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PROFESSOR: So we are going
to switch directions.
00:00:25.990 --> 00:00:29.020
Rather than thinking
about binary variables,
00:00:29.020 --> 00:00:35.400
these Ising variables,
that were discrete,
00:00:35.400 --> 00:00:36.990
think again about a lattice.
00:00:40.240 --> 00:00:50.290
But now at each site we put a
spin that has unit magnitude,
00:00:50.290 --> 00:00:52.520
but m component.
00:00:52.520 --> 00:01:03.440
That is, Si has
components 1, 2, to n.
00:01:03.440 --> 00:01:12.150
And the constraint that this
sum over alpha Si alpha squared
00:01:12.150 --> 00:01:13.690
is unit.
00:01:13.690 --> 00:01:19.285
So clearly, if I-- let's
put it here explicitly.
00:01:19.285 --> 00:01:23.715
Alpha of 1 to n, so that's
when I look at the case of n
00:01:23.715 --> 00:01:27.220
equals to 1, essentially
I have one component.
00:01:27.220 --> 00:01:31.010
2 squared has to be 1, so
it's either plus or minus.
00:01:31.010 --> 00:01:33.790
We recover the Ising variable.
00:01:33.790 --> 00:01:36.380
For n equals to 2,
it's essentially
00:01:36.380 --> 00:01:40.760
a unit vector who's
angle theta, for example,
00:01:40.760 --> 00:01:43.980
can change in three dimensions
if we were exploring
00:01:43.980 --> 00:01:46.570
the surface of the cube.
00:01:46.570 --> 00:01:49.480
And we always
assume that we have
00:01:49.480 --> 00:01:57.960
a weight that tends to make
our spins to be parallel.
00:01:57.960 --> 00:02:03.310
So we use, essentially, the
same form as the Ising model.
00:02:03.310 --> 00:02:06.680
We sum over near neighbors.
00:02:06.680 --> 00:02:10.325
And the interaction,
rather than sigma I sigma
00:02:10.325 --> 00:02:13.690
j, we put it as this
Si dot Sj, where
00:02:13.690 --> 00:02:19.030
these are the dot
products of two vectors.
00:02:19.030 --> 00:02:23.860
Let's call the dimensionless
interaction in front K0.
00:02:26.910 --> 00:02:35.200
So when we want to calculate
the partition function,
00:02:35.200 --> 00:02:38.900
we need to integrate
over all configurations
00:02:38.900 --> 00:02:46.906
of these spins of this weight.
00:02:52.370 --> 00:02:58.410
Now for each case, we
have to do n components.
00:02:58.410 --> 00:03:01.510
But there is a constraint,
which is this one.
00:03:04.880 --> 00:03:11.620
Now I'm going to be focused
on the ground state.
00:03:11.620 --> 00:03:17.720
So when t equals to 0, we
expect that spontaneously
00:03:17.720 --> 00:03:20.350
the particular configuration
will be chosen.
00:03:20.350 --> 00:03:22.425
Everybody will be aligned
to that configuration.
00:03:25.890 --> 00:03:32.840
Without loss of generality,
let's choose aligned state
00:03:32.840 --> 00:03:36.840
to point along the
last component.
00:03:36.840 --> 00:03:41.170
That is, all of the
Si at t equal to 0
00:03:41.170 --> 00:03:47.100
will be of the form 0,0, except
that the last component is
00:03:47.100 --> 00:03:51.810
pointing along some
particular direction.
00:03:51.810 --> 00:03:58.260
So if it was two components, the
y component would always be 1.
00:03:58.260 --> 00:04:02.470
It would be aligned
along the y direction.
00:04:02.470 --> 00:04:05.790
Yes, question?
00:04:05.790 --> 00:04:08.700
AUDIENCE: What dimensionality
is the lattice?
00:04:08.700 --> 00:04:09.950
PROFESSOR: It can be anything.
00:04:09.950 --> 00:04:13.910
So basically we have two
parameters, as usual.
00:04:13.910 --> 00:04:16.880
n is the dimensionality
of spin, and d
00:04:16.880 --> 00:04:21.339
would be the dimensionality
of our lattice.
00:04:21.339 --> 00:04:24.830
In practice for what
the calculations
00:04:24.830 --> 00:04:26.950
that we are going
to be doing, we
00:04:26.950 --> 00:04:29.900
will be focusing in
d that is close to 2.
00:04:36.880 --> 00:04:48.020
Now if the odd fluctuations
at finite t, what happens
00:04:48.020 --> 00:04:51.900
is that the state of the
vector is going to change.
00:04:51.900 --> 00:04:58.920
So this Si at finite
temperature would no longer
00:04:58.920 --> 00:05:01.150
be pointing along
the last component.
00:05:01.150 --> 00:05:04.270
It will start to
have fluctuations.
00:05:04.270 --> 00:05:08.770
Those fluctuations will change
the 0 from the ground state
00:05:08.770 --> 00:05:12.900
to some value I'll call
pi 1, the next one pi 2,
00:05:12.900 --> 00:05:17.570
all of the big pi n minus 1.
00:05:17.570 --> 00:05:21.970
And since the whole entire
thing is a unit vector,
00:05:21.970 --> 00:05:26.410
the last component has to
shrink to adjust for that.
00:05:26.410 --> 00:05:31.020
So we would indicate the
last component by sigma.
00:05:31.020 --> 00:05:37.050
So essentially this subspace of
fluctuations around the ground
00:05:37.050 --> 00:05:40.795
state is captured
through this vector
00:05:40.795 --> 00:05:45.790
pi that is n minus
1 dimensional.
00:05:45.790 --> 00:05:49.910
And this corresponds
to the transverse modes
00:05:49.910 --> 00:05:52.500
that we're looking
at when we were doing
00:05:52.500 --> 00:05:56.070
the expansion of the
Landau-Ginzburg model
00:05:56.070 --> 00:06:00.120
around its symmetry
broken state.
00:06:00.120 --> 00:06:02.920
In this case, the
longitudinal mode,
00:06:02.920 --> 00:06:06.900
essentially, is
infinitely stiff.
00:06:06.900 --> 00:06:09.870
You don't have the
ability to stretch along
00:06:09.870 --> 00:06:12.500
the longitudinal mode
because of the constraint
00:06:12.500 --> 00:06:14.530
that we have put over here.
00:06:14.530 --> 00:06:19.710
So if you think back,
we had this wine bottle,
00:06:19.710 --> 00:06:22.450
or Mexican hat potential.
00:06:22.450 --> 00:06:25.380
And the Goldstone
modes corresponded
00:06:25.380 --> 00:06:29.000
to going along the
bottom, and how easy
00:06:29.000 --> 00:06:32.770
it was to climb
this Mexican hat was
00:06:32.770 --> 00:06:36.010
determined by the
longitudinal mode.
00:06:36.010 --> 00:06:40.170
In this case, the Mexican hat
has become very, very stiff
00:06:40.170 --> 00:06:41.870
to climb on the sides.
00:06:41.870 --> 00:06:44.320
So you don't have the
longitudinal mode.
00:06:44.320 --> 00:06:48.070
You just have these
Goldstone modes.
00:06:48.070 --> 00:06:51.150
The cost to pay
for that is that I
00:06:51.150 --> 00:06:54.100
have to be very careful in
calculating the partition
00:06:54.100 --> 00:06:55.430
function.
00:06:55.430 --> 00:06:59.400
If I'm integrating
over the n components
00:06:59.400 --> 00:07:06.865
of some particular spin,
I have to make sure
00:07:06.865 --> 00:07:14.610
that I remember that this sum
of all of these components is 1.
00:07:18.490 --> 00:07:22.720
So I have to integrate
subject to that constraint.
00:07:22.720 --> 00:07:27.740
And the way that I have
broken things down now,
00:07:27.740 --> 00:07:36.310
I'm integrating over the n minus
1 component of this vector pi.
00:07:36.310 --> 00:07:41.010
And this additional
direction, d Sigma,
00:07:41.010 --> 00:07:44.310
but I can't do both
of them independently.
00:07:44.310 --> 00:07:46.460
Because there's a
delta function that
00:07:46.460 --> 00:07:51.720
enforces that sigma squared
plus pi squared equals to 1.
00:07:51.720 --> 00:07:54.140
The pi squared corresponds
to the magnitude
00:07:54.140 --> 00:07:58.490
of this n minus 1
component vector.
00:07:58.490 --> 00:08:05.610
And essentially, I can solve
for this delta function,
00:08:05.610 --> 00:08:08.490
and really replace
this sigma over here
00:08:08.490 --> 00:08:11.525
with square root of
1 minus pi squared.
00:08:14.740 --> 00:08:19.690
But I have to be a little bit
careful in my integrations.
00:08:19.690 --> 00:08:21.800
Because this delta
function I can
00:08:21.800 --> 00:08:24.800
write as a delta
function of sigma
00:08:24.800 --> 00:08:28.433
plus or minus square root
of 1 minus pi squared.
00:08:35.190 --> 00:08:41.059
And there is a rule that if
I use this delta function
00:08:41.059 --> 00:08:45.410
to set sigma to be equal to
square root of 1 minus pi
00:08:45.410 --> 00:08:48.210
squared, like I
have done over here,
00:08:48.210 --> 00:08:53.600
I have to be careful that the
delta function of a times x
00:08:53.600 --> 00:08:58.670
is actually a delta function
of x divided by modulus of a.
00:08:58.670 --> 00:09:01.540
So essentially, I have to
substitute something here.
00:09:01.540 --> 00:09:06.950
So this is, in fact, equal
to the integration in the pi
00:09:06.950 --> 00:09:11.300
directions because of the
use of this delta function
00:09:11.300 --> 00:09:14.520
to set the value
of sigma, I have
00:09:14.520 --> 00:09:17.670
to divide by the square
root of 1 minus pi squared.
00:09:21.970 --> 00:09:24.070
Which actually
shortly we will write
00:09:24.070 --> 00:09:25.360
this in the following way.
00:09:35.020 --> 00:09:37.740
I guess there's an
overall factor of 1/2
00:09:37.740 --> 00:09:40.670
but it doesn't really matter.
00:09:40.670 --> 00:09:42.410
So yes?
00:09:42.410 --> 00:09:44.290
AUDIENCE: So what do
you do with the fact
00:09:44.290 --> 00:09:48.190
that there are two places where
the delta function is done?
00:09:48.190 --> 00:09:51.020
PROFESSOR: I'm
continuously connecting
00:09:51.020 --> 00:09:53.780
to the solution that
starts at 0 temperature
00:09:53.780 --> 00:09:55.210
with a particular state.
00:09:55.210 --> 00:09:59.210
So I have removed that
ambiguity by the starting
00:09:59.210 --> 00:10:02.740
points of my cube.
00:10:02.740 --> 00:10:08.070
But if I was integrating
over all possibilities,
00:10:08.070 --> 00:10:10.720
then I should really
add that on too.
00:10:10.720 --> 00:10:15.010
And really just make
the partition function
00:10:15.010 --> 00:10:17.150
with the sum of two
equivalent terms--
00:10:17.150 --> 00:10:21.740
one around this ground state,
one around another state.
00:10:21.740 --> 00:10:24.400
AUDIENCE: The product is
supposed to be for a lattice
00:10:24.400 --> 00:10:28.390
site-- for the integration
variable, not for the--
00:10:28.390 --> 00:10:31.130
PROFESSOR: Right.
00:10:31.130 --> 00:10:34.080
So I did something bad here.
00:10:34.080 --> 00:10:39.602
So here I should have written--
so this is an n component
00:10:39.602 --> 00:10:43.762
integration that I have
to do on each side.
00:10:43.762 --> 00:10:46.070
Now let's pick one of the sides.
00:10:46.070 --> 00:10:49.850
So let's say we
pick the side pi.
00:10:49.850 --> 00:10:54.224
For that, I have a small
n integration to do.
00:11:04.120 --> 00:11:05.090
What does it say?
00:11:05.090 --> 00:11:10.520
It basically says
that if, for example,
00:11:10.520 --> 00:11:15.130
I am looking at the
case of n equals to 2,
00:11:15.130 --> 00:11:20.570
then I have started
with a state that
00:11:20.570 --> 00:11:23.600
points along this direction.
00:11:23.600 --> 00:11:26.410
But now I'm allowing
fluctuations pi
00:11:26.410 --> 00:11:29.650
in this direction.
00:11:29.650 --> 00:11:34.560
And I can't simply say that the
amount of these fluctuations,
00:11:34.560 --> 00:11:38.640
pi, is going let's say from
minus infinity to infinity.
00:11:38.640 --> 00:11:42.820
Because how much
pi I have changes
00:11:42.820 --> 00:11:45.730
actually whether it is
small or whether it is large
00:11:45.730 --> 00:11:48.050
when I'm down here.
00:11:48.050 --> 00:11:53.170
And so there's constraints
let's say on how big pi can be.
00:11:53.170 --> 00:11:55.530
Pi cannot be larger than 1.
00:11:55.530 --> 00:11:58.550
And essentially, a
particular magnitude
00:11:58.550 --> 00:12:03.064
of pi-- how much weight does
it has, it is captured by this.
00:12:08.790 --> 00:12:12.710
So that's one thing
to remember when
00:12:12.710 --> 00:12:16.040
we are dealing with
integration over unit spins,
00:12:16.040 --> 00:12:19.700
and we want to look
at the fluctuations.
00:12:19.700 --> 00:12:24.610
The other choice of notation
that I would like to do
00:12:24.610 --> 00:12:26.040
is the following.
00:12:26.040 --> 00:12:32.140
I said that my
starting [INAUDIBLE]
00:12:32.140 --> 00:12:37.949
is a 0 sum over all nearest
neighbors Si dot Sj.
00:12:41.653 --> 00:12:45.870
Now in the state
where all of the spins
00:12:45.870 --> 00:12:50.070
are pointing in one direction,
this factor is unity.
00:12:50.070 --> 00:12:56.110
So the 0 temperature state gets
a factor of 1 here on each one.
00:12:56.110 --> 00:12:58.420
Let's say we are in a
hyper cubic lattice.
00:12:58.420 --> 00:13:00.610
There are d bonds per site.
00:13:00.610 --> 00:13:04.690
So at 0 temperature, I
would have NdK0, basically--
00:13:04.690 --> 00:13:07.980
the value of this ground state.
00:13:07.980 --> 00:13:12.660
And then if I have
fluctuations from that state,
00:13:12.660 --> 00:13:17.920
I can capture that as
follows, as minus K0 over 2.
00:13:17.920 --> 00:13:27.596
It's a reduction in this energy,
as sum over ij Si minus Sj
00:13:27.596 --> 00:13:28.096
squared.
00:13:33.530 --> 00:13:35.330
And you can check that.
00:13:35.330 --> 00:13:39.072
If I square these
terms, I'm going
00:13:39.072 --> 00:13:43.060
to get 1, 1, 1/2, which
basically reproduces
00:13:43.060 --> 00:13:45.900
this, which actually
goes over her.
00:13:45.900 --> 00:13:54.410
And the dot product, minus 2 Si
dot Sj, cancels this minus 1/2,
00:13:54.410 --> 00:13:56.628
basically gives
you this exactly.
00:13:59.610 --> 00:14:02.700
So the reason I write
it in this fashion
00:14:02.700 --> 00:14:08.700
is because very shortly, I want
to switch from going and doing
00:14:08.700 --> 00:14:12.490
things on a lattice to
going to a continuum.
00:14:12.490 --> 00:14:16.650
And you can see that
this form, summing over
00:14:16.650 --> 00:14:19.340
the difference between
near neighbors,
00:14:19.340 --> 00:14:24.500
I very nicely can go
to a gradient squared.
00:14:24.500 --> 00:14:27.220
So essentially that's
what I want to do.
00:14:27.220 --> 00:14:33.040
Whenever I have a
sum over a site,
00:14:33.040 --> 00:14:39.260
I want to replace it with
an integral over a space.
00:14:39.260 --> 00:14:42.350
And I guess to keep
things dimensionless,
00:14:42.350 --> 00:14:45.450
I have to divide by a to get d.
00:14:45.450 --> 00:14:51.180
So I can call that the density
that I have to include,
00:14:51.180 --> 00:14:54.040
which is also the same thing
as the number of lattice
00:14:54.040 --> 00:14:56.342
points in the body of the box.
00:15:00.710 --> 00:15:06.560
So my minus beta
H in the continuum
00:15:06.560 --> 00:15:10.350
goes over to whatever
the contribution
00:15:10.350 --> 00:15:13.480
of the completely
aligned state is.
00:15:13.480 --> 00:15:18.770
And then whatever the
difference of the spins
00:15:18.770 --> 00:15:22.620
is, because of the
small fluctuations,
00:15:22.620 --> 00:15:28.080
I will capture through an
integration of gradient of S
00:15:28.080 --> 00:15:28.580
squared.
00:15:33.380 --> 00:15:39.550
And I call the original
coupling that we're basically
00:15:39.550 --> 00:15:43.790
the strength of the
interaction divide by KdK0.
00:15:43.790 --> 00:15:47.080
Clearly in order
to get the coupling
00:15:47.080 --> 00:15:49.650
that I have in the
continuum, I have
00:15:49.650 --> 00:15:52.175
to have this fact of a to the d.
00:15:52.175 --> 00:15:58.000
But then in the gradient, I
also have to divide by distance.
00:15:58.000 --> 00:16:02.660
So there's something here, a
factor of a to the 2 minus d
00:16:02.660 --> 00:16:06.860
that relates these two factors.
00:16:06.860 --> 00:16:09.310
It should be the
other way around.
00:16:22.140 --> 00:16:24.359
It doesn't matter.
00:16:24.359 --> 00:16:25.150
AUDIENCE: Question.
00:16:25.150 --> 00:16:27.850
PROFESSOR: Yes.
00:16:27.850 --> 00:16:32.178
AUDIENCE: This step is
only valid for a cubicle--
00:16:32.178 --> 00:16:33.480
PROFESSOR: Yes.
00:16:33.480 --> 00:16:36.250
So if it was something
like a triangular lattice,
00:16:36.250 --> 00:16:39.390
or something, there would be
some miracle factors here.
00:16:39.390 --> 00:16:40.874
AUDIENCE: But I
mean like writing
00:16:40.874 --> 00:16:42.290
the difference of
the spin squared
00:16:42.290 --> 00:16:44.504
as the gradient squared.
00:16:44.504 --> 00:16:46.620
Like if it were a
triangular lattice?
00:16:46.620 --> 00:16:48.100
PROFESSOR: Yeah,
so the statement
00:16:48.100 --> 00:16:52.000
is that whatever
lattice you have,
00:16:52.000 --> 00:16:54.960
what am I doing at the
level of the lattice,
00:16:54.960 --> 00:16:56.970
I'm trying to keep
things that are
00:16:56.970 --> 00:16:59.700
close to each other aligned.
00:16:59.700 --> 00:17:03.380
So when I go to the continuum,
how is this captured,
00:17:03.380 --> 00:17:06.720
it's a term like a
gradient squared.
00:17:06.720 --> 00:17:10.980
Now on the hyper cubic
lattices, the relationship
00:17:10.980 --> 00:17:12.950
between what you
put on the bonds
00:17:12.950 --> 00:17:16.270
of the hyper cubic lattice and
what we've got in the continuum
00:17:16.270 --> 00:17:18.359
is immediately apparent.
00:17:18.359 --> 00:17:20.609
If you try to do it on
the triangular lattice,
00:17:20.609 --> 00:17:21.630
you still can.
00:17:21.630 --> 00:17:23.876
And you'll find that
at the end of the day,
00:17:23.876 --> 00:17:25.709
you will get the factor
of square root of 3,
00:17:25.709 --> 00:17:26.890
or something like that.
00:17:26.890 --> 00:17:29.700
So there's some miracle
factor that comes into play.
00:17:36.400 --> 00:17:39.640
And then at the end
of the day, I also
00:17:39.640 --> 00:17:43.200
want to replace these
gradient of a squared
00:17:43.200 --> 00:17:50.230
naturally in terms of
essentially S has n components.
00:17:50.230 --> 00:17:53.680
N minus one of them are pi,
and one of them is sigma.
00:17:53.680 --> 00:17:59.680
So this would be minus
K/2 integral d dx.
00:17:59.680 --> 00:18:02.596
I have gradient of
the pi component,
00:18:02.596 --> 00:18:05.998
and a gradient of the
sigma component squared.
00:18:10.830 --> 00:18:31.080
So after integrating sigma
using the delta functions,
00:18:31.080 --> 00:18:33.155
I'm going to the
continuum limit.
00:18:38.540 --> 00:18:43.490
The partition function
that we have to evaluate up
00:18:43.490 --> 00:18:50.430
to various non-singular
factors, such as this constant
00:18:50.430 --> 00:18:56.680
over here, is obtained
by integrating over
00:18:56.680 --> 00:19:01.030
all configurations
of our pi field,
00:19:01.030 --> 00:19:05.636
now regarded as a
continuously varying object
00:19:05.636 --> 00:19:07.310
under the dimensional lattice.
00:19:10.150 --> 00:19:22.060
And the weight,
which is as follows,
00:19:22.060 --> 00:19:30.590
there is a gradient of
pi squared, essentially
00:19:30.590 --> 00:19:33.490
this term over here.
00:19:33.490 --> 00:19:37.970
There is the gradient
the other term.
00:19:37.970 --> 00:19:42.200
The other term, however,
if I use the delta function
00:19:42.200 --> 00:19:50.780
in square root of 1
minus pi squared squared.
00:19:50.780 --> 00:19:54.940
And then there's this
factor from the integration
00:19:54.940 --> 00:19:58.490
that I have to be careful
of, which I can also
00:19:58.490 --> 00:20:04.580
take to the exponent, and write
as, again, this density times
00:20:04.580 --> 00:20:07.310
log of 1 minus pi squared.
00:20:07.310 --> 00:20:09.632
There's, I think, a
factor of [INAUDIBLE].
00:20:23.820 --> 00:20:30.350
So the weight that I
had started with, with S
00:20:30.350 --> 00:20:36.660
dot S was kind of
very simple-looking.
00:20:36.660 --> 00:20:40.610
But because of the
constraints, was
00:20:40.610 --> 00:20:48.010
hiding a number of conditions.
00:20:48.010 --> 00:20:52.890
And if we explicitly
look at those conditions
00:20:52.890 --> 00:20:59.056
and ask what is the
weight of the fluctuations
00:20:59.056 --> 00:21:03.430
that I have to put
around the ground
00:21:03.430 --> 00:21:05.550
state, these
Goldstone modes, that
00:21:05.550 --> 00:21:08.780
is captured with
this Hamiltonian.
00:21:08.780 --> 00:21:13.580
Part of it is this
old contribution
00:21:13.580 --> 00:21:16.120
from Goldstone
modes, the transfer
00:21:16.120 --> 00:21:18.290
modes that we had seen.
00:21:18.290 --> 00:21:23.720
But now being more careful, we
see that these Goldstone modes,
00:21:23.720 --> 00:21:27.380
I have to be careful about
integrating over them
00:21:27.380 --> 00:21:33.650
because of the additional terms
that capture, essentially,
00:21:33.650 --> 00:21:38.520
the original full symmetry,
full rotational symmetry,
00:21:38.520 --> 00:21:42.715
that was present in
integration over S. Yes?
00:21:42.715 --> 00:21:45.355
AUDIENCE: The integration--
the functional integration,
00:21:45.355 --> 00:21:50.880
pi should be linked
up to a sphere of 1.
00:21:50.880 --> 00:21:53.970
PROFESSOR: This will keep.
00:21:53.970 --> 00:21:59.300
So I put that
constraint over here.
00:21:59.300 --> 00:22:03.110
And it's not just that it
is limited to something,
00:22:03.110 --> 00:22:05.360
but for a particular
value of pi,
00:22:05.360 --> 00:22:06.823
it gets this additional weight.
00:22:12.250 --> 00:22:15.470
So if you like, once I
try to take my integrals
00:22:15.470 --> 00:22:19.808
outside that region, that
factor says the weight as usual.
00:22:28.610 --> 00:22:32.205
So this entity is called
the non-linear sigma model.
00:22:34.840 --> 00:22:40.065
And I never understood why they
don't call it a non-linear pi
00:22:40.065 --> 00:22:40.908
model.
00:22:40.908 --> 00:22:45.090
Because we integrate
immediately with sigma.
00:22:45.090 --> 00:22:46.363
That's how it is.
00:22:49.470 --> 00:22:56.730
So what you're going to do
if we had, essentially, stuff
00:22:56.730 --> 00:22:59.870
that the first term not included
any of the other things,
00:22:59.870 --> 00:23:02.730
we will have had the
analysis of Goldstone modes
00:23:02.730 --> 00:23:06.370
that we had done previously.
00:23:06.370 --> 00:23:08.190
The effect of these
things, you can
00:23:08.190 --> 00:23:12.360
see if I start making
expansion in powers of pi,
00:23:12.360 --> 00:23:16.000
is to generate
interactions that will
00:23:16.000 --> 00:23:18.650
be non-linear terms
among the parts.
00:23:18.650 --> 00:23:21.440
So these Goldstone modes
that we were previously
00:23:21.440 --> 00:23:26.480
dealing with as independent
modes of the system,
00:23:26.480 --> 00:23:29.270
are actually
non-linearly coupled.
00:23:29.270 --> 00:23:31.440
And we want to know
what the effect of that
00:23:31.440 --> 00:23:33.990
is on the behavior
of the entire system.
00:23:36.660 --> 00:23:41.240
So whenever we're faced
with a non-linear theory,
00:23:41.240 --> 00:23:45.080
we have to do some kind of
a preservative analysis.
00:23:45.080 --> 00:23:50.680
And the first thing that
you may be tempted to do
00:23:50.680 --> 00:23:54.100
is to expand the
powers of pi, and then
00:23:54.100 --> 00:23:57.070
look at the Gaussian part,
and then the higher order
00:23:57.070 --> 00:23:58.900
parts, etc.
00:23:58.900 --> 00:24:00.370
That's a way of doing it.
00:24:00.370 --> 00:24:04.700
But there's actually another
way that is more consistent,
00:24:04.700 --> 00:24:09.820
which is to organize
the terms in this weight
00:24:09.820 --> 00:24:14.150
according to powers
of temperature.
00:24:14.150 --> 00:24:17.710
Because after all, I started
with a zero temperature
00:24:17.710 --> 00:24:19.000
configuration.
00:24:19.000 --> 00:24:21.610
And I'm hoping
that I'm expanding
00:24:21.610 --> 00:24:23.540
for small fluctuation.
00:24:23.540 --> 00:24:26.710
So my idea is to-- I
know the ground state.
00:24:26.710 --> 00:24:30.560
I want to see what happens
if I go slightly beyond that.
00:24:30.560 --> 00:24:33.490
And the reason for
fluctuations is temperature,
00:24:33.490 --> 00:24:45.410
so organize terms in this
effective Hamiltonian
00:24:45.410 --> 00:24:50.540
for the pis in powers
of temperature.
00:24:58.190 --> 00:25:02.980
And temperature, by this I mean
the inverse of this coupling
00:25:02.980 --> 00:25:06.460
constant, K,
because even, again,
00:25:06.460 --> 00:25:09.040
if I go through
my old derivation,
00:25:09.040 --> 00:25:11.620
you can see that
I go minus beta H,
00:25:11.620 --> 00:25:14.850
so K0 should be inversely
proportional to temperature.
00:25:14.850 --> 00:25:16.470
K is proportion to K0.
00:25:16.470 --> 00:25:19.320
It should be inversely
proportional to temperature.
00:25:19.320 --> 00:25:23.110
So to some overall
coefficient, let's just
00:25:23.110 --> 00:25:26.760
define temperature [INAUDIBLE].
00:25:26.760 --> 00:25:33.040
Now we see that at the
level that we were looking
00:25:33.040 --> 00:25:37.220
at things before,
from this term it's
00:25:37.220 --> 00:25:40.730
kind of like a
Gaussian form, where
00:25:40.730 --> 00:25:44.470
I have something like K, which
is the inverse temperature
00:25:44.470 --> 00:25:47.460
pi squared.
00:25:47.460 --> 00:25:54.730
So just on dimensional grounds,
up to functional forms, etc.
00:25:54.730 --> 00:26:00.630
we expect pi squared to be
proportional to temperature
00:26:00.630 --> 00:26:02.280
at the 0 order, if you like.
00:26:04.850 --> 00:26:07.260
Because, again, if
temperature goes to 0,
00:26:07.260 --> 00:26:10.380
there's not going
to no fluctuations.
00:26:10.380 --> 00:26:14.350
As I go away from 0 temperature,
the average fluctuations
00:26:14.350 --> 00:26:14.960
will be 0.
00:26:14.960 --> 00:26:17.630
Average squared will be
proportional to temperature.
00:26:17.630 --> 00:26:19.930
It all makes sense.
00:26:19.930 --> 00:26:25.230
So then if I look
at this term, I
00:26:25.230 --> 00:26:28.420
see that dimensionally,
it is inverse temperature
00:26:28.420 --> 00:26:30.670
pi squared is of the
order of temperature.
00:26:30.670 --> 00:26:35.830
So this is dimensionally
t to the 0.
00:26:35.830 --> 00:26:38.920
Whereas if I start
to expand this,
00:26:38.920 --> 00:26:44.260
this log I can start to
expand as minus pi squared
00:26:44.260 --> 00:26:52.210
plus pi to the 4th over 2 pi to
the 6th over 3, and so forth.
00:26:52.210 --> 00:26:56.730
You can see that subsequent
terms in this series
00:26:56.730 --> 00:27:01.070
are higher and higher
order in this temperature.
00:27:01.070 --> 00:27:03.270
This will be the order of
temperature-- temperature
00:27:03.270 --> 00:27:05.030
squared, temperature cubed.
00:27:05.030 --> 00:27:10.585
And already we can see
that this term is small
00:27:10.585 --> 00:27:12.950
compared to this term.
00:27:12.950 --> 00:27:16.610
So although this
is a Gaussian term,
00:27:16.610 --> 00:27:19.780
and I would've
maybe been tempted
00:27:19.780 --> 00:27:22.550
to put it in the 0
order Hamiltonian,
00:27:22.550 --> 00:27:24.530
If I'm organizing
things according
00:27:24.530 --> 00:27:29.420
to orders of temperature, my
0-th order will remain this.
00:27:29.420 --> 00:27:31.690
This will be the
contribution to first order,
00:27:31.690 --> 00:27:33.780
2nd order, 3rd order.
00:27:33.780 --> 00:27:37.870
And similarly, I can
start expanding this.
00:27:37.870 --> 00:27:43.610
Square root is 1 minus
pi squared over 2.
00:27:43.610 --> 00:27:47.395
So then I take the gradient
of minus pi squared over 2.
00:27:47.395 --> 00:27:50.290
I will get pi gradient of pi.
00:27:50.290 --> 00:27:54.500
You can see that the lowest
order term in this expansion
00:27:54.500 --> 00:27:59.040
will be pi gradient of
pi squared, and then
00:27:59.040 --> 00:28:02.350
higher order terms.
00:28:02.350 --> 00:28:07.020
And this is something that
is order of pi to the 4th,
00:28:07.020 --> 00:28:12.170
so it gives the order of
temperature squared multiplied
00:28:12.170 --> 00:28:13.710
by inverse temperatures.
00:28:13.710 --> 00:28:17.066
So this is a term that is
contributing to order of T
00:28:17.066 --> 00:28:19.680
to 0, T to the 1.
00:28:19.680 --> 00:28:27.220
So basically, at order of T
to 0, I have as my beta H0
00:28:27.220 --> 00:28:34.161
just the integral d dx K/2
gradient of pi squared.
00:28:38.000 --> 00:28:43.830
While at order of
T the 1st power,
00:28:43.830 --> 00:28:50.170
I will have a correction
which has two types of terms.
00:28:50.170 --> 00:28:59.350
One term is this K/2 integral
d dx pi gradient of pi squared,
00:28:59.350 --> 00:29:04.190
coming from what was the
gradient of sigma squared.
00:29:04.190 --> 00:29:08.220
And then from here, I
will get a minus rho
00:29:08.220 --> 00:29:13.020
over 2 integral d dx pi squared.
00:29:13.020 --> 00:29:16.860
And then there will be other
terms like order of T squared,
00:29:16.860 --> 00:29:22.140
U2, and so forth.
00:29:22.140 --> 00:29:28.430
So I just re-organized terms
in this interacting Hamiltonian
00:29:28.430 --> 00:29:33.300
in what I expected to be
powers of this temperature.
00:29:33.300 --> 00:29:39.290
Now here we-- one of the
first things that we will do
00:29:39.290 --> 00:29:44.260
is to look at this and
realize that we can decompose
00:29:44.260 --> 00:29:46.955
into modes by going
to previous space,
00:29:46.955 --> 00:29:49.180
I do a Fourier transform.
00:29:49.180 --> 00:29:57.710
This thing becomes K/2 integral
dd q divided by 2 pi to the d q
00:29:57.710 --> 00:30:03.230
squared pi theta of q squared.
00:30:03.230 --> 00:30:06.190
So let's write it as
pi theta [INAUDIBLE].
00:30:10.970 --> 00:30:16.850
And again, as usual,
we will end up
00:30:16.850 --> 00:30:22.987
needing to calculate averages
with this Gaussian rate.
00:30:22.987 --> 00:30:29.655
And what we have here is that
pi alpha of q1 pi beta of q2,
00:30:29.655 --> 00:30:33.150
we get this 0-th ordered rate.
00:30:33.150 --> 00:30:36.660
The components have
to be the same.
00:30:36.660 --> 00:30:42.062
The sum of the two
momenta has to be 0.
00:30:42.062 --> 00:30:46.972
And if so, I just
get K q squared.
00:30:56.301 --> 00:31:01.040
Now I can similarly
Fourier transform the terms
00:31:01.040 --> 00:31:04.310
that I have over here.
00:31:04.310 --> 00:31:08.790
So the interactions
to one-- first one
00:31:08.790 --> 00:31:13.020
becomes rather complicated.
00:31:13.020 --> 00:31:16.630
We saw that when we
have something that's is
00:31:16.630 --> 00:31:19.200
four powers of a field.
00:31:19.200 --> 00:31:21.990
And when we go to
Fourier space, rather
00:31:21.990 --> 00:31:24.220
than having one
integral over x, we
00:31:24.220 --> 00:31:26.750
ended up with
multiple integrals.
00:31:26.750 --> 00:31:29.460
So I will have, essentially,
Fourier transform
00:31:29.460 --> 00:31:32.640
of four factors of pi.
00:31:32.640 --> 00:31:35.580
For each one of them I
will have an integration.
00:31:35.580 --> 00:31:42.160
So I will have dd
q1, dd q2, dd q3.
00:31:42.160 --> 00:31:45.620
And the reason I
don't have the 4th one
00:31:45.620 --> 00:31:51.990
is because of the integration
over x, forcing the four q's
00:31:51.990 --> 00:31:55.350
to be added up to 0.
00:31:55.350 --> 00:32:07.885
So I will have pi alpha
of q1, pi alpha of q2,
00:32:07.885 --> 00:32:14.130
Now note that this
high gradient of pi
00:32:14.130 --> 00:32:16.670
came from a gradient
of pi squared,
00:32:16.670 --> 00:32:20.490
which means that the two
pis that go with this,
00:32:20.490 --> 00:32:23.290
carry the same index.
00:32:23.290 --> 00:32:27.620
Whereas for the next
factor, pi gradient of pi,
00:32:27.620 --> 00:32:29.318
they came from different ones.
00:32:29.318 --> 00:32:35.662
So I have pi q3 I beta
minus q1 minus q2 minus q3.
00:32:41.540 --> 00:32:44.490
Now if I just written
this, this would've
00:32:44.490 --> 00:32:49.800
been the Fourier transform of
my usual 4th order interaction.
00:32:49.800 --> 00:32:51.950
But that's not what
I have because I
00:32:51.950 --> 00:32:55.510
have two additional gradients.
00:32:55.510 --> 00:33:02.060
And so for two of
these factors actually
00:33:02.060 --> 00:33:04.620
I had to take the
gradient first.
00:33:04.620 --> 00:33:07.410
And every time I take a
gradient in Fourier space,
00:33:07.410 --> 00:33:10.476
I will bring a factor of I q.
00:33:10.476 --> 00:33:18.430
So I will have here I q1
dotted with Iq let's say 3.
00:33:24.820 --> 00:33:30.750
So the Fourier transform of
the leading quartic interaction
00:33:30.750 --> 00:33:36.272
that I have, is actually the
form that I have over here.
00:33:36.272 --> 00:33:38.090
There is a trivial
term that comes
00:33:38.090 --> 00:33:40.070
from Fourier transforming.
00:33:40.070 --> 00:33:44.358
It's pi squared because
then I Fourier transform
00:33:44.358 --> 00:33:52.976
that, I get simply pi
alpha of q squared.
00:33:57.080 --> 00:33:59.304
Yes?
00:33:59.304 --> 00:34:00.720
AUDIENCE: Does it
matter which q's
00:34:00.720 --> 00:34:04.120
you're pulling out
as the gradient?
00:34:04.120 --> 00:34:07.170
PROFESSOR: You can see that
these four pis over here
00:34:07.170 --> 00:34:10.449
in Fourier space appear
completely interchangeably.
00:34:10.449 --> 00:34:12.280
So it really doesn't matter, no.
00:34:12.280 --> 00:34:15.865
Because by permutation and
re-ordering these integration,
00:34:15.865 --> 00:34:18.290
you can move it
into something else.
00:34:21.026 --> 00:34:26.270
No, there is-- I shouldn't--
I'll draw a diagram that
00:34:26.270 --> 00:34:30.634
corresponds to that that will
make one constraint apparent.
00:34:30.634 --> 00:34:37.239
So when I was drawing
interaction terms for m
00:34:37.239 --> 00:34:39.389
to the 4th tier for
Landau-Ginzburg,
00:34:39.389 --> 00:34:44.070
and I have something
that has 4 interactions,
00:34:44.070 --> 00:34:47.370
I would draw something
that has 2 lines.
00:34:47.370 --> 00:34:50.670
But the 2 lines had 2 branches.
00:34:50.670 --> 00:34:54.704
And the branching was supposed
to indicate that 2 of them
00:34:54.704 --> 00:34:57.420
were carrier 1
index, and 2 of them
00:34:57.420 --> 00:35:00.790
were carrying the same index.
00:35:00.790 --> 00:35:03.815
Now I have to make
sure that I indicate
00:35:03.815 --> 00:35:05.540
that the branches
of these things
00:35:05.540 --> 00:35:08.140
additionally have
these gradients
00:35:08.140 --> 00:35:11.100
for the Iq's
associated with them.
00:35:11.100 --> 00:35:14.750
And I make a
convention the branch,
00:35:14.750 --> 00:35:20.600
or the q, that has the gradient
on it, I will put a line.
00:35:20.600 --> 00:35:23.720
Now you can see
that if I go back
00:35:23.720 --> 00:35:28.720
and look at the origin of
this, that one of the gradients
00:35:28.720 --> 00:35:32.530
acts on one pair of
pis, and the other acts
00:35:32.530 --> 00:35:35.000
of the other pairs of pis.
00:35:35.000 --> 00:35:38.902
So the other dashed line I
cannot put on the same branch,
00:35:38.902 --> 00:35:41.620
but I have to put over here.
00:35:41.620 --> 00:35:45.110
So the one constraint that
I have to be careful of
00:35:45.110 --> 00:35:48.900
is that these Iq' should
pick one from alpha and one
00:35:48.900 --> 00:35:49.882
from beta.
00:35:58.007 --> 00:35:59.590
This is the diagrammatic
presentation.
00:36:09.630 --> 00:36:16.000
So what I can do is to now
start doing perturbation
00:36:16.000 --> 00:36:17.740
in these interaction.
00:36:17.740 --> 00:36:20.880
You want to do the
lowest order to see
00:36:20.880 --> 00:36:25.630
what the first correction
because of fluctuations
00:36:25.630 --> 00:36:30.590
and interaction of
these Goldstone modes.
00:36:30.590 --> 00:36:34.210
But rather than do
things in two steps,
00:36:34.210 --> 00:36:37.510
first doing perturbation,
encountering difficulty,
00:36:37.510 --> 00:36:41.190
and then converting things
to a normalization group,
00:36:41.190 --> 00:36:44.820
which we've already seen that
happen, that story, in dealing
00:36:44.820 --> 00:36:49.380
with the Landau-Ginzburg
model, Let's immediately
00:36:49.380 --> 00:36:51.670
do the perturbative
renormalization group
00:36:51.670 --> 00:36:54.310
of this model.
00:36:54.310 --> 00:36:56.940
So what I'm supposed
to do things
00:36:56.940 --> 00:37:01.600
is to note that all
of these theories
00:37:01.600 --> 00:37:04.590
came from some
underlying lattice model.
00:37:04.590 --> 00:37:07.640
I was carefully drawing
for you the first lattice
00:37:07.640 --> 00:37:09.410
model originally.
00:37:09.410 --> 00:37:14.910
Which means that there
is some cut off here,
00:37:14.910 --> 00:37:16.950
some lattice cut off.
00:37:16.950 --> 00:37:20.020
Which means that when
I go to Fourier space,
00:37:20.020 --> 00:37:29.640
there is always some kind of a
range of wave numbers or wave
00:37:29.640 --> 00:37:32.460
vectors that I have
to integrate with.
00:37:32.460 --> 00:37:43.010
So essentially, my
pi's are limited
00:37:43.010 --> 00:37:46.350
after I do a little
bit of averaging,
00:37:46.350 --> 00:37:53.470
if you like that there is
some shortest wavelength,
00:37:53.470 --> 00:37:58.200
and the corresponding
largest wave number, lambda,
00:37:58.200 --> 00:38:01.310
in [INAUDIBLE].
00:38:01.310 --> 00:38:04.820
And the procedure for
RG, the first one,
00:38:04.820 --> 00:38:10.960
was to think about
all of these pi modes,
00:38:10.960 --> 00:38:14.460
and brake them into two pieces.
00:38:14.460 --> 00:38:19.310
One's that we're responding
to the short wavelength
00:38:19.310 --> 00:38:22.265
fluctuations that we
want to get rid of,
00:38:22.265 --> 00:38:26.312
and the ones that correspond
to long wavelength fluctuations
00:38:26.312 --> 00:38:30.800
that we would like to keep.
00:38:30.800 --> 00:38:36.895
So my task is as follows, that
I have to really calculate
00:38:36.895 --> 00:38:41.010
the partition function over
here, which in it's Fourier
00:38:41.010 --> 00:38:47.880
representation indicates
averaging over all modes
00:38:47.880 --> 00:38:51.310
that's are in this orange.
00:38:51.310 --> 00:38:54.100
But those modes I'm
going to represent
00:38:54.100 --> 00:38:58.570
as D pi lesser, as
well as D pi greater.
00:38:58.570 --> 00:39:00.240
Each one of these
pi's is, of course,
00:39:00.240 --> 00:39:03.550
an n minus on component vector.
00:39:03.550 --> 00:39:09.650
And I have a rate
that i obtained
00:39:09.650 --> 00:39:14.792
by substituting pi lesser and
pi greater in the expressions
00:39:14.792 --> 00:39:17.820
that I have up there.
00:39:17.820 --> 00:39:24.510
And we can see already that
the 0-th order terms, as usual,
00:39:24.510 --> 00:39:28.124
nicely separates out
into a contribution
00:39:28.124 --> 00:39:35.480
that we have for pi lesser,
a contribution that we have
00:39:35.480 --> 00:39:43.080
for pi greater, and that the
interaction terms will then
00:39:43.080 --> 00:39:49.520
involve both of these modes.
00:39:49.520 --> 00:39:52.540
And in principle, I
could proceed and include
00:39:52.540 --> 00:39:53.920
higher and higher orders.
00:39:57.140 --> 00:40:03.697
Now I want to get rid of all
of the modes that are here.
00:40:03.697 --> 00:40:08.170
So that I have an effective
theory governing the modes
00:40:08.170 --> 00:40:11.150
that are the longer wavelengths,
once I have gotten rid
00:40:11.150 --> 00:40:14.710
of the short wavelength
fluctuations.
00:40:14.710 --> 00:40:19.400
So formally, once I have
integrated over pi greater
00:40:19.400 --> 00:40:22.715
in this double
integral, I will be
00:40:22.715 --> 00:40:27.080
left with the integration
over the pi lesser field.
00:40:31.517 --> 00:40:35.620
And the exponential gets
modified as follows.
00:40:35.620 --> 00:40:41.230
First of all, if I were
to ignore the interactions
00:40:41.230 --> 00:40:45.850
at the lowest order,
the effect of doing
00:40:45.850 --> 00:40:49.663
the integration of the Gaussian
modes that are out here,
00:40:49.663 --> 00:40:51.928
will, as usual,
be a contribution
00:40:51.928 --> 00:40:58.230
to the free energy of the
system coming from the modes
00:40:58.230 --> 00:41:00.250
that I integrated out.
00:41:00.250 --> 00:41:04.526
And clearly it also
depends, I forgot to say,
00:41:04.526 --> 00:41:11.510
that the range of integration
is now between lambda over b
00:41:11.510 --> 00:41:18.330
lambda, where b is my
renormalization factor.
00:41:18.330 --> 00:41:20.520
Yes?
00:41:20.520 --> 00:41:23.355
AUDIENCE: Because you're
coming from a lattice, does
00:41:23.355 --> 00:41:26.620
the particular shape of
the Brillouin zone matter
00:41:26.620 --> 00:41:28.840
more now, or still not really?
00:41:28.840 --> 00:41:32.050
PROFESSOR: It is in no
way different from what
00:41:32.050 --> 00:41:35.170
we were doing before in
the Landau-Ginzburg model.
00:41:35.170 --> 00:41:38.230
In the Landau-Ginzburg model,
I could have also started
00:41:38.230 --> 00:41:40.360
by putting spins,
or whatever degrees
00:41:40.360 --> 00:41:41.900
of freedom on a lattice.
00:41:41.900 --> 00:41:45.204
And let's say if I was
in hyper cubic lattice,
00:41:45.204 --> 00:41:48.340
I would've had Brillouin
zones, such as this.
00:41:48.340 --> 00:41:50.730
And the first thing
that we always said
00:41:50.730 --> 00:41:57.540
was that integrating
all of these things
00:41:57.540 --> 00:42:01.775
gives you an additional
totally harmless component
00:42:01.775 --> 00:42:05.890
to the energy that has
no similar part in it.
00:42:05.890 --> 00:42:12.030
So we're always searching
for the singularities that
00:42:12.030 --> 00:42:15.440
arise at the core
of this integration.
00:42:15.440 --> 00:42:17.600
Whatever you do
with the boundaries,
00:42:17.600 --> 00:42:24.037
no matter how complicated shapes
they have, they don't matter.
00:42:31.290 --> 00:42:36.230
So going back to here.
00:42:36.230 --> 00:42:40.660
If we had ignored
the interactions,
00:42:40.660 --> 00:42:43.620
integrating over
pi greater would've
00:42:43.620 --> 00:42:46.620
giving me this contribution
to the free energy.
00:42:46.620 --> 00:42:54.200
And, of course, beta H0 of
pi lesser would've remained.
00:42:56.870 --> 00:43:02.500
But now the effect of having
the interactions, as usual,
00:43:02.500 --> 00:43:05.450
it is like integrating
into the minus
00:43:05.450 --> 00:43:10.160
u with the rate over here.
00:43:10.160 --> 00:43:15.120
So I would have an
average such as this.
00:43:15.120 --> 00:43:19.030
And we do the cumulant
expansion, as usual.
00:43:19.030 --> 00:43:23.588
And the first term I
would get is the average
00:43:23.588 --> 00:43:32.550
of this quantity with
respect to the Gaussian rate,
00:43:32.550 --> 00:43:38.770
integrating out the high
component modes, high frequency
00:43:38.770 --> 00:43:40.702
modes, and high
order corrections.
00:43:44.560 --> 00:43:45.170
Yes?
00:43:45.170 --> 00:43:48.130
AUDIENCE: So right here you're
doing two expansions kind
00:43:48.130 --> 00:43:49.230
of simultaneously.
00:43:49.230 --> 00:43:52.147
One is you have non-linear
model that you're
00:43:52.147 --> 00:43:54.630
expanding different
powers and temperature.
00:43:54.630 --> 00:43:56.655
And then you
further on expand it
00:43:56.655 --> 00:44:01.670
to cumulants to be able
to account for that.
00:44:01.670 --> 00:44:08.293
PROFESSOR: No, because I
can organize this expansion
00:44:08.293 --> 00:44:12.500
in cumulants in
powers of temperature.
00:44:12.500 --> 00:44:18.420
So this u has an expansion
that is u1, u2, etc.
00:44:18.420 --> 00:44:20.490
organized in powers
of temperature.
00:44:20.490 --> 00:44:21.480
AUDIENCE: OK.
00:44:21.480 --> 00:44:24.780
PROFESSOR: And then when
I take the first cumulant,
00:44:24.780 --> 00:44:27.627
you can see that the average,
the lowest order term,
00:44:27.627 --> 00:44:28.127
will be--
00:44:28.127 --> 00:44:31.606
AUDIENCE: The first cumulant
is linear in temperature,
00:44:31.606 --> 00:44:32.886
and that's what you want?
00:44:32.886 --> 00:44:33.594
PROFESSOR: Right.
00:44:39.070 --> 00:44:44.040
So I'm being consistent
also with the perturbation
00:44:44.040 --> 00:44:48.730
that I had originally stated.
00:44:48.730 --> 00:44:53.650
Actually, since I drew a
diagram for the first term,
00:44:53.650 --> 00:44:57.920
I should state that this term,
since we are now also thinking
00:44:57.920 --> 00:45:03.495
of it as a correction
in u1, I have
00:45:03.495 --> 00:45:06.190
to regard it as 2 factors of pi.
00:45:06.190 --> 00:45:12.650
So I could potentially represent
it by a diagram such as this.
00:45:12.650 --> 00:45:23.620
So diagrammatically, my u1
that I have to take the average
00:45:23.620 --> 00:45:27.130
is composed of
these two entities.
00:45:38.940 --> 00:45:48.270
So what I need to do is to take
the average of that expression.
00:45:48.270 --> 00:45:55.645
So I can either do
that average over here.
00:45:55.645 --> 00:45:58.320
Take the average
of this expression,
00:45:58.320 --> 00:46:00.760
or do it diagrammatically.
00:46:00.760 --> 00:46:03.348
Let us go by the
diagrammatic route.
00:46:06.090 --> 00:46:09.400
So essentially,
what I'm doing is
00:46:09.400 --> 00:46:18.370
that every line that I see over
there that corresponds to pi,
00:46:18.370 --> 00:46:23.800
I am really decomposing
into two parts.
00:46:23.800 --> 00:46:26.910
One of them I will draw as a
straight line that corresponds
00:46:26.910 --> 00:46:31.070
to the pi lesser
that I am keeping.
00:46:31.070 --> 00:46:33.370
Or I replace it
with a wavy line,
00:46:33.370 --> 00:46:38.510
which is the pi greater that
I would be averaging over.
00:46:41.500 --> 00:46:47.940
So the first diagram I
had essentially something
00:46:47.940 --> 00:46:52.570
like this-- actually,
the second diagram.
00:46:52.570 --> 00:46:56.070
The one that comes
from rho pi squared.
00:46:56.070 --> 00:47:00.770
It's actually trivial, so let's
go through the possibilities.
00:47:00.770 --> 00:47:08.600
I can either have both of
these to be pi lessers-- sorry,
00:47:08.600 --> 00:47:09.570
pi greaters.
00:47:09.570 --> 00:47:11.870
So this is pi
greater, pi greater.
00:47:11.870 --> 00:47:14.500
And when I have
to do an average,
00:47:14.500 --> 00:47:16.275
then I can use
the formula that I
00:47:16.275 --> 00:47:19.580
have in red about the
average of 2 pi greaters.
00:47:22.790 --> 00:47:25.540
And that would
essentially amount
00:47:25.540 --> 00:47:28.190
to closing this thing down.
00:47:28.190 --> 00:47:33.855
And numerically, it would gives
me a factor of minus rho over 2
00:47:33.855 --> 00:47:40.680
integral d dK over 2 pi to the
d in the interval between lambda
00:47:40.680 --> 00:47:42.990
over b, lambda .
00:47:42.990 --> 00:47:46.590
And I have the
average of pi alpha pi
00:47:46.590 --> 00:47:51.060
alpha using a factor
of delta alpha alpha.
00:47:51.060 --> 00:47:55.940
Summing over alpha will give
me a factor of n minus 1.
00:47:55.940 --> 00:48:01.482
And the average would be
something like K k squared.
00:48:01.482 --> 00:48:07.540
So I would have to evaluate
something like this.
00:48:07.540 --> 00:48:09.910
But at the end of the day,
I don't care about it.
00:48:09.910 --> 00:48:11.700
Why don't I care about it?
00:48:11.700 --> 00:48:16.210
Because clearly the result of
doing this is another constant.
00:48:16.210 --> 00:48:19.430
It doesn't depend on pi lesser.
00:48:19.430 --> 00:48:29.260
So this is an addition
to the free energy
00:48:29.260 --> 00:48:34.450
once I integrate modes between
lambda over b to lambda,
00:48:34.450 --> 00:48:36.718
there is a contribution
to the free energy that
00:48:36.718 --> 00:48:38.350
comes from this term.
00:48:38.350 --> 00:48:42.410
It doesn't change
the rate that I
00:48:42.410 --> 00:48:47.650
have to assign to configurations
of the pi lesser field.
00:48:47.650 --> 00:48:49.230
That's another possibility.
00:48:49.230 --> 00:48:53.910
Another possibility is I
have one of them being a pi
00:48:53.910 --> 00:48:57.410
greater, one of them
being a pi lesser.
00:48:57.410 --> 00:49:02.630
Clearly, when I try to get
an average of this form,
00:49:02.630 --> 00:49:05.900
I have an average
of one factor of pi
00:49:05.900 --> 00:49:08.943
with a Gaussian
field that is even.
00:49:08.943 --> 00:49:10.655
So this is 0.
00:49:10.655 --> 00:49:13.190
We don't have to worry about it.
00:49:13.190 --> 00:49:19.640
And finally, I will get a
term, which is like this.
00:49:19.640 --> 00:49:22.190
Which doesn't involve
any integrations,
00:49:22.190 --> 00:49:24.920
and really amounts to
taking that term that I
00:49:24.920 --> 00:49:28.190
have over there, and just
making both of those pi
00:49:28.190 --> 00:49:30.010
to be pi lessers.
00:49:30.010 --> 00:49:33.610
So it's essentially the same
form that will reappear,
00:49:33.610 --> 00:49:38.620
now the integration being
from 0 to lambda over 2.
00:49:38.620 --> 00:49:45.180
So we know exactly what happens
with the term on the right.
00:49:45.180 --> 00:49:51.270
Nothing useful, or important
information emerges from it.
00:49:51.270 --> 00:49:56.510
If I go and look at
this one however,
00:49:56.510 --> 00:50:02.250
depending on where
I choose to put
00:50:02.250 --> 00:50:04.680
the solid lines
or the wavy lines,
00:50:04.680 --> 00:50:07.840
I will have a number
of possibilities.
00:50:07.840 --> 00:50:11.670
One thing that is
clearly going to be there
00:50:11.670 --> 00:50:17.270
is essentially I put pi lesser
for each one of the branches.
00:50:17.270 --> 00:50:21.320
Essentially, when i write
it here like this one,
00:50:21.320 --> 00:50:22.855
it is reproducing
the integration
00:50:22.855 --> 00:50:25.960
that I have over there,
except that, again, it
00:50:25.960 --> 00:50:30.990
only goes between 0 and lambda.
00:50:30.990 --> 00:50:34.080
And now I can start
adding wavy lines.
00:50:34.080 --> 00:50:39.360
Any diagram that has one wavy,
and I can put the wavy line
00:50:39.360 --> 00:50:43.560
either on that type
of branch, or I
00:50:43.560 --> 00:50:46.110
can put it on this
type of branch.
00:50:50.280 --> 00:50:55.930
It has only one factor of pi.
00:50:55.930 --> 00:50:59.340
By symmetry, it will
go to 0, like this.
00:50:59.340 --> 00:51:00.990
There will be things
that will have
00:51:00.990 --> 00:51:03.690
three factors of pi lesser.
00:51:20.650 --> 00:51:25.470
And all of these--
again because I'm
00:51:25.470 --> 00:51:29.440
dealing with an odd
number of factors of pi
00:51:29.440 --> 00:51:34.290
greater that I'm
averaging will give me 0.
00:51:34.290 --> 00:51:39.120
There's one other thing
that is kind of interesting.
00:51:39.120 --> 00:51:46.600
I can have all four
of these lines wavy.
00:51:46.600 --> 00:51:48.670
And if I calculate
that average, there's
00:51:48.670 --> 00:51:51.650
a number of ways of
contracting these four
00:51:51.650 --> 00:51:55.770
pi's that will give
me nontrivial factors.
00:51:55.770 --> 00:52:02.460
But these are also contributions
to the free energy.
00:52:02.460 --> 00:52:07.870
They don't depend on the
pi's that I'm leaving out.
00:52:07.870 --> 00:52:11.630
So they don't have to worry
about any of these diagrams
00:52:11.630 --> 00:52:12.150
so far.
00:52:14.720 --> 00:52:23.230
Now I dealt with the 0,
1, 3, and 4 wavy lines.
00:52:23.230 --> 00:52:26.870
So I'm left with 2 wavy
lines and 2 straight lines.
00:52:26.870 --> 00:52:28.890
So let's go through those.
00:52:28.890 --> 00:52:33.085
I could have one
branch be wavy lines
00:52:33.085 --> 00:52:35.595
and one branch be
straight lines.
00:52:40.010 --> 00:52:44.770
And then I take the
average of this object.
00:52:44.770 --> 00:52:48.210
I have a pi greater--
a pi greater here,
00:52:48.210 --> 00:52:51.230
and therefore I can
do an average of two
00:52:51.230 --> 00:52:53.680
of those pi greaters.
00:52:53.680 --> 00:52:57.315
That average will give me a
factor of 1 over K k squared.
00:53:00.690 --> 00:53:04.790
I have to integrate over that.
00:53:04.790 --> 00:53:08.470
But one of these branches had
this additional dash thing
00:53:08.470 --> 00:53:12.290
that corresponds to
having a factor of k.
00:53:12.290 --> 00:53:15.845
So the integral
that I have to do
00:53:15.845 --> 00:53:20.780
involves something like this.
00:53:20.780 --> 00:53:26.010
And then I integrate over the
entirety of the k integration.
00:53:26.010 --> 00:53:31.356
This is an odd power, and so
that will give me a 0 also.
00:53:31.356 --> 00:53:33.070
So this is also 0.
00:53:37.460 --> 00:53:38.940
And there's another
one that's is
00:53:38.940 --> 00:53:44.375
like this where I go like this.
00:53:52.740 --> 00:53:57.320
And although I do the
same thing now with two
00:53:57.320 --> 00:54:00.820
different branches, the k
integration is the same.
00:54:00.820 --> 00:54:02.225
And that vanishes too.
00:54:04.960 --> 00:54:08.350
So you say, is there
anything that is left?
00:54:08.350 --> 00:54:09.340
The answer is yes.
00:54:09.340 --> 00:54:13.760
So the things that are
left are the following.
00:54:13.760 --> 00:54:18.180
I can do something like this.
00:54:26.770 --> 00:54:29.331
Or I can do something like this.
00:54:37.440 --> 00:54:46.080
So these are the two things that
survive and will be nontrivial.
00:54:46.080 --> 00:54:48.270
You can see that
this one will be
00:54:48.270 --> 00:54:54.390
proportional to
pi lesser squared,
00:54:54.390 --> 00:55:00.420
while this one is going to
be proportional to gradient
00:55:00.420 --> 00:55:01.630
of pi lesser squared.
00:55:04.250 --> 00:55:10.825
So this one will renormalize,
if you like, this coefficient.
00:55:13.660 --> 00:55:19.830
Whereas this one, we've modified
and renormalize our coupling
00:55:19.830 --> 00:55:20.740
straight.
00:55:20.740 --> 00:55:23.310
So it turns out
that that is really
00:55:23.310 --> 00:55:25.810
the more important point.
00:55:25.810 --> 00:55:29.100
But let's calculate
the other one too.
00:55:29.100 --> 00:55:29.700
Yes?
00:55:29.700 --> 00:55:32.646
AUDIENCE: Why do we connect the
ones down here with the loops,
00:55:32.646 --> 00:55:34.610
but left all the ends
free in the ones.
00:55:34.610 --> 00:55:38.510
Was that just a matter of the
case of how to write diagram,
00:55:38.510 --> 00:55:41.235
or does that signify something?
00:55:41.235 --> 00:55:42.610
PROFESSOR: Could
you repeat that?
00:55:42.610 --> 00:55:43.780
I'm not sure I understand.
00:55:43.780 --> 00:55:47.906
AUDIENCE: So when we had two
wavy line, both coming out
00:55:47.906 --> 00:55:50.930
one of the diagram, this
line, they just stop.
00:55:50.930 --> 00:55:53.211
We connected them
together when we
00:55:53.211 --> 00:55:55.640
were writing the ones
on the bottom line.
00:55:55.640 --> 00:55:59.480
PROFESSOR: So basically, I
start with an entity that
00:55:59.480 --> 00:56:07.180
has two solid lines
and two wavy lines.
00:56:07.180 --> 00:56:11.640
And what I'm supposed to do
is to do an integration--
00:56:11.640 --> 00:56:19.070
an average of this
over these pi greaters.
00:56:19.070 --> 00:56:26.920
Now the process of
averaging essentially joins
00:56:26.920 --> 00:56:30.150
the two branches.
00:56:30.150 --> 00:56:33.616
If I had the momentum here,
q1, and a momentum here,
00:56:33.616 --> 00:56:36.440
q2, if I had an
index here, alpha,
00:56:36.440 --> 00:56:40.940
and an index here, beta,
that process of averaging
00:56:40.940 --> 00:56:45.150
is equivalent to saying the
same momentum has to go through,
00:56:45.150 --> 00:56:47.640
the same index
has to go through.
00:56:47.640 --> 00:56:50.996
There is no averaging that is
being done on the solid lines,
00:56:50.996 --> 00:56:53.365
so there is-- meaningless
to do anything.
00:57:07.720 --> 00:57:18.410
So this entity
means the following.
00:57:18.410 --> 00:57:29.980
I have K/ 2 let's call
it legs 1, 2, 3, and 4.
00:57:29.980 --> 00:57:39.660
Integral q1 and q2, but q1
and q2 you can see explicitly
00:57:39.660 --> 00:57:40.936
are solid.
00:57:40.936 --> 00:57:46.648
So these are the integration
from 0 to lambda over b.
00:57:46.648 --> 00:57:58.805
I have an integration over
q3, which is over a wavy line.
00:57:58.805 --> 00:58:02.200
So it's between lambda
over b and lambda.
00:58:06.080 --> 00:58:13.390
If I call this branch alpha
and this branch beta, from here
00:58:13.390 --> 00:58:24.290
I have actually pi lesser alpha
of q1 pi lesser beta of q2.
00:58:24.290 --> 00:58:26.300
I should have put them
outside the integration,
00:58:26.300 --> 00:58:28.600
but it doesn't matter.
00:58:28.600 --> 00:58:36.970
And then here I had pi
alpha of q3, pi beta of q4.
00:58:36.970 --> 00:58:41.575
But these also had these
lines associated with them.
00:58:41.575 --> 00:58:49.840
So I have here actually
an i q3, an i q4.
00:58:49.840 --> 00:58:59.210
Again, q4 has to stand for minus
q1 minus q2 minus q3 from this.
00:58:59.210 --> 00:59:04.450
And then I had the pi
pi here, which give me,
00:59:04.450 --> 00:59:09.150
because of the averaging,
delta alpha beta.
00:59:09.150 --> 00:59:14.120
And then I will
have an integration
00:59:14.120 --> 00:59:21.820
that forces q2 plus q4 to be 0.
00:59:21.820 --> 00:59:24.450
And then I have K q3 squared.
00:59:30.700 --> 00:59:36.382
Now q3 plus q4 is the same thing
as minus q1 minus q3, if you
00:59:36.382 --> 00:59:39.530
like, because of
that constraint.
00:59:39.530 --> 00:59:41.800
So I can take that
outside the integration.
00:59:41.800 --> 00:59:44.840
There's no problem.
00:59:44.840 --> 00:59:51.520
I have one integration left,
which is 1 over K q3 squared,
00:59:51.520 --> 00:59:54.530
but these two then
become the same.
00:59:54.530 --> 00:59:56.155
These pi's I will take outside.
00:59:58.760 --> 01:00:02.940
I note that because of this
constraint, q1 and q2 being
01:00:02.940 --> 01:00:10.004
the same, these two really
become one integration that
01:00:10.004 --> 01:00:13.430
goes between 0
and lambda over b.
01:00:13.430 --> 01:00:17.010
And these indices have
been made to be the same.
01:00:17.010 --> 01:00:21.995
So I have pi alpha of
q, this q, squared.
01:00:28.530 --> 01:00:33.380
Then I have the integration
from lambda over b to lambda.
01:00:33.380 --> 01:00:40.270
D d q3 2 pi to the d.
01:00:40.270 --> 01:00:43.270
Here I have i q3 i q4.
01:00:43.270 --> 01:00:47.530
But q4 was said to be minus q3.
01:00:47.530 --> 01:00:50.990
So the two i's and the
minus cancel each other.
01:00:50.990 --> 01:00:54.910
And I will get a
factor of q3 squared.
01:00:54.910 --> 01:00:58.425
And then here I have a
factor of K q3 squared.
01:01:03.420 --> 01:01:08.680
So the overall thing
is just that we
01:01:08.680 --> 01:01:13.370
can see that the K's cancel.
01:01:13.370 --> 01:01:21.840
I have one factor integral
dd q 2 pi to the d.
01:01:21.840 --> 01:01:25.590
I have pi alpha of q squared.
01:01:25.590 --> 01:01:29.380
And these are q lessers.
01:01:29.380 --> 01:01:32.600
I integrated out this quantity.
01:01:32.600 --> 01:01:39.910
The q3's vanish, so I really
have the integral of q3
01:01:39.910 --> 01:01:44.026
over 2 pi to the d.
01:01:44.026 --> 01:01:54.200
Now if I had done the
integral of q3 2 pi
01:01:54.200 --> 01:02:03.370
to the d, all the way from 0 to
lambda, what would I have done?
01:02:03.370 --> 01:02:05.970
If I multiply this
volume here, that
01:02:05.970 --> 01:02:08.564
would be the number of modes.
01:02:08.564 --> 01:02:14.720
So this is, in fact, N/V,
which is the quantity
01:02:14.720 --> 01:02:16.646
that I have called the density.
01:02:19.520 --> 01:02:21.590
But what I'm doing
is, in fact, doing
01:02:21.590 --> 01:02:27.010
just a fraction of this integral
from 0 to lambda over b.
01:02:27.010 --> 01:02:31.075
So if I do the fraction
from 0 to lambda over b,
01:02:31.075 --> 01:02:36.100
then I will get 1
minus b to the minus d.
01:02:36.100 --> 01:02:42.320
Sorry, from lambda
over b to lambda.
01:02:42.320 --> 01:02:46.480
Then if I had done all of
the way from 0 to lambda,
01:02:46.480 --> 01:02:49.600
I will have had one,
but I'm subtracting
01:02:49.600 --> 01:02:51.350
this fraction of it.
01:02:51.350 --> 01:02:57.650
So the answer is rho 1
minus b to the minus d.
01:02:57.650 --> 01:03:03.950
The overall thing here gets
multiplied by rho 1 minus
01:03:03.950 --> 01:03:05.904
b to the minus d.
01:03:09.890 --> 01:03:14.720
It just would correct that
factor of density that we have.
01:03:14.720 --> 01:03:17.810
We'll see shortly it's not
something to worry about.
01:03:17.810 --> 01:03:21.934
The next one is really the
more interesting thing.
01:03:21.934 --> 01:03:34.870
So here we have this diagram,
which is K/2 integral from 0
01:03:34.870 --> 01:03:36.910
to lambda over b.
01:03:40.140 --> 01:03:43.400
Essentially, I will
get the same structure.
01:03:48.730 --> 01:03:51.760
This time let me write
the pi alpha lesser
01:03:51.760 --> 01:04:00.030
of q1 pi beta lesser of q2
outside the last configuration.
01:04:00.030 --> 01:04:07.070
I have the integral from lambda
over b to lambda dd of q3 2
01:04:07.070 --> 01:04:09.860
pi to the 3d.
01:04:09.860 --> 01:04:13.460
And I will have the same
delta function structure,
01:04:13.460 --> 01:04:21.600
except that now these factors
of i q become i q1 times i q2.
01:04:21.600 --> 01:04:23.320
So I can put them
outside already.
01:04:27.208 --> 01:04:30.210
And then I have here
the delta function.
01:04:43.620 --> 01:04:47.520
So the only difference is
that previously the q squared
01:04:47.520 --> 01:04:50.740
was inside the integration.
01:04:50.740 --> 01:04:54.140
Now the q squared is
outside the integration.
01:04:54.140 --> 01:04:59.810
So the final answer will
be K/2 integral 0 to lambda
01:04:59.810 --> 01:05:05.232
over d d d q 2 pi to the d.
01:05:05.232 --> 01:05:08.650
I have a q squared.
01:05:08.650 --> 01:05:15.724
pi of q lesser squared.
01:05:15.724 --> 01:05:18.560
And the coefficient
of that would
01:05:18.560 --> 01:05:21.850
look like what I had before,
except without this factor.
01:05:21.850 --> 01:05:24.830
So it's the integral
from lambda over b
01:05:24.830 --> 01:05:31.873
to lambda dd of q3 divided by 2
pi to the d 1 over Kq3 squared.
01:05:47.920 --> 01:05:54.370
So once I do explicitly
this calculation,
01:05:54.370 --> 01:06:00.400
the answer is going to be
a weight that only depends
01:06:00.400 --> 01:06:04.360
on this pi lesser
that I'm keeping
01:06:04.360 --> 01:06:09.450
and I would indicate that
by beta H tilde, as usual.
01:06:09.450 --> 01:06:13.690
That depends of this pi lesser.
01:06:13.690 --> 01:06:18.410
And we now have all
of the terms that
01:06:18.410 --> 01:06:20.840
contribute to this beta H tilde.
01:06:20.840 --> 01:06:23.740
So let's write them down.
01:06:23.740 --> 01:06:26.300
There's a number of
terms that correspond
01:06:26.300 --> 01:06:30.110
to changes in the free energy.
01:06:30.110 --> 01:06:36.750
So we have a V delta f p at
the 0-th order contribution
01:06:36.750 --> 01:06:40.960
of delta f p at the first
order that are essentially
01:06:40.960 --> 01:06:46.410
a bunch of diagrams, both from
here as well as from here.
01:06:46.410 --> 01:06:48.416
But we don't really
care about them.
01:06:51.560 --> 01:07:03.720
Then we have types of
terms that look like this.
01:07:03.720 --> 01:07:08.180
I can write them already
after Fourier transformation
01:07:08.180 --> 01:07:09.090
in real space.
01:07:09.090 --> 01:07:12.080
So let me do that.
01:07:12.080 --> 01:07:17.990
I have integral dd
x in real space,
01:07:17.990 --> 01:07:22.315
realizing that my cut
off has been changed
01:07:22.315 --> 01:07:27.730
to ba-- things that are
proportional to gradient
01:07:27.730 --> 01:07:31.722
of pi lesser squared.
01:07:31.722 --> 01:07:34.470
Now gradient of
pi lesser squared
01:07:34.470 --> 01:07:36.820
is the things I in
Fourier space becomes
01:07:36.820 --> 01:07:40.810
q squared pi lesser squared.
01:07:40.810 --> 01:07:42.470
And what is the coefficient?
01:07:45.050 --> 01:07:51.910
I had K/2, which
comes from here.
01:07:51.910 --> 01:07:57.680
If I don't do anything, I
just have the 0-th order modes
01:07:57.680 --> 01:08:00.530
acting on these.
01:08:00.530 --> 01:08:02.920
But I just calculated
a correction
01:08:02.920 --> 01:08:09.440
to that that is
something like this.
01:08:09.440 --> 01:08:14.060
So in addition to
what I had before,
01:08:14.060 --> 01:08:21.747
I have this correction,
K/2 times this integral.
01:08:21.747 --> 01:08:27.106
And I'm going to call the
result of this integral
01:08:27.106 --> 01:08:39.439
to be I sub d-- see
that this integral is
01:08:39.439 --> 01:08:43.580
proportional to 1/K.
01:08:43.580 --> 01:08:46.750
And when the integration--
let's call the result
01:08:46.750 --> 01:08:49.380
of the integral
Id of b because it
01:08:49.380 --> 01:08:52.720
depends on both
dimension of integration,
01:08:52.720 --> 01:08:55.870
as well as the factor
V through here.
01:08:55.870 --> 01:08:59.455
So I have 1/K Id of b.
01:09:04.760 --> 01:09:10.050
So that's one type of
that I have generated.
01:09:10.050 --> 01:09:14.950
I started from this
0-th order form.
01:09:14.950 --> 01:09:19.620
And I saw that once I make the
expansion of this to the lowest
01:09:19.620 --> 01:09:22.640
order, I will get a
correction to that.
01:09:22.640 --> 01:09:25.964
And actually, just to
sort of think about it
01:09:25.964 --> 01:09:29.250
in terms of formulae,
you see what
01:09:29.250 --> 01:09:34.140
happened was that the first
term that I had over here
01:09:34.140 --> 01:09:39.490
was pi gradient of
pi repeated twice.
01:09:42.200 --> 01:09:45.600
And what I did was
I essentially did
01:09:45.600 --> 01:09:51.010
an average of these two pi's and
got the correction to gradient
01:09:51.010 --> 01:09:57.785
of pi squared, which is what was
computed here and [INAUDIBLE].
01:09:57.785 --> 01:10:06.190
The next term that I have
is this term by itself,
01:10:06.190 --> 01:10:12.860
not calculated with the pi
lessers only-- so this object.
01:10:12.860 --> 01:10:24.458
So I will write it as K/2
I gradient of pi squared.
01:10:24.458 --> 01:10:26.266
There's no correction.
01:10:26.266 --> 01:10:32.040
The final term that
I have is this term.
01:10:32.040 --> 01:10:41.600
So I have minus rho over
2 pi lesser squared.
01:10:41.600 --> 01:10:49.380
And then the correction that I
have was exactly the same form.
01:10:49.380 --> 01:10:55.232
It was 1 minus 1 minus
b to the minus b.
01:10:55.232 --> 01:10:59.130
The rho over 2 is going to
be hung onto both of them.
01:10:59.130 --> 01:11:02.200
You can see that there's
a rho and there's a 2.
01:11:02.200 --> 01:11:05.910
And so basically I have
two add this to that.
01:11:05.910 --> 01:11:11.342
And to first order, this is
the entire thing of the thing
01:11:11.342 --> 01:11:12.324
that I will get.
01:11:21.180 --> 01:11:26.520
So this however is
just a course gradient.
01:11:26.520 --> 01:11:29.930
It's the first step of the RG.
01:11:29.930 --> 01:11:34.680
And it has to be followed by
the next two steps of the RG.
01:11:34.680 --> 01:11:38.000
Where here, you
look at you field
01:11:38.000 --> 01:11:41.340
and you can see that the
field is much coarser
01:11:41.340 --> 01:11:45.310
because the short distance
cut off rather than being a,
01:11:45.310 --> 01:11:47.541
has been switched to ba.
01:11:47.541 --> 01:11:53.710
So you define your x prime
to x/b, so that you shrink.
01:11:53.710 --> 01:12:01.160
You get the same course pixel
size that you had before.
01:12:01.160 --> 01:12:07.380
And you also have to
do a change in pi.
01:12:07.380 --> 01:12:13.150
So you replace pi lesser
with some factor zeta pi
01:12:13.150 --> 01:12:19.210
prime, so that the
contrast will look fine.
01:12:19.210 --> 01:12:23.309
So once I do that, you
can see that this effect
01:12:23.309 --> 01:12:27.240
of this transformation
is that this coupling
01:12:27.240 --> 01:12:32.280
K will change to K prime.
01:12:32.280 --> 01:12:36.990
Because of the change
of x with b x prime,
01:12:36.990 --> 01:12:41.330
from the integration,
I will get b to the d.
01:12:41.330 --> 01:12:45.200
From the two derivatives, I
will get V to the minus 2.
01:12:49.280 --> 01:12:53.130
From the fact that I have
replace two pi lessers with pi
01:12:53.130 --> 01:12:58.420
primes, I will get a
factor of zeta squared.
01:12:58.420 --> 01:13:07.410
And then I will have this
factor K 1 plus 1/K Id of b.
01:13:14.370 --> 01:13:25.950
So this is the recursion formula
that we will be dealing with.
01:13:25.950 --> 01:13:29.400
Now there is some
subtleties that
01:13:29.400 --> 01:13:33.570
go with this formula that
are worth thinking about.
01:13:33.570 --> 01:13:42.380
Our original system had really
one coupling parameter K
01:13:42.380 --> 01:13:46.310
that because of the constraints
of the full symmetry
01:13:46.310 --> 01:13:52.640
of this field, S, part of it
became the quadratic part that
01:13:52.640 --> 01:13:54.340
was the free field theory.
01:13:54.340 --> 01:13:57.970
But part of it made
the interactions.
01:13:57.970 --> 01:14:00.020
But because of this
vertical symmetry,
01:14:00.020 --> 01:14:02.740
to form of that
interaction was fixed
01:14:02.740 --> 01:14:08.030
and had to be
proportional to K. Now
01:14:08.030 --> 01:14:13.950
if we do our normalization group
correctly, to full symmetry
01:14:13.950 --> 01:14:19.720
that we had has to be
maintained at all levels.
01:14:19.720 --> 01:14:24.260
Which means that the functional
form that I should end up with
01:14:24.260 --> 01:14:28.630
should have the same property
in that the higher order
01:14:28.630 --> 01:14:32.640
coefficient should be
related to the lower order
01:14:32.640 --> 01:14:39.150
coefficient, exactly the same
way as we had over there.
01:14:39.150 --> 01:14:45.380
And at least at this stage, it
looks like that did not happen.
01:14:45.380 --> 01:14:48.730
That is, we got the
correction to this term,
01:14:48.730 --> 01:14:53.110
but we didn't get the
correction to this term.
01:14:53.110 --> 01:14:54.970
We shouldn't be worried
about that right
01:14:54.970 --> 01:15:01.950
here because we calculated
things consistently corrections
01:15:01.950 --> 01:15:06.040
to order of T. And
this was already
01:15:06.040 --> 01:15:12.400
a term that was order
of T. So the real check
01:15:12.400 --> 01:15:17.100
is if you go and calculate
the next order correction,
01:15:17.100 --> 01:15:21.070
you better get a correction
to this term at next order
01:15:21.070 --> 01:15:23.820
that matches exactly this.
01:15:23.820 --> 01:15:25.770
People have done that,
and have checked that.
01:15:25.770 --> 01:15:27.690
And that indeed is the case.
01:15:27.690 --> 01:15:31.316
So one is consistent with this.
01:15:31.316 --> 01:15:34.240
There are other kinds
of consistency checks
01:15:34.240 --> 01:15:36.700
that have happened
all over the place,
01:15:36.700 --> 01:15:40.756
like the fact that this
came back 1 minus 1
01:15:40.756 --> 01:15:45.940
came out to be b to the minus
d, so that the density is
01:15:45.940 --> 01:15:49.030
the same as before,
consistent with the fact
01:15:49.030 --> 01:15:52.490
that you shrunk the
lattice after RG so
01:15:52.490 --> 01:15:56.110
that the pixel size was
the same as before is
01:15:56.110 --> 01:15:58.450
a consequence of that.
01:15:58.450 --> 01:16:01.780
You may worry that that's
not entirely the case
01:16:01.780 --> 01:16:04.610
because when I do
this, I will have also
01:16:04.610 --> 01:16:07.370
a factor of zeta squared.
01:16:07.370 --> 01:16:12.000
But it turns out that zeta is
1 plus order of temperature,
01:16:12.000 --> 01:16:13.640
as we will shortly see.
01:16:13.640 --> 01:16:16.360
So I gain consistent--
everything's
01:16:16.360 --> 01:16:20.750
consistent at this level
that we've calculated things.
01:16:20.750 --> 01:16:25.555
And the only change
is this factor.
01:16:25.555 --> 01:16:28.376
Now the one thing that
we haven't calculated
01:16:28.376 --> 01:16:32.310
is what this zeta is.
01:16:32.310 --> 01:16:36.878
So to calculate zeta,
I note the following
01:16:36.878 --> 01:16:42.465
that I start with
a unit vector that
01:16:42.465 --> 01:16:44.770
is pointing at 0 temperature
along this direction.
01:16:47.530 --> 01:16:50.660
Now because of
fluctuations, this
01:16:50.660 --> 01:16:55.230
is going to be kind of
rotating around this.
01:16:55.230 --> 01:16:59.750
So there is this vector
that is rotating.
01:16:59.750 --> 01:17:04.160
If you average it over some
time, what you will see
01:17:04.160 --> 01:17:08.630
is that the average in all
of these direction is 0.
01:17:08.630 --> 01:17:12.020
The variance is not 0,
but the average is 0.
01:17:12.020 --> 01:17:13.750
But because of
those fluctuations,
01:17:13.750 --> 01:17:17.730
the effective length that
you see in this direction
01:17:17.730 --> 01:17:18.370
has shrunk.
01:17:21.100 --> 01:17:26.940
How much has it shrunk by
is related to this rescaling
01:17:26.940 --> 01:17:29.520
factor that I should chose.
01:17:29.520 --> 01:17:32.250
And so it's essentially
average of something
01:17:32.250 --> 01:17:34.820
like 1 minus pi squared.
01:17:38.590 --> 01:17:41.730
But really it is the
pi lesser squares
01:17:41.730 --> 01:17:44.740
that I'm averaging over.
01:17:44.740 --> 01:17:49.820
Which the lowest order is 1
minus 1/2 the average of pi
01:17:49.820 --> 01:17:58.660
lesser squared, which
is 1 minus 1/2 .
01:17:58.660 --> 01:18:02.980
Now this is an N minus
1 component vector.
01:18:02.980 --> 01:18:08.210
So each one of the components
will give you one contribution.
01:18:08.210 --> 01:18:13.426
The contribution that
you get for one of these
01:18:13.426 --> 01:18:17.275
is simply the average
of pi squared, which
01:18:17.275 --> 01:18:26.330
is 1 over K k squared, which
I have to integrate over K's
01:18:26.330 --> 01:18:29.138
that lie between lambda
over b and lambda.
01:18:32.810 --> 01:18:39.570
And you can see that this is
1 minus 1/2 N and minus 1.
01:18:39.570 --> 01:18:43.080
It is inversely
proportional to K.
01:18:43.080 --> 01:18:45.990
And then the integration
that I have to do
01:18:45.990 --> 01:18:49.330
is precisely the same
integration as here.
01:18:49.330 --> 01:18:52.725
So it is, again, this Id of b.
01:19:01.460 --> 01:19:06.750
So let me write the answer,
say a couple of words about it,
01:19:06.750 --> 01:19:09.720
and then we will deal
with it next time.
01:19:09.720 --> 01:19:13.875
So K prime is going
to be b to the d
01:19:13.875 --> 01:19:19.002
minus 2-- a new
interaction parameters.
01:19:19.002 --> 01:19:26.486
It is one factor of
1 plus 1/K Id of b.
01:19:29.320 --> 01:19:33.600
And then there's one factor
of this square of zeta.
01:19:33.600 --> 01:19:44.110
So that gives you N minus
1 over K Id of b times K.
01:19:44.110 --> 01:19:49.320
So we'll analyze this
more next time around.
01:19:49.320 --> 01:19:55.830
But I thought I would give you
the physical reason for how
01:19:55.830 --> 01:19:59.110
this interaction
parameter changes.
01:19:59.110 --> 01:20:00.870
Let's say we are
in two dimensions.
01:20:00.870 --> 01:20:04.454
So let's forget
about this factor.
01:20:04.454 --> 01:20:08.410
In two dimensions, we can see
that there is one factor that
01:20:08.410 --> 01:20:13.390
says at finite temperature,
we are going to get weaker.
01:20:13.390 --> 01:20:16.400
The interaction is
going to get weaker.
01:20:16.400 --> 01:20:19.100
And the reason for
that is precisely what
01:20:19.100 --> 01:20:21.340
I was explaining over here.
01:20:21.340 --> 01:20:25.110
That is, you have some
kind of a unit vector,
01:20:25.110 --> 01:20:28.050
but because of its
fluctuations, you
01:20:28.050 --> 01:20:30.410
will see that it
will loop shorter.
01:20:30.410 --> 01:20:33.650
And it is less
likely to be ordered.
01:20:33.650 --> 01:20:36.720
The more components
it has to fluctuate,
01:20:36.720 --> 01:20:39.185
the shorter it will look like.
01:20:39.185 --> 01:20:41.260
So there is that term.
01:20:41.260 --> 01:20:44.950
So if this was the
only effect, then K
01:20:44.950 --> 01:20:46.950
will become weaker and weaker.
01:20:46.950 --> 01:20:49.950
And it will have disorder.
01:20:49.950 --> 01:20:52.720
But this effect says
that it actually
01:20:52.720 --> 01:20:56.470
gets stronger because
of the interactions
01:20:56.470 --> 01:20:58.750
that you have among the modes.
01:20:58.750 --> 01:21:02.830
And to show you this to you,
we can experiment with it
01:21:02.830 --> 01:21:04.190
yourself.
01:21:04.190 --> 01:21:06.780
So this is a sheet of paper.
01:21:06.780 --> 01:21:09.560
This bend is an example
of a Goldstone mode
01:21:09.560 --> 01:21:11.990
because I could have
rotated this sheet
01:21:11.990 --> 01:21:14.200
without any cost of energy.
01:21:14.200 --> 01:21:19.520
So this bend is a Goldstone mode
that costs very little energy.
01:21:19.520 --> 01:21:22.840
Now this paper has the
kinds of constraints
01:21:22.840 --> 01:21:24.660
that we have over here.
01:21:24.660 --> 01:21:27.460
And because of
those constraints,
01:21:27.460 --> 01:21:30.530
if I make a mode
in this direction,
01:21:30.530 --> 01:21:33.970
I'm not going to be able to
bend it in the other directions.
01:21:33.970 --> 01:21:37.630
So clearly the mode that
you have this direction,
01:21:37.630 --> 01:21:39.840
and this direction, are coupled.
01:21:39.840 --> 01:21:43.960
That's kind of an example
of something like this.
01:21:43.960 --> 01:21:50.530
Now while it is easy to do this
bend because of this coupling,
01:21:50.530 --> 01:21:54.250
if thermal fluctuations
have created modes
01:21:54.250 --> 01:21:57.330
that are shorter wavelength,
and I have already
01:21:57.330 --> 01:22:00.310
created those modes
over here, then you
01:22:00.310 --> 01:22:01.720
can experiment yourself.
01:22:01.720 --> 01:22:06.700
You'll see that this is harder
to bend compared to this.
01:22:06.700 --> 01:22:10.480
You can see this already.
01:22:10.480 --> 01:22:14.520
So that's the effect
that you have over here.
01:22:14.520 --> 01:22:17.510
So it's the competition
between these two.
01:22:17.510 --> 01:22:19.660
And depending on which
one wins, and you
01:22:19.660 --> 01:22:22.810
can see that that
depends on whether n
01:22:22.810 --> 01:22:26.274
is larger than 2 or less than 2.
01:22:26.274 --> 01:22:29.340
So essentially for n
that is larger than 2,
01:22:29.340 --> 01:22:31.090
you'll find that this term wins.
01:22:31.090 --> 01:22:33.740
And you get disorder
in two dimension.
01:22:33.740 --> 01:22:36.130
And is less than 2,
you will get order
01:22:36.130 --> 01:22:40.490
like we know the Ising
model can be order.
01:22:40.490 --> 01:22:41.950
But there's other
things that can
01:22:41.950 --> 01:22:45.736
be captured by this expression,
that we will look at next time.