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PROFESSOR: OK, let's start.
00:00:26.160 --> 00:00:30.410
So looking for a
way to understand
00:00:30.410 --> 00:00:33.920
the universality of
phase transitions,
00:00:33.920 --> 00:00:36.550
we arrived at the
simplest model that
00:00:36.550 --> 00:00:37.935
should capture some of that.
00:00:37.935 --> 00:00:50.700
That was the Ising model, where
at each site of a lattice--
00:00:50.700 --> 00:00:52.820
and for a while
I will be talking
00:00:52.820 --> 00:00:56.250
in this lecture about
the square lattice--
00:00:56.250 --> 00:00:58.920
you put a binary variable.
00:00:58.920 --> 00:01:06.840
Let's call it sigma i that
takes two values, minus plus 1,
00:01:06.840 --> 00:01:08.540
on each of the N sites.
00:01:08.540 --> 00:01:11.480
So we have a total of 2 to
the N possible configurations.
00:01:15.600 --> 00:01:19.820
And we subject that
to an energy cost
00:01:19.820 --> 00:01:25.920
that tries to make nearest
neighbors to be the same.
00:01:25.920 --> 00:01:29.830
So this symbol stands for
sum over all pairs of nearest
00:01:29.830 --> 00:01:31.950
neighbors on whatever
lattice you have.
00:01:31.950 --> 00:01:34.470
Here the square lattice.
00:01:34.470 --> 00:01:40.320
And the tendency for them
to be parallel as opposed
00:01:40.320 --> 00:01:42.910
to anti-parallel
is captured through
00:01:42.910 --> 00:01:49.050
this dimensionless energy
divided by Kt parameter K.
00:01:49.050 --> 00:01:54.940
So then in principle, as
we change K we could also
00:01:54.940 --> 00:01:57.420
potentially add
a magnetic field.
00:01:57.420 --> 00:02:00.450
There could be a phase
transition in the system.
00:02:00.450 --> 00:02:03.235
And that should be captured
by looking at the partition
00:02:03.235 --> 00:02:07.130
function, which is obtained
by summing over the 2
00:02:07.130 --> 00:02:11.760
to the N possibilities
of this weight that
00:02:11.760 --> 00:02:16.690
is e to the sum over
ij K sigma i sigma j.
00:02:20.300 --> 00:02:23.710
So that's easier said than done.
00:02:23.710 --> 00:02:28.130
And the question was, well,
how can we proceed with this?
00:02:28.130 --> 00:02:32.720
And last lecture, we
suggested two routes
00:02:32.720 --> 00:02:35.250
for looking at this system.
00:02:35.250 --> 00:02:38.780
One of them was to
start by looking
00:02:38.780 --> 00:02:43.665
at a low-temperature expansion.
00:02:43.665 --> 00:02:51.820
And here, let's
say, we would start
00:02:51.820 --> 00:02:54.860
with one of two
possible ground states.
00:02:54.860 --> 00:02:58.100
Let' say all of the
spins could, for example,
00:02:58.100 --> 00:02:59.565
be pointing in the
plus direction.
00:03:06.510 --> 00:03:09.580
That is certainly the
largest contribution
00:03:09.580 --> 00:03:11.240
that you would have
to the partition
00:03:11.240 --> 00:03:13.480
function at 0 temperature.
00:03:13.480 --> 00:03:20.360
And that contribution is all
of the bonds being satisfied.
00:03:20.360 --> 00:03:24.650
Each one of them giving
a factor of e to the K.
00:03:24.650 --> 00:03:26.700
Let's say we are on
a square lattice.
00:03:26.700 --> 00:03:29.510
On a square lattice,
each site has
00:03:29.510 --> 00:03:32.070
two bonds associated with it.
00:03:32.070 --> 00:03:34.890
So this will go like 2N.
00:03:34.890 --> 00:03:39.240
Since we have N sites,
we will 2N bonds.
00:03:39.240 --> 00:03:41.940
There is actually, of
course, two possibilities.
00:03:41.940 --> 00:03:46.060
We can have either all of them
plus or all of them minus.
00:03:46.060 --> 00:03:50.080
So there is a kind of
trivial degeneracy of 2,
00:03:50.080 --> 00:03:53.470
which doesn't really make
too much of a difference.
00:03:53.470 --> 00:03:58.110
And then we can start looking
at excitations around this.
00:03:58.110 --> 00:04:02.060
And so we said that the
first type of excitation
00:04:02.060 --> 00:04:05.480
is somewhere on the lattice
we make one of these pluses
00:04:05.480 --> 00:04:06.650
into a minus.
00:04:06.650 --> 00:04:10.530
And once we do
that, we have made
00:04:10.530 --> 00:04:15.140
4 bonds that go out of
this minus site unhappy.
00:04:15.140 --> 00:04:20.560
So the cost of going from
plus k to minus K, which
00:04:20.560 --> 00:04:27.700
is minus 2k, in this case
is repeated four times.
00:04:27.700 --> 00:04:30.070
And this particular
excitation can
00:04:30.070 --> 00:04:34.070
be placed in any one of N
locations on the lattice.
00:04:34.070 --> 00:04:38.120
So this was this
kind of excitation.
00:04:38.120 --> 00:04:42.400
And then we could go and
have the next excitation
00:04:42.400 --> 00:04:44.666
where two of them are flipped.
00:04:44.666 --> 00:04:47.740
And we would have a
situation such as this.
00:04:47.740 --> 00:04:53.530
And since this dimer can point
in the x- or the y-direction
00:04:53.530 --> 00:04:58.300
on the square lattice,
it has a degeneracy of 2.
00:04:58.300 --> 00:05:01.870
I have e to the minus 2K.
00:05:01.870 --> 00:05:04.850
And how many bonds
have I broken?
00:05:04.850 --> 00:05:07.810
One, two, three,
four, five, six.
00:05:07.810 --> 00:05:10.000
Times 6.
00:05:10.000 --> 00:05:12.270
And I can go on.
00:05:12.270 --> 00:05:17.800
So this was a
situation such as this.
00:05:17.800 --> 00:05:21.520
A general term in
the series would
00:05:21.520 --> 00:05:26.210
correspond to creating some
kind of an island of minus
00:05:26.210 --> 00:05:27.455
in this sea of pluses.
00:05:30.600 --> 00:05:35.540
And the contribution
would be e to the minus 2K
00:05:35.540 --> 00:05:39.550
times the perimeter
of this island.
00:05:48.660 --> 00:05:51.685
So that would be
a way to calculate
00:05:51.685 --> 00:05:58.550
the low-temperature expansion
we discussed last time around.
00:05:58.550 --> 00:06:01.670
We also said that I could do
a high temperature expansion.
00:06:09.610 --> 00:06:12.960
And for this, we use the trick.
00:06:12.960 --> 00:06:16.140
We said I can write
the partition function
00:06:16.140 --> 00:06:20.370
as a sum over all these
2 to the N configuration.
00:06:20.370 --> 00:06:21.285
Yes.
00:06:21.285 --> 00:06:24.080
AUDIENCE: Is there a reason we
don't have separate islands?
00:06:24.080 --> 00:06:25.080
PROFESSOR: Oh, we do.
00:06:25.080 --> 00:06:25.700
We do.
00:06:25.700 --> 00:06:30.990
So in general, in this picture
I could have multiple islands.
00:06:30.990 --> 00:06:32.830
Yes.
00:06:32.830 --> 00:06:35.970
And what would be
interesting is certainly
00:06:35.970 --> 00:06:40.490
when I take the log of
Z, then the log of Z
00:06:40.490 --> 00:06:43.470
here I would have NK.
00:06:43.470 --> 00:06:45.440
I would have log 2.
00:06:45.440 --> 00:06:49.280
And then there would
be a bunch of terms.
00:06:49.280 --> 00:06:53.300
And what we saw was
those bunches of terms,
00:06:53.300 --> 00:06:57.500
starting with that
one, can be captured
00:06:57.500 --> 00:07:00.970
into a series where
the n comes out front
00:07:00.970 --> 00:07:05.380
and the terms in the series
would be functions of e
00:07:05.380 --> 00:07:07.170
to the minus 2K.
00:07:07.170 --> 00:07:10.970
And indeed, when we
exponentiate this,
00:07:10.970 --> 00:07:13.650
this would have
single islands only.
00:07:13.650 --> 00:07:15.630
The exponential will
have multiple islands
00:07:15.630 --> 00:07:17.482
that we have for the
partition functions.
00:07:20.290 --> 00:07:23.800
So these terms, as we see,
are higher powers of n
00:07:23.800 --> 00:07:25.452
because you would
be able to move them
00:07:25.452 --> 00:07:26.960
in different directions.
00:07:26.960 --> 00:07:28.970
When you take the log,
only terms that are
00:07:28.970 --> 00:07:30.300
linear and can survive.
00:07:35.310 --> 00:07:39.710
So this sum over
bond configurations
00:07:39.710 --> 00:07:44.570
I can write as a
product over bonds.
00:07:44.570 --> 00:07:48.330
And the factor of e
to the K sigma i sigma
00:07:48.330 --> 00:07:53.260
j, we saw that we could
capture slightly differently.
00:07:53.260 --> 00:07:55.830
So e to the K sigma
i sigma j I can
00:07:55.830 --> 00:07:59.120
write as hyperbolic
cosine of K 1
00:07:59.120 --> 00:08:05.660
plus hyperbolic tanh
of K sigma i sigma j.
00:08:05.660 --> 00:08:11.670
And this t is going to be my
symbol for hyperbolic tanh
00:08:11.670 --> 00:08:17.150
of K, so that I don't have to
repeat it all over the place.
00:08:17.150 --> 00:08:20.220
So this is just rewriting
of that exponential,
00:08:20.220 --> 00:08:23.260
recognizing that it
has two possibilities.
00:08:23.260 --> 00:08:25.315
We took the cosh
K to the outside.
00:08:28.150 --> 00:08:31.430
And if I'm, again, on
the square lattice,
00:08:31.430 --> 00:08:35.909
I have 2N bonds so there
will be 2N factors of cosh K
00:08:35.909 --> 00:08:38.470
that I will take
into the outside.
00:08:38.470 --> 00:08:41.780
And then I would
have a series, which
00:08:41.780 --> 00:08:47.140
would be these terms that I can
start expanding in powers of t.
00:08:47.140 --> 00:08:50.580
And the lowest order I have 1.
00:08:50.580 --> 00:08:53.410
And then we discussed what
kinds of terms are allowed.
00:09:01.510 --> 00:09:10.140
We saw that if I take just one
factor of t sigma i sigma j,
00:09:10.140 --> 00:09:14.200
then I have a sigma i sitting
here and sigma j sitting here.
00:09:14.200 --> 00:09:19.050
I sum over the two possibilities
of sigma being plus or minus.
00:09:19.050 --> 00:09:21.180
It will give me 0.
00:09:21.180 --> 00:09:24.920
So there is a contribution
order of t if I expand that,
00:09:24.920 --> 00:09:30.070
but summing over sigma i
and sigma j would give me 0.
00:09:30.070 --> 00:09:31.830
So the only choice
that I have is
00:09:31.830 --> 00:09:36.810
that this sigma that is sitting
out here I should square.
00:09:36.810 --> 00:09:39.220
And I can square it
by putting a bond that
00:09:39.220 --> 00:09:42.380
corresponds to this
sigma and that sigma.
00:09:42.380 --> 00:09:43.700
This became sigma squared.
00:09:43.700 --> 00:09:46.200
I don't have to
worry about that.
00:09:46.200 --> 00:09:49.080
Then I can complete
this so that that's
00:09:49.080 --> 00:09:52.600
squared and this so
that that's squared.
00:09:52.600 --> 00:09:55.530
And so this was a
diagram that contributed
00:09:55.530 --> 00:09:58.390
N this quantity t to the fourth.
00:10:01.090 --> 00:10:05.790
Well, what's the next type
of graph that I could draw?
00:10:05.790 --> 00:10:07.510
I could do something like this.
00:10:11.730 --> 00:10:15.120
And that is, again,
something that I
00:10:15.120 --> 00:10:18.783
can orient along the x-direction
or along the y-direction.
00:10:18.783 --> 00:10:22.950
So there is a factor of
2 from the orientation.
00:10:22.950 --> 00:10:26.040
And that's how
many factors of t.
00:10:26.040 --> 00:10:26.600
It is 6.
00:10:32.400 --> 00:10:35.890
And so in general,
what do I have?
00:10:35.890 --> 00:10:39.130
I will have to draw
some kind of a graph
00:10:39.130 --> 00:10:47.275
on the lattice where at
each site, I either have 0,
00:10:47.275 --> 00:10:49.760
2, or potentially 4.
00:10:49.760 --> 00:10:54.350
There is no difficult with 4
bonds emanating from a site
00:10:54.350 --> 00:10:56.700
because that sigma
to the fourth.
00:10:56.700 --> 00:11:00.800
Basically, those kinds
of things actually
00:11:00.800 --> 00:11:04.850
will give me then the sum
over-- possibilities of sigma
00:11:04.850 --> 00:11:09.040
will give me 2 to
the number of sites.
00:11:09.040 --> 00:11:12.470
And then these
graphs, basically will
00:11:12.470 --> 00:11:17.689
get a factor which is 2 to
the number of bonds in graph.
00:11:24.100 --> 00:11:25.830
And then again, you
can see that here I
00:11:25.830 --> 00:11:29.790
could have multiple loops, just
like we discussed over there.
00:11:29.790 --> 00:11:34.210
Multiple loops will go
with larger factors of N.
00:11:34.210 --> 00:11:38.310
The thing that we are
interested is log Z,
00:11:38.310 --> 00:11:49.660
which is N log 2 cosine
squared K. And then I
00:11:49.660 --> 00:11:52.643
have to take the log
of this expression,
00:11:52.643 --> 00:11:58.170
and I'll call it g of t.
00:11:58.170 --> 00:11:59.974
Yes.
00:11:59.974 --> 00:12:05.710
AUDIENCE: How did we get
rid of [INAUDIBLE] bonds?
00:12:05.710 --> 00:12:06.840
PROFESSOR: OK.
00:12:06.840 --> 00:12:16.790
So this e to the K sigma i
sigma j has two possible values.
00:12:16.790 --> 00:12:23.410
It is either e to the
K or e to the minus K.
00:12:23.410 --> 00:12:25.690
And I can write
it, therefore, as e
00:12:25.690 --> 00:12:31.610
to the K plus e to the minus
K over 2 plus e to the K minus
00:12:31.610 --> 00:12:37.240
e to the minus K over 2
sigma i sigma j, right?
00:12:37.240 --> 00:12:38.475
AUDIENCE: My question was--
00:12:38.475 --> 00:12:40.710
PROFESSOR: Yeah, I know.
00:12:40.710 --> 00:12:45.390
And then, this became
cosh K 1 plus what
00:12:45.390 --> 00:12:48.990
I call sigma i sigma j.
00:12:48.990 --> 00:12:54.760
So when I draw my
lattice, for the bond that
00:12:54.760 --> 00:13:01.900
goes between sites i and j,
in the partition function
00:13:01.900 --> 00:13:03.060
I have this contribution.
00:13:06.060 --> 00:13:10.850
This contribution is
completely equivalent to this.
00:13:10.850 --> 00:13:12.700
And this is two terms.
00:13:12.700 --> 00:13:14.860
The cosh K we took outside.
00:13:14.860 --> 00:13:21.250
The two terms is
either 1 or 1 line.
00:13:21.250 --> 00:13:24.220
There isn't anything
that is multiple lines.
00:13:24.220 --> 00:13:27.110
And as I said, you
can make separately
00:13:27.110 --> 00:13:33.770
an expansion in powers of K.
This corresponds to nothing,
00:13:33.770 --> 00:13:37.990
or going forward and backward,
or going forward, backward,
00:13:37.990 --> 00:13:41.959
et cetera, if you make an
expansion in powers of K. This
00:13:41.959 --> 00:13:43.250
term corresponds to going once.
00:13:47.030 --> 00:13:52.140
Essentially, this captures all
of the things that stay back
00:13:52.140 --> 00:13:56.045
to the same site and this is
a re-summation of everything
00:13:56.045 --> 00:13:58.780
that steps you forward.
00:13:58.780 --> 00:14:03.920
So everything that steps you
forward and carries information
00:14:03.920 --> 00:14:07.820
from this sites to that
site has been appropriately
00:14:07.820 --> 00:14:09.060
taken care of.
00:14:09.060 --> 00:14:10.135
And it occurs once.
00:14:12.800 --> 00:14:15.835
And that's one reason.
00:14:15.835 --> 00:14:18.320
And again, if you have
three things coming
00:14:18.320 --> 00:14:20.890
at a particular site,
it's a sigma i q.
00:14:20.890 --> 00:14:23.710
So these are completely
the only thing that happen.
00:14:26.930 --> 00:14:34.770
But in principle, this is
one diagrammatic series.
00:14:34.770 --> 00:14:38.870
This is one diagrammatic series.
00:14:38.870 --> 00:14:41.640
But you stare at
them a little bit
00:14:41.640 --> 00:14:46.890
and you'll see why I put
the same g for both of them.
00:14:46.890 --> 00:14:50.600
On the square lattice,
they're identify series.
00:14:50.600 --> 00:14:55.550
See that series had N
something to the fourth power,
00:14:55.550 --> 00:14:58.500
2N something to the
sixth power, N something
00:14:58.500 --> 00:15:00.980
to the fourth power, 2N
something to the sixth power.
00:15:00.980 --> 00:15:04.150
The first two terms
are identical.
00:15:04.150 --> 00:15:07.710
You can convince yourself that
all the terms will be identical
00:15:07.710 --> 00:15:11.360
also, including
something complicated,
00:15:11.360 --> 00:15:14.370
such as if I were to
draw-- I don't know.
00:15:14.370 --> 00:15:20.750
A diagram such as this one,
which has 2 or 4 per site.
00:15:20.750 --> 00:15:25.780
I can convert that to something
that had spins plus out here,
00:15:25.780 --> 00:15:30.180
then it minus here,
minus here, plus here.
00:15:30.180 --> 00:15:33.420
And it's a completely
consistent diagram
00:15:33.420 --> 00:15:36.562
that I would have had in
the low-temperature series.
00:15:36.562 --> 00:15:38.930
So you can convince
yourself that there
00:15:38.930 --> 00:15:42.493
is a one-to-one correspondence
between these two series.
00:15:42.493 --> 00:15:46.680
They are identical for
the square lattice.
00:15:46.680 --> 00:15:50.770
As we will discuss, this is a
property of the square lattice.
00:15:50.770 --> 00:15:53.360
So you have this
very nice symmetry
00:15:53.360 --> 00:15:57.380
that you conclude that the
partition function per site,
00:15:57.380 --> 00:16:00.820
the part that is
interesting, either
00:16:00.820 --> 00:16:08.366
I can get it from here as
K plus this function of e
00:16:08.366 --> 00:16:15.580
to the minus 2K or from the high
temperatures series as log 2
00:16:15.580 --> 00:16:19.020
hyperbolic cosine
squared K. Plus exactly
00:16:19.020 --> 00:16:25.630
the same function of tanh K.
00:16:25.630 --> 00:16:27.920
This part of it,
actually, I don't really
00:16:27.920 --> 00:16:31.870
care because these are
analytical functions.
00:16:31.870 --> 00:16:34.435
I expect this model to
have a phase transition.
00:16:34.435 --> 00:16:37.197
The low-temperature and the
high-temperature behavior
00:16:37.197 --> 00:16:38.030
should be different.
00:16:38.030 --> 00:16:41.550
It should be order at low
temperature, [INAUDIBLE]
00:16:41.550 --> 00:16:42.860
at high temperature.
00:16:42.860 --> 00:16:45.570
There should be a phase
transition and a singularity
00:16:45.570 --> 00:16:47.270
between those two cases.
00:16:47.270 --> 00:16:50.790
The singular part
must be captured here.
00:16:50.790 --> 00:16:54.260
So these are the
singular functions.
00:16:54.260 --> 00:16:57.490
So the singular part
of the free energy
00:16:57.490 --> 00:17:01.640
has this interesting property
that you can evaluate it
00:17:01.640 --> 00:17:06.819
for some parameter K, which is
large in the low-temperature
00:17:06.819 --> 00:17:10.250
phase, or some
parameter t, which
00:17:10.250 --> 00:17:13.589
is small in the
high-temperature phase.
00:17:13.589 --> 00:17:20.069
And the two will be related
completely to each other.
00:17:20.069 --> 00:17:22.530
They are essentially
the same thing.
00:17:22.530 --> 00:17:25.970
So this property
is called duality.
00:17:29.650 --> 00:17:33.400
And so what I have
said, first of all, is
00:17:33.400 --> 00:17:39.460
that there is a relationship
between, say, the coupling tanh
00:17:39.460 --> 00:17:47.910
K and a coupling that I can
separately call K tilde, such
00:17:47.910 --> 00:17:51.900
that if I evaluate the
high-temperature series at K,
00:17:51.900 --> 00:17:55.430
it is like evaluating the
low-temperature series at K
00:17:55.430 --> 00:18:02.270
tilde, where K tilde
of K is minus 1/2 log
00:18:02.270 --> 00:18:07.710
of the hyperbolic tanh at K.
00:18:07.710 --> 00:18:10.350
I can plot for you what
this function looks like.
00:18:13.272 --> 00:18:18.740
So this is K. This is what
K tilde of K looks like.
00:18:18.740 --> 00:18:22.700
And it is something like this.
00:18:22.700 --> 00:18:27.360
Basically, strong coupling,
or low temperature,
00:18:27.360 --> 00:18:30.295
gets mapped to weak coupling,
or high temperature,
00:18:30.295 --> 00:18:31.495
and vice versa.
00:18:35.120 --> 00:18:39.795
So it's kind of like
this, that there
00:18:39.795 --> 00:18:46.490
is this axes of the
strength of K going
00:18:46.490 --> 00:18:52.610
from low temperature, strong,
to high temperature, weak.
00:18:52.610 --> 00:18:55.730
And what I have shown is that
if I start with somewhere
00:18:55.730 --> 00:19:02.040
out here, it is mapped
to somewhere down here.
00:19:02.040 --> 00:19:06.060
If I start from
somewhere here, it
00:19:06.060 --> 00:19:09.140
would be mapping
to somewhere here.
00:19:15.100 --> 00:19:22.440
So one question to ask is,
well, OK, I start from here.
00:19:22.440 --> 00:19:24.820
I go here.
00:19:24.820 --> 00:19:30.890
If I put that value of k in
here, do I go to a third point
00:19:30.890 --> 00:19:33.550
or do I come back to here?
00:19:33.550 --> 00:19:36.560
And I will show you
that it is, in fact,
00:19:36.560 --> 00:19:41.620
a mapping that goes both ways.
00:19:41.620 --> 00:19:48.410
And a way to show
that is like this.
00:19:48.410 --> 00:19:52.150
Let me look at the
hyperbolic sine of 2K.
00:19:54.750 --> 00:19:58.950
Hyperbolic sines of twice
the angles, or twice
00:19:58.950 --> 00:20:05.500
the hyperbolic sines of
K hyperbolic cosine of K.
00:20:05.500 --> 00:20:07.915
All my answers are
in terms of tanh K,
00:20:07.915 --> 00:20:12.650
so I can make this sine to
become a tanh by dividing
00:20:12.650 --> 00:20:17.640
by cosh, and then making
this cosh squared.
00:20:17.640 --> 00:20:22.400
So that is 2
hyperbolic tanh of K.
00:20:22.400 --> 00:20:27.730
And then, there's various
ways to sort of remember
00:20:27.730 --> 00:20:31.350
the identity for
hyperbolic cosine squared
00:20:31.350 --> 00:20:34.210
minus sine squared is 1.
00:20:34.210 --> 00:20:41.320
If I divide by c squared,
it becomes 1 minus t squared
00:20:41.320 --> 00:20:43.680
is 1 over c squared.
00:20:43.680 --> 00:20:46.780
So the hyperbolic
cosine squared here
00:20:46.780 --> 00:20:54.990
is the inverse of 1 minus
hyperbolic tanh squared of K.
00:20:54.990 --> 00:21:00.710
Now, for tanh K, we have this
identity, is 2 e to the minus
00:21:00.710 --> 00:21:02.640
2K tilde.
00:21:02.640 --> 00:21:04.700
1 minus the square of that.
00:21:08.010 --> 00:21:12.490
If I multiply both sides
by-- e to the numerator
00:21:12.490 --> 00:21:20.398
and denominator by e
to the plus 2K tilde,
00:21:20.398 --> 00:21:28.666
hopefully you recognize this
as 1 over hyperbolic sine of K
00:21:28.666 --> 00:21:30.466
tilde 2K tilde.
00:21:33.800 --> 00:21:42.000
So the identity here that was
kind of not very transparent,
00:21:42.000 --> 00:21:44.880
if I had made the
change of variables
00:21:44.880 --> 00:21:48.950
to the hyperbolic sine
of twice the angle,
00:21:48.950 --> 00:21:49.860
had this simple form.
00:21:55.830 --> 00:21:59.375
So then the symmetry
between K and K tilde
00:21:59.375 --> 00:22:01.300
is immediately obvious.
00:22:01.300 --> 00:22:06.520
You pick one value of K, or
sine K, and then the inverse.
00:22:06.520 --> 00:22:09.980
And the inverse of the inverse,
you are back to where you are.
00:22:09.980 --> 00:22:14.640
So it is clear that this is
kind of like an x to y0 1 over x
00:22:14.640 --> 00:22:15.670
mapping.
00:22:15.670 --> 00:22:20.050
x to 1 over x mapping would also
kind of look exactly like this.
00:22:20.050 --> 00:22:22.070
In fact, if instead
of K tilde and K
00:22:22.070 --> 00:22:26.190
I had plotted hyperbolic sine
versus hyperbolic sine of twice
00:22:26.190 --> 00:22:29.056
the angle, it would have
been just the 1 over x curve.
00:22:34.900 --> 00:22:44.600
Now, just to give you another
example, if I had the function
00:22:44.600 --> 00:22:50.740
f of x, which is x
1 plus x squared.
00:22:50.740 --> 00:22:53.290
This function, if I
divide by x squared,
00:22:53.290 --> 00:23:00.170
becomes x inverse 1
plus x inverse squared.
00:23:00.170 --> 00:23:03.560
So this is, again,
f of x inverse.
00:23:03.560 --> 00:23:08.950
So if I evaluate this
function for any value like 5,
00:23:08.950 --> 00:23:11.980
then I know the value
exactly for 1/5.
00:23:11.980 --> 00:23:16.780
If I evaluate it for 200, I know
it for 1/200, and vice versa.
00:23:16.780 --> 00:23:20.110
Our g function is
kind of like that.
00:23:22.940 --> 00:23:26.070
Now, this function you
can see that starts
00:23:26.070 --> 00:23:32.520
to go linearly increases with
x, and then eventually it
00:23:32.520 --> 00:23:37.920
comes down like this
must have one maximum.
00:23:37.920 --> 00:23:39.160
Where is the maximum?
00:23:42.360 --> 00:23:43.390
It has to be 1.
00:23:43.390 --> 00:23:46.950
I don't have to take derivatives
of everything, et cetera.
00:23:46.950 --> 00:23:49.665
If there is one point which
corresponds to the maximum,
00:23:49.665 --> 00:23:51.450
it's the point that
maps to itself.
00:23:54.700 --> 00:23:58.250
Now, this function
for the Ising model,
00:23:58.250 --> 00:23:59.730
I know it has a
phase transition.
00:23:59.730 --> 00:24:02.190
Or, I guess it has
a phase transition.
00:24:02.190 --> 00:24:04.920
So there is one point,
hopefully one point,
00:24:04.920 --> 00:24:06.240
at which it becomes singular.
00:24:06.240 --> 00:24:08.080
I don't know, maybe
it's three point.
00:24:08.080 --> 00:24:12.360
But let's say it's one point
at which it becomes singular.
00:24:12.360 --> 00:24:16.590
Then I should be able to figure
its singularity by precisely
00:24:16.590 --> 00:24:19.090
the same argument.
00:24:19.090 --> 00:24:25.400
So the function that corresponds
to x going to 1 over x
00:24:25.400 --> 00:24:26.830
is this hyperbolic sine.
00:24:30.180 --> 00:24:35.840
So if there is a point which
is the unique point that
00:24:35.840 --> 00:24:38.310
corresponds to the
singularity, it
00:24:38.310 --> 00:24:42.860
has to be the point that is
self-dual-- maps on to itself,
00:24:42.860 --> 00:24:44.790
just like 1.
00:24:44.790 --> 00:24:49.380
So sine of 2 Kc should be 1.
00:24:49.380 --> 00:24:50.520
And what is this?
00:24:50.520 --> 00:24:53.950
Hyperbolic sine we can
write as e to the 2 Kc
00:24:53.950 --> 00:24:57.750
minus e to the
minus 2 Kc over 2.
00:24:57.750 --> 00:25:03.610
We can manipulate
this equation slightly
00:25:03.610 --> 00:25:12.940
to e to the 4 Kc minus 2 e to
the 2 Kc minus 1 equals to 0,
00:25:12.940 --> 00:25:17.310
which is a quadratic
equation for e to the 2 Kc.
00:25:17.310 --> 00:25:23.478
So I can immediately solve
for e to the 2 Kc is--
00:25:23.478 --> 00:25:29.305
This has a 2, so I can say
1 minus plus square root
00:25:29.305 --> 00:25:32.535
of square of that plus this.
00:25:32.535 --> 00:25:35.930
So that is a square root of 2.
00:25:35.930 --> 00:25:38.380
The exponential
better be positive,
00:25:38.380 --> 00:25:40.180
so I can't pick the
negative solution.
00:25:43.310 --> 00:25:44.660
And so we know--
00:25:44.660 --> 00:25:46.070
AUDIENCE: Isn't it [INAUDIBLE]?
00:25:52.190 --> 00:25:54.020
PROFESSOR: Where did I-- OK.
00:25:54.020 --> 00:25:57.445
Multiply by e to the 2 Kc.
00:25:57.445 --> 00:26:00.300
AUDIENCE: Taking that as
correct, I didn't check it--
00:26:00.300 --> 00:26:02.140
PROFESSOR: Yes.
00:26:02.140 --> 00:26:03.595
OK.
00:26:03.595 --> 00:26:09.340
x squared minus 2b
plus-- x plus c equals.
00:26:09.340 --> 00:26:12.220
Well, actually,
this is-- so x is
00:26:12.220 --> 00:26:18.460
b minus plus square root
of b squared minus ac.
00:26:18.460 --> 00:26:21.350
Our c is negative.
00:26:21.350 --> 00:26:22.380
So it's 1 plus 1.
00:26:26.310 --> 00:26:33.485
So the critical coupling
that we have is 1/2 log of 1
00:26:33.485 --> 00:26:37.390
plus square root of 2.
00:26:37.390 --> 00:26:40.950
So we know that the critical
point of the Ising model
00:26:40.950 --> 00:26:44.135
occurs at this value, which
you can put on your calculator.
00:26:44.135 --> 00:26:46.775
And it is something like this.
00:26:50.060 --> 00:26:51.430
There is one assumption.
00:26:51.430 --> 00:26:54.790
Of course, there is essentially
only one singularity.
00:26:54.790 --> 00:26:57.770
And there is one singularity
in this free energy.
00:26:57.770 --> 00:27:01.661
But if it is, we have
solved it for this case.
00:27:10.550 --> 00:27:20.390
So I think to emphasize
this was discovered
00:27:20.390 --> 00:27:26.500
around '50s by Wannier,
this idea of duality.
00:27:26.500 --> 00:27:30.530
And suddenly, you had exact
solution for something
00:27:30.530 --> 00:27:33.830
like the square Ising model.
00:27:33.830 --> 00:27:38.350
Question is, how much
information does it give you?
00:27:38.350 --> 00:27:42.880
First of all, the
property of self-duality
00:27:42.880 --> 00:27:44.590
is that of the square lattice.
00:27:54.720 --> 00:27:59.050
So if I had done this on
the triangular lattice,
00:27:59.050 --> 00:28:03.800
you would've seen that
the low-temperature and
00:28:03.800 --> 00:28:06.900
high-temperature
expansions don't match.
00:28:06.900 --> 00:28:11.010
It turns out that in
order to construct
00:28:11.010 --> 00:28:14.420
the dual of any lattice,
what you have to do
00:28:14.420 --> 00:28:20.620
is to put, let's say, points
in the center of the units
00:28:20.620 --> 00:28:26.970
that you have and see what
lattice these centers make.
00:28:26.970 --> 00:28:29.820
So when you try to do that
for the triangular lattice,
00:28:29.820 --> 00:28:32.790
you will see that the centers
form, actually, hexagonal
00:28:32.790 --> 00:28:35.550
lattice, and vice versa.
00:28:35.550 --> 00:28:40.060
However, there is a
trick using duality
00:28:40.060 --> 00:28:43.380
that you can still calculate
critical points of hexagonal
00:28:43.380 --> 00:28:45.770
and triangular lattice.
00:28:45.770 --> 00:28:48.130
And that you will do in
one of the problem sets.
00:28:50.710 --> 00:28:52.960
OK, secondly.
00:28:52.960 --> 00:28:59.870
Again, that trick allows you
to go beyond square lattice,
00:28:59.870 --> 00:29:02.630
but it turns out
that for reasons
00:29:02.630 --> 00:29:05.950
that we will see
shortly, it is limited.
00:29:05.950 --> 00:29:09.080
And you can only do
these kinds of dualities
00:29:09.080 --> 00:29:12.150
to yourself for
two-dimensional lattices.
00:29:17.880 --> 00:29:23.320
And what these kinds
of mappings in general
00:29:23.320 --> 00:29:26.170
give for
two-dimensional lattices
00:29:26.170 --> 00:29:30.990
is potentially, but not always,
the critical value of Kc.
00:29:30.990 --> 00:29:33.520
And again, one of the
things that you will see
00:29:33.520 --> 00:29:40.470
is that you can do this for
other models in two dimension.
00:29:40.470 --> 00:29:43.590
For example, the
Potts model we can
00:29:43.590 --> 00:29:50.750
calculate the critical point
through this kind of procedure.
00:29:50.750 --> 00:29:53.440
However, it doesn't
tell you anything
00:29:53.440 --> 00:29:56.820
about the nature
of the singularity.
00:29:56.820 --> 00:30:00.560
So essentially, what we've
shown is that on the K-axis,
00:30:00.560 --> 00:30:05.060
there is some point maybe
that describes the singularity
00:30:05.060 --> 00:30:07.700
that you are going to have.
00:30:07.700 --> 00:30:11.390
But the shape of this
singularity, the exponent
00:30:11.390 --> 00:30:12.890
can be anything.
00:30:12.890 --> 00:30:17.160
And this mapping does not
tell you anything about that.
00:30:17.160 --> 00:30:19.130
It does tell you one thing.
00:30:19.130 --> 00:30:23.130
We also mentioned that
the ratio of amplitudes
00:30:23.130 --> 00:30:27.780
above and below for
various singular quantities
00:30:27.780 --> 00:30:30.740
is something that
is universal because
00:30:30.740 --> 00:30:33.090
of these mappings
from high temperatures
00:30:33.090 --> 00:30:34.260
to low temperatures.
00:30:34.260 --> 00:30:38.780
Although I don't know what the
nature of the singularity is,
00:30:38.780 --> 00:30:42.860
I know that the amplitude
ratio is [INAUDIBLE].
00:30:42.860 --> 00:30:46.460
So there is some
universal information
00:30:46.460 --> 00:30:50.210
that one gains beyond the
non-universal location
00:30:50.210 --> 00:30:53.292
of the critical point,
but not that much more.
00:30:57.861 --> 00:30:58.360
OK.
00:31:01.280 --> 00:31:03.498
Any questions?
00:31:03.498 --> 00:31:07.466
AUDIENCE: Is it possible
to extract from this line
00:31:07.466 --> 00:31:08.954
a differential equation for g?
00:31:15.910 --> 00:31:17.110
PROFESSOR: Yes.
00:31:17.110 --> 00:31:22.710
And indeed, that
differential relation
00:31:22.710 --> 00:31:24.280
you will use in
one of the problem
00:31:24.280 --> 00:31:27.050
sets that I forgot
to mention, and will
00:31:27.050 --> 00:31:30.900
be used to derive the value
of the derivative, which
00:31:30.900 --> 00:31:34.196
is related to the energy of the
system at the critical point.
00:31:34.196 --> 00:31:34.946
But you are right.
00:31:43.110 --> 00:31:45.710
This is such a beautiful
thing that maybe we
00:31:45.710 --> 00:31:49.630
can try to force it to
work in higher dimensions.
00:31:49.630 --> 00:31:52.775
So let's see if we
were to try to go
00:31:52.775 --> 00:31:56.945
with this approach for the 3D
Ising model what would happen.
00:32:02.885 --> 00:32:04.330
So what did we do?
00:32:04.330 --> 00:32:08.105
We wrote the low-temperature
series, high-temperature series
00:32:08.105 --> 00:32:10.420
and compared them.
00:32:10.420 --> 00:32:14.980
Again, let's do
the cubic lattice,
00:32:14.980 --> 00:32:17.690
which I will not
really attempt to draw.
00:32:24.330 --> 00:32:29.090
That's the system that
we want to calculate.
00:32:29.090 --> 00:32:30.990
So let's do the low T-series.
00:32:33.590 --> 00:32:37.830
Our partition function is
going to start with the state
00:32:37.830 --> 00:32:41.010
where every spin
is, let's say, up.
00:32:41.010 --> 00:32:43.720
Three bonds per site
on the cubic lattice.
00:32:43.720 --> 00:32:46.520
So it's 3 NK.
00:32:46.520 --> 00:32:51.550
Again, the trivial degeneracy of
2 for the two possible all plus
00:32:51.550 --> 00:32:54.960
or all minus states.
00:32:54.960 --> 00:32:57.700
The first excitation
is to flip a spin.
00:32:57.700 --> 00:33:02.526
So any one of N sites could have
been flipped, creating a cube.
00:33:02.526 --> 00:33:08.595
A cube has 6 faces that go
out, so there is essentially
00:33:08.595 --> 00:33:10.460
6 bonds that are broken.
00:33:10.460 --> 00:33:13.420
So basically,
there is this minus
00:33:13.420 --> 00:33:17.750
that is in a box
surrounded by plus.
00:33:17.750 --> 00:33:24.240
And as you can see, 6 plus
minus 1 that go out of that.
00:33:24.240 --> 00:33:27.290
The next one would be
when we have 2 minuses.
00:33:30.280 --> 00:33:36.001
And that can be oriented three
ways in three dimensions.
00:33:36.001 --> 00:33:39.410
e to the minus 2K.
00:33:39.410 --> 00:33:41.220
1, 2 times 4.
00:33:41.220 --> 00:33:42.160
8 plus 2.
00:33:42.160 --> 00:33:42.900
Times 10.
00:33:45.630 --> 00:33:49.460
And so the general
term in the series I
00:33:49.460 --> 00:33:59.100
have to draw some droplet of
minuses in a sea of pluses.
00:33:59.100 --> 00:34:04.420
And then I would have e to the
minus 2K times the boundary
00:34:04.420 --> 00:34:06.014
or the area of this droplet.
00:34:10.100 --> 00:34:12.380
Actually, droplets
because there could
00:34:12.380 --> 00:34:15.179
be multiple droplets
as we've seen.
00:34:15.179 --> 00:34:16.766
There's no problem with that.
00:34:20.933 --> 00:34:25.949
If I do the high T, follow
exactly the procedure
00:34:25.949 --> 00:34:29.489
I had described before,
partition function is going
00:34:29.489 --> 00:34:31.875
to be 2 to the number of sites.
00:34:31.875 --> 00:34:36.179
Cosh K to the power of the
number of bonds and there
00:34:36.179 --> 00:34:38.540
are 3 bonds per site.
00:34:38.540 --> 00:34:41.250
So there's 3N there.
00:34:41.250 --> 00:34:45.739
And then we start to
draw our diagrams.
00:34:45.739 --> 00:34:47.860
The first diagram
is just exactly
00:34:47.860 --> 00:34:49.550
like what we had before.
00:34:49.550 --> 00:34:51.933
I have to make a square.
00:34:56.290 --> 00:35:04.760
And this square can be placed
on any face of the cube.
00:35:04.760 --> 00:35:09.950
And there are 3 faces
that are equivalent.
00:35:09.950 --> 00:35:14.810
The next type of diagram that
I can draw has 6 bonds in it.
00:35:14.810 --> 00:35:18.940
So this could be
an example of that.
00:35:18.940 --> 00:35:24.710
And if you do the counting,
there are 18N of those.
00:35:24.710 --> 00:35:26.790
And so you go.
00:35:26.790 --> 00:35:28.760
And the genetic
term in the series
00:35:28.760 --> 00:35:31.965
is going to be some
find of a loop.
00:35:31.965 --> 00:35:37.860
Again, even number of bonds
per site is the operative term.
00:35:37.860 --> 00:35:46.050
And then I have t to the
power of the number of bonds
00:35:46.050 --> 00:35:48.478
making this closed loop.
00:35:48.478 --> 00:35:49.390
Or loops.
00:35:54.970 --> 00:35:56.840
So you stare at
the series and you
00:35:56.840 --> 00:35:59.790
see immediately that
there is no correspondence
00:35:59.790 --> 00:36:00.910
like we saw before.
00:36:00.910 --> 00:36:04.280
The coefficient here are 1N, 3N.
00:36:04.280 --> 00:36:06.480
Here, there are 1, 3, and 18N.
00:36:06.480 --> 00:36:08.580
Powers are 6, 10.
00:36:08.580 --> 00:36:10.520
Here are 4, 6.
00:36:10.520 --> 00:36:14.070
There's no correspondence
between these two.
00:36:14.070 --> 00:36:18.770
So there is nothing
that one could say.
00:36:18.770 --> 00:36:21.520
But you say, I really like this.
00:36:21.520 --> 00:36:26.590
So maybe I'll phrase the
question differently.
00:36:26.590 --> 00:36:31.170
Can I consider some
other model whose
00:36:31.170 --> 00:36:34.845
high-temperature
expansion reproduces
00:36:34.845 --> 00:36:41.120
this low-temperature
expansion of the Ising model?
00:36:41.120 --> 00:36:46.430
So this is the
question, can we find
00:36:46.430 --> 00:36:59.640
a model whose high T
expansion reproduces
00:36:59.640 --> 00:37:03.210
low T of 3D Ising model?
00:37:10.260 --> 00:37:15.140
So rather than knowing
what the model is,
00:37:15.140 --> 00:37:18.040
now we are going to
kind of work backward
00:37:18.040 --> 00:37:22.490
from this graphical
picture that we have.
00:37:22.490 --> 00:37:28.440
So what would have been
the analog thing over here,
00:37:28.440 --> 00:37:33.880
let's say that I had
this picture of droplets
00:37:33.880 --> 00:37:36.050
in the 2D Ising model.
00:37:36.050 --> 00:37:41.000
I recognize that I need to
make these perimeters out
00:37:41.000 --> 00:37:42.640
of something.
00:37:42.640 --> 00:37:49.170
And I know that I can make these
things that are joined together
00:37:49.170 --> 00:37:55.820
to a procedure such as the
one that we have over here.
00:37:55.820 --> 00:38:01.410
But the unit thing,
there it was the elements
00:38:01.410 --> 00:38:04.280
that I had along the perimeter.
00:38:04.280 --> 00:38:06.580
What is the
corresponding unit that I
00:38:06.580 --> 00:38:13.640
have for the low temperature
series of the Ising model?
00:38:13.640 --> 00:38:21.350
I have e to the minus 2K to the
power of the number of faces.
00:38:21.350 --> 00:38:25.140
So first thing is
unit has to be a face.
00:38:31.970 --> 00:38:35.430
So basically, what
I need to do is
00:38:35.430 --> 00:38:44.970
to have a series, which is an
expansion in terms of faces,
00:38:44.970 --> 00:38:49.140
and then somehow I can
glue these faces together,
00:38:49.140 --> 00:38:51.825
like I glued these
bonds together.
00:38:54.820 --> 00:38:59.740
So we found our unit.
00:38:59.740 --> 00:39:02.910
The next thing that we
need is some kind of a glue
00:39:02.910 --> 00:39:07.120
to put all of these
LEGO faces together.
00:39:07.120 --> 00:39:10.400
So how did we join
things together here?
00:39:10.400 --> 00:39:16.660
We had these sigmas that
were sitting by themselves.
00:39:16.660 --> 00:39:18.970
And then putting
two sigmas together,
00:39:18.970 --> 00:39:22.970
I ensured that when
I summed over sigma,
00:39:22.970 --> 00:39:27.170
I had to glue two
of the T's together.
00:39:27.170 --> 00:39:29.720
Can I do the same
thing over here?
00:39:29.720 --> 00:39:34.100
If I put the sigmas on the
corners of these faces,
00:39:34.100 --> 00:39:39.140
you can see it doesn't work
because here I have three.
00:39:39.140 --> 00:39:49.010
So I'm forced to put the
sigmas on the lines that
00:39:49.010 --> 00:39:49.982
join the faces.
00:39:52.640 --> 00:39:56.030
So what I need to
do is, therefore,
00:39:56.030 --> 00:40:00.510
to have a variable
such as this where
00:40:00.510 --> 00:40:07.490
I have these sigmas
sitting on the-- let's call
00:40:07.490 --> 00:40:10.270
this a plaquette, p.
00:40:10.270 --> 00:40:12.680
And this plaquette
will be having
00:40:12.680 --> 00:40:16.485
around it four different bonds.
00:40:19.330 --> 00:40:24.071
And if I have the product
of these four bonds--
00:40:24.071 --> 00:40:29.311
again, these sigmas
being minus plus 1--
00:40:29.311 --> 00:40:34.380
I am forced to glue
these sigmas in pairs.
00:40:34.380 --> 00:40:37.640
And I can join these things
together, these squares,
00:40:37.640 --> 00:40:41.520
to make whatever shape that
I like that would correspond
00:40:41.520 --> 00:40:45.060
to the shapes that
I have over there.
00:40:45.060 --> 00:40:52.620
So what I need to do is to have
for each face a factor of this.
00:41:01.137 --> 00:41:07.330
So this is the analog of this
factor that I have over here.
00:41:07.330 --> 00:41:10.960
And then what I
need to do is to do
00:41:10.960 --> 00:41:15.310
a product over all plaquettes.
00:41:15.310 --> 00:41:18.820
And I sum over all sigma tildes.
00:41:21.800 --> 00:41:27.112
And this would be the partition
function of some other system.
00:41:27.112 --> 00:41:30.210
In this other
system, you can see
00:41:30.210 --> 00:41:32.910
that if I make its
expansion, there
00:41:32.910 --> 00:41:35.610
will be a one-to-one
correspondence
00:41:35.610 --> 00:41:38.860
between the terms in the
expansion of this partition
00:41:38.860 --> 00:41:42.857
function and the 3D Ising
model partition function.
00:41:49.190 --> 00:41:51.960
Again, this kind of
term we have seen.
00:41:51.960 --> 00:41:55.230
If I had put factors of cosh
here, which don't really
00:41:55.230 --> 00:42:07.330
do much, I can re-express as
e to the something-- k tilde
00:42:07.330 --> 00:42:13.140
sigma 1p sigma 2p
sigma 3p sigma 4p.
00:42:13.140 --> 00:42:16.360
Essentially every time
you see 1 plus t times
00:42:16.360 --> 00:42:19.814
some binary variable, you can
rewrite it into this fashion.
00:42:23.190 --> 00:42:28.510
So what we have come up
with is the following.
00:42:28.510 --> 00:42:32.390
That in order to
construct the dual
00:42:32.390 --> 00:42:35.830
of the three-dimensional
Ising model, what we do
00:42:35.830 --> 00:42:39.980
is you go all over your cube.
00:42:39.980 --> 00:42:46.630
On each bond of it, you put a
variable that is minus plus 1.
00:42:46.630 --> 00:42:49.060
So previously for
the Ising model,
00:42:49.060 --> 00:42:52.350
the variables were
sitting on the sites.
00:42:52.350 --> 00:42:59.030
So Ising, these
were site variables.
00:42:59.030 --> 00:43:05.200
Whereas, this dual Ising,
these are the bond variables
00:43:05.200 --> 00:43:06.920
that are minus plus 1.
00:43:10.870 --> 00:43:14.990
In the case of the
Ising, the interactions
00:43:14.990 --> 00:43:20.410
were the product of
sites making on a bond.
00:43:20.410 --> 00:43:24.010
Whereas, for the dual
Ising, the interactions
00:43:24.010 --> 00:43:25.530
are around the face.
00:43:25.530 --> 00:43:28.274
There's four of them
that go around the face.
00:43:35.780 --> 00:43:42.720
But whatever this
new theory is, we
00:43:42.720 --> 00:43:46.960
know that its free energy
because of this relation
00:43:46.960 --> 00:43:50.375
is related to the free energy
of the three-dimensional Ising
00:43:50.375 --> 00:43:50.874
model.
00:43:53.540 --> 00:43:56.030
Also, we know that the
three-dimensional Ising model
00:43:56.030 --> 00:43:59.640
has a phase transition
between a disordered phase
00:43:59.640 --> 00:44:03.290
and the magnetized phase
at low temperature.
00:44:03.290 --> 00:44:05.110
There is a singularity.
00:44:05.110 --> 00:44:09.470
As I span the parameter K of the
three-dimensional Ising model,
00:44:09.470 --> 00:44:11.206
there is a Kc.
00:44:11.206 --> 00:44:14.210
Now, I can find
out what that Kc is
00:44:14.210 --> 00:44:16.380
because I don't
have self-duality.
00:44:16.380 --> 00:44:20.830
But I know that as I span the
parameter K of the Ising model,
00:44:20.830 --> 00:44:26.020
I'm also spanning the parameter
K tilde of this new theory.
00:44:26.020 --> 00:44:29.230
And since the original model
has a phase transition,
00:44:29.230 --> 00:44:33.030
this new model must also
have a phase transition.
00:44:33.030 --> 00:44:35.840
So there exists a
Kc for both models.
00:44:43.470 --> 00:44:44.210
You say, OK.
00:44:44.210 --> 00:44:46.185
Fine.
00:44:46.185 --> 00:44:49.150
But there is some complicated
kind of Ising model
00:44:49.150 --> 00:44:53.830
that you have devised and
it has a phase transition.
00:44:53.830 --> 00:44:55.810
What's the big deal?
00:44:55.810 --> 00:44:57.980
Well, the big deal is
that this model is not
00:44:57.980 --> 00:45:03.690
supposed to have a phase
transition because it
00:45:03.690 --> 00:45:07.060
has a different
type of symmetry.
00:45:07.060 --> 00:45:09.740
The symmetry that we
have for the Ising model
00:45:09.740 --> 00:45:13.490
is a global symmetry.
00:45:13.490 --> 00:45:23.660
That is, the energy
of a particular state
00:45:23.660 --> 00:45:28.080
is the energy of the state
in which all of the spins
00:45:28.080 --> 00:45:29.910
are reversed.
00:45:29.910 --> 00:45:33.240
Because the form of
the energy is bilinear.
00:45:33.240 --> 00:45:36.700
If I take all of the sigmas
from one configuration
00:45:36.700 --> 00:45:39.390
and make them minus
in that configuration,
00:45:39.390 --> 00:45:41.680
the energy will not change.
00:45:41.680 --> 00:45:43.410
But I have to do that globally.
00:45:43.410 --> 00:45:46.380
It's a global symmetry.
00:45:46.380 --> 00:45:52.660
Now, this model has a local
symmetry because what I can do
00:45:52.660 --> 00:45:56.310
is I can pick one of the sites.
00:45:56.310 --> 00:46:00.810
And out of this site,
there are six off
00:46:00.810 --> 00:46:04.200
these bonds that are
going out on which
00:46:04.200 --> 00:46:07.400
there is one of
these sigma tilde.
00:46:07.400 --> 00:46:12.830
If I pick this site and I change
the sign of all of these six
00:46:12.830 --> 00:46:19.470
that emanate from this site,
the energy will not change.
00:46:19.470 --> 00:46:23.630
Because the energy gets
contributions from faces.
00:46:23.630 --> 00:46:26.910
And you can see that for
any one of the faces,
00:46:26.910 --> 00:46:30.220
there are two sigmas
that have changed.
00:46:30.220 --> 00:46:33.110
So the energy, which is the
product of all four of them,
00:46:33.110 --> 00:46:33.860
has not changed.
00:46:37.160 --> 00:46:41.920
So this model has
a different form,
00:46:41.920 --> 00:46:44.730
which is a local symmetry.
00:46:44.730 --> 00:46:49.640
And in fact, it is very much
related to gauge theories.
00:46:49.640 --> 00:46:54.300
It's a kind of discrete
version of the gauge theories
00:46:54.300 --> 00:46:56.970
that you have seen
in electromagnetism.
00:46:56.970 --> 00:46:58.550
Since there are
two possibilities,
00:46:58.550 --> 00:47:00.240
it's sometimes called
a Z2 gauge theory.
00:47:04.160 --> 00:47:09.760
Now, the thing about the gauge
theories is that there is
00:47:09.760 --> 00:47:21.390
a theorem which states that
local or these gauge theories,
00:47:21.390 --> 00:47:28.730
gauge symmetries, cannot
be spontaneously broken.
00:47:39.570 --> 00:47:42.030
So for the case of
the Ising model,
00:47:42.030 --> 00:47:46.130
we have this symmetry between
sigma going to minus sigma.
00:47:46.130 --> 00:47:49.310
But yet, we know that if
I go to low temperature,
00:47:49.310 --> 00:47:53.740
I will have a state in which
globally all of the spins
00:47:53.740 --> 00:47:57.140
are either plus or minus.
00:47:57.140 --> 00:48:00.870
So there is a
symmetry broken state
00:48:00.870 --> 00:48:04.340
which is what we
have been discussing.
00:48:04.340 --> 00:48:08.210
Now, the reason that that
cannot take place in these gauge
00:48:08.210 --> 00:48:11.490
theories, I will just
sketch what is happening.
00:48:14.900 --> 00:48:16.570
Essentially, we
have been thinking
00:48:16.570 --> 00:48:19.560
in terms of this
broken symmetries
00:48:19.560 --> 00:48:24.590
by putting an infinitesimal
magnetic field.
00:48:24.590 --> 00:48:29.930
And we saw that, basically,
if I'm at temperatures of 1
00:48:29.930 --> 00:48:35.870
over K's that are below
some critical value,
00:48:35.870 --> 00:48:40.090
then if h is plus,
everybody would be plus.
00:48:40.090 --> 00:48:44.140
If h is minus, everybody
would be minus.
00:48:44.140 --> 00:48:50.640
And the reason as you approach
h goes to 0 from one site
00:48:50.640 --> 00:48:57.000
that you don't get average of
0 is because the difference
00:48:57.000 --> 00:48:59.610
between the energy of
this state and that state
00:48:59.610 --> 00:49:05.640
as you go to 0 temperature
is proportional to N times h.
00:49:05.640 --> 00:49:09.600
So although h is going to
0 with N being very large,
00:49:09.600 --> 00:49:15.800
the influence of infinitesimal
h is magnified enormously.
00:49:15.800 --> 00:49:19.680
Now, for the case of these
local gauge theories,
00:49:19.680 --> 00:49:22.150
you cannot have a
similar argument.
00:49:22.150 --> 00:49:26.790
Because if I pick this spin,
let's say one of these bond
00:49:26.790 --> 00:49:28.490
spins.
00:49:28.490 --> 00:49:35.440
And let's say is its
average-- what is its average?
00:49:35.440 --> 00:49:38.382
As I said, the h going to 0.
00:49:38.382 --> 00:49:42.270
Well, the difference
between a state
00:49:42.270 --> 00:49:46.010
in which it is plus or the
state in which it is minus
00:49:46.010 --> 00:49:47.650
is, in fact, 6h.
00:49:47.650 --> 00:49:49.670
Because all I need
to do is to pick
00:49:49.670 --> 00:49:52.180
a site that is
close to that bond
00:49:52.180 --> 00:49:55.060
and flip all of the spins
that are close to that.
00:49:55.060 --> 00:49:58.230
All of the K's are
equally satisfied.
00:49:58.230 --> 00:49:59.890
The difference
between that state
00:49:59.890 --> 00:50:04.030
and the one where there
is a flip is just 6h.
00:50:04.030 --> 00:50:06.460
So that remains
finite as h goes to 0.
00:50:06.460 --> 00:50:11.480
There is no barrier towards
flipping those spins.
00:50:11.480 --> 00:50:16.400
So there is no broken
symmetry in this system.
00:50:16.400 --> 00:50:21.810
So this can be proven very
nicely and rigorously.
00:50:21.810 --> 00:50:28.670
So we have now two
statements about this Ising
00:50:28.670 --> 00:50:30.330
version of a gauge theory.
00:50:30.330 --> 00:50:34.480
First of all, we know
that at low temperatures,
00:50:34.480 --> 00:50:38.150
still the average
value of each bond
00:50:38.150 --> 00:50:41.730
is equally likely
to be plus or minus.
00:50:41.730 --> 00:50:46.430
From that perspective of local
values of these bond spins,
00:50:46.430 --> 00:50:50.710
it is as disordered as the
highest-temperature phase.
00:50:50.710 --> 00:50:54.990
Yet, because of its duality
to the three-dimensional Ising
00:50:54.990 --> 00:50:58.650
model, we know that
it undergoes some kind
00:50:58.650 --> 00:51:01.785
of a singularity going
from high temperatures
00:51:01.785 --> 00:51:03.450
to low temperatures.
00:51:03.450 --> 00:51:07.070
So there is probably some
kind of a phase transition,
00:51:07.070 --> 00:51:09.930
but it has to be very
different from any of the phase
00:51:09.930 --> 00:51:13.270
transitions that we have
discussed so far because there
00:51:13.270 --> 00:51:18.440
is no spontaneous
symmetry breaking.
00:51:18.440 --> 00:51:20.200
So what's going on?
00:52:00.750 --> 00:52:04.720
Now, later on in
the course, we will
00:52:04.720 --> 00:52:09.520
see another example of
this that is much less
00:52:09.520 --> 00:52:12.480
exotic than a gauge
theory, but it
00:52:12.480 --> 00:52:16.010
has the same kind of
principle applicable to it.
00:52:16.010 --> 00:52:20.690
There will be a phase transition
without local symmetry breaking
00:52:20.690 --> 00:52:26.930
in something like a
superfluid in two dimensions.
00:52:26.930 --> 00:52:30.820
So one thing that that phase
transition and this one
00:52:30.820 --> 00:52:34.220
have in common is,
again, the lack
00:52:34.220 --> 00:52:42.190
of this local-order parameter
from symmetry breaking.
00:52:42.190 --> 00:52:46.560
And both of them
share something that
00:52:46.560 --> 00:52:52.050
was pointed out by Wegner
once this puzzle emerged,
00:52:52.050 --> 00:53:01.510
which is that one has to look
at some kind of a global-- well,
00:53:01.510 --> 00:53:03.030
I shouldn't even call it global.
00:53:08.016 --> 00:53:09.890
It is something that is
called a Wilson loop.
00:53:26.760 --> 00:53:29.440
So the idea is the following.
00:53:29.440 --> 00:53:32.580
That we have these
variable sigma tilde
00:53:32.580 --> 00:53:36.280
that are sitting on
the bonds of a lattice.
00:53:36.280 --> 00:53:42.810
Now, the problem is that with
this local transformations,
00:53:42.810 --> 00:53:46.740
I can very easily make this
sigma go to minus sigma.
00:53:46.740 --> 00:53:52.410
So that is not a good
thing to consider.
00:53:52.410 --> 00:53:56.680
However, what was the problem?
00:53:56.680 --> 00:53:58.390
Let's say I pick this site.
00:53:58.390 --> 00:54:00.820
All of the sigmas that
went out of that site
00:54:00.820 --> 00:54:03.030
I changed to minus themselves.
00:54:03.030 --> 00:54:04.700
That was the gauge
transformation,
00:54:04.700 --> 00:54:07.280
but it became minus.
00:54:07.280 --> 00:54:12.950
But if I multiply the sigma
with another sigma that goes out
00:54:12.950 --> 00:54:16.450
of that site, then I
have cured that problem.
00:54:16.450 --> 00:54:19.750
If I changed this
to minus itself,
00:54:19.750 --> 00:54:21.840
this changes, this changes.
00:54:21.840 --> 00:54:25.000
The product remains [INAUDIBLE].
00:54:25.000 --> 00:54:27.100
But then I have
the problem here.
00:54:27.100 --> 00:54:34.410
So what I do is I
make a long loop.
00:54:34.410 --> 00:54:38.060
I look at the expectation value.
00:54:38.060 --> 00:54:41.820
So this Wilson
loop is the product
00:54:41.820 --> 00:54:46.830
of sigma tilde around a loop.
00:54:51.720 --> 00:54:56.720
And what I can do is I
can look at the average
00:54:56.720 --> 00:54:58.304
of that quantity.
00:54:58.304 --> 00:55:01.540
The average of that
quantity is something
00:55:01.540 --> 00:55:05.260
that is clearly invariant
to this kind of gauge
00:55:05.260 --> 00:55:06.770
transformation.
00:55:06.770 --> 00:55:10.610
So the signatures of a
potential phase transition
00:55:10.610 --> 00:55:14.580
could potentially be revealed by
looking at something like this.
00:55:17.680 --> 00:55:19.880
But clearly, that
is a quantity that
00:55:19.880 --> 00:55:24.030
is always also going
to be positive.
00:55:24.030 --> 00:55:25.990
So the thing that
I am looking at
00:55:25.990 --> 00:55:29.627
is not that this quantity
is, say, positive
00:55:29.627 --> 00:55:32.800
in one phase and 0
in the other phase.
00:55:32.800 --> 00:55:36.790
It is on how this
quantity depends
00:55:36.790 --> 00:55:40.345
on the shape and
characteristics of these loop.
00:55:42.960 --> 00:55:47.950
So what I can do is I can
calculate this average, both
00:55:47.950 --> 00:55:53.410
in high temperatures and low
temperatures, and compare them.
00:55:53.410 --> 00:56:05.193
So if I look at-- let's starts
with, yeah, high temperatures.
00:56:05.193 --> 00:56:09.500
The high-temperature expansion.
00:56:09.500 --> 00:56:12.430
So I want to calculate
the expectation
00:56:12.430 --> 00:56:18.550
value of the product of
sigma tilde, let's call it i,
00:56:18.550 --> 00:56:22.580
where i belongs to
some kind of a loop c.
00:56:22.580 --> 00:56:27.370
So c is all of these bonds.
00:56:27.370 --> 00:56:31.210
I want to calculate
that expectation value.
00:56:31.210 --> 00:56:34.240
How do I calculate
that expectation value?
00:56:34.240 --> 00:56:40.440
Well, I have to sum
over all configurations
00:56:40.440 --> 00:56:48.900
of this product with a weight.
00:56:48.900 --> 00:56:50.760
What is my weight?
00:56:50.760 --> 00:56:55.800
My weight is this
factor of product
00:56:55.800 --> 00:57:02.202
over all plaquettes of 1 plus
t sigma tilde sigma tilde sigma
00:57:02.202 --> 00:57:05.191
tilde for the plaquette.
00:57:11.090 --> 00:57:13.400
So this is the
weight that I have.
00:57:13.400 --> 00:57:16.560
I can put the hyperbolic
cosine or not put it,
00:57:16.560 --> 00:57:18.970
it doesn't matter.
00:57:18.970 --> 00:57:21.715
But then this weight has
to be properly normalized.
00:57:21.715 --> 00:57:25.510
It means that I have
to divide by something
00:57:25.510 --> 00:57:28.830
in the denominator,
which basically does not
00:57:28.830 --> 00:57:31.461
include the quantity
that I am averaging.
00:57:46.400 --> 00:57:50.080
So the graphs that
occur in the denominator
00:57:50.080 --> 00:57:52.960
are the things that we
have been discussing.
00:57:52.960 --> 00:57:56.020
Essentially, I start with 1.
00:57:56.020 --> 00:57:59.520
The next term is to put
these faces together
00:57:59.520 --> 00:58:02.860
to make a cube, and then
more complicated shapes.
00:58:02.860 --> 00:58:05.520
That, essentially,
every bond is going
00:58:05.520 --> 00:58:09.194
to be having some
kind of a complement.
00:58:09.194 --> 00:58:12.280
Well, but what about the
terms in the numerator?
00:58:12.280 --> 00:58:14.550
For the terms in
the numerator, I
00:58:14.550 --> 00:58:17.970
have these factors
of sigma tilde
00:58:17.970 --> 00:58:22.410
that are lying all
around this loop.
00:58:22.410 --> 00:58:25.540
And I'm summing over
the two possible values.
00:58:25.540 --> 00:58:30.610
So in order that summing
over this does not give me 0,
00:58:30.610 --> 00:58:34.970
I better make sure that there
is a complement to that.
00:58:34.970 --> 00:58:38.630
The complement to that
can only come from here.
00:58:38.630 --> 00:58:42.610
So for example, I would
put a face over here
00:58:42.610 --> 00:58:46.760
that ensures that that is
squared and that is squared.
00:58:46.760 --> 00:58:48.305
But then I have this one.
00:58:48.305 --> 00:58:49.840
Then I will put
another one here.
00:58:49.840 --> 00:58:52.155
I will put another
one, et cetera.
00:58:55.400 --> 00:59:00.240
And you can see that the
lowest-order term that I would
00:59:00.240 --> 00:59:08.110
get is this factor that
characterize each one of them.
00:59:10.870 --> 00:59:15.688
Raised to the power of
the area of this loop c.
00:59:19.150 --> 00:59:21.680
You can ask higher-order terms.
00:59:21.680 --> 00:59:25.550
You can kind of build
a hat on top of this.
00:59:25.550 --> 00:59:27.790
This you can put
anywhere, again,
00:59:27.790 --> 00:59:30.530
along this area of this thing.
00:59:30.530 --> 00:59:33.280
So the next correction in
this series you can see
00:59:33.280 --> 00:59:37.450
is also going to be something
that will be 1 plus t
00:59:37.450 --> 00:59:40.020
to the fourth times the area.
00:59:40.020 --> 00:59:43.880
The point is that as you
add more and more terms,
00:59:43.880 --> 00:59:48.410
you preserve the structure
that the whole thing is going
00:59:48.410 --> 00:59:53.180
to be proportional
to the area of loop
00:59:53.180 --> 00:59:56.484
times some function
of this parameter t.
01:00:01.180 --> 01:00:07.300
So what we know is that
if I take the expectation
01:00:07.300 --> 01:00:11.340
value of this entity,
then it's logarithm
01:00:11.340 --> 01:00:14.470
will be proportional
in high temperatures
01:00:14.470 --> 01:00:16.144
to the area of this loop.
01:00:21.470 --> 01:00:25.680
Now, what happens if I
try to do a low T-series
01:00:25.680 --> 01:00:26.700
for the same quantity?
01:00:34.090 --> 01:00:38.530
So I have to start
with a configuration
01:00:38.530 --> 01:00:45.850
at low temperature that
minimizes the energy.
01:00:50.930 --> 01:00:55.200
One configuration clearly is
one where all of the sigmas
01:00:55.200 --> 01:00:55.970
are plus.
01:00:59.730 --> 01:01:03.570
And that will give me a term.
01:01:03.570 --> 01:01:07.180
If I am calculating
the partition function
01:01:07.180 --> 01:01:09.630
in the denominator,
there will be
01:01:09.630 --> 01:01:13.310
a term that will be
proportional to e to whatever
01:01:13.310 --> 01:01:18.720
this K tilde is per face.
01:01:18.720 --> 01:01:22.112
And there are three
faces of a cube.
01:01:22.112 --> 01:01:23.570
There are N cubed,
so there will be
01:01:23.570 --> 01:01:30.210
3K tilde N for the
configuration that is all plus.
01:01:30.210 --> 01:01:34.290
But this is not the only
low-temperature configuration.
01:01:34.290 --> 01:01:36.470
That is what we were discussing.
01:01:36.470 --> 01:01:40.780
Because I can pick
a site out of N site
01:01:40.780 --> 01:01:42.630
and the 6 bonds
that go out of it,
01:01:42.630 --> 01:01:45.110
I can make minus themselves.
01:01:45.110 --> 01:01:48.020
And the energy would
be exactly the same.
01:01:48.020 --> 01:01:53.650
So whereas the Ising model,
I had a multiplicity of 2.
01:01:53.650 --> 01:02:01.670
Here, there is a
multiplicity of 2 to the N.
01:02:01.670 --> 01:02:06.950
So that's the lowest term in
the low-temperature expansion
01:02:06.950 --> 01:02:08.500
of the partition function.
01:02:08.500 --> 01:02:10.090
I'm doing the
partition function,
01:02:10.090 --> 01:02:11.381
which is the denominator first.
01:02:13.770 --> 01:02:16.090
Then, what can I do?
01:02:16.090 --> 01:02:19.520
Then, let's say I start with a
configuration where all of them
01:02:19.520 --> 01:02:20.930
are pluses.
01:02:20.930 --> 01:02:23.450
There are, of course, 2 to
the N gauge copies of that.
01:02:23.450 --> 01:02:25.780
So whatever I do to
this configuration,
01:02:25.780 --> 01:02:28.610
I can do the analog
in all of the others.
01:02:28.610 --> 01:02:32.740
But let's keep the copy where
all of the sigmas are plus,
01:02:32.740 --> 01:02:37.270
and then I flip one of
the sigmas to minus.
01:02:37.270 --> 01:02:40.190
Then, it's essentially--
think of a cube.
01:02:40.190 --> 01:02:43.850
There is a line that was
plus and I made it minus.
01:02:43.850 --> 01:02:48.840
There are four faces going out
of that that were previously
01:02:48.840 --> 01:02:52.920
plus K, now become
minus K. So I will
01:02:52.920 --> 01:02:58.890
have 2K tilde times 4
because of these four things
01:02:58.890 --> 01:03:01.270
that are going out.
01:03:01.270 --> 01:03:04.966
And the bond I can orient
in x-, y-, or z-direction.
01:03:04.966 --> 01:03:09.000
So there are 3N possibilities.
01:03:09.000 --> 01:03:13.750
And so I could have a series
such as this in the denominator
01:03:13.750 --> 01:03:17.660
where subsequent terms would
be to put more and more
01:03:17.660 --> 01:03:19.390
minus in this
particular [INAUDIBLE].
01:03:22.100 --> 01:03:26.670
Now, let's see how these
series would affect the sum
01:03:26.670 --> 01:03:31.160
that I would have to do in order
to calculate this expectation
01:03:31.160 --> 01:03:32.660
value.
01:03:32.660 --> 01:03:39.900
For any one of these
configurations,
01:03:39.900 --> 01:03:42.330
since I am kind of looking
at the ground state,
01:03:42.330 --> 01:03:45.120
let's say they're all pluses.
01:03:45.120 --> 01:03:51.270
Clearly, the contribution to
this product will be unity.
01:03:51.270 --> 01:03:53.870
That does not change.
01:03:53.870 --> 01:03:56.540
But now, let's think
about the configurations
01:03:56.540 --> 01:04:01.260
in which one of the
bonds is made to flip.
01:04:01.260 --> 01:04:05.320
As long as that
bond does not touch
01:04:05.320 --> 01:04:09.340
any of the bonds that
are part of the loop,
01:04:09.340 --> 01:04:12.540
the value of the loop
will remain the same.
01:04:12.540 --> 01:04:16.280
So let's say that
the loop has P sites.
01:04:16.280 --> 01:04:22.590
So this is number of bonds in c.
01:04:22.590 --> 01:04:23.953
Let's call it Pc.
01:04:28.590 --> 01:04:38.047
For these, with this weight,
the value of Wilson loop
01:04:38.047 --> 01:04:38.880
would still be plus.
01:04:41.520 --> 01:04:45.050
But for the times where I
have picked one of these
01:04:45.050 --> 01:04:50.180
to become minus, then the
product becomes minus.
01:04:50.180 --> 01:04:55.730
So for the remaining Pc
times of this factor,
01:04:55.730 --> 01:04:59.390
e to the minus 2K
tilde times 4, rather
01:04:59.390 --> 01:05:00.870
than having plus
I will have minus.
01:05:06.410 --> 01:05:09.870
So you can see
that-- what is this?
01:05:09.870 --> 01:05:11.190
N should be up here.
01:05:13.860 --> 01:05:20.340
That the difference between
what is in the numerator
01:05:20.340 --> 01:05:25.480
and what is in the denominator
of this low-temperature series
01:05:25.480 --> 01:05:29.200
has to do with the
bonds that have
01:05:29.200 --> 01:05:34.070
been sitting as part
of the Wilson loop.
01:05:34.070 --> 01:05:38.890
And if I imagine that
this is a small quantity
01:05:38.890 --> 01:05:41.520
and write these as
exponentials, you
01:05:41.520 --> 01:05:44.210
can see that this
is going to start
01:05:44.210 --> 01:05:49.960
with e to the minus
2K tilde times 4 times
01:05:49.960 --> 01:05:53.605
the perimeter of this cluster.
01:05:53.605 --> 01:05:54.960
Of this loop.
01:05:57.680 --> 01:06:02.250
And you can go and look at
higher and higher order terms.
01:06:02.250 --> 01:06:06.670
The point is that
in high temperature,
01:06:06.670 --> 01:06:11.710
the property of the shape
of the loop that determines
01:06:11.710 --> 01:06:15.806
this expectation
value is its area.
01:06:15.806 --> 01:06:20.987
Whereas, in the low-temperature
expansion, it is its perimeter.
01:06:25.090 --> 01:06:27.390
So you could, for
example, calculate
01:06:27.390 --> 01:06:33.240
the-- for a large loop,
the log of this quantity
01:06:33.240 --> 01:06:36.680
and divide it by the perimeter.
01:06:36.680 --> 01:06:39.910
And in the low-temperature
phase, it would be finite.
01:06:39.910 --> 01:06:41.755
In the high-temperature
phase, it
01:06:41.755 --> 01:06:44.055
would go to 0 because
the area scale
01:06:44.055 --> 01:06:46.580
is bigger than the perimeter.
01:06:46.580 --> 01:06:49.900
So we have found something
that is an analog of an older
01:06:49.900 --> 01:06:53.930
parameter and can distinguish
the different phases,
01:06:53.930 --> 01:06:57.144
and it is reflected in the
way that the correlations take
01:06:57.144 --> 01:06:57.644
place.
01:07:33.710 --> 01:07:38.730
Let's try to sort of think about
some potential physics that
01:07:38.730 --> 01:07:41.860
could be related to this.
01:07:41.860 --> 01:07:47.230
Let's start with the gauge
theory aspect of this.
01:07:47.230 --> 01:07:51.210
Well, the one gauge theory
that you probably know
01:07:51.210 --> 01:07:56.010
is quantum electrodynamics,
whose action
01:07:56.010 --> 01:07:59.540
you would write in
the following way.
01:07:59.540 --> 01:08:02.845
The action would involve
an integration over space
01:08:02.845 --> 01:08:05.760
as well as time.
01:08:05.760 --> 01:08:10.380
And by appropriate
rescalings, you
01:08:10.380 --> 01:08:14.000
can write the energy that is
in the electromagnetic field
01:08:14.000 --> 01:08:20.899
as d mu A mu minus d
mu A mu squared, where
01:08:20.899 --> 01:08:33.450
A is the 4-vector potential
out of which you can construct
01:08:33.450 --> 01:08:37.930
the electric field
and magnetic field.
01:08:37.930 --> 01:08:41.880
And the reason this
is a gauge theory
01:08:41.880 --> 01:08:55.750
is because if I take A and
add to it some function of phi
01:08:55.750 --> 01:09:01.399
of x and t, as long as
I take the derivative
01:09:01.399 --> 01:09:04.660
of this function, you can
see that the change here
01:09:04.660 --> 01:09:08.540
would be d mu d mu
minus d mu by d mu.
01:09:08.540 --> 01:09:10.160
There is really no change.
01:09:10.160 --> 01:09:14.500
And we know that
basically you can
01:09:14.500 --> 01:09:20.010
choose whatever
value of this phase--
01:09:20.010 --> 01:09:24.073
this gauge fixing
potential over here.
01:09:27.321 --> 01:09:30.880
Now, this is the
electromagnetic field by itself.
01:09:30.880 --> 01:09:34.479
If you want to couple it
to something like matter
01:09:34.479 --> 01:09:37.420
or electrons, you
write something
01:09:37.420 --> 01:09:46.300
like i d bar, which
is some derivative,
01:09:46.300 --> 01:09:50.864
and then you have e A bar.
01:09:50.864 --> 01:09:55.254
If there's a mass to this
object, you would put it here.
01:09:55.254 --> 01:09:58.320
This would be something
like psi bar psi.
01:09:58.320 --> 01:10:01.380
This would be something
that would describe
01:10:01.380 --> 01:10:05.510
the coupling of this
electromagnetic field
01:10:05.510 --> 01:10:08.801
to some charged particle,
such as the electron.
01:10:14.720 --> 01:10:19.680
And this entire thing satisfies
the gauge symmetry provided
01:10:19.680 --> 01:10:24.420
that once you do this, you
also replace psi with e
01:10:24.420 --> 01:10:30.260
to the ie phi psi.
01:10:30.260 --> 01:10:37.140
So the same phi, if it appears
in both, essentially the change
01:10:37.140 --> 01:10:40.190
in A that you would
have from here
01:10:40.190 --> 01:10:42.390
will be compensated
by the change
01:10:42.390 --> 01:10:45.590
that you would get from the
derivative acting on this phase
01:10:45.590 --> 01:10:46.760
factor.
01:10:46.760 --> 01:10:51.030
And so the whole
thing is not affected.
01:10:51.030 --> 01:10:55.470
What we have constructed
in this model
01:10:55.470 --> 01:10:59.550
is kind of an Ising
analog of this theory.
01:10:59.550 --> 01:11:03.350
Because the Hamiltonian
that we have,
01:11:03.350 --> 01:11:07.730
which carries the weight
after exponentiate
01:11:07.730 --> 01:11:11.740
of the different
configurations, has a part
01:11:11.740 --> 01:11:19.540
which is the sum over all of the
plaquettes of this sigma tilde
01:11:19.540 --> 01:11:25.020
sigma tilde sigma
tilde sigma tilde,
01:11:25.020 --> 01:11:28.910
the four bonds
around the plaquette.
01:11:28.910 --> 01:11:32.870
We could put some kind of a
coupling here if we want to.
01:11:32.870 --> 01:11:36.980
And the analog of
this transformation
01:11:36.980 --> 01:11:41.650
that we have-- well, maybe it
will become more apparent if I
01:11:41.650 --> 01:11:47.500
add the next term, which is
the analog of the coupling
01:11:47.500 --> 01:11:48.290
to matter.
01:11:48.290 --> 01:11:55.110
If I put a spin and
then sigma tilde ij sj.
01:11:55.110 --> 01:12:01.940
So again, imagine that we
have our cubic lattice,
01:12:01.940 --> 01:12:07.100
or some other lattice, in which
we have these variables sigma
01:12:07.100 --> 01:12:10.680
tilde that are
sitting on the bonds.
01:12:10.680 --> 01:12:14.730
And the first term is the
product around the face.
01:12:14.730 --> 01:12:17.840
And the second term
I put these variables
01:12:17.840 --> 01:12:19.620
s that are sitting here.
01:12:24.770 --> 01:12:30.350
And I have made a coupling
between these two s's.
01:12:30.350 --> 01:12:32.830
So if the sigma
tildes were not there,
01:12:32.830 --> 01:12:36.730
I could make an Ising model
with s's being plus or minus,
01:12:36.730 --> 01:12:40.080
which are coupled across
nearest neighbors.
01:12:40.080 --> 01:12:42.830
What I do is that the
strength of that coupling I
01:12:42.830 --> 01:12:47.010
make to be plus or
minus, depending
01:12:47.010 --> 01:12:51.770
on the value of the
gauge field, if you like.
01:12:51.770 --> 01:12:55.310
This Ising gauge field
that is sitting over there.
01:12:55.310 --> 01:13:00.030
Now, the analog of these
symmetries that we have for QED
01:13:00.030 --> 01:13:01.700
is as follows.
01:13:01.700 --> 01:13:08.860
I can pick a particular s i
and change its value to minus.
01:13:08.860 --> 01:13:15.725
And I can pick all of the sigma
tildes that go out of that i
01:13:15.725 --> 01:13:22.390
to the neighbours and
simultaneously make them minus.
01:13:22.390 --> 01:13:25.580
And then this energy
would not change.
01:13:25.580 --> 01:13:30.240
First of all, let's
say if I pick this site
01:13:30.240 --> 01:13:33.970
and change its face from
being plus or minus,
01:13:33.970 --> 01:13:38.795
then the bonds that-- the
sigma tildes that go out of it
01:13:38.795 --> 01:13:42.460
will change their values
to minus themselves.
01:13:42.460 --> 01:13:46.820
Since each one of the
faces contains two of them,
01:13:46.820 --> 01:13:51.880
the value of the energy
from here is not changed.
01:13:51.880 --> 01:13:57.400
Since the couplings to the
neighboring s's involve
01:13:57.400 --> 01:13:59.800
the sigma tildes that
sit between them,
01:13:59.800 --> 01:14:02.170
and I have flipped
both s and sigma tilde,
01:14:02.170 --> 01:14:05.070
those do not change
irrespective of what
01:14:05.070 --> 01:14:08.240
I do with the face of
all the other ones.
01:14:08.240 --> 01:14:13.970
So I have made an Ising,
or binary version,
01:14:13.970 --> 01:14:16.260
of this transformation,
constructed
01:14:16.260 --> 01:14:21.280
a model that has, except
being Ising symmetry,
01:14:21.280 --> 01:14:23.735
a lot of the properties
that you would
01:14:23.735 --> 01:14:25.230
have for this kind of action.
01:14:29.930 --> 01:14:37.530
Now again, continuing with that,
this difference between why
01:14:37.530 --> 01:14:42.080
we see an area rule
or a perimeter rule
01:14:42.080 --> 01:14:47.600
has some physical consequence
that is worth mentioning.
01:14:47.600 --> 01:14:51.430
And this, again,
has not much to do
01:14:51.430 --> 01:14:53.700
with the main thrust
of this course,
01:14:53.700 --> 01:14:57.690
but just a matter of
overall education.
01:14:57.690 --> 01:14:59.290
It is useful to know.
01:15:02.500 --> 01:15:07.160
So in this picture, where
one of the dimensions
01:15:07.160 --> 01:15:14.990
corresponds to time,
imagine that you
01:15:14.990 --> 01:15:21.320
create a kind of a
Wilson loop which
01:15:21.320 --> 01:15:25.640
is very long in one
direction that I
01:15:25.640 --> 01:15:29.670
want to think of as being time.
01:15:29.670 --> 01:15:36.680
And the analog of this
action that we have discussed
01:15:36.680 --> 01:15:43.425
is to create a pair of charges,
separate them by a distance x,
01:15:43.425 --> 01:15:50.350
propagate them for a long
time t, and then [INAUDIBLE].
01:15:50.350 --> 01:15:53.430
And ask, what is the
contribution of a configuration
01:15:53.430 --> 01:15:58.940
such as this to their
action on average?
01:15:58.940 --> 01:16:06.420
And so you would say that if
particles are at a distance x,
01:16:06.420 --> 01:16:11.130
they are subject to some
kind of a potential v of x.
01:16:11.130 --> 01:16:13.720
And if this potential
has been propagated
01:16:13.720 --> 01:16:17.090
in time for a
length or a duration
01:16:17.090 --> 01:16:24.710
that I will call T, that
the effect that it has
01:16:24.710 --> 01:16:33.910
on the system is to have
an interaction such as this
01:16:33.910 --> 01:16:39.310
in the action propagated
over a time T.
01:16:39.310 --> 01:16:42.274
So I should have
something like this.
01:16:45.230 --> 01:16:48.730
So this should
somehow be related
01:16:48.730 --> 01:16:54.540
to this average of the
Wilson loop in manner
01:16:54.540 --> 01:16:58.530
to not be made precise,
but very rough just
01:16:58.530 --> 01:17:01.190
to get the general idea.
01:17:01.190 --> 01:17:07.970
And so what we have said is that
the value of this Wilson loop
01:17:07.970 --> 01:17:10.313
has different
behaviors at high T
01:17:10.313 --> 01:17:15.780
and low T in its
dependence on shape.
01:17:15.780 --> 01:17:20.480
And that high temperatures, it
is proportional to the area.
01:17:20.480 --> 01:17:24.790
So it should be
proportional to xT.
01:17:24.790 --> 01:17:28.820
Whereas, in low temperature, it
is proportional to perimeter.
01:17:28.820 --> 01:17:36.600
So it should be
proportional to x plus T.
01:17:36.600 --> 01:17:43.285
So if I read off the form of V
of x from these two dependents,
01:17:43.285 --> 01:17:48.360
you will see that V of
x goes proportionately
01:17:48.360 --> 01:17:53.660
to x in one regime.
01:17:53.660 --> 01:17:58.190
And once I divide
by T, the leading
01:17:58.190 --> 01:18:01.130
coefficients-- so
this is essentially
01:18:01.130 --> 01:18:07.460
I want to look at the limit
where T is becoming very large.
01:18:07.460 --> 01:18:11.460
So this thing when I divide
by T goes to a constant.
01:18:15.900 --> 01:18:19.720
And I can very
roughly interpret this
01:18:19.720 --> 01:18:22.120
as the interaction
between particles
01:18:22.120 --> 01:18:29.660
that are separated by x
via this kind of theory.
01:18:29.660 --> 01:18:31.890
And I see that
for particles that
01:18:31.890 --> 01:18:36.900
are separated by x in
that kind of theory,
01:18:36.900 --> 01:18:42.070
there is a weak coupling--
high temperature
01:18:42.070 --> 01:18:49.430
corresponds to weak coupling--
where the further apart I
01:18:49.430 --> 01:18:55.310
go, the potential that
is bringing them together
01:18:55.310 --> 01:18:59.510
becomes linearly stronger.
01:18:59.510 --> 01:19:01.490
So this is what is
called confinement.
01:19:07.000 --> 01:19:13.970
Whereas, in the other
limit of low temperatures,
01:19:13.970 --> 01:19:18.950
or strong coupling,
what you find
01:19:18.950 --> 01:19:22.990
is that the interaction
between them
01:19:22.990 --> 01:19:25.175
essentially goes to a constant.
01:19:25.175 --> 01:19:27.050
The potential goes
to a constant,
01:19:27.050 --> 01:19:29.210
so the force would go to 0.
01:19:29.210 --> 01:19:31.720
So they are asymptotically free.
01:19:38.500 --> 01:19:43.800
So if I start with
this kind of theory
01:19:43.800 --> 01:19:47.770
and try to interpret it in
the language of quantum field
01:19:47.770 --> 01:19:52.050
theory as something that
describes interaction
01:19:52.050 --> 01:19:57.690
between particles, I find that
it has potentially two phases.
01:19:57.690 --> 01:20:01.110
One phase in which the
particles of the theory
01:20:01.110 --> 01:20:04.950
are strongly bind
together, like quarks
01:20:04.950 --> 01:20:07.830
that are inside the nucleus.
01:20:07.830 --> 01:20:09.940
And you can try to
separate the quarks,
01:20:09.940 --> 01:20:11.390
but they would snap back.
01:20:11.390 --> 01:20:13.680
You can't have free quarks.
01:20:13.680 --> 01:20:19.120
And then there is another phase
where essentially the particles
01:20:19.120 --> 01:20:20.930
don't see each other.
01:20:20.930 --> 01:20:24.810
And indeed, quarks
right inside the nucleus
01:20:24.810 --> 01:20:26.260
are essentially free.
01:20:26.260 --> 01:20:29.430
We can sort of regard
them as free particles.
01:20:29.430 --> 01:20:33.780
So this theory
actually has aspects
01:20:33.780 --> 01:20:38.730
of what is known as confinement
and asymptotic freedom
01:20:38.730 --> 01:20:41.550
within quantum chromodynamics.
01:20:41.550 --> 01:20:43.590
The difference is
that in this theory,
01:20:43.590 --> 01:20:47.720
there is a phase transition
and the two behaviors
01:20:47.720 --> 01:20:49.625
are separated from each other.
01:20:49.625 --> 01:20:53.590
Whereas in QCD, it's
essentially a crossover
01:20:53.590 --> 01:20:55.980
from one behavior
to another behavior
01:20:55.980 --> 01:20:58.790
without the phase transition.
01:20:58.790 --> 01:21:04.140
So we started by thinking
about these Ising models.
01:21:04.140 --> 01:21:06.620
And we kind of
branched into theories
01:21:06.620 --> 01:21:10.170
that describe loops, theories
that describe droplets,
01:21:10.170 --> 01:21:13.930
theories that describe
gauge couplings, et cetera.
01:21:13.930 --> 01:21:17.850
So you can see that that nice,
simple line of the partition
01:21:17.850 --> 01:21:22.180
function that I have written
for you has within it
01:21:22.180 --> 01:21:25.010
a lot of interesting complexity.
01:21:25.010 --> 01:21:27.870
We kind of went
off the direction
01:21:27.870 --> 01:21:31.260
that we wanted to go
with phase transitions,
01:21:31.260 --> 01:21:33.910
so we will remedy
that next time, coming
01:21:33.910 --> 01:21:39.350
back to thinking about how to
think in terms of the Ising
01:21:39.350 --> 01:21:43.975
model, and try to do more with
understanding the behavior
01:21:43.975 --> 01:21:47.550
and singularities of
this partition function.