momentum that
00:28:17.840 --> 00:28:21.720
goes in here, q, will have
to just go through here
00:28:21.720 --> 00:28:25.120
and there is no influence
on it on the momentum
00:28:25.120 --> 00:28:27.620
that I'm integrating.
00:28:27.620 --> 00:28:30.170
Whereas, if you look
at this diagram,
00:28:30.170 --> 00:28:32.730
you will find that it
is possible to have
00:28:32.730 --> 00:28:37.970
a momentum that going in here
and gets a contribution over
00:28:37.970 --> 00:28:38.780
here.
00:28:38.780 --> 00:28:43.350
And so the calculation, if I
were to do at higher order,
00:28:43.350 --> 00:28:46.800
will have in the
denominator a product of two
00:28:46.800 --> 00:28:48.890
of these factors,
but one of them
00:28:48.890 --> 00:28:51.780
will explicitly depend on q.
00:28:51.780 --> 00:28:54.390
And if I expanded
in powers of q,
00:28:54.390 --> 00:28:58.100
I will get a correction
that will appear here.
00:28:58.100 --> 00:28:59.945
It will not change
our life as we
00:28:59.945 --> 00:29:03.835
will see what it's good
to know that it is there.
00:29:03.835 --> 00:29:07.702
And there be higher-order
corrections here, too.
00:29:07.702 --> 00:29:08.630
AUDIENCE: Question.
00:29:08.630 --> 00:29:10.460
PROFESSOR: Yes.
00:29:10.460 --> 00:29:14.510
AUDIENCE: Are the
functions a1 and a2
00:29:14.510 --> 00:29:18.875
just labeled as such to
make our lives easier
00:29:18.875 --> 00:29:24.240
or because they don't have any
sort of universality with them?
00:29:24.240 --> 00:29:25.550
PROFESSOR: Both.
00:29:25.550 --> 00:29:29.500
They don't carry at this
order in the expansion
00:29:29.500 --> 00:29:32.320
any information for
what we will need,
00:29:32.320 --> 00:29:35.810
but I have to show
it to you explicitly.
00:29:35.810 --> 00:29:39.740
So right now, I keep
them as placeholders.
00:29:39.740 --> 00:29:41.800
If, at the end of
the day, we find
00:29:41.800 --> 00:29:45.660
that our answers will
depend on these quantities,
00:29:45.660 --> 00:29:48.270
then we have to go back
and calculate them.
00:29:48.270 --> 00:29:52.170
But ultimately, the reason
that we won't need them
00:29:52.170 --> 00:29:53.940
is what I described last time.
00:29:53.940 --> 00:29:59.320
That we will calculate
exponents only to lowest order
00:29:59.320 --> 00:30:03.220
in 4 minus epsilon-- 4
minus d, which is epsilon.
00:30:03.220 --> 00:30:06.460
And that all of these
u's at the fixed point
00:30:06.460 --> 00:30:08.510
will be of the order of epsilon.
00:30:08.510 --> 00:30:10.650
So both of these
terms are terms that
00:30:10.650 --> 00:30:13.206
are order of epsilon
squared and will
00:30:13.206 --> 00:30:14.715
be ignorable at
level of epsilon.
00:30:18.640 --> 00:30:21.850
But so far I haven't talked
anything about epsilon,
00:30:21.850 --> 00:30:25.060
so I may as well keep it.
00:30:25.060 --> 00:30:29.050
And similarly, l prime
would be something
00:30:29.050 --> 00:30:31.360
that goes with q
to the fourth if I
00:30:31.360 --> 00:30:33.450
were to Fourier transform it.
00:30:33.450 --> 00:30:35.670
So this would be b
to the d minus 4.
00:30:35.670 --> 00:30:37.550
It is still z squared.
00:30:37.550 --> 00:30:40.010
It will be proportional to l.
00:30:40.010 --> 00:30:44.350
It will have exactly these
kinds of corrections also.
00:30:48.160 --> 00:30:52.645
And I can keep going with the
list of all second-order terms.
00:30:57.700 --> 00:31:00.980
Now, then we got to u prime.
00:31:00.980 --> 00:31:04.050
u prime was a fourth
order, fourth power of m.
00:31:04.050 --> 00:31:06.600
So it carried z to the fourth.
00:31:06.600 --> 00:31:09.640
It involved three
derivatives in q space,
00:31:09.640 --> 00:31:14.330
so it gave me b to the minus 3d.
00:31:14.330 --> 00:31:20.300
And then to the lowest order
when I did the expansion
00:31:20.300 --> 00:31:23.460
from u, one of the
terms that I got
00:31:23.460 --> 00:31:26.570
was the original
potential evaluated
00:31:26.570 --> 00:31:29.570
for m tilde rather
than the original m.
00:31:29.570 --> 00:31:33.360
So I always will get this term.
00:31:33.360 --> 00:31:36.100
And then I noticed that when
I go and calculate things
00:31:36.100 --> 00:31:39.100
at second order--
and we explicitly
00:31:39.100 --> 00:31:45.110
did that-- we got a
term that was minus 4 u
00:31:45.110 --> 00:31:51.425
squared n plus 8 integral
lambda over b to lambda
00:31:51.425 --> 00:31:58.560
d dk 2 pi to the d 1
over t plus k k squared.
00:31:58.560 --> 00:32:02.780
And presumably, these
series also continue.
00:32:02.780 --> 00:32:03.830
The whole thing squared.
00:32:03.830 --> 00:32:08.010
It was a squared propagator
that was appearing over here.
00:32:08.010 --> 00:32:11.310
And again, reminding
you that these
00:32:11.310 --> 00:32:18.330
came from diagrams that were--
some of it was like this.
00:32:18.330 --> 00:32:22.230
There was a loop that
gave us the factor of n.
00:32:22.230 --> 00:32:31.840
And then there were things
like this one or this one.
00:32:41.850 --> 00:32:42.350
Yeah.
00:32:48.350 --> 00:32:56.470
And those gave out this huge
number compared to this.
00:32:56.470 --> 00:33:03.740
Now, if I had included this
term that is proportional to v,
00:33:03.740 --> 00:33:07.270
presumably I would
have gotten corrections
00:33:07.270 --> 00:33:10.070
that are order of uv.
00:33:10.070 --> 00:33:12.560
If I go to higher
orders, I will certainly
00:33:12.560 --> 00:33:15.665
get things that are
of the order of u6.
00:33:15.665 --> 00:33:21.200
I will certainly get things
that are of the order of u cubed
00:33:21.200 --> 00:33:23.080
and so forth.
00:33:23.080 --> 00:33:26.270
So there is a whole
bunch of corrections
00:33:26.270 --> 00:33:29.140
that in principle,
if I am supposed
00:33:29.140 --> 00:33:31.920
to include everything and
keep track of everything,
00:33:31.920 --> 00:33:34.038
I should include.
00:33:34.038 --> 00:33:40.160
Now, the v itself, v prime,
has two additional derivatives
00:33:40.160 --> 00:33:41.800
with respect to u.
00:33:41.800 --> 00:33:45.770
So it will be b to
the minus 3d minus 2.
00:33:45.770 --> 00:33:48.540
It's a z to the
fourth type of term.
00:33:48.540 --> 00:33:54.500
It is v, and then it will
certainly get corrections at,
00:33:54.500 --> 00:33:57.430
say, at order of
uv and so forth.
00:34:04.680 --> 00:34:08.130
And what else did I
write down in the series?
00:34:08.130 --> 00:34:12.449
I can write as many as we
like. u prime to the 6.
00:34:12.449 --> 00:34:17.659
This is something that
goes with 6 powers of z
00:34:17.659 --> 00:34:20.975
and will have 5 derivatives.
00:34:20.975 --> 00:34:23.760
Sorry, 5 integrations in q.
00:34:23.760 --> 00:34:26.320
So it will give me
b to the minus 5d.
00:34:26.320 --> 00:34:31.150
And again, presumably I
will have u6 minus order
00:34:31.150 --> 00:34:35.489
of something like u squared
v and all kinds of things.
00:34:40.350 --> 00:34:42.090
All right.
00:34:42.090 --> 00:34:43.492
Yes.
00:34:43.492 --> 00:34:46.648
AUDIENCE: So in the calculation
of terms like k prime and l
00:34:46.648 --> 00:34:50.534
prime, you have factors
like b minus d minus 2.
00:34:50.534 --> 00:34:53.427
Is it minus 2 or plus 2?
00:34:53.427 --> 00:34:54.010
PROFESSOR: OK.
00:34:54.010 --> 00:34:59.870
So these came from Fourier
transforming this entity.
00:34:59.870 --> 00:35:03.860
When I Fourier transform,
I get integral dd q.
00:35:03.860 --> 00:35:07.640
I will have t plus
k q squared plus l q
00:35:07.640 --> 00:35:10.010
to the fourth, et cetera.
00:35:10.010 --> 00:35:12.820
And my task is that
whenever I see q,
00:35:12.820 --> 00:35:16.380
I replace it with
p inverse q prime.
00:35:16.380 --> 00:35:18.710
So this would be
b to the minus d.
00:35:18.710 --> 00:35:21.410
This would be b to the minus 2.
00:35:21.410 --> 00:35:23.150
This would be b to the minus 4.
00:35:27.100 --> 00:35:28.199
So that's how it comes.
00:35:28.199 --> 00:35:28.740
AUDIENCE: OK.
00:35:28.740 --> 00:35:29.240
Thank you.
00:35:33.749 --> 00:35:34.790
PROFESSOR: Anything else?
00:35:40.390 --> 00:35:42.220
OK.
00:35:42.220 --> 00:35:54.030
So then we had to choose
what this factor of z is.
00:35:54.030 --> 00:35:58.590
And we said, let's
choose it such
00:35:58.590 --> 00:36:05.392
that k prime is the same as k.
00:36:08.730 --> 00:36:16.410
But k prime over k we can see
is z squared b to the minus
00:36:16.410 --> 00:36:18.620
d minus 2.
00:36:18.620 --> 00:36:24.740
If I divide through by
this k, then I will get 1,
00:36:24.740 --> 00:36:29.350
and then something here
which is order of u squared.
00:36:35.540 --> 00:36:42.040
Now, we will justify later
why u in order to be small
00:36:42.040 --> 00:36:45.320
so that I can make
a construction that
00:36:45.320 --> 00:36:49.910
is perturbative in u will
be of the order of epsilon.
00:36:49.910 --> 00:36:53.500
But in any case, if I
want to in some sense
00:36:53.500 --> 00:36:56.540
keep the lowest order
in u, at this order
00:36:56.540 --> 00:37:01.250
I am justified to
get rid of this term.
00:37:01.250 --> 00:37:04.100
And when I do that
to this order,
00:37:04.100 --> 00:37:09.220
I will find that z is b
to the 1 plus d over 2.
00:37:09.220 --> 00:37:11.870
And probably in
principle, corrections
00:37:11.870 --> 00:37:19.010
that will be of the order of
this epsilon to the squared.
00:37:21.930 --> 00:37:27.570
So I do that choice.
00:37:27.570 --> 00:37:32.340
Secondly, I'll make my
b to be infinitesimal.
00:37:35.100 --> 00:37:39.530
And so that means that
mu prime at scale b,
00:37:39.530 --> 00:37:42.380
the set of parameters--
each parameter
00:37:42.380 --> 00:37:49.170
would be basically mu plus
a small shift d mu by dl.
00:37:49.170 --> 00:37:56.120
And then I can recast
these jumps by factors of b
00:37:56.120 --> 00:38:00.820
that I have up there
to flow equations.
00:38:00.820 --> 00:38:02.960
And so what do I get?
00:38:02.960 --> 00:38:07.350
For the first one,
we got dt by dl.
00:38:07.350 --> 00:38:12.380
And I had chosen z squared to
be b to the minus d minus 2.
00:38:12.380 --> 00:38:14.500
So compared to the
original one, it's
00:38:14.500 --> 00:38:16.820
just two more factors of b.
00:38:16.820 --> 00:38:21.780
Two more factors of
b will give me 2t.
00:38:21.780 --> 00:38:26.720
And then I will have to deal
with that integration evaluated
00:38:26.720 --> 00:38:28.810
when b is very
small, which means
00:38:28.810 --> 00:38:32.140
that I have to just
evaluate it on the shell.
00:38:32.140 --> 00:38:43.750
So I will have 4u n plus 2
kd lambda to the d divided
00:38:43.750 --> 00:38:48.900
by t plus k lambda squared
plus higher-orders terms
00:38:48.900 --> 00:38:52.050
in this propagator.
00:38:52.050 --> 00:38:54.540
And then I will
have, presumably,
00:38:54.540 --> 00:38:58.260
some a1-looking
quantity, but evaluated
00:38:58.260 --> 00:39:02.160
on the shell that
depends on u squared,
00:39:02.160 --> 00:39:04.230
and then I will have
higher-order terms.
00:39:12.340 --> 00:39:13.272
Yes.
00:39:13.272 --> 00:39:15.660
AUDIENCE: I'm kind
of curious on why
00:39:15.660 --> 00:39:18.010
we choose k equals
k prime instead
00:39:18.010 --> 00:39:20.510
of the constant in front of any
of the other gradient terms.
00:39:20.510 --> 00:39:24.967
Why is k equals k prime better
than l equals l prime or--
00:39:24.967 --> 00:39:25.550
PROFESSOR: OK.
00:39:25.550 --> 00:39:30.220
We discussed this in the
context of the Gaussian model.
00:39:30.220 --> 00:39:32.570
So what we saw for
the Gaussian model is
00:39:32.570 --> 00:39:36.150
that if I choose
l prime to be l,
00:39:36.150 --> 00:39:39.840
then I will have k
prime being b squared k
00:39:39.840 --> 00:39:42.215
and t prime will be
b to the fourth t.
00:39:42.215 --> 00:39:47.190
So I will have two
relevant directions.
00:39:47.190 --> 00:39:51.120
So I want to have, in some
sense, the minimal levels
00:39:51.120 --> 00:39:54.770
of direction guided by
the experimental fact
00:39:54.770 --> 00:39:57.170
that you do whatever you
like and you see the phase
00:39:57.170 --> 00:40:01.530
transition, except that you
have to change one parameter.
00:40:01.530 --> 00:40:03.120
There could be something else.
00:40:03.120 --> 00:40:06.310
There could very well--
somebody comes to me later
00:40:06.310 --> 00:40:08.680
and describes some kind
of a phase transition
00:40:08.680 --> 00:40:12.010
that requires two
relevant directions.
00:40:12.010 --> 00:40:16.210
And the physics of it may guide
me to make the other choice.
00:40:16.210 --> 00:40:18.840
But for the problem that
I'm telling you right now,
00:40:18.840 --> 00:40:20.860
the physics guides me
to make this choice.
00:40:26.450 --> 00:40:28.950
There is no equation
for k because we already
00:40:28.950 --> 00:40:31.930
set that to be 0.
00:40:31.930 --> 00:40:33.665
Let's write the equation for u.
00:40:36.430 --> 00:40:41.460
So four u, z to the
fourth becomes 4 plus 2d.
00:40:41.460 --> 00:40:50.120
And then minus 3d
becomes 4 minus d u.
00:40:50.120 --> 00:40:52.980
And then the next
order term becomes
00:40:52.980 --> 00:41:02.930
minus 4 u squared n plus 8 kd
lambda to the d t plus k lambda
00:41:02.930 --> 00:41:08.440
squared and so
fourth squared when
00:41:08.440 --> 00:41:10.840
I evaluate that
integral on the shell.
00:41:10.840 --> 00:41:12.670
And then I will have
higher-order terms.
00:41:21.560 --> 00:41:28.710
So this is where this idea
of making an expansion
00:41:28.710 --> 00:41:31.660
in dimensions come into play.
00:41:31.660 --> 00:41:34.450
Because we want to have
these sets of equations
00:41:34.450 --> 00:41:38.040
somehow under control, we
need to have a small parameter
00:41:38.040 --> 00:41:40.850
in which we are
making an expansion.
00:41:40.850 --> 00:41:44.610
And ultimately, we will be
looking at the fixed point.
00:41:44.610 --> 00:41:46.510
And the fixed point
occurs at u star.
00:41:46.510 --> 00:41:49.950
That is, of the
order of 4 minus d.
00:41:49.950 --> 00:41:54.380
Otherwise, there is no
small control parameter.
00:41:54.380 --> 00:41:58.170
So the suggestion, actually,
that goes to Fisher
00:41:58.170 --> 00:42:07.470
was to organize the expansion as
a power series in this quantity
00:42:07.470 --> 00:42:09.340
epsilon.
00:42:09.340 --> 00:42:14.600
And eventually then, ask what
the properties of these series
00:42:14.600 --> 00:42:15.950
are as a function of epsilon.
00:42:19.410 --> 00:42:21.870
So then I have all those others.
00:42:21.870 --> 00:42:28.230
I forgot, actually, to
write l, by dt by dl.
00:42:28.230 --> 00:42:31.880
Well, compared to k, it
has two additional factor
00:42:31.880 --> 00:42:38.680
of gradients, which means that
it will start with minus 2l.
00:42:38.680 --> 00:42:41.330
And then we said that
it will get corrections
00:42:41.330 --> 00:42:44.135
that are of the
order of u squared,
00:42:44.135 --> 00:42:46.210
and uv, and such things.
00:42:49.180 --> 00:42:50.220
dv by dl.
00:42:54.540 --> 00:42:58.440
I mean, compared to this
term, compared to u,
00:42:58.440 --> 00:43:02.150
it has two more gradients
in the construction.
00:43:02.150 --> 00:43:07.740
So it's dimension will
be minus 2 plus epsilon.
00:43:07.740 --> 00:43:10.975
And then we'll get directions
of the order of, presumably,
00:43:10.975 --> 00:43:12.441
u squared, uv, [INAUDIBLE].
00:43:16.370 --> 00:43:18.870
And then, what else did I write?
00:43:18.870 --> 00:43:24.710
I wrote something
about d u6 by dl.
00:43:24.710 --> 00:43:31.565
d u6 by dl, I have to substitute
for z 1 plus d over 2.
00:43:31.565 --> 00:43:34.400
Subtract 5d.
00:43:34.400 --> 00:43:38.290
Rewrite d as 4 minus epsilon.
00:43:38.290 --> 00:43:41.400
Once you do that, you will
find that it becomes minus 2
00:43:41.400 --> 00:43:44.360
plus 2 epsilon.
00:43:48.889 --> 00:43:55.725
Let me just make sure that I
am not saying something wrong.
00:43:55.725 --> 00:44:02.451
Yeah, u6 plus order
of uv and so forth.
00:44:05.820 --> 00:44:10.930
So there is this whole
set of parameters
00:44:10.930 --> 00:44:24.280
that are being changed as a
function of-- going away--
00:44:24.280 --> 00:44:29.065
changing the rescaling
by a factor of b
00:44:29.065 --> 00:44:32.050
is 1 plus an infinitesimal.
00:44:32.050 --> 00:44:37.130
So this is the flow of
parameter in this space.
00:44:37.130 --> 00:44:42.540
So then to confirm
the ideas of Kadanoff,
00:44:42.540 --> 00:44:44.180
we have to find the fixed point.
00:44:53.440 --> 00:44:55.200
And there is clearly
a fixed point
00:44:55.200 --> 00:44:57.410
when all of these
parameters are 0.
00:44:57.410 --> 00:44:59.530
If they are 0, nothing changes.
00:44:59.530 --> 00:45:03.800
And I'm back to
the Gaussian model,
00:45:03.800 --> 00:45:07.790
which is described
by just gradient of m
00:45:07.790 --> 00:45:09.910
squared type of theory.
00:45:09.910 --> 00:45:12.610
So this is the fixed
point that corresponds
00:45:12.610 --> 00:45:17.406
to t star, u star, l star,
v star, all of the things
00:45:17.406 --> 00:45:20.290
that I can think
of, are equal to 0.
00:45:20.290 --> 00:45:22.650
It's a perfectly good fixed
point of the transformation.
00:45:26.300 --> 00:45:28.940
It doesn't suit us
because it actually
00:45:28.940 --> 00:45:32.310
has still two
relevant directions.
00:45:32.310 --> 00:45:36.610
It's obvious that if I
make a small change in u,
00:45:36.610 --> 00:45:42.660
then in dimensions less than
4, u is a relevant direction
00:45:42.660 --> 00:45:44.220
and t is a relevant direction.
00:45:44.220 --> 00:45:49.580
Two directions does not describe
the physics that I want.
00:45:49.580 --> 00:45:53.160
But there is fortunately
another fixed point,
00:45:53.160 --> 00:45:57.020
the one that we call the
O n fixed point because it
00:45:57.020 --> 00:46:00.150
explicitly depends
on the parameter n.
00:46:00.150 --> 00:46:06.860
And what I need to do is
to set this equal to 0.
00:46:06.860 --> 00:46:10.830
And if I set that equal
to 0, what do I get?
00:46:10.830 --> 00:46:16.470
I get u star just
manipulating this.
00:46:16.470 --> 00:46:18.110
It is proportional to epsilon.
00:46:18.110 --> 00:46:20.500
1u drops out.
00:46:20.500 --> 00:46:25.710
It is proportional to
epsilon The coefficient has
00:46:25.710 --> 00:46:33.350
a factor of 1 over 4 n plus 8.
00:46:33.350 --> 00:46:35.910
Basically, the inverse of this.
00:46:35.910 --> 00:46:38.320
And then also, the
inverse of all of that.
00:46:38.320 --> 00:46:45.970
So I have t star plus k lambda
squared and so forth squared
00:46:45.970 --> 00:46:48.625
divided by kd lambda to the d.
00:46:56.060 --> 00:47:02.550
Then, what I need to do is to
set the second equation to 0.
00:47:02.550 --> 00:47:04.360
You can see that
this is a term that
00:47:04.360 --> 00:47:07.310
is order of epsilon squared now.
00:47:07.310 --> 00:47:10.380
Whereas, this is a term
that is order of epsilon.
00:47:10.380 --> 00:47:13.470
So for calculating the
position of the fixed point,
00:47:13.470 --> 00:47:15.560
I don't need this parameter.
00:47:15.560 --> 00:47:16.680
And what do I get?
00:47:16.680 --> 00:47:31.010
I will get that t star is minus
2 n plus 2 kd lambda to the d
00:47:31.010 --> 00:47:35.730
divided by t star plus k
lambda squared and so forth.
00:47:35.730 --> 00:47:38.450
Times u star.
00:47:38.450 --> 00:47:46.290
u star is epsilon 4 n plus
8 t star plus k lambda
00:47:46.290 --> 00:47:48.440
squared and so forth squared.
00:47:48.440 --> 00:47:52.006
Divided by kd lambda to the d.
00:47:56.040 --> 00:47:59.770
Again, I'm calculating
everything correctly
00:47:59.770 --> 00:48:01.920
to order of epsilon.
00:48:01.920 --> 00:48:04.350
So since t star is
order of epsilon,
00:48:04.350 --> 00:48:06.510
I can drop it over here.
00:48:06.510 --> 00:48:14.020
So my u star is, in fact,
epsilon divided by 4 n plus 8.
00:48:14.020 --> 00:48:17.470
And then this
combination k lambda
00:48:17.470 --> 00:48:24.270
squared and so forth squared
divided by kd lambda to the d.
00:48:24.270 --> 00:48:26.590
And doing the same
thing up here,
00:48:26.590 --> 00:48:36.515
my t star is epsilon n plus
2 divided by 2 n plus 8.
00:48:36.515 --> 00:48:40.680
It has an overall minus sign.
00:48:40.680 --> 00:48:43.460
The kd parts cancel.
00:48:43.460 --> 00:48:46.050
And one of these
factors cancel, so I
00:48:46.050 --> 00:48:50.285
will get k lambda
squared squared.
00:48:50.285 --> 00:48:51.940
Sorry, no square here.
00:48:51.940 --> 00:48:53.610
And both of these
will get corrections
00:48:53.610 --> 00:48:56.250
that are order of
epsilon squared
00:48:56.250 --> 00:48:57.694
that I haven't calculated.
00:49:03.410 --> 00:49:06.630
Now, let's make sure that
it was justified for me
00:49:06.630 --> 00:49:12.390
to focus on these two parameters
and look at everything else
00:49:12.390 --> 00:49:15.576
as being not important before.
00:49:15.576 --> 00:49:18.390
Well, look at these equations.
00:49:18.390 --> 00:49:21.650
This equation says that
if I had a term that
00:49:21.650 --> 00:49:25.890
was order of u squared
evaluated at a fixed point,
00:49:25.890 --> 00:49:27.830
it would be epsilon squared.
00:49:27.830 --> 00:49:32.220
So l star would be of the
order of epsilon squared.
00:49:32.220 --> 00:49:34.320
You can check that
v star would be
00:49:34.320 --> 00:49:37.310
of the order of epsilon squared.
00:49:37.310 --> 00:49:40.625
A lot of those things will be
of the order of epsilon squared.
00:49:44.670 --> 00:49:47.390
v star.
00:49:47.390 --> 00:49:49.520
And actually, if you
look at it carefully,
00:49:49.520 --> 00:49:52.850
you'll find that things
like u6 will be even worse.
00:49:52.850 --> 00:49:56.473
They would start at order of
epsilon cubed and so forth.
00:50:00.550 --> 00:50:06.330
So quite systematically in this
small parameter that Fisher
00:50:06.330 --> 00:50:14.400
introduced, we see that what has
happened is that we have a huge
00:50:14.400 --> 00:50:18.770
set of parameters, these mu's.
00:50:18.770 --> 00:50:22.090
But we can focus
on the projection
00:50:22.090 --> 00:50:28.290
in the parameter space t and u.
00:50:28.290 --> 00:50:33.030
And in that parameter
space, we certainly always
00:50:33.030 --> 00:50:36.320
have the Gaussian fixed point.
00:50:36.320 --> 00:50:42.040
But as long as I am in
dimensions less than 4,
00:50:42.040 --> 00:50:45.360
the Gaussian fixed
point is not only
00:50:45.360 --> 00:50:51.440
relevant in the t-direction
by a factor of 2,
00:50:51.440 --> 00:50:56.190
but it also is relevant
in another direction.
00:50:56.190 --> 00:51:00.900
There is an eigen-direction
that is slightly shifted
00:51:00.900 --> 00:51:04.350
with respect to t equals to 0.
00:51:04.350 --> 00:51:06.280
It's not just the u-axis.
00:51:06.280 --> 00:51:09.550
Along that direction,
it moves away.
00:51:09.550 --> 00:51:11.240
Here you have an
eigenvalue of 2.
00:51:11.240 --> 00:51:15.370
Here you have an
eigenvalue of epsilon.
00:51:15.370 --> 00:51:17.115
So that's the
Gaussian fixed point.
00:51:20.030 --> 00:51:23.400
But now we found
another fixed point,
00:51:23.400 --> 00:51:25.550
which is occurring
for some positive u
00:51:25.550 --> 00:51:29.050
star and some negative u star.
00:51:29.050 --> 00:51:30.705
This is the o n fixed point.
00:51:39.350 --> 00:51:43.170
Kind of just by
the continuity, you
00:51:43.170 --> 00:51:47.420
would expect that if
things are going into here,
00:51:47.420 --> 00:51:50.560
it probably makes sense that
it should be going like here
00:51:50.560 --> 00:51:54.280
and this should be a
negative eigenvalue.
00:51:54.280 --> 00:51:57.260
But one can
explicitly check that.
00:51:57.260 --> 00:52:01.250
So basically, the
procedure to check that
00:52:01.250 --> 00:52:04.610
is to do what I told you.
00:52:04.610 --> 00:52:09.550
I have to construct a
linearized matrix that
00:52:09.550 --> 00:52:19.390
relates delta mu going
away from this fixed point
00:52:19.390 --> 00:52:23.795
to what happens under rescaling.
00:52:26.320 --> 00:52:35.250
So basically, under rescaling
I will find that if I set my mu
00:52:35.250 --> 00:52:41.270
to be mu star plus a
small change delta mu,
00:52:41.270 --> 00:52:43.890
then it will be moving away.
00:52:43.890 --> 00:52:47.250
And I can look at how,
let's say, delta t changes,
00:52:47.250 --> 00:52:50.880
how delta u changes,
how delta l changes,
00:52:50.880 --> 00:52:54.950
the whole list of parameters
that I have over here.
00:52:54.950 --> 00:52:59.190
The linearized matrix
will relate them
00:52:59.190 --> 00:53:02.850
to the vector that
corresponds to delta t,
00:53:02.850 --> 00:53:04.660
delta u, blah, blah.
00:53:09.120 --> 00:53:15.390
So I have to go back to
these recursion relations,
00:53:15.390 --> 00:53:19.380
make small changes in
all of the parameters,
00:53:19.380 --> 00:53:23.370
linearize the result,
construct that matrix,
00:53:23.370 --> 00:53:27.250
and then evaluate the
eigenvalues of that matrix.
00:53:27.250 --> 00:53:32.090
Again, consistency to the
order that I have done things.
00:53:32.090 --> 00:53:36.360
And for example, one of the
things that we saw last time
00:53:36.360 --> 00:53:38.720
is that there will
be an element here
00:53:38.720 --> 00:53:40.740
that corresponds
to the change in u
00:53:40.740 --> 00:53:43.430
if I make a change in delta t.
00:53:43.430 --> 00:53:45.370
There is such a contribution.
00:53:45.370 --> 00:53:49.430
If I make a change delta t with
respect to the fixed point,
00:53:49.430 --> 00:53:52.780
I will get a
derivative from here.
00:53:52.780 --> 00:53:56.930
But that derivative
multiplies u squared.
00:53:56.930 --> 00:53:58.950
Evaluated at the
fixed point means
00:53:58.950 --> 00:54:01.040
that I will get a
term down here that
00:54:01.040 --> 00:54:02.610
is order of epsilon squared.
00:54:06.420 --> 00:54:11.080
And then the second
element here,
00:54:11.080 --> 00:54:14.630
what happens if I
make a change in u?
00:54:14.630 --> 00:54:18.060
Well, I will get a epsilon here.
00:54:18.060 --> 00:54:22.110
And then I get a
subtraction from here.
00:54:22.110 --> 00:54:25.530
And this subtraction
we evaluated last time
00:54:25.530 --> 00:54:30.190
and it turned out to be
epsilon minus 2 epsilon.
00:54:30.190 --> 00:54:33.960
So the relevance
that we had over here
00:54:33.960 --> 00:54:36.820
became an irrelevance
that I wanted.
00:54:39.560 --> 00:54:43.690
So there is some matrix
element in this corner.
00:54:43.690 --> 00:54:47.770
But since this is 0, as we
discussed if I look at this 2
00:54:47.770 --> 00:54:52.380
by 2 block, it doesn't
affect this eigenvalue.
00:54:52.380 --> 00:54:56.250
Since I did not evaluate
this eigenvalue last time,
00:54:56.250 --> 00:54:58.110
I'll do it now.
00:54:58.110 --> 00:55:01.650
So in order to
calculate the yt, what
00:55:01.650 --> 00:55:03.740
I need to do is to
see what happens
00:55:03.740 --> 00:55:06.740
if I change t to t plus delta t.
00:55:06.740 --> 00:55:09.500
So I have to take a
derivative with respect to t.
00:55:09.500 --> 00:55:11.980
From the first
one, I will get 2.
00:55:11.980 --> 00:55:19.200
From the second term, I will
get minus 4 u n plus 2 kd
00:55:19.200 --> 00:55:23.310
lambda to the d-- what is
in the denominator squared.
00:55:23.310 --> 00:55:29.540
So t plus k lambda squared
and so forth squared.
00:55:29.540 --> 00:55:33.720
But I have to evaluate
this at the fixed point.
00:55:33.720 --> 00:55:36.420
So I put a u star here.
00:55:36.420 --> 00:55:38.560
I put a u star here.
00:55:38.560 --> 00:55:41.600
Since u star is already
order of epsilon,
00:55:41.600 --> 00:55:45.700
this order of epsilon
term I can ignore.
00:55:45.700 --> 00:55:54.170
And so what I have here is 2
minus 4 n plus 2 epsi-- OK.
00:55:54.170 --> 00:55:56.210
Now, let's put u star.
00:55:56.210 --> 00:55:58.310
u star I have up here.
00:55:58.310 --> 00:56:06.600
It is epsilon divided
by 4 n plus 8.
00:56:06.600 --> 00:56:11.380
I have k lambda squared
and so forth squared.
00:56:11.380 --> 00:56:14.846
kd lambda to the d.
00:56:14.846 --> 00:56:18.020
So I substituted the u star.
00:56:18.020 --> 00:56:19.260
I had the n plus 2.
00:56:19.260 --> 00:56:24.540
So now I have the
kd lambda to the d.
00:56:24.540 --> 00:56:28.310
And I have this
whole thing squared.
00:56:31.460 --> 00:56:37.370
And you see that all of
these things cancel out.
00:56:37.370 --> 00:56:43.040
And the answer is
simply 2 minus n
00:56:43.040 --> 00:56:48.260
plus 2 epsilon
divided by n plus 8.
00:56:48.260 --> 00:56:52.630
And somehow, I feel that
I made a factor of-- no,
00:56:52.630 --> 00:56:53.890
I think that's fine.
00:56:56.680 --> 00:56:57.295
Double check.
00:57:06.315 --> 00:57:06.815
Yep.
00:57:12.640 --> 00:57:13.230
All right.
00:57:26.730 --> 00:57:29.990
So let's see what happened.
00:57:29.990 --> 00:57:37.080
We have identified two
fixed points, Gaussian
00:57:37.080 --> 00:57:40.457
and in dimensions
less than 4, the O n.
00:57:43.140 --> 00:57:46.230
Associated with
this are a number
00:57:46.230 --> 00:57:50.370
of operators that tell me-- or
eigen-directions that tell me
00:57:50.370 --> 00:57:56.750
if I go away from the fixed
point, whether I would go back
00:57:56.750 --> 00:57:58.980
or I would go away.
00:57:58.980 --> 00:58:03.860
And so for the
Gaussian, these just
00:58:03.860 --> 00:58:08.940
have the names of the parameters
that we have to set non-zero.
00:58:08.940 --> 00:58:19.860
So their names are things like
t, u, l, v, u6, and so forth.
00:58:19.860 --> 00:58:23.820
And the value of these
exponents we can actually
00:58:23.820 --> 00:58:28.250
get without doing anything
because what is happening here
00:58:28.250 --> 00:58:30.560
is just dimensional analysis.
00:58:30.560 --> 00:58:35.010
So if I simply replace
m by something,
00:58:35.010 --> 00:58:38.430
m prime, gradients or
integrations with some power
00:58:38.430 --> 00:58:41.760
of distance b, I can
very easily figure out
00:58:41.760 --> 00:58:44.740
what the dimensions of
these quantities are.
00:58:44.740 --> 00:58:54.250
They are 2, epsilon, minus 2,
minus 2 plus epsilon, minus 2
00:58:54.250 --> 00:58:57.900
plus 2 epsilon, and so forth.
00:58:57.900 --> 00:59:00.840
So this is simple
dimensional analysis.
00:59:00.840 --> 00:59:06.020
And in some sense, these
correspond to their dimensions
00:59:06.020 --> 00:59:10.480
of the theory of the
variables that you have.
00:59:10.480 --> 00:59:14.390
Problem with it as a description
of what I see in experiments
00:59:14.390 --> 00:59:18.460
is the presence of two
relevant directions.
00:59:18.460 --> 00:59:22.550
Now, we found a new
fixed point that
00:59:22.550 --> 00:59:27.320
is under control to
order of epsilon.
00:59:27.320 --> 00:59:30.310
And what we find is
that this exponent
00:59:30.310 --> 00:59:38.340
for what was analogous to t
shifted to be 2 minus n plus 2
00:59:38.340 --> 00:59:39.860
over n plus 8 epsilon.
00:59:43.460 --> 00:59:47.010
While the one that was
epsilon shifted by minus 2
00:59:47.010 --> 00:59:49.310
epsilon and became
minus epsilon.
00:59:54.380 --> 00:59:57.560
What I see is a pattern
that essentially all that
00:59:57.560 --> 01:00:01.590
can happen, since I'm doing
a perturbation in epsilon,
01:00:01.590 --> 01:00:05.800
is that these quantities
can at most change
01:00:05.800 --> 01:00:07.760
by order of epsilon.
01:00:07.760 --> 01:00:12.360
So this was minus 2 becomes
minus 2 plus order of epsilon.
01:00:12.360 --> 01:00:15.730
This becomes minus 2
plus order of epsilon.
01:00:15.730 --> 01:00:18.385
It will not necessarily
be minus 2 plus epsilon.
01:00:18.385 --> 01:00:21.980
It could be minus 2 minus
7 epsilon plus 11 epsilon.
01:00:21.980 --> 01:00:24.610
Maybe even epsilon
squared, I don't know.
01:00:24.610 --> 01:00:32.340
But the point is
that clearly, even
01:00:32.340 --> 01:00:35.070
if I put all the
infinity of parameters,
01:00:35.070 --> 01:00:40.320
as long as I am in 3.999
dimension, at this fixed point
01:00:40.320 --> 01:00:45.540
I only have one
relevant direction.
01:00:45.540 --> 01:00:50.330
So it does describe the
physics that I want, at least
01:00:50.330 --> 01:00:53.074
in this perturbative sense
of the epsilon expansion.
01:00:56.770 --> 01:01:02.460
And so I have my yt.
01:01:02.460 --> 01:01:06.000
Actually, in order to
get all of the exponents,
01:01:06.000 --> 01:01:07.771
I really need two.
01:01:07.771 --> 01:01:10.820
I need yt and maybe yh.
01:01:10.820 --> 01:01:12.310
But yh is very simple.
01:01:12.310 --> 01:01:20.560
If I were to add to this
magnetic field term,
01:01:20.560 --> 01:01:27.840
then in Fourier representation
it just goes and sits over here
01:01:27.840 --> 01:01:30.170
at q equals to 0.
01:01:30.170 --> 01:01:32.970
And when I do all
of my rescalings,
01:01:32.970 --> 01:01:35.650
et cetera, the only
thing that happens to it
01:01:35.650 --> 01:01:37.930
is that it just picks
the factor of z.
01:01:41.930 --> 01:01:47.120
And we've shown that z is
b to the 1 plus d over 2
01:01:47.120 --> 01:01:50.590
plus order of epsilon squared.
01:01:50.590 --> 01:01:55.380
And so essentially,
we also have our yh.
01:01:55.380 --> 01:01:57.980
So I can even add
it to this table.
01:01:57.980 --> 01:01:59.900
There is a yh.
01:01:59.900 --> 01:02:01.710
There is a magnetic field.
01:02:01.710 --> 01:02:06.080
The corresponding yh is 1 plus
d over 2 for the Gaussian.
01:02:06.080 --> 01:02:14.235
It is 1 plus d over 2 plus order
of epsilon squared for the O n
01:02:14.235 --> 01:02:14.734
model.
01:02:17.470 --> 01:02:21.880
So now we have
everything that we need.
01:02:21.880 --> 01:02:26.840
We can compute
things in principle.
01:02:26.840 --> 01:02:29.730
We find, first of
all, that if I look
01:02:29.730 --> 01:02:35.780
at the divergence of the
correlation length, essentially
01:02:35.780 --> 01:02:42.310
we saw that under
rescaling yt tells us
01:02:42.310 --> 01:02:47.050
that if our magnetic field
is 0, how I get thrown away
01:02:47.050 --> 01:02:49.870
from the fixed point over here.
01:02:49.870 --> 01:02:51.870
There is a relevant
direction out here
01:02:51.870 --> 01:02:57.770
that we've discovered whose
eigenvalue here is no longer 2.
01:02:57.770 --> 01:03:02.540
It is 2 minus this formula
that we've calculated.
01:03:02.540 --> 01:03:06.450
And presumably again,
if I go and look
01:03:06.450 --> 01:03:11.070
at my set of parameters,
what I have is that
01:03:11.070 --> 01:03:16.245
in this infinite dimensional
space, as I reduce temperature,
01:03:16.245 --> 01:03:21.090
I will be going from,
say, one point here,
01:03:21.090 --> 01:03:23.350
then lower temperature
would be here,
01:03:23.350 --> 01:03:25.220
lower temperature would be here.
01:03:25.220 --> 01:03:30.020
So there would be a trajectory
as a function of shifting
01:03:30.020 --> 01:03:36.100
temperature, which at some
point that trajectory hits
01:03:36.100 --> 01:03:40.990
the basin of attraction of this
O n fixed point that we found.
01:03:40.990 --> 01:03:45.680
And then being
away from here will
01:03:45.680 --> 01:03:48.780
have a projection
along this axis.
01:03:48.780 --> 01:03:52.020
And we can relate that to
the divergence of, say,
01:03:52.020 --> 01:03:54.810
the correlation length
to the free energy.
01:03:54.810 --> 01:03:57.550
Our u was 1 over yt.
01:03:57.550 --> 01:04:01.550
It is the inverse
of this object.
01:04:01.550 --> 01:04:07.550
So let's say this
is-- if I divide,
01:04:07.550 --> 01:04:12.900
I have 1/2 1 minus
n plus 2 over 2 n
01:04:12.900 --> 01:04:18.320
plus 8 epsilon raised
to the minus 1 power.
01:04:18.320 --> 01:04:21.260
And to be consistent, I
should really only expand this
01:04:21.260 --> 01:04:23.310
to the order of epsilon.
01:04:23.310 --> 01:04:31.420
So I have 1/2 plus 1/4 n
plus 2 n plus 8 epsilon.
01:04:35.080 --> 01:04:37.500
So what does it tell me?
01:04:37.500 --> 01:04:40.600
Well, it tells me that
Gaussian fixed point,
01:04:40.600 --> 01:04:43.010
the correlation length
exponent was 1/2.
01:04:43.010 --> 01:04:44.990
We already saw that.
01:04:44.990 --> 01:04:47.780
We see that when we
go to this O n model,
01:04:47.780 --> 01:04:51.820
the correlation length exponent
becomes larger than 1/2.
01:04:51.820 --> 01:04:54.460
I guess that agrees
with our table.
01:04:54.460 --> 01:04:57.690
And I guess we can
try to estimate
01:04:57.690 --> 01:05:02.630
what values we would get if
we were to put n equals to 1,
01:05:02.630 --> 01:05:05.300
n equals to 2, et cetera.
01:05:05.300 --> 01:05:09.460
So this is n equals to 1.
01:05:09.460 --> 01:05:15.100
If I put epsilon equals to
1, what do I get for mu?
01:05:15.100 --> 01:05:20.580
I will get 1/2 plus 1/4 of 3/9.
01:05:20.580 --> 01:05:22.460
So that's 1/12.
01:05:22.460 --> 01:05:30.152
So that would give me
something like 0.58.
01:05:30.152 --> 01:05:31.060
All right.
01:05:31.060 --> 01:05:33.030
Not bad for a
low-order expansion
01:05:33.030 --> 01:05:36.680
coming from 4 to something
that's in three dimensions.
01:05:36.680 --> 01:05:40.140
What happens if I
go to n equals to 2?
01:05:40.140 --> 01:05:44.090
OK, so correction is
4/10 divided by 4.
01:05:44.090 --> 01:05:45.090
So it's 0.1.
01:05:45.090 --> 01:05:49.280
So I would get 0.6.
01:05:49.280 --> 01:05:51.930
What happens if I
put n equals to 3?
01:05:51.930 --> 01:05:56.040
I will get 5 divided by 44.
01:05:56.040 --> 01:06:01.860
And I believe that gives
me something like 61.
01:06:06.380 --> 01:06:12.250
So it gets worse when I
go to larger values of n,
01:06:12.250 --> 01:06:14.690
but it does capture a trend.
01:06:14.690 --> 01:06:18.740
Experimentally, we
see that mu becomes
01:06:18.740 --> 01:06:23.960
larger as you go from 1 to 2
or 3-component order parameter.
01:06:23.960 --> 01:06:29.920
That trend is already captured
by this low-order expansion.
01:06:29.920 --> 01:06:35.990
Once you have mu, you can,
for example, calculate alpha.
01:06:35.990 --> 01:06:39.480
Alpha is 2 minus d mu.
01:06:39.480 --> 01:06:43.530
So you do 2 minus your
d is 4 minus epsilon.
01:06:43.530 --> 01:06:54.795
Your mu is this 1/2 1 plus
1/8 n plus 2 n plus 8 epsilon.
01:06:59.860 --> 01:07:02.050
1/2, sorry.
01:07:02.050 --> 01:07:05.420
And you do the algebra
and I'll write the answer.
01:07:05.420 --> 01:07:11.199
It is 4 minus n epsilon
divided by 2 n plus 8.
01:07:16.089 --> 01:07:17.556
OK, and let me check.
01:07:32.200 --> 01:07:36.480
So if I now substitute
epsilon equals
01:07:36.480 --> 01:07:40.890
to 1 for these different
values of n, what I get
01:07:40.890 --> 01:07:56.220
are for alpha 0.17, 0.11, 0.06.
01:07:56.220 --> 01:07:59.200
I don't know, maybe I have
a factor of 2 missing,
01:07:59.200 --> 01:07:59.830
or whatever.
01:07:59.830 --> 01:08:04.710
But these numbers, I
think, are correct.
01:08:04.710 --> 01:08:10.590
So you can see that in
reality alpha is positive
01:08:10.590 --> 01:08:13.495
for the liquid gas
system n equals to 1.
01:08:13.495 --> 01:08:15.220
It is more or less 0.
01:08:15.220 --> 01:08:19.310
This is the logarithmic
lambda point for superfluids.
01:08:19.310 --> 01:08:24.260
And then it becomes negative,
clearly, for magnets.
01:08:24.260 --> 01:08:27.560
The formula that we have
predicts all of these numbers
01:08:27.560 --> 01:08:30.779
to be positive, but it
gets the right trend
01:08:30.779 --> 01:08:34.470
that as you go to
larger values of n,
01:08:34.470 --> 01:08:37.990
the value of the
exponent alpha calculated
01:08:37.990 --> 01:08:43.430
at this order in epsilon
expansion becomes lower.
01:08:43.430 --> 01:08:46.090
So that trend is captured.
01:08:46.090 --> 01:08:51.210
So at this stage,
I guess I would
01:08:51.210 --> 01:08:54.882
say that the
problem that I posed
01:08:54.882 --> 01:08:58.100
is solved in the same
sense that I would
01:08:58.100 --> 01:09:05.399
say we have solved for the
energy levels of the helium
01:09:05.399 --> 01:09:06.560
atom.
01:09:06.560 --> 01:09:09.930
Certainly, you can sort
of ignore the interaction
01:09:09.930 --> 01:09:13.990
between electrons and
calculate hydrogenic energies.
01:09:13.990 --> 01:09:16.790
And then you can do
perturbation in the strength
01:09:16.790 --> 01:09:21.330
of the interaction and
get corrections to that.
01:09:21.330 --> 01:09:24.810
So essentially, you know the
trends and you know everything.
01:09:24.810 --> 01:09:28.310
And we have been
able to sort of find
01:09:28.310 --> 01:09:31.609
the physical structure
that would give us
01:09:31.609 --> 01:09:36.229
a root to calculating
what these exponents are.
01:09:36.229 --> 01:09:38.170
We see that the
exponents are really
01:09:38.170 --> 01:09:41.920
a function of dimensionality
and the symmetry of the order
01:09:41.920 --> 01:09:43.130
parameter.
01:09:43.130 --> 01:09:47.410
All of the trends are captured,
but the numerical values,
01:09:47.410 --> 01:09:50.920
not surprisingly,
at this low order
01:09:50.920 --> 01:09:54.060
have not been
captured very well.
01:09:54.060 --> 01:09:56.540
So presumably, you would
need to do the same thing
01:09:56.540 --> 01:09:58.270
that you would do
for the helium atom.
01:09:58.270 --> 01:10:01.360
You could do to higher and
higher order calculations.
01:10:01.360 --> 01:10:02.670
You could do simulations.
01:10:02.670 --> 01:10:05.320
You could do all
kinds of other things.
01:10:05.320 --> 01:10:08.360
But the conceptual
foundation is basically
01:10:08.360 --> 01:10:10.364
what we have laid out here.
01:10:12.970 --> 01:10:19.210
OK, there is one thing
that-- well, many things that
01:10:19.210 --> 01:10:22.350
remain to be answered.
01:10:22.350 --> 01:10:25.730
One of them is--
well, how do you
01:10:25.730 --> 01:10:29.710
know that there isn't a
fixed point somewhere else?
01:10:29.710 --> 01:10:33.185
You calculated things
perturbatively.
01:10:33.185 --> 01:10:38.590
The answer is that once you
do higher-order calculations,
01:10:38.590 --> 01:10:41.880
et cetera, you find that
your results converge,
01:10:41.880 --> 01:10:45.970
more or less, better and better
to the results of simulations
01:10:45.970 --> 01:10:48.030
or experiments, et cetera.
01:10:48.030 --> 01:10:51.990
So there is no
evidence from whatever
01:10:51.990 --> 01:10:55.640
we know that there
is need for something
01:10:55.640 --> 01:10:58.970
that I would call
a strong coupling,
01:10:58.970 --> 01:11:01.090
non-perturbative fixed point.
01:11:01.090 --> 01:11:02.240
It's not a proof.
01:11:02.240 --> 01:11:04.880
We can't prove that
there isn't such a thing.
01:11:04.880 --> 01:11:09.270
But there is apparently
no need for such a thing
01:11:09.270 --> 01:11:12.450
to discuss what is
observed experimentally.
01:11:12.450 --> 01:11:14.712
Yes.
01:11:14.712 --> 01:11:18.250
AUDIENCE: You told us last time
that in order for the mu fixed
01:11:18.250 --> 01:11:21.056
point to make sense, we must
have epsilon very small.
01:11:21.056 --> 01:11:22.520
PROFESSOR: Yes.
01:11:22.520 --> 01:11:26.920
AUDIENCE: But now we're
putting epsilon back to 1.
01:11:26.920 --> 01:11:28.370
PROFESSOR: OK.
01:11:28.370 --> 01:11:32.790
So when people inevitably
ask me this question,
01:11:32.790 --> 01:11:37.670
I give them the following
two functions of epsilon.
01:11:37.670 --> 01:11:41.010
One of them is e to
the epsilon over 100
01:11:41.010 --> 01:11:44.960
and the other is e
to the 100 epsilon.
01:11:47.760 --> 01:11:53.310
Do I know if I put epsilon
of 1 a priori whether or not
01:11:53.310 --> 01:11:57.720
putting epsilon equals to 1 is
a good thing for the expansion
01:11:57.720 --> 01:11:58.830
or not?
01:11:58.830 --> 01:12:00.050
I don't.
01:12:00.050 --> 01:12:03.690
And so I don't know
whether it is bad
01:12:03.690 --> 01:12:05.910
and I don't know
whether it is good,
01:12:05.910 --> 01:12:09.150
unless I calculate many
more terms in the series
01:12:09.150 --> 01:12:13.284
and discuss what the
convergence of the series is.
01:12:13.284 --> 01:12:16.670
AUDIENCE: Epsilon is 4 minus d
is supposed to be an integer.
01:12:16.670 --> 01:12:17.670
We just--
01:12:17.670 --> 01:12:20.440
PROFESSOR: Oh, you're worried
about its integerness as
01:12:20.440 --> 01:12:25.150
opposed to treating
it as a continuum?
01:12:25.150 --> 01:12:26.066
OK.
01:12:26.066 --> 01:12:31.898
AUDIENCE: It's OK when you
first assume it [INAUDIBLE].
01:12:31.898 --> 01:12:35.300
At the end of the day,
it must be an integer.
01:12:35.300 --> 01:12:36.120
PROFESSOR: OK.
01:12:36.120 --> 01:12:39.800
So the example somebody was
asking me also last time
01:12:39.800 --> 01:12:43.310
that I have in mind is
you've learned n factorial
01:12:43.310 --> 01:12:47.020
to be the product of 1, 2, 3, 4.
01:12:47.020 --> 01:12:50.360
And you know it for whatever
integer that you would like,
01:12:50.360 --> 01:12:52.770
you do the multiplication.
01:12:52.770 --> 01:12:57.610
But we've also established that
n factorial is an integral 0
01:12:57.610 --> 01:13:03.120
to infinity dx x to
the n e to the minus x.
01:13:03.120 --> 01:13:07.340
And so the question
is now, can I
01:13:07.340 --> 01:13:11.770
talk about 4.111
factorial or not?
01:13:11.770 --> 01:13:16.340
Can I expand 4.11
factorial close
01:13:16.340 --> 01:13:21.580
to what I have 4 assuming the
form of this so-called gamma
01:13:21.580 --> 01:13:23.390
function?
01:13:23.390 --> 01:13:29.510
So the gamma function
is a function of n
01:13:29.510 --> 01:13:34.140
that at integer values it
falls on whatever we know,
01:13:34.140 --> 01:13:37.240
but it has a perfect
analytic continuation
01:13:37.240 --> 01:13:41.750
and I can in principle evaluate
4 factorial by expanding
01:13:41.750 --> 01:13:45.750
around 3 and the derivatives
of the gamma function
01:13:45.750 --> 01:13:47.386
evaluated at 3.
01:13:47.386 --> 01:13:48.885
AUDIENCE: But these
kind of integers
01:13:48.885 --> 01:13:51.700
don't have anything
with dimensionality.
01:13:51.700 --> 01:13:54.770
PROFESSOR: OK, where did our
dimensionality come from?
01:13:54.770 --> 01:13:57.460
Our dimensionality
appears in our expressions
01:13:57.460 --> 01:14:00.070
because we have to do
integrals of this form.
01:14:05.746 --> 01:14:07.660
And what do we do?
01:14:07.660 --> 01:14:10.780
We replace this with a
surface area, which actually
01:14:10.780 --> 01:14:13.460
involves the
factorial by the way.
01:14:13.460 --> 01:14:17.620
And then we have k
to the d minus 1 dk.
01:14:17.620 --> 01:14:22.600
So these integrals are
functions of dimension
01:14:22.600 --> 01:14:25.740
that have exactly the same
properties as the gamma
01:14:25.740 --> 01:14:27.740
function and the n factorial.
01:14:27.740 --> 01:14:31.690
They're perfectly
well expandable.
01:14:31.690 --> 01:14:33.300
And they do have singularities.
01:14:33.300 --> 01:14:36.060
Actually, it turns out
the gamma functions also
01:14:36.060 --> 01:14:39.340
have singularities at
minus 1, things like that.
01:14:39.340 --> 01:14:42.991
Our functions have singularities
at two dimensions and so forth.
01:14:46.470 --> 01:14:51.300
But the issue of convergence
is very important.
01:14:51.300 --> 01:14:55.610
So let's say that there were
some powerful field theories.
01:14:55.610 --> 01:14:59.020
And in order to do
calculations at higher orders,
01:14:59.020 --> 01:15:02.273
you need to go and
do field theory.
01:15:02.273 --> 01:15:07.350
And you calculate
the exponent gamma.
01:15:07.350 --> 01:15:12.220
And I will write the gamma
exponent for the case of n
01:15:12.220 --> 01:15:14.610
equals to 1.
01:15:14.610 --> 01:15:20.580
And the series for that is 1
plus-- if we sort of go and do
01:15:20.580 --> 01:15:25.510
all of our calculations to
lowest order, gamma is 2 mu.
01:15:25.510 --> 01:15:29.690
So it will simply be twice
what we have over here.
01:15:29.690 --> 01:15:35.065
And the first correction is,
indeed, 167 times epsilon.
01:15:40.580 --> 01:15:49.160
The next one is 0.077
epsilon squared.
01:15:49.160 --> 01:15:53.870
The next one is
minus-- problematic--
01:15:53.870 --> 01:15:58.080
049 epsilon cubed.
01:15:58.080 --> 01:16:04.130
Next one, 180 epsilon
to the fourth.
01:16:04.130 --> 01:16:06.550
Next one-- and I think
this is as far as people
01:16:06.550 --> 01:16:10.220
have calculated things--
epsilon to the fifth.
01:16:13.680 --> 01:16:17.900
Then, let's put
epsilon equals to 1
01:16:17.900 --> 01:16:21.180
and see what we get
at the various orders.
01:16:21.180 --> 01:16:23.250
So clearly, I start with 1.
01:16:23.250 --> 01:16:27.750
Next order I will get 1.167.
01:16:27.750 --> 01:16:34.355
At the next order,
I will get 1.244.
01:16:34.355 --> 01:16:36.880
It's getting there, huh?
01:16:36.880 --> 01:16:43.850
And the next order
I will get 1.195.
01:16:43.850 --> 01:16:47.990
Then I will get 1.375.
01:16:47.990 --> 01:16:51.270
And then I will get 0.96.
01:16:51.270 --> 01:16:54.820
[LAUGHTER]
01:16:54.820 --> 01:17:01.730
So this is the signature of what
is called an asymptotic series,
01:17:01.730 --> 01:17:06.230
something that as you evaluate
more therms gets closer
01:17:06.230 --> 01:17:08.740
to the expected
result, but then starts
01:17:08.740 --> 01:17:11.880
to move away and oscillate.
01:17:11.880 --> 01:17:14.400
Yet, there are tricks.
01:17:14.400 --> 01:17:18.585
And if you know your tricks,
you can put epsilon equals to 1
01:17:18.585 --> 01:17:22.070
in that series and do
clever enough terms
01:17:22.070 --> 01:17:35.730
to get 1.2385 minus plus 0.0025.
01:17:35.730 --> 01:17:39.760
So the trick is called
Borel summation.
01:17:45.770 --> 01:17:53.470
So one can show that if you go
to high orders in this series,
01:17:53.470 --> 01:17:57.770
asymptotically the terms in
the series scale like, say,
01:17:57.770 --> 01:18:04.670
the p-th term in the series will
scale as p factorial, something
01:18:04.670 --> 01:18:08.980
like a to the power of p and
some coefficient in front.
01:18:08.980 --> 01:18:13.250
So if I write the general
term in this series
01:18:13.250 --> 01:18:17.540
f sub p epsilon to
the p, my statement
01:18:17.540 --> 01:18:23.620
is that the magnitude of f sub
p asymptotically for p much
01:18:23.620 --> 01:18:26.270
larger than 1 going to
infinity has this form.
01:18:29.570 --> 01:18:33.570
So clearly, because
of this p factorial,
01:18:33.570 --> 01:18:34.935
this is growing too rapidly.
01:18:38.490 --> 01:18:44.750
But what you can do is you
can rewrite this series, which
01:18:44.750 --> 01:18:52.230
is sum over p f of p epsilon
of p using this integral that I
01:18:52.230 --> 01:18:54.835
had over here for p factorial.
01:18:58.430 --> 01:19:02.380
So I multiply and
divide by p factorial.
01:19:02.380 --> 01:19:09.110
So it becomes a sum over p f
of p epsilon of p integral 0
01:19:09.110 --> 01:19:14.100
to infinity dx x to
the p e to the minus x
01:19:14.100 --> 01:19:18.080
divided by p factorial.
01:19:18.080 --> 01:19:22.940
And the fp over p factorial
gets rid of this factor.
01:19:22.940 --> 01:19:25.930
So then you can recast
this as the integral
01:19:25.930 --> 01:19:30.400
from 0 to infinity
dx e to the minus x.
01:19:30.400 --> 01:19:35.310
And what you do is you sum
the series f of p divided by p
01:19:35.310 --> 01:19:39.250
factorial epsilon x
raised to the power of p.
01:19:42.560 --> 01:19:50.060
And it turns out that this is
called some kind of a Borel
01:19:50.060 --> 01:19:53.900
function corresponding
to this series.
01:19:53.900 --> 01:19:58.890
And as long as the terms in your
series only diverge this badly,
01:19:58.890 --> 01:20:04.180
people can make sense of this
function, Borel function.
01:20:04.180 --> 01:20:06.310
And then you perform
the integration,
01:20:06.310 --> 01:20:07.906
and then you come
up with this number.
01:20:12.690 --> 01:20:17.760
So that's one thing to note.
01:20:17.760 --> 01:20:25.310
The other thing
to note is I said
01:20:25.310 --> 01:20:29.680
that what I want to do
for my perturbation theory
01:20:29.680 --> 01:20:35.270
to make sense is for
this u star to be small.
01:20:35.270 --> 01:20:38.100
And I said that the
knob that we have
01:20:38.100 --> 01:20:41.120
is for epsilon to be small.
01:20:41.120 --> 01:20:44.810
But there is, if you look at
that expression, another knob.
01:20:44.810 --> 01:20:49.380
I can make n going to infinity.
01:20:49.380 --> 01:20:54.930
So if n becomes very
large-- so that also
01:20:54.930 --> 01:20:57.940
can make the thing small.
01:20:57.940 --> 01:21:01.180
So there is an
alternative expansion.
01:21:01.180 --> 01:21:03.410
Rather than going with
epsilon going to 0,
01:21:03.410 --> 01:21:06.670
you go to what is called
a spherical model.
01:21:06.670 --> 01:21:08.800
That is, an infinite
number of components,
01:21:08.800 --> 01:21:11.540
and then do in
expansion in 1 over n.
01:21:11.540 --> 01:21:15.795
And so then you
basically-- what you
01:21:15.795 --> 01:21:19.920
are interested is things
that are happening
01:21:19.920 --> 01:21:24.455
as a function of d and n.
01:21:24.455 --> 01:21:28.960
And you have-- above
4, you know that you
01:21:28.960 --> 01:21:30.320
are in the Gaussian world.
01:21:36.100 --> 01:21:43.790
At n goes to infinity, you
have this O n type of models.
01:21:43.790 --> 01:21:47.670
And you find that these models
actually only make sense
01:21:47.670 --> 01:21:52.520
in dimensions that
are larger than 2.
01:21:52.520 --> 01:21:58.640
So you can then perturbatively
come either from here
01:21:58.640 --> 01:22:01.780
or you can come from
here and you try to get
01:22:01.780 --> 01:22:05.930
to the exponents you
are interested over here
01:22:05.930 --> 01:22:09.060
or over here.
01:22:09.060 --> 01:22:13.850
So basically, that's the story.
01:22:13.850 --> 01:22:16.330
And for this work,
I think, as I said,
01:22:16.330 --> 01:22:21.620
Wilson did this pertubative RG.
01:22:21.620 --> 01:22:24.680
Michael Fisher
was the person who
01:22:24.680 --> 01:22:28.250
focused it into an
epsilon expansion.
01:22:28.250 --> 01:22:34.140
And in 1982, Wilson got the
Nobel Prize for the work.
01:22:34.140 --> 01:22:35.850
Potentially, it
could have been also
01:22:35.850 --> 01:22:39.590
awarded to Fisher and Kadanoff
for their contributions
01:22:39.590 --> 01:22:41.680
to this whole story.
01:22:41.680 --> 01:22:45.590
So that's the end of
this part of the course.
01:22:45.590 --> 01:22:48.700
And now that we have
established this background,
01:22:48.700 --> 01:22:54.500
we will try to get the exponents
and the statistical behavior
01:22:54.500 --> 01:22:57.410
by a number of
other perspectives.
01:22:57.410 --> 01:23:00.050
So basically, this
was a perspective
01:23:00.050 --> 01:23:03.020
and a route that gave an answer.
01:23:03.020 --> 01:23:05.960
And hopefully, we'll be
able to complement it
01:23:05.960 --> 01:23:09.143
with other ways of
looking at the story.