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PROFESSOR: OK.
00:00:22.510 --> 00:00:23.245
Let's start.
00:00:26.730 --> 00:00:29.560
So for several
lectures, we've been
00:00:29.560 --> 00:00:36.840
talking about the
XY model, which
00:00:36.840 --> 00:00:44.340
is a collection of
two component spins,
00:00:44.340 --> 00:00:46.120
for example defined
on a square lattice,
00:00:46.120 --> 00:00:50.190
but it could be any
type of lattice.
00:00:50.190 --> 00:00:53.680
And today we are going
to compare and contrast
00:00:53.680 --> 00:00:56.530
that with the case of a solid.
00:00:59.440 --> 00:01:03.730
More specifically for
simplicity, an isotropic solid.
00:01:09.180 --> 00:01:13.070
And a typical example of
an isotropic solid in two
00:01:13.070 --> 00:01:16.970
dimension is the
triangular lattice.
00:01:16.970 --> 00:01:18.210
So something like this.
00:01:29.090 --> 00:01:32.530
What both of these
systems have in common
00:01:32.530 --> 00:01:37.350
is that there is an underlying
continuous symmetry that
00:01:37.350 --> 00:01:39.020
is broken.
00:01:39.020 --> 00:01:45.290
In the case of the XY
model, at low temperatures
00:01:45.290 --> 00:01:48.920
all of the spins would
be pointing more or less
00:01:48.920 --> 00:01:49.865
in the same direction.
00:01:53.890 --> 00:01:57.900
Potentially with
fluctuations, let's say, go
00:01:57.900 --> 00:02:00.090
around the direction
where they're
00:02:00.090 --> 00:02:02.180
supposed to be
pointing-- being, let's
00:02:02.180 --> 00:02:04.800
say, here in the
vertical-- characterized
00:02:04.800 --> 00:02:06.550
by some kind of an angle.
00:02:06.550 --> 00:02:08.000
Let's call it theta.
00:02:11.500 --> 00:02:18.440
And essentially the uniform
state at 0 temperature
00:02:18.440 --> 00:02:21.740
can be characterized
by any theta.
00:02:21.740 --> 00:02:25.740
And because of the
deformations, it's
00:02:25.740 --> 00:02:30.950
captured through these low
energy Goldstone modes.
00:02:30.950 --> 00:02:35.100
And the probability of
some particular deformation
00:02:35.100 --> 00:02:40.430
will be proportional at low
temperature to the integral,
00:02:40.430 --> 00:02:43.920
let's say, in d
dimensions of gradient,
00:02:43.920 --> 00:02:48.290
the change in theta within
different spins squared.
00:02:48.290 --> 00:02:51.580
Of course, with some
considerations as
00:02:51.580 --> 00:02:54.830
we discussed about topological
defects, et cetera,
00:02:54.830 --> 00:02:58.450
being buried in
this description.
00:02:58.450 --> 00:03:03.620
Now, similarly, away
from 0 temperature
00:03:03.620 --> 00:03:08.910
the atoms of a solid
will fluctuate around
00:03:08.910 --> 00:03:14.500
the position that
corresponds to the minimum.
00:03:14.500 --> 00:03:17.050
And so the actual
picture that you
00:03:17.050 --> 00:03:20.230
would see at any
finite temperature
00:03:20.230 --> 00:03:24.860
would be distorted with respect
to the perfect configuration
00:03:24.860 --> 00:03:29.190
and there would be some kind
of a vector [INAUDIBLE] u
00:03:29.190 --> 00:03:36.160
at each location that would
describe this fluctuation.
00:03:36.160 --> 00:03:43.020
Again, there is no cost in
making a uniform distortion
00:03:43.020 --> 00:03:46.020
theta that is the
same across space.
00:03:46.020 --> 00:03:48.750
And similarly here,
there is no cost
00:03:48.750 --> 00:03:52.320
in uniformly translating things.
00:03:52.320 --> 00:03:56.300
So these would both imply that
this row can [? continue ?]
00:03:56.300 --> 00:03:59.590
[INAUDIBLE] leads
to Goldstone modes.
00:03:59.590 --> 00:04:02.920
And the corresponding
energy cost
00:04:02.920 --> 00:04:09.050
in the case of the
isotropically distorted system
00:04:09.050 --> 00:04:12.770
is typically returned
as an integral involving
00:04:12.770 --> 00:04:15.460
elastic moduli, as we've seen.
00:04:15.460 --> 00:04:18.769
And the traditional
way to write that
00:04:18.769 --> 00:04:32.920
is in this form where
the same reason here
00:04:32.920 --> 00:04:35.210
that it wasn't theta
that was appearing
00:04:35.210 --> 00:04:38.110
but gradient of theta,
because a uniform theta
00:04:38.110 --> 00:04:40.140
does not make any cross.
00:04:40.140 --> 00:04:46.060
These strain fields are related
derivatives of the distortion.
00:04:46.060 --> 00:04:48.280
But now we remember
that this u is a vector
00:04:48.280 --> 00:04:56.890
and this strain field uij it
was one half of the derivative
00:04:56.890 --> 00:05:00.110
of uj in the i direction
derivative of ui
00:05:00.110 --> 00:05:01.630
in the j direction symmetrized.
00:05:05.770 --> 00:05:10.450
Actually, probably better
to write this symmetrically
00:05:10.450 --> 00:05:12.590
as beta h equals
to this quantity.
00:05:15.630 --> 00:05:21.270
And then the typical
type of calculation
00:05:21.270 --> 00:05:25.500
that we did for the
XY model was to go
00:05:25.500 --> 00:05:29.930
in terms of the independent
spin wave modes.
00:05:29.930 --> 00:05:31.650
So we go to Fourier space.
00:05:31.650 --> 00:05:36.322
This becomes an integral
ddq 2 pi to the d.
00:05:36.322 --> 00:05:39.900
The derivative here
gave q squared.
00:05:39.900 --> 00:05:45.357
And then I had something like
the Fourier transformed theta
00:05:45.357 --> 00:05:45.856
squared.
00:05:48.940 --> 00:05:53.880
Now I can also more
clearly represent
00:05:53.880 --> 00:05:57.520
this in the Fourier description.
00:05:57.520 --> 00:06:04.930
I will get something like 1/2
integral vdq 2 pi to the d.
00:06:04.930 --> 00:06:11.510
And a little bit of work
shows that this form Fourier
00:06:11.510 --> 00:06:16.190
transforms to something
that is proportional to mu.
00:06:16.190 --> 00:06:18.940
Again, it has to be
proportionality to q squared
00:06:18.940 --> 00:06:20.770
because the energy
should go to 0
00:06:20.770 --> 00:06:24.310
as q goes to 0 for very
long wavelength modes.
00:06:24.310 --> 00:06:30.140
And so we'll have something
like q squared mu tilde of q
00:06:30.140 --> 00:06:30.880
squared.
00:06:30.880 --> 00:06:33.250
This is a vector squared.
00:06:33.250 --> 00:06:35.260
u is a vector.
00:06:35.260 --> 00:06:36.910
q is a vector.
00:06:36.910 --> 00:06:43.040
And there is another component
that gives you mu plus lambda.
00:06:43.040 --> 00:06:47.300
And consistent with
rotational symmetry,
00:06:47.300 --> 00:06:52.350
we can have a term that is
q dotted with u tilde of q,
00:06:52.350 --> 00:06:53.540
the whole thing squared.
00:06:58.480 --> 00:07:01.630
And again, you can
reason that you
00:07:01.630 --> 00:07:05.350
can describe the isotropic
system in terms of just two
00:07:05.350 --> 00:07:09.310
elastic moduli is that the
only rotational invariant
00:07:09.310 --> 00:07:12.710
quantities are q squared,
u squared, and q.u,
00:07:12.710 --> 00:07:17.180
that those are the only terms
that you can [? find. ?]
00:07:17.180 --> 00:07:19.210
Now the next thing.
00:07:19.210 --> 00:07:23.220
Once you know this form, you
can immediately say something
00:07:23.220 --> 00:07:25.110
about the fluctuations.
00:07:25.110 --> 00:07:30.940
So you say that expectation
value of theta tilde q theta
00:07:30.940 --> 00:07:38.180
tilde q prime, if I assume
that this is a Gaussian theory
00:07:38.180 --> 00:07:40.310
and these are the
only fluctuations
00:07:40.310 --> 00:07:42.620
that I'm considering,
the answer is
00:07:42.620 --> 00:07:47.970
going to be 2 pi to the d
delta function q plus q prime.
00:07:47.970 --> 00:07:53.410
And then I will get a
factor of 1 over kq squared.
00:07:59.330 --> 00:08:04.900
The corresponding thing here
is a bit more complicated.
00:08:04.900 --> 00:08:09.490
Because as we said, this
quantity u tilde is a vector.
00:08:09.490 --> 00:08:12.990
And I can look at the
correlation between, say,
00:08:12.990 --> 00:08:16.380
the i-th component of
that vector along mode
00:08:16.380 --> 00:08:21.740
q, the j-th component of
that vector along mu 2 prime
00:08:21.740 --> 00:08:24.750
and ask what this is.
00:08:24.750 --> 00:08:29.900
And again, only the
same values of q
00:08:29.900 --> 00:08:33.350
are coupled together, so we get
the usual formula over here.
00:08:37.620 --> 00:08:43.809
If this term was the only
term in the story, that
00:08:43.809 --> 00:08:49.040
is if we were dealing
with vectors q and u that
00:08:49.040 --> 00:08:50.850
are orthogonal to
each other, then we
00:08:50.850 --> 00:08:55.680
would just get the
1 over q squared.
00:08:55.680 --> 00:09:02.030
And so there is actually
a term that is like that.
00:09:02.030 --> 00:09:06.690
But because of this other
term, there is the possibility
00:09:06.690 --> 00:09:11.060
that q and u are in the same
direction, in which case,
00:09:11.060 --> 00:09:13.470
these two costs
will add up and I
00:09:13.470 --> 00:09:15.960
will get something that
is proportional to 2
00:09:15.960 --> 00:09:16.820
mu plus lambda.
00:09:16.820 --> 00:09:26.980
And so then I will get 2
mu plus lambda q squared.
00:09:26.980 --> 00:09:34.360
And here I will have qi qj.
00:09:34.360 --> 00:09:36.965
This becomes q to the 4th.
00:09:36.965 --> 00:09:41.835
Let me make sure that I did
not write this incorrectly.
00:09:50.580 --> 00:09:53.100
Yes, I did write it incorrectly.
00:09:53.100 --> 00:09:57.025
It turns out that there is a
mu plus lambda out here too.
00:10:03.330 --> 00:10:07.120
But essentially there's
just a small complication
00:10:07.120 --> 00:10:13.710
that we get because of the
fact that you have a vector u.
00:10:13.710 --> 00:10:17.200
But overall the
scaling is something
00:10:17.200 --> 00:10:21.090
that goes like 1 over q squared,
which is characteristics
00:10:21.090 --> 00:10:30.710
of Goldstone modes here with
additional vectorial things
00:10:30.710 --> 00:10:32.990
to worry about.
00:10:32.990 --> 00:10:36.141
Let me make sure I
separate these two.
00:10:39.990 --> 00:10:44.410
Now one of the things that
we have been concerned with
00:10:44.410 --> 00:10:51.170
is whether these fluctuations
destroy long range order, so
00:10:51.170 --> 00:10:55.180
for long range order the
information that we would like
00:10:55.180 --> 00:10:59.150
to have is that the
entirety of the system
00:10:59.150 --> 00:11:01.980
is roughly pointing
in the same direction,
00:11:01.980 --> 00:11:04.610
meaning that if I know
that this beam here
00:11:04.610 --> 00:11:07.010
is pointed in the
vertical direction,
00:11:07.010 --> 00:11:11.300
if I go very far away,
it is still more or less
00:11:11.300 --> 00:11:15.030
pointed out-- pointing
in the same direction.
00:11:15.030 --> 00:11:18.600
So for that, we could
look, for example,
00:11:18.600 --> 00:11:22.350
as the degree which
[? with ?] these two spins?
00:11:22.350 --> 00:11:23.175
Yes, question?
00:11:23.175 --> 00:11:25.496
AUDIENCE: [INAUDIBLE]
q squared [INAUDIBLE]?
00:11:29.488 --> 00:11:30.720
PROFESSOR: This is 4.
00:11:30.720 --> 00:11:32.485
Is that what you're
worried about?
00:11:32.485 --> 00:11:33.195
AUDIENCE: Yeah.
00:11:33.195 --> 00:11:34.195
It should be 2, I guess.
00:11:34.195 --> 00:11:37.244
PROFESSOR: No, because I have
two powers of q out front.
00:11:37.244 --> 00:11:37.910
AUDIENCE: Right.
00:11:37.910 --> 00:11:39.326
PROFESSOR: So I
think in the notes
00:11:39.326 --> 00:11:42.540
what I have is that
I have actually
00:11:42.540 --> 00:11:44.110
written things this way.
00:11:44.110 --> 00:11:47.710
I put the factor of mu
q squared out front.
00:11:47.710 --> 00:11:53.190
And then I had this
as qi qj over q.
00:11:53.190 --> 00:11:54.290
And I think--
00:11:54.290 --> 00:11:57.540
AUDIENCE: [INAUDIBLE]
q squared [INAUDIBLE]?
00:11:57.540 --> 00:11:59.980
PROFESSOR: q squared.
00:11:59.980 --> 00:12:05.160
Let me one more time check to
make sure that that's not--
00:12:05.160 --> 00:12:06.610
actually there's a minus sign.
00:12:09.410 --> 00:12:11.790
All right.
00:12:11.790 --> 00:12:13.945
AUDIENCE: Wouldn't there
be another factor of mu
00:12:13.945 --> 00:12:17.474
to compensate for
the factoring out?
00:12:17.474 --> 00:12:18.330
PROFESSOR: Yeah.
00:12:18.330 --> 00:12:24.000
So let's check this in
the following fashion.
00:12:24.000 --> 00:12:31.730
So I can look at modes
where u and q are
00:12:31.730 --> 00:12:34.300
perpendicular to each other.
00:12:34.300 --> 00:12:36.980
When u and q are
perpendicular to each other,
00:12:36.980 --> 00:12:40.025
I don't have this there.
00:12:40.025 --> 00:12:43.040
And so then what I should
have is just a cost
00:12:43.040 --> 00:12:45.900
that is 1 over mu q squared.
00:12:45.900 --> 00:12:46.400
OK.
00:12:46.400 --> 00:12:48.710
So let's see if
that comes about.
00:12:57.100 --> 00:13:01.000
What I can do is
let's say imagine
00:13:01.000 --> 00:13:09.720
that I'm looking at a q that is
oriented along the x direction.
00:13:09.720 --> 00:13:15.370
And I look at u's that
are in the y direction.
00:13:15.370 --> 00:13:22.390
And I correlate two
u's in the y direction.
00:13:22.390 --> 00:13:25.660
So I will definitely
get a delta ij here.
00:13:25.660 --> 00:13:28.170
I would have a 1.
00:13:28.170 --> 00:13:34.720
Here I would get a
factor of nothing,
00:13:34.720 --> 00:13:37.740
because this term is absent.
00:13:37.740 --> 00:13:40.250
The q's that I'm
looking at don't
00:13:40.250 --> 00:13:44.700
have any component along the
direction of u that I'm having.
00:13:44.700 --> 00:13:47.920
So this is absent
and I will get a 1
00:13:47.920 --> 00:13:51.710
over mu q squared, which
is consistent with that.
00:13:51.710 --> 00:13:59.250
Now let's look at the case
where I look at u's that
00:13:59.250 --> 00:14:02.095
are aligned with the vector q.
00:14:02.095 --> 00:14:04.720
Let's say again in
the x direction.
00:14:04.720 --> 00:14:09.180
Then, as far as the
energy is concerned,
00:14:09.180 --> 00:14:12.620
both terms are
present and the cost
00:14:12.620 --> 00:14:15.850
should be 2 mu plus lambda.
00:14:15.850 --> 00:14:18.160
So the answer that I
should ultimately get
00:14:18.160 --> 00:14:20.560
should be 1 over
2 mu plus lambda.
00:14:20.560 --> 00:14:22.970
So let's check.
00:14:22.970 --> 00:14:28.130
So here q squared-- and
cancels with this q squared.
00:14:28.130 --> 00:14:30.920
All of the q's are in the
direction of the indices
00:14:30.920 --> 00:14:32.620
that I'm looking at.
00:14:32.620 --> 00:14:34.130
So that cancels.
00:14:34.130 --> 00:14:35.020
This is 1.
00:14:35.020 --> 00:14:39.685
So I will get 1 minus mu plus
lambda over 2 mu plus lambda.
00:14:39.685 --> 00:14:47.110
So then I will get 2 mu plus
lambda minus mu minus lambda.
00:14:47.110 --> 00:14:50.110
So I will get a mu in
front that cancels this.
00:14:50.110 --> 00:14:53.040
And I will be left with
just the 2 mu plus lambda.
00:14:53.040 --> 00:14:55.725
So we've checked
that this is correct.
00:15:00.850 --> 00:15:01.540
All right.
00:15:01.540 --> 00:15:05.790
So this is a proof, if
you like, by induction.
00:15:05.790 --> 00:15:09.830
You knew that the only
forms that are consistent
00:15:09.830 --> 00:15:13.860
are delta ij and qi
qj over q squared.
00:15:13.860 --> 00:15:17.110
So we can give them two
coefficients, a and b,
00:15:17.110 --> 00:15:21.720
and then adjust a and b so
that these two limits that I
00:15:21.720 --> 00:15:24.250
considered are reproduced.
00:15:24.250 --> 00:15:27.073
So-- OK.
00:15:27.073 --> 00:15:30.300
I didn't have to
resort to my notes
00:15:30.300 --> 00:15:34.550
if I was willing to spend
the corresponding three
00:15:34.550 --> 00:15:37.520
minutes here.
00:15:37.520 --> 00:15:42.400
So going back here, in order
to see the degree of alignment
00:15:42.400 --> 00:15:46.370
of the two spins that we have
over here, what we need to do
00:15:46.370 --> 00:15:49.260
is to do is to look something
like cosine of, say,
00:15:49.260 --> 00:15:54.022
theta at location x
minus theta at location x
00:15:54.022 --> 00:16:01.390
prime, which is non
other than the real part
00:16:01.390 --> 00:16:06.220
of the expectation value of
e to the theta at location
00:16:06.220 --> 00:16:08.810
x minus theta at
location x prime.
00:16:12.680 --> 00:16:20.180
And assuming that everything is
governed by this Gaussian rate,
00:16:20.180 --> 00:16:25.240
the answer is going to obtain
by computing this Gaussian
00:16:25.240 --> 00:16:29.590
as exponential of
minus 1/2 the average
00:16:29.590 --> 00:16:32.560
of theta x minus
theta x prime squared.
00:16:35.680 --> 00:16:39.270
So then my task is
to replace theta x
00:16:39.270 --> 00:16:44.870
and theta x prime in terms of
these theta tilde in Fourier
00:16:44.870 --> 00:16:46.210
space.
00:16:46.210 --> 00:16:49.640
And then this will give
me some expectation
00:16:49.640 --> 00:16:52.340
that involves this average.
00:16:52.340 --> 00:16:55.100
And when I complete
that, eventually I
00:16:55.100 --> 00:16:58.563
need to Fourier transform
1 over q squared.
00:16:58.563 --> 00:17:01.970
And we've seen that the Fourier
transform of the 1 over q
00:17:01.970 --> 00:17:05.060
squared is the
Coulomb interaction.
00:17:05.060 --> 00:17:10.849
And so ultimately this becomes
exponential of minus of 1
00:17:10.849 --> 00:17:11.920
over k.
00:17:11.920 --> 00:17:15.770
The Coulomb interaction
as a function
00:17:15.770 --> 00:17:20.950
of x minus x prime with
the appropriate cutoff
00:17:20.950 --> 00:17:23.040
included in this expression.
00:17:27.560 --> 00:17:31.740
And then the statement
that we always make
00:17:31.740 --> 00:17:36.620
is that this entity
at large distances,
00:17:36.620 --> 00:17:40.690
the Coulomb interaction saying
three dimension, falls off as 1
00:17:40.690 --> 00:17:42.520
over separation.
00:17:42.520 --> 00:17:45.830
So in three dimensions,
this at large distances
00:17:45.830 --> 00:17:49.210
goes to a constant 1
over k is something that
00:17:49.210 --> 00:17:50.930
is proportional to temperature.
00:17:50.930 --> 00:17:52.810
So after doing
enough temperature,
00:17:52.810 --> 00:17:57.650
this entity at large
distances goes to a constant.
00:17:57.650 --> 00:18:01.810
So there is some knowledge
about the correlation
00:18:01.810 --> 00:18:04.440
between these
things that is left.
00:18:04.440 --> 00:18:08.200
Whereas in two
dimensions and below,
00:18:08.200 --> 00:18:11.440
when I go to sufficiently
large distances
00:18:11.440 --> 00:18:14.720
the Coulomb
interaction diverges.
00:18:14.720 --> 00:18:20.900
This whole thing goes to
0 and information is lost.
00:18:20.900 --> 00:18:25.740
So this is the in content
that Goldstone modes
00:18:25.740 --> 00:18:30.860
destroy through long range order
in two dimensions and below.
00:18:30.860 --> 00:18:34.560
Except that we saw
that in two dimensions,
00:18:34.560 --> 00:18:38.430
because the Coulomb interaction
grows logarithmically,
00:18:38.430 --> 00:18:41.340
this correlation
fall off according
00:18:41.340 --> 00:18:49.440
to this description only as
a power law with an exponent
00:18:49.440 --> 00:18:51.610
connected to k.
00:18:51.610 --> 00:18:55.710
And then the discussion
that we had for the XY model
00:18:55.710 --> 00:18:57.850
was that at high
temperatures you
00:18:57.850 --> 00:19:00.740
have exponential decay
in two dimensions.
00:19:00.740 --> 00:19:04.240
At low temperature you
have power law decay.
00:19:04.240 --> 00:19:06.310
Although there is no
true long range order,
00:19:06.310 --> 00:19:09.160
there has to be a
phase transition.
00:19:09.160 --> 00:19:11.040
And we saw that
this transition was
00:19:11.040 --> 00:19:16.360
described by the unbinding
of these topological defects.
00:19:16.360 --> 00:19:21.360
So we would like to state
that something similar to that
00:19:21.360 --> 00:19:22.780
is going on over here.
00:19:25.580 --> 00:19:29.000
The only reason I
went through this
00:19:29.000 --> 00:19:35.100
was to figure out what
the analog of this entity
00:19:35.100 --> 00:19:39.180
should be in the
case of the solid.
00:19:39.180 --> 00:19:44.765
So the analog of our theta is
this vector u, the distortion.
00:19:47.790 --> 00:19:50.480
What should be the analog
of e to the i theta?
00:19:54.150 --> 00:20:01.540
And the things that you should
keep in mind is that theta,
00:20:01.540 --> 00:20:03.390
I can't really tell
it's difference
00:20:03.390 --> 00:20:09.090
between theta and theta plus 2
pi, theta of 4 pi, et cetera.
00:20:09.090 --> 00:20:13.230
So this e to the i
theta captures that.
00:20:13.230 --> 00:20:16.510
If theta goes to theta
plus a multiple of 2 pi, e
00:20:16.510 --> 00:20:19.750
to the i theta
remains invariant.
00:20:19.750 --> 00:20:24.470
So it's a good
measure of ordering.
00:20:24.470 --> 00:20:27.680
It's better than theta itself.
00:20:27.680 --> 00:20:31.550
So similarly, there's
a better measure
00:20:31.550 --> 00:20:34.910
of deviation from
perfect order here
00:20:34.910 --> 00:20:38.510
for the case of the lattice.
00:20:38.510 --> 00:20:42.980
You say, well, why
should I worry about
00:20:42.980 --> 00:20:48.836
that since I can measure what
u is from the perfect position?
00:20:48.836 --> 00:20:52.880
Well, my answer is
this point here.
00:20:52.880 --> 00:20:56.280
Did it come from
here or from there?
00:20:56.280 --> 00:21:00.185
Should you use for u this
or should you use that?
00:21:00.185 --> 00:21:02.910
You don't know.
00:21:02.910 --> 00:21:08.680
So the u that you have the
to use similar to the theta
00:21:08.680 --> 00:21:12.920
that you're using is
somewhat arbitrary.
00:21:12.920 --> 00:21:18.440
And the arbitrariness of
it is given by unit vectors
00:21:18.440 --> 00:21:21.080
along the original lattice.
00:21:21.080 --> 00:21:26.570
That is, u and any u that
is increased or decreased
00:21:26.570 --> 00:21:31.300
by some lattice vector
of the original lattice
00:21:31.300 --> 00:21:32.360
is the same thing.
00:21:32.360 --> 00:21:35.230
You can't take them apart
in the same sense that theta
00:21:35.230 --> 00:21:38.890
and multiple of 2 pi you cannot.
00:21:38.890 --> 00:21:43.980
So then that suggest that I can
construct some kind of an order
00:21:43.980 --> 00:21:50.330
parameter, I call it row
G, which is like this.
00:21:50.330 --> 00:21:52.790
This e to i something.
00:21:52.790 --> 00:21:57.200
It's e to the iu
but u is a vector.
00:21:57.200 --> 00:22:04.490
And what I do I multiple by a
G, which is an inverse lattice
00:22:04.490 --> 00:22:06.560
vector.
00:22:06.560 --> 00:22:10.830
So G is an inverse
lattice vector.
00:22:14.950 --> 00:22:19.020
And essentially the
inverse lattice vectors
00:22:19.020 --> 00:22:24.210
are defined to have the property
that when you multiply them
00:22:24.210 --> 00:22:30.800
with any of the original lattice
vectors-- which are, let's say,
00:22:30.800 --> 00:22:34.930
parametrized by two
integers, m and n--
00:22:34.930 --> 00:22:37.600
the answer is going to
be some multiple of 2 pi.
00:22:44.300 --> 00:22:48.030
So you can see that
irrespective of whether I choose
00:22:48.030 --> 00:22:53.120
u as distortion from one lattice
position or any other lattice
00:22:53.120 --> 00:22:56.330
position that
happens to be nearby,
00:22:56.330 --> 00:23:00.800
this measure is going to be
giving you the same face.
00:23:03.940 --> 00:23:09.720
So in the same way as
before for the spins where
00:23:09.720 --> 00:23:13.010
I asked whether spins
at large distances
00:23:13.010 --> 00:23:18.270
are correlated long range,
I can do the same thing here
00:23:18.270 --> 00:23:32.600
and define a long range
correlations for a solid
00:23:32.600 --> 00:23:42.340
by looking at e to the iG dotted
with, let's say, u at position
00:23:42.340 --> 00:23:45.190
x minus u at position x prime.
00:23:45.190 --> 00:23:49.340
And x and x prime are two
points on the lattice.
00:23:49.340 --> 00:23:54.980
And again, the u's that I
have are Gaussian distributed.
00:23:54.980 --> 00:23:56.960
So this I will
complete the square.
00:23:56.960 --> 00:24:00.310
I will have
exponential minus 1/2.
00:24:00.310 --> 00:24:04.020
Well, OK, let me be a little
bit careful with this.
00:24:04.020 --> 00:24:08.920
I have G alpha G
beta over 2 because I
00:24:08.920 --> 00:24:11.490
have two factors of G dot u.
00:24:11.490 --> 00:24:15.670
I will write them using
Einstein's summation convention
00:24:15.670 --> 00:24:27.015
as G alpha times u alpha
component times the u beta
00:24:27.015 --> 00:24:27.515
component.
00:24:36.160 --> 00:24:42.210
So once more what we
can do is to then write
00:24:42.210 --> 00:24:47.170
these u's in terms of
their Fourier vectors.
00:24:47.170 --> 00:24:51.180
So what I will have is
exponential minus G alpha G
00:24:51.180 --> 00:24:52.480
beta over 2.
00:24:55.260 --> 00:25:02.675
I will need to
integrate over q's.
00:25:06.830 --> 00:25:13.790
Because of these
factors, I will get
00:25:13.790 --> 00:25:21.200
something like 2 minus 2
cosine of qx minus x prime.
00:25:29.590 --> 00:25:34.930
So whenever you go to Fourier
space, you will get e to the iq
00:25:34.930 --> 00:25:38.100
x minus e to the iq x prime.
00:25:38.100 --> 00:25:40.080
Different one for
here with different q
00:25:40.080 --> 00:25:42.350
primes, but when you
take the average,
00:25:42.350 --> 00:25:45.360
q prime is set to
minus q so then you
00:25:45.360 --> 00:25:47.180
get this form as usual.
00:25:47.180 --> 00:25:51.390
And then I will have the average
Fourier transform use, which
00:25:51.390 --> 00:25:53.150
is what I wrote over there.
00:25:53.150 --> 00:26:02.890
I will get a 1 over mu q square
delta i delta alpha beta minus
00:26:02.890 --> 00:26:12.938
q alpha q beta q squared mu
plus lambda over mu plus lambda.
00:26:26.870 --> 00:26:32.090
There is a little
bit of subtlety here.
00:26:32.090 --> 00:26:36.170
So I will do something
this is not quite kosher
00:26:36.170 --> 00:26:38.740
but gives me the right answer.
00:26:38.740 --> 00:26:46.110
Certainly I can take this
G alpha a beta inside here.
00:26:46.110 --> 00:26:49.200
When it multiplies
delta alpha beta,
00:26:49.200 --> 00:26:50.870
I will simply get G squared.
00:26:53.600 --> 00:26:55.410
I really G alpha G alpha.
00:26:55.410 --> 00:26:56.210
Sum over alpha.
00:26:56.210 --> 00:26:58.990
It will give me G squared.
00:26:58.990 --> 00:27:05.360
Here what I will
get is a G dot q
00:27:05.360 --> 00:27:11.110
squared because I will have
two Gq's divided by q squared.
00:27:15.130 --> 00:27:20.270
Ultimately I have to integrate
over all values of q, including
00:27:20.270 --> 00:27:21.170
all angle values.
00:27:23.990 --> 00:27:27.150
So there will be, as far
as the angle is concerned,
00:27:27.150 --> 00:27:30.930
something like a
cosine squared here.
00:27:30.930 --> 00:27:34.450
And the rather
unconventional thing to do
00:27:34.450 --> 00:27:37.230
is to angular
averages that first
00:27:37.230 --> 00:27:39.700
and write this as
G squared over 2.
00:27:44.100 --> 00:27:46.520
Once you do that,
then essentially
00:27:46.520 --> 00:27:51.730
you can see that this structure
just gives you a constant.
00:27:51.730 --> 00:27:55.210
And what you will need to
do is the usual Fourier
00:27:55.210 --> 00:27:58.990
transformation of
1 over q squared.
00:27:58.990 --> 00:28:03.300
So eventually you can
see that the answer
00:28:03.300 --> 00:28:09.080
is going to be proportional
to minus G squared.
00:28:09.080 --> 00:28:15.440
I will get a factor
of 2 mu u from here.
00:28:15.440 --> 00:28:20.220
Once I have written this
as G squared over 2,
00:28:20.220 --> 00:28:24.330
I will get 1 minus
1/2 of this quantity.
00:28:24.330 --> 00:28:28.230
So that becomes 2
mu plus lambda--
00:28:28.230 --> 00:28:33.130
or 4 mu plus lambda
minus mu minus lambda.
00:28:33.130 --> 00:28:41.880
So I will get a 3 mu plus lambda
divided by 2 2 mu plus lambda.
00:28:41.880 --> 00:28:45.490
And so there's a
factor of 4 here.
00:28:45.490 --> 00:28:48.830
So that's just the numerics
that goes out front.
00:28:48.830 --> 00:28:52.750
The key to the whole thing will
be the usual integration of 1
00:28:52.750 --> 00:28:54.160
over q squared.
00:28:54.160 --> 00:28:58.270
Just as we had before, I will
get again the same Coulomb
00:28:58.270 --> 00:29:01.710
interaction as a
function of x minus x
00:29:01.710 --> 00:29:04.920
prime with appropriate
lattice cut-off
00:29:04.920 --> 00:29:06.920
introduced so that
the short distance
00:29:06.920 --> 00:29:08.075
behavior is controlled.
00:29:14.350 --> 00:29:16.810
This whole thing has a name.
00:29:16.810 --> 00:29:18.595
It's called a
Debye-Waller factor.
00:29:25.160 --> 00:29:25.710
Well, almost.
00:29:28.561 --> 00:29:29.060
In 3D.
00:29:34.030 --> 00:29:36.050
And the point is
that if I'm looking
00:29:36.050 --> 00:29:39.056
at this whole thing
in three dimensions,
00:29:39.056 --> 00:29:44.026
at large enough separation
this will go to a constant,
00:29:44.026 --> 00:29:46.290
just as we had before.
00:29:46.290 --> 00:29:51.410
And I will conclude
that this entity, as I
00:29:51.410 --> 00:29:55.000
look at two points that are
very far apart, at least that's
00:29:55.000 --> 00:29:58.500
sufficiently low temperature
and all of my mus and lambdas
00:29:58.500 --> 00:30:00.280
are scaled inversely
with temperature,
00:30:00.280 --> 00:30:03.150
so this whole thing is
proportional to temperature.
00:30:03.150 --> 00:30:07.450
Essentially, you can see
that this information
00:30:07.450 --> 00:30:10.870
at large distance goes
to a finite constant,
00:30:10.870 --> 00:30:14.250
telling me that there's
long range orders preserved.
00:30:14.250 --> 00:30:16.880
Again, when I go
to two dimensions,
00:30:16.880 --> 00:30:22.350
you would say that this,
at large distances,
00:30:22.350 --> 00:30:24.510
grows logarithmically.
00:30:24.510 --> 00:30:27.780
And so the correlations
will fall off,
00:30:27.780 --> 00:30:32.630
but fall of as a power law.
00:30:32.630 --> 00:30:38.780
Now for the case
of a solid, usually
00:30:38.780 --> 00:30:47.960
what you do in order to
probe for the order is
00:30:47.960 --> 00:30:56.400
you shine x-rays and
you see what comes out.
00:30:56.400 --> 00:31:00.170
So essentially, you
will see that there
00:31:00.170 --> 00:31:03.180
will be some scattering.
00:31:03.180 --> 00:31:08.740
And the way that the wave
vector of the light has changed
00:31:08.740 --> 00:31:11.630
is characterized by a vector q.
00:31:11.630 --> 00:31:15.755
And we discussed this already,
that the structure factor
00:31:15.755 --> 00:31:24.340
s of q is going to be related to
expectation value of the phase
00:31:24.340 --> 00:31:26.280
that you get from
the different points.
00:31:26.280 --> 00:31:31.070
So I will get a factor
of e to the i q.
00:31:31.070 --> 00:31:37.900
the location of the different
atoms, let's call them r, of x.
00:31:37.900 --> 00:31:40.140
I have to do a sum over x.
00:31:40.140 --> 00:31:43.010
I have to scatter
from all of the atoms.
00:31:43.010 --> 00:31:45.740
There's some overall form
factor that if I'm not
00:31:45.740 --> 00:31:48.070
really interested,
but this whole thing
00:31:48.070 --> 00:31:49.245
has to be first squished.
00:31:53.850 --> 00:31:59.620
So once I square this, I
will be getting averages
00:31:59.620 --> 00:32:02.060
between pairs of points.
00:32:02.060 --> 00:32:04.650
Because of
translational symmetry
00:32:04.650 --> 00:32:08.770
I can relate to that
sum over pairs of points
00:32:08.770 --> 00:32:11.370
to a center of
mass that typically
00:32:11.370 --> 00:32:16.410
would pick up the number
of points that I have.
00:32:16.410 --> 00:32:19.990
And then the
expectation value of e
00:32:19.990 --> 00:32:23.960
to the something like
sum over all points, e
00:32:23.960 --> 00:32:34.930
to the i q.r r of x
minus r of 0 average.
00:32:34.930 --> 00:32:36.990
This average I can
certainly take out here.
00:32:42.435 --> 00:32:43.409
Yes?
00:32:43.409 --> 00:32:44.170
STUDENT: Sorry.
00:32:44.170 --> 00:32:46.230
Is r what you're
calling u before?
00:32:46.230 --> 00:32:47.600
PROFESSOR: No.
00:32:47.600 --> 00:32:52.910
So r is the actual location of
where the particle is location.
00:32:52.910 --> 00:32:58.130
So r, let's say for a particle
that at zero temperature
00:32:58.130 --> 00:33:00.500
was sitting at lattice
point-- that is,
00:33:00.500 --> 00:33:05.520
let's say, in two dimensions
labeled by m and n--
00:33:05.520 --> 00:33:09.090
it is shifted by an
amount that is u that I
00:33:09.090 --> 00:33:11.320
would assign to that point.
00:33:11.320 --> 00:33:16.900
So the r's that I put here are
the actual physical position
00:33:16.900 --> 00:33:18.270
of the atom.
00:33:18.270 --> 00:33:20.210
Which is composed
of where it was
00:33:20.210 --> 00:33:23.143
sitting at 0 temperature
plus a little bit more.
00:33:30.880 --> 00:33:38.295
So when I do this, indeed I will
get a sum over all positions.
00:33:41.850 --> 00:33:48.555
This could be a label mn,
whatever, of e to the iq.
00:33:51.300 --> 00:33:53.890
The difference
between the two r's
00:33:53.890 --> 00:33:57.930
would be, first of all, the
difference between these two
00:33:57.930 --> 00:34:02.570
at 0 temperature, which
would be some lattice
00:34:02.570 --> 00:34:05.060
vector of some variety.
00:34:05.060 --> 00:34:06.705
Let's call r 0 of x.
00:34:09.940 --> 00:34:15.230
And that quantity has
[INAUDIBLE] fluctuations.
00:34:15.230 --> 00:34:23.080
The fluctuations go into e to
the iq.u of x minus mu of 0.
00:34:32.170 --> 00:34:35.870
Now if I'm doing this at 0
temperature or very close to 0
00:34:35.870 --> 00:34:41.900
temperature where this is either
1 at 0 temperature or not too
00:34:41.900 --> 00:34:44.900
much fluctuating at
high temperatures,
00:34:44.900 --> 00:34:47.929
I will essentially be
adding a lot of e to iqr's.
00:34:51.520 --> 00:34:53.900
These are essentially phases.
00:34:53.900 --> 00:34:58.210
They can be positive, negative,
all over the complex plane.
00:34:58.210 --> 00:35:02.400
And typically you would expect
that for a randomly chosen q,
00:35:02.400 --> 00:35:08.410
the answer from adding all
of these things will be 0.
00:35:08.410 --> 00:35:12.510
So the only time we're
close to 0 temperature,
00:35:12.510 --> 00:35:18.450
you expect this to be
something that is significant.
00:35:18.450 --> 00:35:36.470
Scattering is when q is one of
these inverse lattice vectors.
00:35:36.470 --> 00:35:41.670
So if I was really doing this at
0 temperature, this would be 1.
00:35:41.670 --> 00:35:45.030
And the answer that I
would get is essentially
00:35:45.030 --> 00:35:49.350
I would get these delta
function back peaks
00:35:49.350 --> 00:35:53.665
at the locations that correspond
to the inverse lattice spacing.
00:35:53.665 --> 00:35:56.215
And that's presuming
something that you have seen
00:35:56.215 --> 00:35:58.470
in crystallography or whatever.
00:35:58.470 --> 00:36:02.120
You take a solid and you
scatter light from it
00:36:02.120 --> 00:36:03.885
and you will back points.
00:36:13.120 --> 00:36:19.350
So what I expect
is that when I have
00:36:19.350 --> 00:36:26.750
a q that is roughly close to
G but not necessarily sitting
00:36:26.750 --> 00:36:30.870
exactly at that point,
what I would get
00:36:30.870 --> 00:36:35.270
is a scattering that these
proportional to an integral
00:36:35.270 --> 00:36:41.060
over the entire
lattice of something
00:36:41.060 --> 00:36:50.590
like how far I am away
from the right position.
00:36:50.590 --> 00:36:55.395
And I should really write this
as a sum over lattice points,
00:36:55.395 --> 00:37:00.250
but I can approximate
this by an integral.
00:37:00.250 --> 00:37:04.190
The integral again has the
property that it will give me 0
00:37:04.190 --> 00:37:07.700
unless q is exactly
sitting at G.
00:37:07.700 --> 00:37:11.160
And then for here, I
evaluate this quantity
00:37:11.160 --> 00:37:14.240
when q is close to
G according to what
00:37:14.240 --> 00:37:18.120
I have calculated up there.
00:37:18.120 --> 00:37:23.600
So I will get
exponential of minus G
00:37:23.600 --> 00:37:29.160
squared-- What do I have?
00:37:29.160 --> 00:37:35.770
Over 4 mu, three mu plus
lambda, 2 mu plus lambda.
00:37:35.770 --> 00:37:40.140
The Coulomb interaction
as a function of x.
00:37:45.981 --> 00:37:46.480
Yes?
00:37:46.480 --> 00:37:49.012
STUDENT: Are you missing
an overall factor of m?
00:37:49.012 --> 00:37:51.510
PROFESSOR: That's why I
wrote the proportionality.
00:37:51.510 --> 00:37:56.340
Because what I'm really
interested is what you see.
00:37:56.340 --> 00:38:02.160
And what you see is how things
are varying as a function of q.
00:38:02.160 --> 00:38:04.230
So I'll tell you
what the answer is.
00:38:04.230 --> 00:38:07.410
So you do the Fourier
transform from this.
00:38:07.410 --> 00:38:14.640
And in d equals to 3, what
you find at 0 temperature,
00:38:14.640 --> 00:38:18.700
as we said, you
will get Bragg spots
00:38:18.700 --> 00:38:21.830
at the locations of
G's that correspond
00:38:21.830 --> 00:38:29.990
to the inverse lattice vectors,
which actually themselves
00:38:29.990 --> 00:38:31.030
form a lattice.
00:38:36.420 --> 00:38:40.040
So this is essentially
the space of q.
00:38:40.040 --> 00:38:45.130
And these are the
different values of G.
00:38:45.130 --> 00:38:49.290
Now what happens is that
the strength of each one
00:38:49.290 --> 00:38:53.210
of these delta functions
is then modulated
00:38:53.210 --> 00:38:57.620
by this factor evaluated
at the corresponding G.
00:38:57.620 --> 00:39:00.890
And since it is
proportional to G squared,
00:39:00.890 --> 00:39:03.750
you find that the spots
that are close to the origin
00:39:03.750 --> 00:39:04.580
are well defined.
00:39:08.360 --> 00:39:15.210
As you go further and further
away, the become very weaker.
00:39:15.210 --> 00:39:17.630
And at some point you
cease to see them.
00:39:17.630 --> 00:39:21.290
So that's the-- the
Debye-Waller factor
00:39:21.290 --> 00:39:25.910
describes how these
things vanish as you
00:39:25.910 --> 00:39:27.380
go further and further away.
00:39:27.380 --> 00:39:30.420
And essentially, you find
that they go and diminish
00:39:30.420 --> 00:39:33.400
as e to the minus G squared.
00:39:33.400 --> 00:39:39.960
But clearly, this kind of
scattering from a crystal
00:39:39.960 --> 00:39:42.100
is very different
from the scattering
00:39:42.100 --> 00:39:44.460
that you expect from
an liquid, which
00:39:44.460 --> 00:39:51.860
is essentially you will have
a bright spot at q close to 0
00:39:51.860 --> 00:39:54.320
and things that would
symmetrically fall off,
00:39:54.320 --> 00:39:56.170
maybe with a couple
of oscillations
00:39:56.170 --> 00:39:59.600
due to the size of the
particles, at cetera.
00:40:04.120 --> 00:40:10.920
Now what happens in d equals
to 2 is kind of interesting.
00:40:10.920 --> 00:40:15.520
Because in d equals
to 2, I'm doing a two
00:40:15.520 --> 00:40:19.510
dimensional integral,
and this thing
00:40:19.510 --> 00:40:21.980
is growing as a power law.
00:40:21.980 --> 00:40:24.300
So basically what
I end up having
00:40:24.300 --> 00:40:31.410
to deal with is something
like e to the i q minus G x.
00:40:31.410 --> 00:40:37.270
And something that falls off
as a over x to some power
00:40:37.270 --> 00:40:46.120
that I will simply cause eta G.
So related to that combination.
00:40:46.120 --> 00:40:50.170
And when you
integrate this, just
00:40:50.170 --> 00:40:52.450
dimensionally you
can see that it
00:40:52.450 --> 00:40:58.140
is an integral that scales as
x to the power of 2 minus eta.
00:40:58.140 --> 00:41:03.200
x is inverse to q minus G. So
this is going to scale as 1
00:41:03.200 --> 00:41:07.893
over q minus g to the
power of 2 minus eta.
00:41:11.280 --> 00:41:18.930
So as you go along some
particular direction
00:41:18.930 --> 00:41:23.750
and ask what you see,
you find that will
00:41:23.750 --> 00:41:30.860
see peaks close to the locations
of the lattice vectors.
00:41:30.860 --> 00:41:36.410
So let's say they are equally
spaced along the particular
00:41:36.410 --> 00:41:38.420
axis that I'm picking.
00:41:38.420 --> 00:41:41.750
In three dimension, you would
have been seeing delta function
00:41:41.750 --> 00:41:45.980
at each one of these positions
whose strength is governed
00:41:45.980 --> 00:41:48.010
by this Debye-Waller factor.
00:41:48.010 --> 00:41:50.040
In two dimensions,
you will essentially
00:41:50.040 --> 00:41:54.752
see power laws close
to each one of them.
00:41:54.752 --> 00:42:00.970
But as you go further, the
exponent of the power law
00:42:00.970 --> 00:42:02.580
changes.
00:42:02.580 --> 00:42:06.030
So at some point you even
cease to have a divergence.
00:42:06.030 --> 00:42:10.960
But you can essentially
ignore anything
00:42:10.960 --> 00:42:12.360
like that [? happening. ?]
00:42:12.360 --> 00:42:16.400
So you can see that because
we don't have true long range
00:42:16.400 --> 00:42:19.710
order into two
dimensions because
00:42:19.710 --> 00:42:21.610
of these logarithmic
divergences,
00:42:21.610 --> 00:42:24.440
et cetera, the
traditional picture
00:42:24.440 --> 00:42:28.155
that we have of a solid that
is characterized by Bragg
00:42:28.155 --> 00:42:32.950
scattering and delta
function peaks is modified,
00:42:32.950 --> 00:42:34.790
but still it is
something that looks
00:42:34.790 --> 00:42:37.850
very different from the liquid.
00:42:37.850 --> 00:42:42.680
So we expect that as
we raise temperature
00:42:42.680 --> 00:42:46.240
there is a phase transition
between scattering
00:42:46.240 --> 00:42:51.480
of this form and something
that looks like liquid.
00:42:51.480 --> 00:42:55.120
And you can roughly
guess that essentially it
00:42:55.120 --> 00:42:59.670
is going to be related
with these etas becoming
00:42:59.670 --> 00:43:06.630
larger and larger so that
these divergences disappear.
00:43:06.630 --> 00:43:08.130
But what is the mechanism?
00:43:10.860 --> 00:43:14.980
So let's go back and see what
the mechanism was that we
00:43:14.980 --> 00:43:17.630
discovered for the XY model.
00:43:17.630 --> 00:43:23.050
So we are going to look at
d equals to 2 from now on.
00:43:23.050 --> 00:43:26.780
That discussion so
far was general.
00:43:26.780 --> 00:43:32.260
In d equals to 2, we said that
we have topological defect.
00:43:38.980 --> 00:43:44.330
And the story here was that,
as I have been discussing,
00:43:44.330 --> 00:43:47.870
the angle is
undefined up to 2 pi.
00:43:47.870 --> 00:43:50.785
So it would configuration of
angles that kind of radiate
00:43:50.785 --> 00:43:54.230
away from the origin.
00:43:57.640 --> 00:44:05.750
And the idea was that
if I take a circuit
00:44:05.750 --> 00:44:09.720
and integrate around
this circuit--
00:44:09.720 --> 00:44:18.820
let's call this the s gradient
of theta-- the answer does not
00:44:18.820 --> 00:44:21.560
need to come back to 0.
00:44:21.560 --> 00:44:25.194
It will be some
multiple of 2 pi .
00:44:25.194 --> 00:44:26.110
Some integer multiple.
00:44:30.371 --> 00:44:35.100
And similarly here, we
said that the value of u
00:44:35.100 --> 00:44:42.430
is undefined up to
a lattice spacing.
00:44:42.430 --> 00:44:45.660
That leads to topological
defects, which
00:44:45.660 --> 00:44:49.280
I find it very difficult to
draw for the triangular lattice,
00:44:49.280 --> 00:44:51.745
so I'll draw it for
the square lattice.
00:45:17.980 --> 00:45:22.500
So at each one of
these positions
00:45:22.500 --> 00:45:23.980
there's a particle sitting.
00:45:26.710 --> 00:45:29.195
And you look out here.
00:45:29.195 --> 00:45:31.550
You have a perfect
square lattice.
00:45:31.550 --> 00:45:33.080
You look out here.
00:45:33.080 --> 00:45:35.690
You have a perfect
square lattice.
00:45:35.690 --> 00:45:38.770
But clearly it is not a
perfect square lattice.
00:45:38.770 --> 00:45:42.670
And the analog of the
circuit that I drew over here
00:45:42.670 --> 00:45:45.980
that encloses a
singlularty-- there's clearly
00:45:45.980 --> 00:45:49.540
some kind of a defect
sitting over here--
00:45:49.540 --> 00:45:54.190
is that I can start with
some point on the lattice
00:45:54.190 --> 00:45:57.310
and perform what is
called a Burgers Circuit.
00:46:05.650 --> 00:46:09.640
What Burgers Circuit
is is some walk
00:46:09.640 --> 00:46:12.150
that you would take
on a lattice, which
00:46:12.150 --> 00:46:14.590
in a perfect lattice
would bring you back
00:46:14.590 --> 00:46:16.690
to your starting point.
00:46:16.690 --> 00:46:20.460
So for example-- that was
not a good starting point.
00:46:20.460 --> 00:46:24.520
Let me start from here.
00:46:24.520 --> 00:46:30.720
Let's say I take four steps
up-- one, two, three, four.
00:46:30.720 --> 00:46:33.700
I take five steps to the right.
00:46:33.700 --> 00:46:36.910
One, two, three, four, five.
00:46:36.910 --> 00:46:38.920
I reverse my four steps.
00:46:38.920 --> 00:46:40.620
So I go four steps down.
00:46:40.620 --> 00:46:42.800
One, two, three, four.
00:46:42.800 --> 00:46:47.200
I reverse my five steps
and I go five steps left.
00:46:47.200 --> 00:46:51.180
One, two, three, four, five.
00:46:51.180 --> 00:46:55.100
And you see that I did
not end up where I was.
00:47:00.000 --> 00:47:03.570
I ended up here.
00:47:03.570 --> 00:47:07.580
And a failure to
close my circuit
00:47:07.580 --> 00:47:09.370
has to be a lattice vector.
00:47:12.710 --> 00:47:26.310
And this is called d and
it's called a Burgers Vector
00:47:26.310 --> 00:47:30.210
that characterize this defect,
which is called a dislocation.
00:47:38.250 --> 00:47:45.070
So the uncertainty that I have
in assigning a value for u
00:47:45.070 --> 00:47:50.800
because of being on a lattice
or deforming a lattice
00:47:50.800 --> 00:47:55.430
gives rise to these
kinds of ambiguity
00:47:55.430 --> 00:47:58.630
that are similar to
the topological defects
00:47:58.630 --> 00:48:00.710
that you had over here.
00:48:00.710 --> 00:48:05.690
Whereas the defects here were
characterized by an integer,
00:48:05.690 --> 00:48:07.760
the defects here
are characterized
00:48:07.760 --> 00:48:10.410
by a lattice vector.
00:48:10.410 --> 00:48:15.310
And here the simplest things
were, say, plus or minus.
00:48:15.310 --> 00:48:17.470
Clearly here on
the square lattice
00:48:17.470 --> 00:48:21.530
the simplest things are plus b,
minus b along the x direction
00:48:21.530 --> 00:48:24.280
and plus b minus b
along the y direction.
00:48:24.280 --> 00:48:26.160
For example, on the
triangular lattice
00:48:26.160 --> 00:48:30.560
it would be plus minus at
any of the three directions
00:48:30.560 --> 00:48:32.010
that would define the lattice.
00:48:35.330 --> 00:48:41.470
So calculating the distortion
field here was actually
00:48:41.470 --> 00:48:48.750
an easy matter because we said
that the gradient of theta
00:48:48.750 --> 00:48:52.620
was something like 1 over r.
00:48:52.620 --> 00:49:04.110
And so the gradient of
theta, which was 1 over r,
00:49:04.110 --> 00:49:09.370
I could in fact write--
we saw as vector
00:49:09.370 --> 00:49:14.640
that is orthogonal to the vector
that is going out of the plane.
00:49:14.640 --> 00:49:17.420
So we call that a z hat.
00:49:17.420 --> 00:49:24.660
It was also orthogonal to
the direction of motion
00:49:24.660 --> 00:49:28.760
away from the defect,
which we could characterize
00:49:28.760 --> 00:49:32.410
by looking at the gradient.
00:49:32.410 --> 00:49:35.330
The gradient vector
for this distortion
00:49:35.330 --> 00:49:39.040
is clearly in the
radial direction.
00:49:39.040 --> 00:49:42.890
And the answer that we
had here for one defect
00:49:42.890 --> 00:49:49.490
was ni log of r minus ri.
00:49:49.490 --> 00:49:51.870
And since I'm taking the
gradient it doesn't matter.
00:49:51.870 --> 00:49:55.860
I can't put or not
put the cut off.
00:49:55.860 --> 00:49:59.596
And if I had multiple ones,
I would simply sum over it.
00:50:02.490 --> 00:50:08.820
So this was the contribution
to the distortion
00:50:08.820 --> 00:50:15.950
that came from a collection
of defects such as this.
00:50:15.950 --> 00:50:20.650
Now you can see that essentially
in some continuum sense,
00:50:20.650 --> 00:50:25.440
all I'm doing here is similar.
00:50:25.440 --> 00:50:28.890
I can be taking a big circuit.
00:50:28.890 --> 00:50:33.530
And as I go along
that big circuit,
00:50:33.530 --> 00:50:39.190
I can take a gradient--
So let's-- yeah.
00:50:39.190 --> 00:50:39.860
The s.
00:50:39.860 --> 00:50:44.500
gradient of the
distortion field.
00:50:44.500 --> 00:50:46.300
Distortion field is,
of course, a vector.
00:50:46.300 --> 00:50:50.350
So it has components that I
can write as the x component
00:50:50.350 --> 00:50:51.180
or the y component.
00:50:51.180 --> 00:50:55.560
So this alpha could be the x
component of the distortion
00:50:55.560 --> 00:50:58.920
or the y component
of the distortion.
00:50:58.920 --> 00:51:04.020
And once I complete the circuit,
here the answer was 2 pi n.
00:51:04.020 --> 00:51:07.140
Here the answer is
a lattice vector,
00:51:07.140 --> 00:51:11.665
which is this Burgers Vector
that could be in any direction
00:51:11.665 --> 00:51:12.447
that you want.
00:51:15.370 --> 00:51:18.570
Now as far as
mathematics is concerned,
00:51:18.570 --> 00:51:25.690
this line and this line for each
component are exactly the same.
00:51:25.690 --> 00:51:29.960
So the solution that I can
write for gradient of u
00:51:29.960 --> 00:51:33.870
I can just copy from
here for each component.
00:51:33.870 --> 00:51:37.580
So I can write that
gradient of the u that
00:51:37.580 --> 00:51:41.710
is due to a collection
of dislocations
00:51:41.710 --> 00:51:46.360
is something like z
hat crossed with curl
00:51:46.360 --> 00:51:54.480
of a sum over all of
the potential locations
00:51:54.480 --> 00:51:56.640
off my defect.
00:51:56.640 --> 00:51:59.605
Rather than n, I have b over 2.
00:51:59.605 --> 00:52:07.611
And it's a vector log of
minus ri over [INAUDIBLE].
00:52:11.760 --> 00:52:17.533
So if I have a collection
of these dislocations
00:52:17.533 --> 00:52:24.150
at different places on my
lattice-- locations ri,
00:52:24.150 --> 00:52:28.010
strengths, b alpha
which is a vector--
00:52:28.010 --> 00:52:31.330
then the distortion field
that they would generate
00:52:31.330 --> 00:52:33.830
is given by this.
00:52:33.830 --> 00:52:36.947
It's very just following the
answers that the [INAUDIBLE].
00:52:43.290 --> 00:52:51.780
So then what we did for the
case of the overall system--
00:52:51.780 --> 00:52:57.210
in order to find how the
different topological defects
00:52:57.210 --> 00:53:02.070
interact with each
was to calculate
00:53:02.070 --> 00:53:12.220
beta h, which was an integral
of gradient of theta squared.
00:53:14.650 --> 00:53:15.150
Yes?
00:53:15.150 --> 00:53:20.131
STUDENT: So the b alpha, you
have an index i [? also? ?]
00:53:20.131 --> 00:53:22.580
PROFESSOR: b alpha
have index i also.
00:53:22.580 --> 00:53:24.068
Like like the ni have index i.
00:53:24.068 --> 00:53:25.930
So maybe I write
it in this fashion.
00:53:25.930 --> 00:53:26.430
Yeah.
00:53:34.710 --> 00:53:38.960
So this is the cost that
we have to evaluate,
00:53:38.960 --> 00:53:41.870
except that we
have to be somewhat
00:53:41.870 --> 00:53:48.702
careful with the meaning
of this gradient of theta
00:53:48.702 --> 00:53:52.160
in that the gradient
of theta, as we said,
00:53:52.160 --> 00:53:58.370
has a contribution that
is from regular spin
00:53:58.370 --> 00:54:03.200
waves, the Goldstone modes,
that can be deformed back
00:54:03.200 --> 00:54:05.320
to everybody pointing
in the same direction
00:54:05.320 --> 00:54:07.910
without topological defect.
00:54:07.910 --> 00:54:17.930
Plus a contribution due to this
topological defect that are
00:54:17.930 --> 00:54:21.708
categorized in that case by ni.
00:54:21.708 --> 00:54:25.520
Now similarly here for
the case of the solid,
00:54:25.520 --> 00:54:27.560
we have the beta h.
00:54:27.560 --> 00:54:31.330
Slightly more complicated
integral d 2x.
00:54:31.330 --> 00:54:38.264
We have mu uij uij plus
lambda over 2 uii ujj.
00:54:42.880 --> 00:54:46.970
And what we can do is to
say that our strain field
00:54:46.970 --> 00:54:55.110
uij has a component that
is like the Goldstone modes
00:54:55.110 --> 00:54:57.500
that we have been
calculating so far,
00:54:57.500 --> 00:55:01.680
essentially treating
everything as Gaussians.
00:55:01.680 --> 00:55:04.200
Except that I will
write it as phi ij.
00:55:07.200 --> 00:55:14.430
And then a part that would come
from the distortion field u
00:55:14.430 --> 00:55:16.480
bar that I calculated.
00:55:16.480 --> 00:55:20.660
So I take that distortion field
and then take derivatives of it
00:55:20.660 --> 00:55:23.120
to symmetrize
appropriately to construct
00:55:23.120 --> 00:55:24.790
the strain that sums from them.
00:55:28.490 --> 00:55:34.750
So we substituted
this form over here.
00:55:34.750 --> 00:55:42.040
And we found that beta h that we
got had a part that was simply
00:55:42.040 --> 00:55:45.080
relate to the Goldstone modes.
00:55:45.080 --> 00:55:49.220
This was the part that could
be treating as a Gaussian.
00:55:49.220 --> 00:55:56.950
And then we had a
part that corresponded
00:55:56.950 --> 00:56:05.620
to the interactions
among these defects.
00:56:05.620 --> 00:56:11.782
And that part we saw
had the character
00:56:11.782 --> 00:56:22.140
of charges ni and
nj plus minus 1
00:56:22.140 --> 00:56:25.750
characterizing these
topological defects
00:56:25.750 --> 00:56:28.350
having a Coulomb interaction
[? between them ?].
00:56:37.500 --> 00:56:42.810
And then, of course,
this i less than j,
00:56:42.810 --> 00:56:46.610
we had to worry about what
was happening when i was j.
00:56:46.610 --> 00:56:51.970
And that we considered
to be the contributions
00:56:51.970 --> 00:56:57.390
of the core energy that,
once exponentiated,
00:56:57.390 --> 00:57:01.350
we described as y.
00:57:01.350 --> 00:57:07.750
So it's beta h which is log
of that would be log of y.
00:57:07.750 --> 00:57:14.400
And actually this k bar, by
the way, was simply 2 pi k.
00:57:14.400 --> 00:57:20.680
And it was 2 pi k because each
one of these charges is 2 pi n.
00:57:20.680 --> 00:57:22.400
So there's 2 pi 2 pi.
00:57:22.400 --> 00:57:24.050
But the Coulomb
interaction really
00:57:24.050 --> 00:57:26.355
should be log divided by 2 pi.
00:57:26.355 --> 00:57:29.896
And so k bar becomes
2 pi k times this.
00:57:32.860 --> 00:57:40.422
So I can do the same
thing over here.
00:57:40.422 --> 00:57:47.740
And what I find happens is
that my beta h gets decomposed
00:57:47.740 --> 00:57:48.890
as follows.
00:57:48.890 --> 00:57:55.690
There will be a part that is
simply the original expression
00:57:55.690 --> 00:58:01.680
now for the field
that is well behaved
00:58:01.680 --> 00:58:03.470
and has no dislocation piece.
00:58:08.290 --> 00:58:10.920
And then you
wouldn't be surprised
00:58:10.920 --> 00:58:13.520
because this structure
is no different
00:58:13.520 --> 00:58:14.770
from the other structure.
00:58:14.770 --> 00:58:17.960
There's essentially a
gradient of this distortion
00:58:17.960 --> 00:58:19.420
field squared.
00:58:19.420 --> 00:58:22.190
And you can see that if I
take a second derivative
00:58:22.190 --> 00:58:27.290
of this distortion field or one
derivative of this, effectively
00:58:27.290 --> 00:58:30.490
I will have two
derivatives of a log, which
00:58:30.490 --> 00:58:32.170
will give me a delta function.
00:58:32.170 --> 00:58:35.730
So essentially these
things do behave
00:58:35.730 --> 00:58:39.370
like what you would get from
Coulomb type of potential.
00:58:39.370 --> 00:58:41.740
[INAUDIBLE] plus [INAUDIBLE]
if the potential is 0.
00:58:41.740 --> 00:58:46.770
So not surprisingly, I
we get a minus k bar sum
00:58:46.770 --> 00:58:52.600
over pairs of where these
dislocations are located.
00:58:52.600 --> 00:59:00.440
And then I will have b at
location i, b at location j.
00:59:00.440 --> 00:59:10.050
And then I would have a log of
ri minus rj with some cut off.
00:59:10.050 --> 00:59:14.260
Except that these b's
are actually vector.
00:59:14.260 --> 00:59:20.510
So this term is a dot
product of the two b's.
00:59:20.510 --> 00:59:24.050
And it turns out that when
you go through the algebra,
00:59:24.050 --> 00:59:28.170
there is another
term, which is bi.
00:59:28.170 --> 00:59:41.820
ri minus rj, bj dotted
with ri minus rj
00:59:41.820 --> 00:59:47.040
divided by ri minus rj squared.
00:59:51.190 --> 00:59:58.370
So the charges that we
had in our original theory
00:59:58.370 --> 01:00:00.070
were scalar quantities.
01:00:00.070 --> 01:00:05.600
These defects were characterized
by a scalar value of n.
01:00:05.600 --> 01:00:08.400
And they were interacting
with something
01:00:08.400 --> 01:00:13.180
that was like the ordinary
Coulomb potential.
01:00:13.180 --> 01:00:15.640
Whereas now we are
looking at a system
01:00:15.640 --> 01:00:20.220
that is characterized by
charges that are vectors.
01:00:20.220 --> 01:00:23.300
And it turns out that
what we have here
01:00:23.300 --> 01:00:27.590
is the vectorial analog
of the Coulomb potential.
01:00:27.590 --> 01:00:32.190
And again, if you think about
an isotropic system and vectors,
01:00:32.190 --> 01:00:35.170
this is a vector
that you can form,
01:00:35.170 --> 01:00:39.280
but b.r is another
vector that you can form.
01:00:39.280 --> 01:00:41.440
And so both of them do appear.
01:00:41.440 --> 01:00:43.180
And once you go through
the whole algebra
01:00:43.180 --> 01:00:46.680
and through the inverse
Fourier transform, et cetera,
01:00:46.680 --> 01:00:49.370
you get this
additional contribution
01:00:49.370 --> 01:00:51.050
to this vector
Coulomb interaction.
01:00:57.470 --> 01:01:00.170
That's really is
the only difference.
01:01:00.170 --> 01:01:01.830
And of course,
you will again get
01:01:01.830 --> 01:01:09.220
a sum over all locations
of the core energies
01:01:09.220 --> 01:01:11.497
for creating these defects.
01:01:17.840 --> 01:01:31.130
And the value of k bar here
is related to these parameters
01:01:31.130 --> 01:01:39.536
by mu mu plus lambda divided
by pi 2u plus lambda.
01:01:44.970 --> 01:01:49.170
So this combination
controls what you have.
01:01:57.790 --> 01:02:03.010
So the next thing
that we did was
01:02:03.010 --> 01:02:08.640
to construct a perturbation.
01:02:08.640 --> 01:02:16.310
We essentially said that if
I have these pairs of charges
01:02:16.310 --> 01:02:19.170
that can spontaneously
appear at an energy
01:02:19.170 --> 01:02:26.790
cost but an entropy gain,
they will effectively weaken
01:02:26.790 --> 01:02:28.850
the overall Coulomb potential.
01:02:28.850 --> 01:02:33.060
For example, if I
have to test charges
01:02:33.060 --> 01:02:36.340
then there will be
some polarization
01:02:36.340 --> 01:02:38.910
of the medium,
some reorientation
01:02:38.910 --> 01:02:42.280
of these charges that
appear, that weakens
01:02:42.280 --> 01:02:43.670
the effective charge here.
01:02:46.220 --> 01:02:48.840
And we could
perturbatively calculate
01:02:48.840 --> 01:02:51.990
what the correction
was, except that we
01:02:51.990 --> 01:02:54.480
found that that
correction, which
01:02:54.480 --> 01:02:59.980
involves an integration over
the separation of these things.
01:02:59.980 --> 01:03:02.460
There was an integration
that was potentially
01:03:02.460 --> 01:03:07.610
divergent for us no matter
how small we made the core
01:03:07.610 --> 01:03:09.910
energies here.
01:03:09.910 --> 01:03:13.770
And so what we did was
eventually to construct an RG.
01:03:17.100 --> 01:03:24.600
And the RG was that the
value of this parameter k
01:03:24.600 --> 01:03:30.010
bar-- actually more
usefully its inverse that
01:03:30.010 --> 01:03:34.640
is related to
temperature-- changed
01:03:34.640 --> 01:03:39.980
as a function of integrating
out short distance
01:03:39.980 --> 01:03:42.580
degrees of freedom.
01:03:42.580 --> 01:03:46.641
It was something like 4 pi--
I think it was 4 pi cubed,
01:03:46.641 --> 01:03:47.890
it doesn't matter-- y squared.
01:03:52.005 --> 01:03:59.460
And that the scaling
of to core energy
01:03:59.460 --> 01:04:05.820
itself was determined
by 2 minus pi k,
01:04:05.820 --> 01:04:09.390
which in terms of this k
bar becomes 2 minus k bar
01:04:09.390 --> 01:04:11.961
over 2 times y.
01:04:17.260 --> 01:04:20.770
And we can do exactly
the same thing over here,
01:04:20.770 --> 01:04:23.670
except for the
complication of having
01:04:23.670 --> 01:04:27.860
to deal with the charges
that are vectorial.
01:04:27.860 --> 01:04:32.630
And you get that the strength
of the Coulomb interaction,
01:04:32.630 --> 01:04:39.010
it's inverse that is
related temperature
01:04:39.010 --> 01:04:41.670
gets re-normalized by
exactly the same process.
01:04:41.670 --> 01:04:45.490
That is, there will be
charges that will appear.
01:04:45.490 --> 01:04:49.120
All the mathematics you
can see is clearly similar.
01:04:49.120 --> 01:04:52.190
And there will be a
reduction, which then you
01:04:52.190 --> 01:04:56.340
can think of as an increase
in temperature, which will
01:04:56.340 --> 01:04:58.840
be proportional to y squared.
01:04:58.840 --> 01:05:02.950
I don't know what the constant
of proportionality is.
01:05:02.950 --> 01:05:07.360
And that dy by dl--
so essentially I
01:05:07.360 --> 01:05:10.230
focus on the simplest
type of dislocations
01:05:10.230 --> 01:05:14.590
that I would have
of unit spacing--
01:05:14.590 --> 01:05:16.750
has exactly the
same [INAUDIBLE],
01:05:16.750 --> 01:05:19.998
2 minus k bar over 2 y.
01:05:25.370 --> 01:05:28.926
It turns out that there
is actually a difference.
01:05:28.926 --> 01:05:31.850
And the difference
is as follows.
01:05:34.360 --> 01:05:39.020
That while we did not
calculate the next correction
01:05:39.020 --> 01:05:44.430
in this series, I said
that this correction came
01:05:44.430 --> 01:05:48.300
from essentially what
would a single dipole do,
01:05:48.300 --> 01:05:51.880
which would be some contribution
that is order of y squared.
01:05:51.880 --> 01:05:53.970
And I expect that there
will be correction
01:05:53.970 --> 01:05:58.430
if I look at a configuration
that has pairs of dipoles.
01:05:58.430 --> 01:06:01.270
And this will change
this by order of y
01:06:01.270 --> 01:06:05.348
to the fourth and this
by order of y cubed.
01:06:09.320 --> 01:06:14.050
Whereas if I look at something
like a triangular lattice,
01:06:14.050 --> 01:06:20.110
you can see that if I have
two dislocations out there
01:06:20.110 --> 01:06:22.450
as test dislocations
pointing out
01:06:22.450 --> 01:06:25.430
in opposite
directions, which would
01:06:25.430 --> 01:06:29.240
have some kind of a
interaction such as this.
01:06:29.240 --> 01:06:31.430
And I ask how that
interaction is
01:06:31.430 --> 01:06:36.696
modified by the presence
of dipoles of dislocation
01:06:36.696 --> 01:06:39.240
in the medium.
01:06:39.240 --> 01:06:42.120
I could put a pair
of dislocations
01:06:42.120 --> 01:06:44.660
integrate over the
separation between them.
01:06:44.660 --> 01:06:47.480
I will get, essentially,
the same type
01:06:47.480 --> 01:06:53.430
of mathematical structure that
would ultimately give me this.
01:06:53.430 --> 01:06:57.010
But then the next order
term in this series
01:06:57.010 --> 01:07:00.100
is not for
dislocations because I
01:07:00.100 --> 01:07:03.500
can have a neutral configuration
of three dislocations,
01:07:03.500 --> 01:07:06.820
such as this.
01:07:06.820 --> 01:07:09.310
And so once you
do that, you will
01:07:09.310 --> 01:07:11.810
find that the next
correction here
01:07:11.810 --> 01:07:16.196
is order of y cubed and
order of y squared here.
01:07:23.340 --> 01:07:31.290
But generically, both of
them have a phase diagram
01:07:31.290 --> 01:07:37.160
for RG flows that involves
the inverse of the interaction
01:07:37.160 --> 01:07:38.210
of these charges.
01:07:38.210 --> 01:07:41.000
That is a
temperature-like quantity.
01:07:41.000 --> 01:07:45.505
And there is a critical value
of that inverse, which is 1/4.
01:07:49.950 --> 01:07:56.640
And what happens is that--
let's see, this other axis is y.
01:08:00.230 --> 01:08:04.260
Anything smaller than
1/4 y tends to go to 0.
01:08:04.260 --> 01:08:07.470
Anything larger, y
tends to get larger.
01:08:07.470 --> 01:08:12.530
And there is going to be
some kind of a separatrix
01:08:12.530 --> 01:08:20.170
that he describes the transition
between flows that go down here
01:08:20.170 --> 01:08:23.729
and the flows that go away.
01:08:30.850 --> 01:08:40.590
And presumably, as I heat up my
original system-- my lattice,
01:08:40.590 --> 01:08:44.180
different types of lattices--
at low temperature,
01:08:44.180 --> 01:08:47.870
I can figure out what this
combination of mu an lambda is.
01:08:47.870 --> 01:08:51.740
That gives me some value of k.
01:08:51.740 --> 01:08:54.439
And there is some kind of the
core energy for the vertices
01:08:54.439 --> 01:08:56.069
that I can calculate.
01:08:56.069 --> 01:08:58.600
So there will be some point
in this phase diagram, which
01:08:58.600 --> 01:09:02.680
at low enough temperature I will
go over here, which corresponds
01:09:02.680 --> 01:09:05.490
to a system that has an
effective logarithmic
01:09:05.490 --> 01:09:08.069
interaction between
dislocations.
01:09:08.069 --> 01:09:11.359
As I change the temperature,
I will proceed around
01:09:11.359 --> 01:09:15.170
some trajectory such as this
because everything will change.
01:09:15.170 --> 01:09:18.800
And presumably, at some
point I will intersect this
01:09:18.800 --> 01:09:22.540
and I will go to
the other phase.
01:09:22.540 --> 01:09:26.700
Now the characteristic
of the other phase
01:09:26.700 --> 01:09:33.270
was that this parameter
k was going to 0.
01:09:33.270 --> 01:09:35.590
It was essentially the
logarithmic interaction
01:09:35.590 --> 01:09:36.820
was disappeared.
01:09:36.820 --> 01:09:44.500
And you can see that that
parameter k going to 0
01:09:44.500 --> 01:09:49.370
means that mu, the shear
modulus, has to go to 0.
01:09:49.370 --> 01:09:55.820
So basically this phase out
here is definitely characterized
01:09:55.820 --> 01:09:57.520
by 0 shear modulus.
01:10:00.820 --> 01:10:04.960
So again, the two
parameters mu and lambda, mu
01:10:04.960 --> 01:10:07.410
is the one that
gives you the cost
01:10:07.410 --> 01:10:09.850
of trying to shear in a system.
01:10:09.850 --> 01:10:14.610
And essentially the presence
of the resistance to shear
01:10:14.610 --> 01:10:17.440
is what defines a solid for you.
01:10:17.440 --> 01:10:22.450
So this is a solid to something
that does not have rigidity
01:10:22.450 --> 01:10:22.950
transition.
01:10:26.390 --> 01:10:29.830
It turns out that there
is one subtlety here.
01:10:29.830 --> 01:10:33.970
I don't know whether it's
good to mention or not.
01:10:33.970 --> 01:10:48.660
But so in the case of alpha
this superfluidity, for example,
01:10:48.660 --> 01:10:52.540
we said that this
coupling strength as you
01:10:52.540 --> 01:10:58.130
approach the transition
rose to a universal value.
01:10:58.130 --> 01:11:01.560
In these units it will be 4.
01:11:01.560 --> 01:11:04.669
But that it approaches
that universal value
01:11:04.669 --> 01:11:05.960
with a logarithmic singularity.
01:11:10.390 --> 01:11:12.500
Sorry, with a square
root singularly.
01:11:12.500 --> 01:11:13.270
Sorry.
01:11:13.270 --> 01:11:16.270
And we saw that this was
experimentally observed
01:11:16.270 --> 01:11:19.210
in films of superfluid film.
01:11:19.210 --> 01:11:22.730
I also said that
on the other side
01:11:22.730 --> 01:11:24.670
you have a finite correlations.
01:11:24.670 --> 01:11:27.740
The correlations will be
decaying exponentially.
01:11:27.740 --> 01:11:33.900
And this correlation length has
this very unusual signature,
01:11:33.900 --> 01:11:40.710
which is that it diverged as
a square root of t minus tc
01:11:40.710 --> 01:11:42.660
in the exponent.
01:11:42.660 --> 01:11:45.190
It was giving rise to these
essentially singularities.
01:11:49.480 --> 01:11:54.425
While everything kind of looks
identical between this RG
01:11:54.425 --> 01:11:58.930
and the other RG, it
turns out that once you
01:11:58.930 --> 01:12:03.780
keep track of these corrections,
these corrections actually
01:12:03.780 --> 01:12:05.790
are important.
01:12:05.790 --> 01:12:09.750
And for the vectorial
version of the Coulomb gas,
01:12:09.750 --> 01:12:17.660
you find that they shear modulus
that is finite in this phase
01:12:17.660 --> 01:12:20.160
reaches its final
value, which is not
01:12:20.160 --> 01:12:22.590
universal because the
thing that the universal is
01:12:22.590 --> 01:12:26.380
that combination that
involves both mu and lambda.
01:12:26.380 --> 01:12:32.410
But rather than stc
minus t to the 1/2,
01:12:32.410 --> 01:12:35.260
stc minus t to
some exponent that
01:12:35.260 --> 01:12:42.566
is called mu bar, which
is 0.36963 [INAUDIBLE].
01:12:42.566 --> 01:12:45.440
So this is a kind
of subtle thing
01:12:45.440 --> 01:12:48.500
that is buried in these
recursion relationships
01:12:48.500 --> 01:12:50.480
once you go to a higher order.
01:12:50.480 --> 01:12:53.410
And accompanying that is
a correlation length that
01:12:53.410 --> 01:13:03.480
also diverges with some behavior
dominated by the same exponent.
01:13:10.780 --> 01:13:16.960
So once people understood
the original [? Koster ?]
01:13:16.960 --> 01:13:20.040
[INAUDIBLE]
transition, it was kind
01:13:20.040 --> 01:13:23.600
of a very natural
next step for them
01:13:23.600 --> 01:13:26.460
to think about these
locations in a two dimensional
01:13:26.460 --> 01:13:30.920
solid and say that
in the same manner
01:13:30.920 --> 01:13:35.750
that the unbinding
of these defects
01:13:35.750 --> 01:13:39.680
got rid of the residual
long range order that
01:13:39.680 --> 01:13:42.900
was left in the XY model.
01:13:42.900 --> 01:13:46.560
That this kind of
ordering that we said
01:13:46.560 --> 01:13:50.030
exists for the two
dimensional solid,
01:13:50.030 --> 01:13:56.240
this appears as a result of
unbinding of dislocations
01:13:56.240 --> 01:13:59.500
and you get a liquid.
01:13:59.500 --> 01:14:02.820
Turns out that
that's not correct.
01:14:02.820 --> 01:14:06.490
And it was pointed
out by Bert Halperin
01:14:06.490 --> 01:14:12.020
I think that if you
look at a solid,
01:14:12.020 --> 01:14:13.465
there's two things that you.
01:14:13.465 --> 01:14:16.890
You have the
translational order,
01:14:16.890 --> 01:14:19.510
but there's also
orientational order.
01:14:19.510 --> 01:14:23.110
That is, you can
look, for example,
01:14:23.110 --> 01:14:27.370
at a collection of
spins here and the bonds
01:14:27.370 --> 01:14:30.800
are pointing along
the x and y directions
01:14:30.800 --> 01:14:33.662
and you can look down here
and the bonds are pointing
01:14:33.662 --> 01:14:34.745
in the x and y directions.
01:14:38.000 --> 01:14:43.410
And from the picture
that I drew over here,
01:14:43.410 --> 01:14:47.420
you can kind of see that once
I insert this dislocation which
01:14:47.420 --> 01:14:50.190
corresponds to an
additional line,
01:14:50.190 --> 01:14:54.110
the positions here
are distorted.
01:14:54.110 --> 01:14:57.660
Once I have many
of these inserted,
01:14:57.660 --> 01:14:59.725
it is kind of
obvious that I will
01:14:59.725 --> 01:15:05.045
lose any idea of how the
position of this lattice point
01:15:05.045 --> 01:15:08.210
and something out
there is related.
01:15:08.210 --> 01:15:10.310
It's not so obvious
that inserting
01:15:10.310 --> 01:15:13.510
lots and lots of
these lines will
01:15:13.510 --> 01:15:19.530
remove the knowledge
that I have about
01:15:19.530 --> 01:15:23.970
the orientation of the bonds.
01:15:23.970 --> 01:15:29.040
And the insight that
we will follow up--
01:15:29.040 --> 01:15:32.790
and maybe it's probably a
good idea to do it next time--
01:15:32.790 --> 01:15:40.700
is that actually once
the dislocations unbind,
01:15:40.700 --> 01:15:45.730
we will lose this
kind of ordering.
01:15:45.730 --> 01:15:47.960
If I do this in two
dimensions, these
01:15:47.960 --> 01:15:51.730
are power law decays
at low temperatures.
01:15:51.730 --> 01:15:54.080
Once the dislocations
unbind, this
01:15:54.080 --> 01:15:58.890
becomes an exponential
decay, as I would expect.
01:15:58.890 --> 01:16:05.500
But I can also define locally
some kind of an orientation.
01:16:05.500 --> 01:16:07.940
Let's again call it theta.
01:16:07.940 --> 01:16:14.510
But theta is, let's say, the
orientation of the bond that
01:16:14.510 --> 01:16:18.150
connects two neighboring spins.
01:16:18.150 --> 01:16:23.520
Now what I can do is to look at
the correlation of orientations
01:16:23.520 --> 01:16:25.700
at different positions.
01:16:25.700 --> 01:16:33.380
And it turns out that
if I'm doing something
01:16:33.380 --> 01:16:44.810
like a triangular lattice,
I can't just pick the angle
01:16:44.810 --> 01:16:49.170
and correlate the angle from
one point to another point
01:16:49.170 --> 01:16:54.980
because let's say on the square
lattice the angle itself,
01:16:54.980 --> 01:16:57.210
I don't know whether
it was coming
01:16:57.210 --> 01:16:59.790
from a bond that was
originally in the x direction
01:16:59.790 --> 01:17:01.490
or in the y direction.
01:17:01.490 --> 01:17:05.310
So this is unknown up to
a factor of 90 degrees.
01:17:05.310 --> 01:17:08.930
In the triangular lattice
it is an unknown up
01:17:08.930 --> 01:17:13.540
to a factor of 60 degrees.
01:17:13.540 --> 01:17:19.670
So what I should do is I should
define e to the 6 i theta
01:17:19.670 --> 01:17:29.090
at each location as a measure
of the orientational order.
01:17:29.090 --> 01:17:33.150
You can see that, again,
if I'm at 0 temperature,
01:17:33.150 --> 01:17:37.470
I can reorient the lattice
any way that I like.
01:17:37.470 --> 01:17:42.070
This phase would be the same
across the entire system.
01:17:42.070 --> 01:17:44.930
So what I'm interested
is to find out
01:17:44.930 --> 01:17:53.460
what is the expectation
value of this object when
01:17:53.460 --> 01:17:57.115
I look at points that
are further apart.
01:17:57.115 --> 01:17:59.990
And again, clearly if
I'm at 0 temperature
01:17:59.990 --> 01:18:02.300
these thetas are the same.
01:18:02.300 --> 01:18:05.960
This will go to 1 at
finite temperature.
01:18:05.960 --> 01:18:12.140
Because of the fluctuations
it will start to move a bit.
01:18:12.140 --> 01:18:16.600
And what we can show-- and
you will do it next time--
01:18:16.600 --> 01:18:19.985
is that as long as you are
in the 0 temperature phase
01:18:19.985 --> 01:18:25.545
you will find that this
goes to a constant,
01:18:25.545 --> 01:18:29.090
even in two dimensions.
01:18:29.090 --> 01:18:34.620
But once the dislocations
unbind, what we will show
01:18:34.620 --> 01:18:37.770
is that it doesn't
decay exponentially,
01:18:37.770 --> 01:18:43.710
but then it starts to decay
as 1 over x to some power,
01:18:43.710 --> 01:18:45.142
let's call it eta theta.
01:18:48.790 --> 01:18:51.480
And then at some
higher temperatures,
01:18:51.480 --> 01:18:53.670
it eventually starts
to decay exponentially.
01:18:58.410 --> 01:19:02.420
So there is the
original solid in two
01:19:02.420 --> 01:19:07.210
dimension that we started
that has true long range
01:19:07.210 --> 01:19:11.560
order in the orientations.
01:19:11.560 --> 01:19:15.810
And that is reflected
in these Fourier
01:19:15.810 --> 01:19:17.570
transforms that we make.
01:19:17.570 --> 01:19:21.900
So we said that when we do
the Fourier transform and look
01:19:21.900 --> 01:19:24.550
at the x-ray scattering,
let's say from a two
01:19:24.550 --> 01:19:30.820
dimensional crystal, you were
seeing these kinds of pictures.
01:19:30.820 --> 01:19:32.900
And then these were
becoming weaker
01:19:32.900 --> 01:19:36.680
as we went further
and further out.
01:19:36.680 --> 01:19:39.700
But now you can see that
this picture clearly
01:19:39.700 --> 01:19:44.720
has a very, very defined
orientational aspect to it.
01:19:44.720 --> 01:19:49.000
Now once we go to this
intermediate phase,
01:19:49.000 --> 01:19:53.490
all of these peaks disappear.
01:19:53.490 --> 01:19:59.960
But what you find is that
when you look at this, rather
01:19:59.960 --> 01:20:02.570
than getting a ring
that is uniform,
01:20:02.570 --> 01:20:09.395
you will a see a ring that has
very well- defined variation.
01:20:09.395 --> 01:20:16.945
One, two, three,
four, five, six.
01:20:16.945 --> 01:20:21.040
Six-fold symmetry to it.
01:20:21.040 --> 01:20:25.490
So this phase has no
knowledge of where
01:20:25.490 --> 01:20:30.490
the particles are located,
but knows the orientations.
01:20:30.490 --> 01:20:33.060
It's a kind of a liquid crystal.
01:20:33.060 --> 01:20:34.500
It actually has a name.
01:20:34.500 --> 01:20:36.290
It's called hexatic.
01:20:36.290 --> 01:20:44.370
But it's one of the family of
different types of materials
01:20:44.370 --> 01:20:47.370
that have no translational
order but some kind
01:20:47.370 --> 01:20:49.390
of orientational order
that are hexatics.
01:20:53.882 --> 01:21:00.000
So the transition between
this hexatic phase
01:21:00.000 --> 01:21:04.580
to this fully disordered phase,
it turns out in two dimension
01:21:04.580 --> 01:21:08.500
to be another one of
these dislocation--
01:21:08.500 --> 01:21:12.120
well, topology defect
unbinding transition.
01:21:12.120 --> 01:21:16.340
So maybe we will finish
with that next time around.