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PROFESSOR: OK, let's start.
00:00:26.390 --> 00:00:30.600
So it's good to
remind ourselves why
00:00:30.600 --> 00:00:34.390
we are doing what
we are doing today.
00:00:34.390 --> 00:00:36.900
So we've seen that
in a number of cases,
00:00:36.900 --> 00:00:41.750
we look at something like
the coexistence line of gas
00:00:41.750 --> 00:00:47.070
and liquid that terminates
at the critical point.
00:00:47.070 --> 00:00:50.620
And that in the vicinity
of this critical point,
00:00:50.620 --> 00:00:54.730
we see various thermodynamic
quantities and correlation
00:00:54.730 --> 00:00:57.570
functions that have
properties that
00:00:57.570 --> 00:01:02.590
are independent of the
materials that are considered.
00:01:02.590 --> 00:01:07.980
So this led to this
concept of universality,
00:01:07.980 --> 00:01:14.720
and we were able to justify
that by looking at properties
00:01:14.720 --> 00:01:17.230
of this statistical field.
00:01:23.650 --> 00:01:28.730
And we ended up with
[INAUDIBLE] normalization
00:01:28.730 --> 00:01:32.320
group procedure,
which classified
00:01:32.320 --> 00:01:35.780
the different universality
classes according
00:01:35.780 --> 00:01:42.130
to the number of components
of the order parameter,
00:01:42.130 --> 00:01:46.380
the thing that categorizes
the coexisting phases,
00:01:46.380 --> 00:01:49.250
and the dimensionality of space.
00:01:49.250 --> 00:01:51.260
And that, in
particular something
00:01:51.260 --> 00:01:56.950
like a liquid-gas system, would
correspond to n equals to 1.
00:01:56.950 --> 00:01:59.400
Another example that
would correspond to that
00:01:59.400 --> 00:02:04.950
would be, for example,
a mixture of two metals
00:02:04.950 --> 00:02:06.320
in a binary alloy.
00:02:06.320 --> 00:02:09.729
You can have the
different components mixed
00:02:09.729 --> 00:02:13.530
or phase separated
from each other.
00:02:13.530 --> 00:02:21.220
So the normalization group
method gave us the reason
00:02:21.220 --> 00:02:24.270
for there is this
universality, but we
00:02:24.270 --> 00:02:26.940
found that calculating
the exponent
00:02:26.940 --> 00:02:30.970
was a hard task coming
from four dimensions.
00:02:30.970 --> 00:02:36.970
So the question is, given that
these models or these numbers,
00:02:36.970 --> 00:02:40.390
the singularities
here are universal,
00:02:40.390 --> 00:02:43.950
can we obtain them from
a different perspective?
00:02:43.950 --> 00:02:48.210
And so let's say we are
focused on this kind
00:02:48.210 --> 00:02:51.780
of liquid-gas system,
which belong to this n
00:02:51.780 --> 00:02:54.860
equals to 1 universality class.
00:02:54.860 --> 00:02:57.440
So we can try to imagine
the simplest model
00:02:57.440 --> 00:02:59.930
that we can try to
solve that belongs
00:02:59.930 --> 00:03:02.220
to that universality class.
00:03:02.220 --> 00:03:06.570
And again, maybe thinking
in terms of a binary alloy,
00:03:06.570 --> 00:03:09.520
something that has
two possible values.
00:03:09.520 --> 00:03:11.550
In the liquid-gas,
it could be cells
00:03:11.550 --> 00:03:15.090
that are either empty or
filled with a particle.
00:03:15.090 --> 00:03:24.080
And so this binary model
is this Ising model,
00:03:24.080 --> 00:03:26.860
where, at each
side of a lattice,
00:03:26.860 --> 00:03:30.380
we put a variable
that is minus plus 1.
00:03:30.380 --> 00:03:34.030
And so the idea is, again, if
I take any one of these Ising
00:03:34.030 --> 00:03:36.790
models and I
coarse-grain them, I
00:03:36.790 --> 00:03:39.190
will end up with the
same statistical field,
00:03:39.190 --> 00:03:42.210
and it would have the
same universality class.
00:03:42.210 --> 00:03:46.370
But if I make a sufficiently
simple version of these models,
00:03:46.370 --> 00:03:50.840
maybe I can do something else
and solve them in a manner
00:03:50.840 --> 00:03:54.450
that these critical
behaviors can come up
00:03:54.450 --> 00:03:57.180
in an easier fashion.
00:03:57.180 --> 00:04:04.660
So let's say we are interested
in two dimensions or three
00:04:04.660 --> 00:04:06.250
dimensions.
00:04:06.250 --> 00:04:09.080
I can draw two
dimensions better.
00:04:09.080 --> 00:04:11.190
We draw a square lattice.
00:04:11.190 --> 00:04:15.580
On each side of it, we put
one of these variables.
00:04:15.580 --> 00:04:19.690
And in order to
capture this tendency
00:04:19.690 --> 00:04:24.150
that there is a possibility
of coexistence where
00:04:24.150 --> 00:04:28.240
you have patches that are
made of liquid or gas,
00:04:28.240 --> 00:04:31.740
or made of copper or
zinc in our binary alloy,
00:04:31.740 --> 00:04:35.740
we need to have a tendency for
things that are close to each
00:04:35.740 --> 00:04:39.700
other to be in the same
state so that we can capture
00:04:39.700 --> 00:04:49.510
by a Hamiltonian, which is a
sum over nearest neighbors, that
00:04:49.510 --> 00:04:54.250
gives an enhanced weight
if they are parallel.
00:04:54.250 --> 00:04:58.210
And whatever that
coupling is, once it
00:04:58.210 --> 00:05:05.660
is rescaled by kT, this
combination, the energy divided
00:05:05.660 --> 00:05:12.830
by kT, we can parametrize
by a dimensionless number k.
00:05:12.830 --> 00:05:15.550
And calculating the
behavior of the system
00:05:15.550 --> 00:05:17.600
as a function of
temperature, as the strength
00:05:17.600 --> 00:05:20.670
of the coupling in
this simplified model,
00:05:20.670 --> 00:05:23.790
amounts to calculating
the partition function
00:05:23.790 --> 00:05:28.740
as a function of a parameter
k, which is a sum over,
00:05:28.740 --> 00:05:32.270
if I'm in a system that
has n sites all to the n
00:05:32.270 --> 00:05:39.490
configurations, of a weight that
tries to make variables that
00:05:39.490 --> 00:05:45.290
are next to each other
to be in the same state.
00:05:45.290 --> 00:05:48.680
So clearly, what
is captured here
00:05:48.680 --> 00:05:53.070
is a competition
between energy-- energy
00:05:53.070 --> 00:05:57.190
would like everybody to
be in the same state--
00:05:57.190 --> 00:05:58.670
versus entropy.
00:05:58.670 --> 00:06:02.770
Entropy wants to have
different states at each site.
00:06:02.770 --> 00:06:06.400
So you'll have a factor
of 2 per site as opposed
00:06:06.400 --> 00:06:10.670
to everybody being aligned,
which is essentially one state.
00:06:10.670 --> 00:06:14.330
And so that
competition potentially
00:06:14.330 --> 00:06:17.520
could lead you to a phase
transition between something
00:06:17.520 --> 00:06:21.080
that has coexistent at low
temperature and something that
00:06:21.080 --> 00:06:24.650
is disordered at
high temperatures.
00:06:24.650 --> 00:06:28.320
So now we have just
recast the problem.
00:06:28.320 --> 00:06:32.725
Rather than having a partition
function which was a functional
00:06:32.725 --> 00:06:36.680
integral over all configurations
of the statistical field,
00:06:36.680 --> 00:06:38.960
I have to do this
partition function, which
00:06:38.960 --> 00:06:41.790
is finite number
of configurations,
00:06:41.790 --> 00:06:43.760
but it's still an
interacting theory.
00:06:43.760 --> 00:06:48.990
I cannot independently move
the variable at each site.
00:06:48.990 --> 00:06:52.300
So the question is,
are there approaches
00:06:52.300 --> 00:06:55.470
by which I can calculate this?
00:06:55.470 --> 00:07:00.310
And one set of approaches
is to start with a limit
00:07:00.310 --> 00:07:04.670
that I can solve and
start expanding on that.
00:07:04.670 --> 00:07:08.660
And these expansions that are
analogous to the perturbation
00:07:08.660 --> 00:07:13.240
expansions that we
learned in 8.333
00:07:13.240 --> 00:07:16.320
about interacting
systems, in this case
00:07:16.320 --> 00:07:20.240
are usually called
series expansions.
00:07:20.240 --> 00:07:22.155
One would perform
them on a lattice.
00:07:25.980 --> 00:07:31.740
Now, I kind of hinted at
two limits of the problem
00:07:31.740 --> 00:07:35.220
that we know exactly
what is happening,
00:07:35.220 --> 00:07:39.360
and those lead to two
different series expansions.
00:07:39.360 --> 00:07:42.245
One of them is the
low-temperature expansions.
00:07:49.980 --> 00:07:54.400
And here the idea
is that I know what
00:07:54.400 --> 00:07:56.660
the system is doing
at T equals to 0.
00:07:56.660 --> 00:08:01.210
At T equals to 0, I have to
find the configuration that
00:08:01.210 --> 00:08:03.360
minimizes the energy.
00:08:03.360 --> 00:08:07.350
T equals to 0 is also equivalent
to k going to infinity.
00:08:07.350 --> 00:08:11.570
I have to find a state
that maximizes this weight,
00:08:11.570 --> 00:08:19.390
and that's obviously the
case where all of the spins
00:08:19.390 --> 00:08:23.500
are either plus or minus.
00:08:23.500 --> 00:08:28.360
So all sigma i equals to
plus 1 or all sigma i equals
00:08:28.360 --> 00:08:29.580
to minus 1.
00:08:32.970 --> 00:08:36.440
But for the sake of
doing one or the other,
00:08:36.440 --> 00:08:44.900
let's imagine that
they are all plus
00:08:44.900 --> 00:08:52.700
and that I am solving the
problem for the generalization
00:08:52.700 --> 00:08:56.560
of square cube to
d-dimensional lattice.
00:08:56.560 --> 00:08:59.790
After all, we were doing
d dimensions in general.
00:08:59.790 --> 00:09:05.990
So in d dimensions, each
spin will have d neighbors.
00:09:05.990 --> 00:09:10.330
And so if I ask, what
is the weight that I
00:09:10.330 --> 00:09:15.100
will get at 0 temperature,
essentially each spin
00:09:15.100 --> 00:09:20.360
would have d factors of k.
00:09:20.360 --> 00:09:26.590
So the weight that I would get
at T equals to 0-- let's call
00:09:26.590 --> 00:09:34.210
that Z of T equals to 0--
is simply e to the dNk.
00:09:36.920 --> 00:09:39.170
There are N sites.
00:09:39.170 --> 00:09:43.155
Each one of them has d
neighbors in d dimensions.
00:09:43.155 --> 00:09:46.550
Of course, each one of
them has 2d neighbours,
00:09:46.550 --> 00:09:49.570
but then I have to count the
number of neighbors per site.
00:09:49.570 --> 00:09:53.140
So basically, this bond is
shared by two neighbors,
00:09:53.140 --> 00:09:56.610
so half of it
contributes to this site.
00:09:56.610 --> 00:09:58.390
And there are two possibilities.
00:09:58.390 --> 00:10:03.700
So the partition function at
T equals to 0 is simply this.
00:10:03.700 --> 00:10:08.000
It's just the contribution
of the two ground states.
00:10:08.000 --> 00:10:14.060
Now, we are interested in
the limit where T goes to 0.
00:10:14.060 --> 00:10:18.260
So at T equals to 0, I
know what is happening.
00:10:18.260 --> 00:10:23.800
Now, what I will get
as I allow temperature
00:10:23.800 --> 00:10:29.880
to be larger, at some
cost I am able to flip
00:10:29.880 --> 00:10:34.270
some of these spins from,
say, the plus to minus.
00:10:34.270 --> 00:10:46.280
And I will get, in this case,
islands of negative spin
00:10:46.280 --> 00:10:47.345
in sea of plus.
00:10:51.500 --> 00:10:58.280
And these islands will
give a contribution
00:10:58.280 --> 00:11:07.690
that is going to be
exponentially small in k
00:11:07.690 --> 00:11:12.750
and something to do with the
bonds that I have broken.
00:11:12.750 --> 00:11:18.930
And by broken, I mean gone
from the high energy, well,
00:11:18.930 --> 00:11:23.630
highly satisfied plus-plus state
to the unsatisfied plus-minus
00:11:23.630 --> 00:11:30.620
state, in fact, 2k times
number of broken bonds.
00:11:40.130 --> 00:11:44.930
So we can very easily write the
first few terms in this series.
00:11:44.930 --> 00:11:52.980
So let's make a list of
the excitation, or island
00:11:52.980 --> 00:11:59.850
that I can make,
how many ways I can
00:11:59.850 --> 00:12:07.300
make this, which I
will call degeneracy,
00:12:07.300 --> 00:12:09.935
and the number of broken bonds.
00:12:20.030 --> 00:12:22.160
So clearly, the
simplest thing that I
00:12:22.160 --> 00:12:29.790
can do in a sea of pluses
is to make one island, which
00:12:29.790 --> 00:12:34.630
is simply a site that
has been previously plus
00:12:34.630 --> 00:12:36.600
and now has gone
to become a minus.
00:12:39.260 --> 00:12:48.790
And this particular excitation
can occur any one of N places
00:12:48.790 --> 00:12:52.290
if I have a lattice
of size N. And I'm
00:12:52.290 --> 00:12:56.470
going to ignore any corrections
that I may have from the edges.
00:12:56.470 --> 00:13:00.350
If you want you can do that,
and that'd be more precise.
00:13:00.350 --> 00:13:02.780
But let's focus,
essentially, on things
00:13:02.780 --> 00:13:06.710
that are proportional to N.
00:13:06.710 --> 00:13:10.660
Then how many bonds
have I broken?
00:13:10.660 --> 00:13:13.850
You can see that
in two dimensions,
00:13:13.850 --> 00:13:16.160
I have broken four bonds.
00:13:16.160 --> 00:13:19.130
In three dimensions,
it would have been six.
00:13:19.130 --> 00:13:23.530
So essentially, it
is twice the number
00:13:23.530 --> 00:13:28.580
of dimensions that
is taking place.
00:13:28.580 --> 00:13:30.760
And so the
contribution to energy
00:13:30.760 --> 00:13:36.105
is going to be e
to the minus 2d.
00:13:38.710 --> 00:13:41.950
And I went from
plus k to minus k,
00:13:41.950 --> 00:13:44.940
so in fact, I would
have to multiply by 2k.
00:13:49.030 --> 00:13:51.840
Now, the next thing
that I can do,
00:13:51.840 --> 00:13:55.810
the lowest energy excitation
is to put two minuses that
00:13:55.810 --> 00:13:58.655
are next to each other
in this sea of pluses.
00:14:03.413 --> 00:14:05.900
OK?
00:14:05.900 --> 00:14:08.780
Now, you can see that
in two dimension,
00:14:08.780 --> 00:14:14.290
I can orient this pair
along the x-direction
00:14:14.290 --> 00:14:16.200
or along the y-direction.
00:14:16.200 --> 00:14:20.210
And in general, there would be d
directions, so I would have dN.
00:14:23.140 --> 00:14:25.480
Roughly, you would say
that the number of bonds
00:14:25.480 --> 00:14:33.590
that you have
broken is twice what
00:14:33.590 --> 00:14:38.300
you had before if the
two were separate.
00:14:38.300 --> 00:14:42.430
But there is this
thing in between
00:14:42.430 --> 00:14:46.000
that is now actually
a satisfied bond.
00:14:46.000 --> 00:14:50.400
So you can convince
yourself that, actually,
00:14:50.400 --> 00:14:53.480
if the two of them
were separate,
00:14:53.480 --> 00:14:58.530
these two minus excitations,
I would have 4d.
00:14:58.530 --> 00:15:04.690
But because I joined them,
essentially I have 2d minus 1
00:15:04.690 --> 00:15:08.020
from each one of them,
and there's two of those.
00:15:08.020 --> 00:15:10.630
And of course, the
next lowest excitation
00:15:10.630 --> 00:15:14.060
would indeed be to
have two minuses that
00:15:14.060 --> 00:15:22.180
have no site, are totally
separate from each other.
00:15:22.180 --> 00:15:25.790
And the contribution--
the number of these,
00:15:25.790 --> 00:15:28.220
well, this is
something to count.
00:15:28.220 --> 00:15:32.070
The first one can
any one of N places.
00:15:32.070 --> 00:15:37.830
The next one can be in any one
of N minus 2d minus 1 places.
00:15:37.830 --> 00:15:41.710
It cannot be on the same
one, and it cannot be in any
00:15:41.710 --> 00:15:43.370
of the 2d neighbors.
00:15:43.370 --> 00:15:45.810
And I should have
double counting,
00:15:45.810 --> 00:15:49.590
so there is a factor of 2 here.
00:15:49.590 --> 00:15:54.230
And the cost of this is simply
twice that, so this is 4d.
00:15:57.540 --> 00:16:05.630
So if I want to start writing
a partition function expanded
00:16:05.630 --> 00:16:08.580
beyond what I have
at 0 temperature,
00:16:08.580 --> 00:16:14.010
what I would have
would be 2e to the dNk.
00:16:14.010 --> 00:16:16.500
There's zero temperature
contribution.
00:16:16.500 --> 00:16:29.894
I would have 1 plus N e
to the minus 4dk plus dN
00:16:29.894 --> 00:16:39.560
e to the minus 4 2d minus 1 k.
00:16:39.560 --> 00:16:43.650
And then from the other
one that I have written,
00:16:43.650 --> 00:16:51.536
N N minus 2d minus 1
over 2e to the minus 8dk.
00:16:51.536 --> 00:16:55.180
And I can keep going and
adding higher and higher order
00:16:55.180 --> 00:16:56.641
terms of the series.
00:16:56.641 --> 00:16:58.750
OK?
00:16:58.750 --> 00:16:59.380
OK.
00:16:59.380 --> 00:17:02.310
Once I have the
partition function,
00:17:02.310 --> 00:17:05.339
I can start calculating
the energy, which
00:17:05.339 --> 00:17:12.159
would be minus d log Z
with respect to d beta.
00:17:14.859 --> 00:17:15.665
What is beta?
00:17:19.710 --> 00:17:26.430
Well, I said that this
factor is something like 1
00:17:26.430 --> 00:17:31.130
over kT, which is beta and J.
00:17:31.130 --> 00:17:35.380
So assuming that I
have a fixed energy
00:17:35.380 --> 00:17:39.600
and I'm changing temperature,
and the variations of k
00:17:39.600 --> 00:17:43.370
are reflecting the
inverse temperature beta,
00:17:43.370 --> 00:17:48.310
then I can certainly
multiply here a J and a J,
00:17:48.310 --> 00:17:50.960
which is a constant.
00:17:50.960 --> 00:17:56.150
And all I need to
do is to take a J d
00:17:56.150 --> 00:18:02.710
by dk of log of the expression
that I have above there.
00:18:02.710 --> 00:18:03.210
OK?
00:18:03.210 --> 00:18:06.380
So let's take the log
of that expression.
00:18:06.380 --> 00:18:16.530
I have log of 2
plus dNk from here.
00:18:16.530 --> 00:18:21.560
And then I have log of
1 plus terms in a series
00:18:21.560 --> 00:18:25.410
that I have
calculated perturbity.
00:18:25.410 --> 00:18:29.500
Now, log of 1 plus
a small quantity I
00:18:29.500 --> 00:18:32.220
can always expand
as a small quantity
00:18:32.220 --> 00:18:34.550
minus x squared over 2.
00:18:34.550 --> 00:18:37.520
You may worry whether or
not, with N ultimately
00:18:37.520 --> 00:18:41.910
going to infinity, this
is a small quantity.
00:18:41.910 --> 00:18:43.970
Neglecting that
for the time being,
00:18:43.970 --> 00:18:48.180
if I look at this as log
of 1 plus a small quantity,
00:18:48.180 --> 00:18:55.190
from here I would get N e
to the minus 4dk plus dN
00:18:55.190 --> 00:19:03.670
e to the minus 4 2d minus
1 k plus N N minus 1
00:19:03.670 --> 00:19:12.380
minus 2d over 2e to the
minus 8dk, and so forth.
00:19:12.380 --> 00:19:18.970
But then remember
that log of 1 plus x
00:19:18.970 --> 00:19:25.570
is x minus x squared over 2 plus
x cubed over 3, and so forth.
00:19:25.570 --> 00:19:29.540
So if x is my small
quantity, I will
00:19:29.540 --> 00:19:33.320
have a correction, which
is minus x squared over 2.
00:19:33.320 --> 00:19:36.280
Let's just do it
for this first term.
00:19:36.280 --> 00:19:43.250
I will get minus N squared
over 2e to the minus 8dk,
00:19:43.250 --> 00:19:48.140
and there will be a whole
bunch of higher-order terms.
00:19:48.140 --> 00:19:50.850
OK?
00:19:50.850 --> 00:19:53.300
Now, where am I going with this?
00:19:53.300 --> 00:19:59.380
Ultimately, I want to calculate
various quantities that
00:19:59.380 --> 00:20:05.300
are extensive in the sense that
they are proportional to N,
00:20:05.300 --> 00:20:07.840
and when I divide by
N I will get something
00:20:07.840 --> 00:20:10.050
like energy per site.
00:20:10.050 --> 00:20:14.960
So if I do that, I have to
divide this whole thing by N,
00:20:14.960 --> 00:20:17.710
I can see that here
I have a term that
00:20:17.710 --> 00:20:22.980
is log T divided by N. In
the N goes to infinity limit,
00:20:22.980 --> 00:20:26.960
it's a term that has order
of 1 over N I can neglect.
00:20:26.960 --> 00:20:30.940
But all of these other
terms are proportional to N.
00:20:30.940 --> 00:20:37.630
And when I divide by N, I
can drop these factors of N.
00:20:37.630 --> 00:20:40.470
Well, except that I
have a couple of terms
00:20:40.470 --> 00:20:43.670
that, if I had
left by themselves,
00:20:43.670 --> 00:20:46.330
potentially could have
been order of N squared.
00:20:46.330 --> 00:20:49.460
I have N squared over
2, but fortunately, you
00:20:49.460 --> 00:20:52.870
can see that it
cancels out over there.
00:20:52.870 --> 00:20:56.690
Now, the reason this
happens, and also
00:20:56.690 --> 00:20:59.560
the reason this
series is legitimate,
00:20:59.560 --> 00:21:03.280
is because we already did
something very similar to that
00:21:03.280 --> 00:21:08.595
in 8.333 when we were doing
these cumulant expansions.
00:21:08.595 --> 00:21:12.000
And when we were doing
these cumulant expansions,
00:21:12.000 --> 00:21:16.660
we obtained the series for,
then, the grand partition
00:21:16.660 --> 00:21:19.900
function, which was a
whole bunch of terms.
00:21:19.900 --> 00:21:26.130
But when we took the log, only
the connected terms survived.
00:21:26.130 --> 00:21:28.280
And the connected
terms were the things
00:21:28.280 --> 00:21:30.190
that, because they
had a center of mass,
00:21:30.190 --> 00:21:34.130
were giving you a factor that
was proportional to volume.
00:21:34.130 --> 00:21:38.230
And here you expect that
ultimately everything
00:21:38.230 --> 00:21:41.250
that I will get here, if I
calculate, let's say, log Z
00:21:41.250 --> 00:21:44.650
properly and then
divide by N, it
00:21:44.650 --> 00:21:48.100
should be something
that is order of 1.
00:21:48.100 --> 00:21:52.240
It shouldn't be order
of N, or N cubed,
00:21:52.240 --> 00:21:54.360
or any of these other terms.
00:21:54.360 --> 00:21:59.030
So essentially, the purpose of
all of these higher-order terms
00:21:59.030 --> 00:22:04.550
is really to subtract
off things such as this
00:22:04.550 --> 00:22:07.710
that would arise in
the counting when
00:22:07.710 --> 00:22:11.080
we look at islands
and excitations that
00:22:11.080 --> 00:22:12.250
are disconnected.
00:22:12.250 --> 00:22:15.310
So I could have something right
here, something right here.
00:22:15.310 --> 00:22:18.930
So this would be,
essentially, a product
00:22:18.930 --> 00:22:23.400
of the contributions of
these different islands.
00:22:23.400 --> 00:22:29.630
As long as they are disconnected
from some term in the series,
00:22:29.630 --> 00:22:33.300
there would be a subtraction
that would get rid of that
00:22:33.300 --> 00:22:36.830
and would ensure that these
additional factors of N,
00:22:36.830 --> 00:22:41.350
because I can move each island
over the entire lattice,
00:22:41.350 --> 00:22:43.360
would disappear.
00:22:43.360 --> 00:22:51.960
So I have this series,
and now I can basically
00:22:51.960 --> 00:22:54.080
take the derivatives.
00:22:54.080 --> 00:22:58.230
So I have minus J,
and I take d by dk
00:22:58.230 --> 00:23:00.760
of the various terms
that have survived.
00:23:00.760 --> 00:23:02.950
The first one is d.
00:23:02.950 --> 00:23:08.030
So dJ is essentially
the energy pair site
00:23:08.030 --> 00:23:09.840
that I would have
at 0 temperature.
00:23:09.840 --> 00:23:12.960
I have strength J, deeper site.
00:23:12.960 --> 00:23:14.980
And then the
excitations will start
00:23:14.980 --> 00:23:17.310
to reduce that and correct that.
00:23:17.310 --> 00:23:25.690
And so from here, I would get
minus 4d e to the minus 4dk.
00:23:25.690 --> 00:23:33.815
From here, I would get
minus 4 2d minus 1 d
00:23:33.815 --> 00:23:39.380
e to the minus 4 2d minus 1 k.
00:23:39.380 --> 00:23:42.800
The N-squared terms disappeared.
00:23:42.800 --> 00:23:46.570
So I would have
2d plus 1 over 2.
00:23:46.570 --> 00:23:51.200
But then it gets multiplied by
8d when I take a derivative.
00:23:51.200 --> 00:24:02.830
So I will get plus 4d 2d
plus 1 e to the minus 8dk,
00:24:02.830 --> 00:24:07.160
and so forth in the series.
00:24:07.160 --> 00:24:08.880
OK?
00:24:08.880 --> 00:24:14.360
So these terms that
are subtraction,
00:24:14.360 --> 00:24:16.720
you can see that you
can really easily
00:24:16.720 --> 00:24:22.200
connect to these
primary excitations.
00:24:22.200 --> 00:24:25.720
If you like, this
term corresponds
00:24:25.720 --> 00:24:30.790
to taking two of these and
colliding them with each other.
00:24:30.790 --> 00:24:32.550
They cannot be on
top of each other.
00:24:32.550 --> 00:24:34.750
They cannot be
next to each other.
00:24:34.750 --> 00:24:37.250
And so there is a
subtraction because a number
00:24:37.250 --> 00:24:39.570
of configurations
are not allowed.
00:24:39.570 --> 00:24:42.710
So this is, in
some sense, a kind
00:24:42.710 --> 00:24:47.710
of expansion in
these excitations
00:24:47.710 --> 00:24:50.882
and the interactions
among these excitations.
00:24:50.882 --> 00:24:52.210
OK?
00:24:52.210 --> 00:24:54.920
Now presumably,
what is happening
00:24:54.920 --> 00:24:57.700
is that at very low
temperature, you
00:24:57.700 --> 00:25:02.415
are going to get these
individual simple excitations
00:25:02.415 --> 00:25:05.790
with a little bit of
interaction between them.
00:25:05.790 --> 00:25:09.870
As you increase the temperature,
the size of these islands
00:25:09.870 --> 00:25:11.280
will get bigger and bigger.
00:25:11.280 --> 00:25:13.850
They start to merge
into each other.
00:25:13.850 --> 00:25:15.800
Configurations
that you would see
00:25:15.800 --> 00:25:19.550
will be big islands in a sea.
00:25:19.550 --> 00:25:23.200
And presumably, the
size of these islands
00:25:23.200 --> 00:25:26.330
is some measure of
the correlation length
00:25:26.330 --> 00:25:29.980
that you have in this
low temperature state.
00:25:29.980 --> 00:25:32.920
Eventually, this
correlation length
00:25:32.920 --> 00:25:36.110
will hit the size of the system.
00:25:36.110 --> 00:25:40.160
And then the starting point,
that you had a sea of pluses
00:25:40.160 --> 00:25:43.140
and you're exciting around
it, it is no longer valid.
00:25:43.140 --> 00:25:48.170
If you like, that vacuum
state has become unstable,
00:25:48.170 --> 00:25:51.830
and this series, the way
that we are constructing it,
00:25:51.830 --> 00:25:55.970
ceases to go beyond that point.
00:25:55.970 --> 00:25:57.760
OK?
00:25:57.760 --> 00:26:01.670
So let's take another step.
00:26:01.670 --> 00:26:04.590
If I've calculated the
energy, I could also
00:26:04.590 --> 00:26:12.310
calculate the heat
capacity, which is d by dT.
00:26:12.310 --> 00:26:16.310
Actually, I expect the heat
capacity to be extensive also,
00:26:16.310 --> 00:26:19.040
so I'll divide by
N. So I will look
00:26:19.040 --> 00:26:23.210
at the heat capacity per site.
00:26:23.210 --> 00:26:26.170
I know that the natural
units of heat capacity
00:26:26.170 --> 00:26:30.830
are kB, which has dimensions of
energy divided by temperature.
00:26:30.830 --> 00:26:33.580
So I divide by kB.
00:26:33.580 --> 00:26:37.546
So here I will have kBT.
00:26:37.546 --> 00:26:47.010
But then I notice that kBT,
these are related inversely
00:26:47.010 --> 00:26:55.000
to K, capital K. It is J over
K. So I can write this as J
00:26:55.000 --> 00:27:06.340
over K, and d by d-- 1 over K
will give me a minus K squared.
00:27:09.010 --> 00:27:14.530
I will have a factor of 1
over J, and the 1 over J
00:27:14.530 --> 00:27:18.330
actually cancels this
factor of J here.
00:27:18.330 --> 00:27:20.670
So all I need to do-- well,
actually, let me write
00:27:20.670 --> 00:27:27.520
it, J d by dK of this E over N.
00:27:27.520 --> 00:27:29.820
So the expression
that I have above, I
00:27:29.820 --> 00:27:32.570
have to take another
derivative with respect
00:27:32.570 --> 00:27:36.567
to K multiplied by
minus K squared over J.
00:27:36.567 --> 00:27:38.990
The J's cancel
out, and so I will
00:27:38.990 --> 00:27:42.630
have a series that will be
proportional to K squared.
00:27:42.630 --> 00:27:45.800
Good, I made everything
dimensionless.
00:27:45.800 --> 00:27:49.200
And then the first term
that will contribute
00:27:49.200 --> 00:27:58.800
will be 16 d squared e to
the minus 4dK, from here.
00:27:58.800 --> 00:28:04.640
And from here, I will
get 16 2d minus 1
00:28:04.640 --> 00:28:11.680
squared d e to the
minus 4 2d minus 1 K.
00:28:11.680 --> 00:28:19.416
And from here, I would
get minus 32d 2d plus 1 e
00:28:19.416 --> 00:28:24.258
to the minus 8dK,
and then so forth.
00:28:24.258 --> 00:28:26.520
OK?
00:28:26.520 --> 00:28:30.910
So you can see that
this is something
00:28:30.910 --> 00:28:34.860
that is a kind of
mechanical process,
00:28:34.860 --> 00:28:38.540
that in the '40s and
'50s, without even
00:28:38.540 --> 00:28:43.420
the need for any computers,
people could sit down and draw
00:28:43.420 --> 00:28:46.390
excitations,
provide these terms,
00:28:46.390 --> 00:28:51.160
and go to higher and higher
order terms in the series.
00:28:51.160 --> 00:28:54.190
Now, the reason that they
were going to do this
00:28:54.190 --> 00:28:57.660
is that the
expectation that if I
00:28:57.660 --> 00:29:01.810
look at this heat capacity
as a function of something
00:29:01.810 --> 00:29:09.600
like temperature, which is e
to the 1 over k, for example,
00:29:09.600 --> 00:29:12.780
then it starts at 0.
00:29:16.250 --> 00:29:21.920
And if we get corrections from
these higher and higher order
00:29:21.920 --> 00:29:24.490
terms in the
series-- I calculated
00:29:24.490 --> 00:29:27.770
the first few--
I don't know what
00:29:27.770 --> 00:29:31.750
will happen if I were to include
higher and higher order terms.
00:29:31.750 --> 00:29:37.210
But my expectation is that,
say, at least at some point
00:29:37.210 --> 00:29:41.270
when this expansion from
low temperature breaks down,
00:29:41.270 --> 00:29:45.460
I will have a divergence, let's
say, of the heat capacity.
00:29:45.460 --> 00:29:47.710
Or maybe I calculated
susceptibility
00:29:47.710 --> 00:29:50.160
or some other
quantity, and I expect
00:29:50.160 --> 00:29:53.670
to have some singularity.
00:29:53.670 --> 00:30:00.940
And maybe by looking and fitting
more terms in the series,
00:30:00.940 --> 00:30:04.990
one can guess what the
exponent and the location
00:30:04.990 --> 00:30:07.890
of the singularity is.
00:30:07.890 --> 00:30:10.370
So you can see that,
actually in this case,
00:30:10.370 --> 00:30:16.530
the natural variable that
I am expanding is not K,
00:30:16.530 --> 00:30:22.050
but e to the minus 2dK--
sorry, e to the minus 2K
00:30:22.050 --> 00:30:26.850
because each excitation will
have a number of broken bonds
00:30:26.850 --> 00:30:28.220
that I have to calculate.
00:30:28.220 --> 00:30:32.590
Each one of them makes a
contribution like this.
00:30:32.590 --> 00:30:36.600
So maybe we can call
this our new variable.
00:30:36.600 --> 00:30:40.830
And we have a series that
has a function of this
00:30:40.830 --> 00:30:44.724
or some other variable,
has a singularity.
00:30:44.724 --> 00:30:46.640
Actually, you should be
able to, first of all,
00:30:46.640 --> 00:30:51.410
convince yourself that the
nature of the singularity
00:30:51.410 --> 00:30:57.450
is not modified by any
mapping that is analytical
00:30:57.450 --> 00:31:00.380
at the point of the singularity.
00:31:00.380 --> 00:31:03.820
So if the heat capacity
as a function of k
00:31:03.820 --> 00:31:07.740
has a particular divergence,
as a function of u
00:31:07.740 --> 00:31:10.435
it will have exactly
the same divergence.
00:31:10.435 --> 00:31:20.140
In particular, we expect
that as u approaches
00:31:20.140 --> 00:31:23.640
some critical value,
the kinds of functions
00:31:23.640 --> 00:31:28.570
that we are interested have a
behavior, a singular behavior,
00:31:28.570 --> 00:31:35.660
that is something like
1 minus u over uC.
00:31:35.660 --> 00:31:37.520
Let's say for the
heat capacity, I
00:31:37.520 --> 00:31:42.820
would expect some kind of
a singularity such as this.
00:31:42.820 --> 00:31:46.552
If I had a pure
function such as this
00:31:46.552 --> 00:31:52.650
and I constructed an
expansion in u, what do I get?
00:31:52.650 --> 00:31:58.240
I will get 1 plus
alpha u over uC
00:31:58.240 --> 00:32:05.860
plus alpha alpha plus 1
over 2 uC squared u squared,
00:32:05.860 --> 00:32:08.110
and so forth.
00:32:08.110 --> 00:32:12.785
It's just a binary
series expanded.
00:32:12.785 --> 00:32:18.970
The l term in the series would
be alpha alpha plus 1 alpha
00:32:18.970 --> 00:32:26.700
plus l minus 1 divided by l
factorial-- that's actually
00:32:26.700 --> 00:32:35.370
2 factorial-- uC to the power
of l u to the l and so forth.
00:32:35.370 --> 00:32:35.870
OK?
00:32:38.730 --> 00:32:43.310
Now, typically, one of the
ways that you look at series
00:32:43.310 --> 00:32:47.340
and decide whether it's a
singular convergent series
00:32:47.340 --> 00:32:50.300
or what the behavior
is is to look
00:32:50.300 --> 00:32:56.540
at the ratio of
subsequent terms.
00:32:56.540 --> 00:33:02.010
So let's say that when
I calculated my function
00:33:02.010 --> 00:33:07.240
C as a function of
u, I constructed
00:33:07.240 --> 00:33:13.540
a series whose terms
had coefficients
00:33:13.540 --> 00:33:15.862
that I will call al.
00:33:15.862 --> 00:33:17.640
OK?
00:33:17.640 --> 00:33:25.050
So here, if you had
exactly this series,
00:33:25.050 --> 00:33:34.740
you would say that the ratio
al divided by al minus 1
00:33:34.740 --> 00:33:39.170
is essentially the ratio of
one of these factors compared
00:33:39.170 --> 00:33:41.190
to the previous one.
00:33:41.190 --> 00:33:43.650
And every time you add
one of these factors,
00:33:43.650 --> 00:33:48.930
you add a term that is like
this alpha plus l minus 1,
00:33:48.930 --> 00:33:53.970
l factorial compared to l minus
1 factorial has a factor of l,
00:33:53.970 --> 00:33:57.090
and then you have uC.
00:33:57.090 --> 00:34:05.820
And I can rewrite this as uC
inverse, l divided by l is 1,
00:34:05.820 --> 00:34:12.559
and then I have minus 1
minus alpha divided by l.
00:34:12.559 --> 00:34:13.059
OK?
00:34:16.420 --> 00:34:22.310
So a pure divergence of the
form that I have over here
00:34:22.310 --> 00:34:25.989
would predict that the
ratio of subsequent terms
00:34:25.989 --> 00:34:28.630
would be something like this.
00:34:28.630 --> 00:34:31.699
And presumably, if
you go sufficiently
00:34:31.699 --> 00:34:36.210
high in the series, in order
to reproduce this divergence
00:34:36.210 --> 00:34:38.335
you must have that form.
00:34:38.335 --> 00:34:44.310
So what you could do
as a test is to plot,
00:34:44.310 --> 00:34:47.550
for your actual series, what
the ratio of these terms
00:34:47.550 --> 00:34:54.179
is as a function of 1 over l.
00:34:54.179 --> 00:34:59.560
So you can start with the ratio
of the second to first term.
00:34:59.560 --> 00:35:02.640
You would be at 1/2.
00:35:02.640 --> 00:35:07.540
Then you would go 1/3,
then you would go 1/4,
00:35:07.540 --> 00:35:10.590
you would have
1/5, and basically
00:35:10.590 --> 00:35:13.290
you would have a set of points.
00:35:13.290 --> 00:35:16.350
And you would plot
what the location
00:35:16.350 --> 00:35:20.580
is for the first term in
the series, the next term
00:35:20.580 --> 00:35:26.410
in the series, the next term
in the series, and so forth.
00:35:26.410 --> 00:35:28.990
And if you are
lucky, you would be
00:35:28.990 --> 00:35:32.980
able to then pass
a straight line
00:35:32.980 --> 00:35:36.950
at large distances
in the series.
00:35:36.950 --> 00:35:45.740
And the intercept of
that extrapolated line
00:35:45.740 --> 00:35:50.660
would be your inverse
of the singular point.
00:35:50.660 --> 00:35:55.000
And the slope of
this line would give
00:35:55.000 --> 00:36:00.090
you 1 minus alpha or
minus 1 minus alpha.
00:36:00.090 --> 00:36:02.220
OK?
00:36:02.220 --> 00:36:10.120
So there is really, a
priori, not much reason
00:36:10.120 --> 00:36:13.280
to hope that that will
happen because you
00:36:13.280 --> 00:36:18.560
can say that if I look at the
series that is A 1 minus u
00:36:18.560 --> 00:36:26.320
over uC to the minus alpha, plus
I add an analytic part, which
00:36:26.320 --> 00:36:36.120
is sum p equals 1 to,
say, 52 of bl u to the l.
00:36:36.120 --> 00:36:43.990
For any bl in this function has
exactly the same singularity
00:36:43.990 --> 00:36:46.070
as the original one.
00:36:46.070 --> 00:36:50.310
And yet the first 52
terms in the series,
00:36:50.310 --> 00:36:53.110
because of this additional
analytical form,
00:36:53.110 --> 00:36:56.750
have nothing to do with
the eventual singularity.
00:36:56.750 --> 00:36:59.340
They're going to
be massing that.
00:36:59.340 --> 00:37:05.740
So there is no reason for you
to expect that this should work.
00:37:05.740 --> 00:37:09.670
But when people do this, and
they find that, let's say,
00:37:09.670 --> 00:37:18.270
for d equals to 2 up to
some jumping up and down,
00:37:18.270 --> 00:37:22.010
they get a reasonable
straight line.
00:37:22.010 --> 00:37:26.980
And the exponent that they get
would correspond very closely
00:37:26.980 --> 00:37:31.060
to the alpha of 0, which is the
logarithmic divergence that one
00:37:31.060 --> 00:37:31.970
gets.
00:37:31.970 --> 00:37:36.520
So this is, for d equals to
2, and then they repeat it,
00:37:36.520 --> 00:37:39.310
let's say, for d
equals to 3, they
00:37:39.310 --> 00:37:44.520
get a different set of points.
00:37:44.520 --> 00:37:46.280
OK?
00:37:46.280 --> 00:37:49.050
Maybe not perfectly
on a straight line,
00:37:49.050 --> 00:37:55.620
but you can still extrapolate
and conclude from that
00:37:55.620 --> 00:38:00.770
that you'll have an alpha
which is roughly 0.11 when
00:38:00.770 --> 00:38:05.160
d equals to 3,
which is quite good.
00:38:05.160 --> 00:38:11.470
So for some reason or
other, these lattice models
00:38:11.470 --> 00:38:16.990
are kind of sufficiently
simple that,
00:38:16.990 --> 00:38:20.060
in an appropriate
expansion, they
00:38:20.060 --> 00:38:24.800
don't seem to give you
that much of a problem.
00:38:24.800 --> 00:38:28.680
And so people have gone
and calculated series,
00:38:28.680 --> 00:38:31.500
let's say, this was
in '50s and '60s,
00:38:31.500 --> 00:38:33.110
just by drawing things on hand.
00:38:33.110 --> 00:38:36.440
And maybe some
primitive computers,
00:38:36.440 --> 00:38:40.930
you can go to order of
20 terms in this series,
00:38:40.930 --> 00:38:43.980
and then extrapolate exponents
for various quantities.
00:38:46.702 --> 00:38:47.202
OK?
00:38:51.060 --> 00:38:54.340
But it's not as simple as that.
00:38:54.340 --> 00:39:00.640
And the reason I calculated
the first three terms for you
00:39:00.640 --> 00:39:06.340
was to show you that what I
told you here was clearly a lie.
00:39:06.340 --> 00:39:08.330
Why is that?
00:39:08.330 --> 00:39:12.530
Because of the three terms that
I explicitly calculated for you
00:39:12.530 --> 00:39:17.204
in that series, the
third one is negative.
00:39:17.204 --> 00:39:18.500
Right?
00:39:18.500 --> 00:39:22.535
So clearly, if I
were to plot that,
00:39:22.535 --> 00:39:25.745
I will get something over here.
00:39:25.745 --> 00:39:27.490
Right?
00:39:27.490 --> 00:39:32.650
So what's going gone there
is a different issue.
00:39:32.650 --> 00:39:37.510
And people have developed
kind of methodologies and ways
00:39:37.510 --> 00:39:40.400
to look at series and
guess what is going on
00:39:40.400 --> 00:39:45.560
and yet continue to
extract exponents.
00:39:45.560 --> 00:39:58.460
So one potential origin
for alternating signs--
00:39:58.460 --> 00:40:03.050
and any series that has a
divergence such as the one
00:40:03.050 --> 00:40:07.790
that I have indicated for you
will have, eventually, signs
00:40:07.790 --> 00:40:13.640
that need to be positive--
has to do with the following.
00:40:13.640 --> 00:40:19.490
Let's say if I take a series,
which is 1 over 1 minus z/2.
00:40:19.490 --> 00:40:20.610
OK?
00:40:20.610 --> 00:40:22.180
This is a very nice series.
00:40:22.180 --> 00:40:28.210
It's 1 plus 0/2 z
squared/4, z cubed/8.
00:40:28.210 --> 00:40:32.520
You could apply this
ratio test to this series
00:40:32.520 --> 00:40:36.815
and conclude that you
have a linear divergence.
00:40:36.815 --> 00:40:41.200
Now, suppose I multiply
that by 1 over 1
00:40:41.200 --> 00:40:45.630
plus z squared, which
is a function that's
00:40:45.630 --> 00:40:49.360
perfectly well-behaved
as a function of z.
00:40:49.360 --> 00:40:53.660
Yet if I multiply it here, I
will get 1 minus z squared plus
00:40:53.660 --> 00:40:58.100
z to the fourth
minus z to the sixth.
00:40:58.100 --> 00:41:05.020
And what it does is
it kind of distorts
00:41:05.020 --> 00:41:07.520
what is happening over here.
00:41:07.520 --> 00:41:09.870
Actually, in this
series you can see
00:41:09.870 --> 00:41:15.285
it kind of becomes ill-defined
when z is of order of 1.
00:41:15.285 --> 00:41:19.260
It changes the signs, et cetera.
00:41:19.260 --> 00:41:25.580
But the function itself has a
perfectly good singularity that
00:41:25.580 --> 00:41:27.910
appears at z equals to 2.
00:41:27.910 --> 00:41:31.050
And starting from
an expansion from 0,
00:41:31.050 --> 00:41:35.110
there should be no
problems along the line
00:41:35.110 --> 00:41:37.910
until you hit z of 2.
00:41:37.910 --> 00:41:41.530
What is the reason for
these alternating signs?
00:41:41.530 --> 00:41:46.460
It is because you should be
looking at the complex z plane.
00:41:46.460 --> 00:41:54.300
And in the complex z plane, you
have poles at plus and minus i
00:41:54.300 --> 00:41:59.286
which are located closer to
the origin than you have at 2.
00:41:59.286 --> 00:42:02.440
So basically, your
series will start
00:42:02.440 --> 00:42:07.100
to have problems by
the time you hit here,
00:42:07.100 --> 00:42:10.710
and that problem is reflected
in the alternating behavior.
00:42:10.710 --> 00:42:13.116
It's also showing up over there.
00:42:13.116 --> 00:42:17.520
Yet it has nothing to do with
going along the real axis
00:42:17.520 --> 00:42:20.986
and encountering the
singularity that you are after.
00:42:20.986 --> 00:42:23.130
OK?
00:42:23.130 --> 00:42:27.330
So one thing that you
can do is to say, well,
00:42:27.330 --> 00:42:31.740
who said I should
use z as my variable?
00:42:31.740 --> 00:42:36.930
Maybe I can choose some
other function v of z.
00:42:36.930 --> 00:42:38.080
OK?
00:42:38.080 --> 00:42:42.730
And then when I choose
the appropriate thing,
00:42:42.730 --> 00:42:48.300
the singularity on the real
axis will be pushed to v of 2.
00:42:48.300 --> 00:42:51.900
But maybe I chose appropriate
function of v of z
00:42:51.900 --> 00:42:53.730
such that the
other singularities
00:42:53.730 --> 00:42:59.030
are pushed very far away so that
the first singularity that I
00:42:59.030 --> 00:43:01.076
encounter is over here.
00:43:01.076 --> 00:43:03.220
OK?
00:43:03.220 --> 00:43:06.010
And it turns out that
if you take this series
00:43:06.010 --> 00:43:11.740
over here and rather than
working with e to the minus k,
00:43:11.740 --> 00:43:16.630
we recast things in terms
of tanh K-- let's call
00:43:16.630 --> 00:43:20.770
that v-- which is e to
the K plus e to the minus
00:43:20.770 --> 00:43:24.120
K-- well actually, tanh
K I can also write as e
00:43:24.120 --> 00:43:28.435
to the 2K minus 1
e to the 2K plus 1.
00:43:28.435 --> 00:43:30.830
I mean, it's just
a transformation.
00:43:30.830 --> 00:43:35.220
So I can replace
e to the minus 2K
00:43:35.220 --> 00:43:40.400
with some function v,
substitute for u in that series,
00:43:40.400 --> 00:43:42.970
and I will have a
different function
00:43:42.970 --> 00:43:46.250
as an expansion in powers of v.
00:43:46.250 --> 00:43:50.340
And once people do that,
same thing happens as here.
00:43:50.340 --> 00:43:53.860
You'll find a function that
all of its terms are, in fact,
00:43:53.860 --> 00:43:58.890
positive, and the things that
I mentioned to you over here
00:43:58.890 --> 00:43:59.800
were applied.
00:43:59.800 --> 00:44:04.920
After such transformation,
you get very nice behaviors.
00:44:04.920 --> 00:44:06.270
OK?
00:44:06.270 --> 00:44:08.590
So there seems to
be some guesswork
00:44:08.590 --> 00:44:11.600
into finding the
appropriate transformation.
00:44:11.600 --> 00:44:14.240
There are other methods
for dealing with series
00:44:14.240 --> 00:44:17.800
and extracting
singularities called
00:44:17.800 --> 00:44:23.680
Pade approximants, et cetera,
which I won't go into.
00:44:23.680 --> 00:44:27.680
But there are kind of, again,
clever mathematical tricks
00:44:27.680 --> 00:44:33.845
for extracting singularity
out of series such as this.
00:44:33.845 --> 00:44:34.345
OK?
00:44:38.770 --> 00:44:45.060
So I'll tell you shortly
why this tanh K is really
00:44:45.060 --> 00:44:48.240
a good expansion factor.
00:44:48.240 --> 00:44:51.590
It turns out that
for Ising models,
00:44:51.590 --> 00:44:55.980
it's actually the
right expansion factor
00:44:55.980 --> 00:44:58.670
if we go to the other
limit of high temperatures.
00:45:03.576 --> 00:45:05.060
OK?
00:45:05.060 --> 00:45:12.220
So basically, now at
T going to infinity,
00:45:12.220 --> 00:45:21.340
you would say that sigma
i is minus or plus 1
00:45:21.340 --> 00:45:22.330
with equal probability.
00:45:27.830 --> 00:45:32.290
As T goes to
infinity, this factor
00:45:32.290 --> 00:45:38.350
that encodes the tendency of
spins to be next to each other
00:45:38.350 --> 00:45:42.280
has been scaled to 0,
so I know exactly what
00:45:42.280 --> 00:45:43.940
is going on at
infinite temperature.
00:45:43.940 --> 00:45:47.550
Basically, at each site, I
have an independent variable
00:45:47.550 --> 00:45:50.440
that is decoupled
from everything else.
00:45:50.440 --> 00:45:56.120
So I can start expanding around
that for, say, the partition
00:45:56.120 --> 00:45:57.970
function.
00:45:57.970 --> 00:46:01.500
Let's think of it for
a general spin system.
00:46:01.500 --> 00:46:04.210
So I will write it
as a trace over,
00:46:04.210 --> 00:46:08.000
let's say, if I have Potts
model rather than two values,
00:46:08.000 --> 00:46:12.220
I would have K values
of something like e
00:46:12.220 --> 00:46:15.000
to the minus beta
H, again, trying
00:46:15.000 --> 00:46:17.630
to be reasonably general.
00:46:17.630 --> 00:46:21.840
And the idea is that as you
go to infinite temperature,
00:46:21.840 --> 00:46:25.260
beta goes to 0,
and this function
00:46:25.260 --> 00:46:31.075
you can expand in a series 1
minus beta H plus beta squared
00:46:31.075 --> 00:46:33.897
H squared over 2, and so forth.
00:46:38.870 --> 00:46:41.820
Now, the trace of
1 is essentially
00:46:41.820 --> 00:46:45.900
summing over all
possible states.
00:46:45.900 --> 00:46:47.440
Let's say the two
states that you
00:46:47.440 --> 00:46:50.010
would have for the Ising
model or however many
00:46:50.010 --> 00:46:55.160
that you have for Potts models
at each site independently.
00:46:55.160 --> 00:46:58.940
So that can give me
some partition function
00:46:58.940 --> 00:47:00.340
that I will call Z0.
00:47:00.340 --> 00:47:05.560
It is simply 2 to the
n for the Ising model.
00:47:05.560 --> 00:47:09.130
But once I factor
that, you can see
00:47:09.130 --> 00:47:12.430
that the rest of the
terms in the series
00:47:12.430 --> 00:47:20.810
can be regarded as expectation
values of this Hamiltonian
00:47:20.810 --> 00:47:23.270
with respect to this
weight in which all
00:47:23.270 --> 00:47:25.920
of the degrees of
freedom are treated
00:47:25.920 --> 00:47:28.530
as independent,
unconstrained variables.
00:47:34.660 --> 00:47:37.090
And of course, the thing
that I'm interested
00:47:37.090 --> 00:47:41.080
is log of the
partition function.
00:47:41.080 --> 00:47:46.350
And so that will
give me log of Z0,
00:47:46.350 --> 00:47:49.850
and then I have the
log of this series.
00:47:49.850 --> 00:47:56.380
And then you can see that that
series is a generating function
00:47:56.380 --> 00:47:59.590
for the moments of
the Hamiltonian.
00:47:59.590 --> 00:48:03.350
So its log will be the
generating function
00:48:03.350 --> 00:48:14.080
for the cumulant, so H
to the l 0, the cumulant.
00:48:14.080 --> 00:48:16.480
So the variance at
the second order
00:48:16.480 --> 00:48:18.334
and appropriate cumulant
at higher orders.
00:48:44.220 --> 00:48:46.390
OK?
00:48:46.390 --> 00:48:51.500
So let's try to calculate this
for the Ising model, where
00:48:51.500 --> 00:49:01.172
my minus beta H is K sum
over i, j sigma i sigma j.
00:49:01.172 --> 00:49:03.420
OK?
00:49:03.420 --> 00:49:06.070
Then at the lowest
order, what do I get?
00:49:06.070 --> 00:49:14.612
The average of beta
H is K sum over i,
00:49:14.612 --> 00:49:22.110
j average of sigma i sigma
j with this zeroed weight.
00:49:22.110 --> 00:49:24.340
But as I emphasized,
at zeroed weight,
00:49:24.340 --> 00:49:27.960
every site independently
can be plus or minus.
00:49:27.960 --> 00:49:33.430
Because of the
independence, I can do this.
00:49:33.430 --> 00:49:37.390
And then since each site
has equal probability to be
00:49:37.390 --> 00:49:40.170
plus or minus, its average is 0.
00:49:40.170 --> 00:49:41.970
So basically, this will be 0.
00:49:46.263 --> 00:49:48.180
OK?
00:49:48.180 --> 00:49:51.670
So the first thing that
can happen in that series--
00:49:51.670 --> 00:49:54.560
if I go to the next order.
00:49:54.560 --> 00:50:00.000
So at next order, beta H
squared would involve K
00:50:00.000 --> 00:50:10.130
squared sum over i, j K, l
sigma i sigma j sigma K sigma l.
00:50:10.130 --> 00:50:13.850
And I have to take
an average of this,
00:50:13.850 --> 00:50:16.690
which means that I have to
take an average of something
00:50:16.690 --> 00:50:19.080
like this.
00:50:19.080 --> 00:50:19.580
OK.
00:50:19.580 --> 00:50:23.630
And you would say, well,
again, everything is 0.
00:50:23.630 --> 00:50:26.630
Well, there is one case
where it won't be 0--
00:50:26.630 --> 00:50:29.340
if these two pairs
are identical.
00:50:29.340 --> 00:50:30.730
Right?
00:50:30.730 --> 00:50:41.540
So this is going to give me
K squared sum over pair i,
00:50:41.540 --> 00:50:44.960
j being the same as K, l.
00:50:44.960 --> 00:50:46.830
Then I will get,
essentially, sigma i
00:50:46.830 --> 00:50:48.780
squared sigma j squared.
00:50:48.780 --> 00:50:50.690
Sigma i squared is 1.
00:50:50.690 --> 00:50:52.420
Sigma j squared is 1.
00:50:52.420 --> 00:50:54.510
So basically, I will get 1.
00:50:54.510 --> 00:50:56.600
And this is going
to give me K squared
00:50:56.600 --> 00:50:59.233
times the number of bonds.
00:50:59.233 --> 00:51:01.000
OK?
00:51:01.000 --> 00:51:02.890
So you can see that
I can start thinking
00:51:02.890 --> 00:51:05.350
of this already graphically.
00:51:05.350 --> 00:51:12.610
Because what I did over here,
I said that on my lattice
00:51:12.610 --> 00:51:19.690
this sum says you pick
one sigma i sigma j.
00:51:19.690 --> 00:51:23.910
If I were to pick the other
sigma i sigma j over here,
00:51:23.910 --> 00:51:25.680
the average would be 0.
00:51:25.680 --> 00:51:30.780
I am forced to put two of
them on top of each other.
00:51:30.780 --> 00:51:34.350
If I go to three,
there is no way
00:51:34.350 --> 00:51:38.580
that I can draw a
diagram that involves
00:51:38.580 --> 00:51:46.730
three pairs in which every
single site occurs twice,
00:51:46.730 --> 00:51:48.120
which is what I need.
00:51:48.120 --> 00:51:53.120
Because a single site appearing
by itself or three times
00:51:53.120 --> 00:51:56.430
will give me sigma i cubed
is the same as sigma i.
00:51:56.430 --> 00:52:00.080
It will average to 0.
00:52:00.080 --> 00:52:05.380
So the next thing that I can
do is to go to level four.
00:52:05.380 --> 00:52:08.260
At the level of
four, I can certainly
00:52:08.260 --> 00:52:09.870
do something like this.
00:52:09.870 --> 00:52:12.830
I can put all four of
them on top of each other,
00:52:12.830 --> 00:52:16.740
and then I get a K to
the fourth contribution.
00:52:16.740 --> 00:52:23.060
Or I could put a pair
here, and if they're here,
00:52:23.060 --> 00:52:27.410
for log Z that would be
unacceptable because that will
00:52:27.410 --> 00:52:30.840
get subtracted out when
I calculate the variance.
00:52:33.400 --> 00:52:34.720
It's not a connected piece.
00:52:34.720 --> 00:52:36.490
It's a disconnected piece.
00:52:36.490 --> 00:52:38.660
But I could have
something like this,
00:52:38.660 --> 00:52:40.610
two of them turned like this.
00:52:40.610 --> 00:52:42.390
So that's four.
00:52:42.390 --> 00:52:45.960
But really, the one that is
nontrivial and interesting
00:52:45.960 --> 00:52:50.050
is when I do something
like this, like a square.
00:52:50.050 --> 00:52:54.270
So I go here sigma 1
sigma 2, sigma 2 sigma 3.
00:52:54.270 --> 00:52:57.070
That sigma 2 has been
repeated twice and becomes
00:52:57.070 --> 00:52:59.560
sigma 2 squared and goes away.
00:52:59.560 --> 00:53:02.770
Sigma 3 sigma 4, sigma
3 repeated twice,
00:53:02.770 --> 00:53:07.246
sigma 4 repeated twice, sigma
1 repeated twice, [INAUDIBLE].
00:53:07.246 --> 00:53:08.070
OK?
00:53:08.070 --> 00:53:11.622
So you can see that
this kind of expansion
00:53:11.622 --> 00:53:16.270
will naturally lead you
into an expansion in terms
00:53:16.270 --> 00:53:19.220
of loops on a lattice.
00:53:19.220 --> 00:53:23.370
So the natural form of
high temperature expansions
00:53:23.370 --> 00:53:26.257
are these closed
strings or loops,
00:53:26.257 --> 00:53:28.340
if you like, that you have
to draw on the lattice.
00:53:31.480 --> 00:53:37.360
Now, it's also clear
that the thing that
00:53:37.360 --> 00:53:45.720
goes between two sites, that I'm
indicating by K, in all cases
00:53:45.720 --> 00:53:48.140
is likely to be
repeated by putting
00:53:48.140 --> 00:53:51.030
more and more things
on top of each other
00:53:51.030 --> 00:53:52.645
without modifying the effect.
00:53:52.645 --> 00:53:57.750
So I can go here to 4 and things
like actually 3 and things
00:53:57.750 --> 00:53:58.760
like that.
00:53:58.760 --> 00:54:01.540
So basically, you can
see that I should really
00:54:01.540 --> 00:54:08.040
do a summation over the
contribution of 2, 4, et
00:54:08.040 --> 00:54:11.620
cetera all on top of
each other, or 1, 3,
00:54:11.620 --> 00:54:16.440
5 on top of each other, and
call them new variables.
00:54:16.440 --> 00:54:23.280
So when we were doing the
cluster expansion for particles
00:54:23.280 --> 00:54:25.870
interacting, we
encountered this thing
00:54:25.870 --> 00:54:29.870
that we thought v was a
good variable to expand it.
00:54:29.870 --> 00:54:32.010
But then because
of these repeats,
00:54:32.010 --> 00:54:35.860
we decided that e to
the minus beta v minus 1
00:54:35.860 --> 00:54:39.190
was a good variable
to expand it.
00:54:39.190 --> 00:54:41.280
So a similar thing happens here.
00:54:41.280 --> 00:54:45.740
And for the Ising model,
it is a very natural thing
00:54:45.740 --> 00:54:51.120
to recast this series in
a slightly different way.
00:54:51.120 --> 00:54:58.120
You see that the contribution
of each bond to the partition
00:54:58.120 --> 00:55:02.350
function, and by a bond I mean
a pair of neighboring sites,
00:55:02.350 --> 00:55:07.030
is a factor e to the
K sigma i sigma j.
00:55:07.030 --> 00:55:07.530
OK?
00:55:10.130 --> 00:55:13.560
Now, since we are dealing
with binary variables,
00:55:13.560 --> 00:55:17.110
this product, sigma i sigma
j, can only take two values.
00:55:17.110 --> 00:55:20.930
It's either plus K
or minus K depending
00:55:20.930 --> 00:55:24.160
on where things are
aligned or misaligned.
00:55:24.160 --> 00:55:26.700
So I can indicate
the binary nature
00:55:26.700 --> 00:55:28.140
of this in the
following fashion.
00:55:28.140 --> 00:55:33.690
I can write this as e to the
K plus e to the minus K over 2
00:55:33.690 --> 00:55:41.760
plus sigma i sigma j e to the K
minus e to the minus K over 2.
00:55:41.760 --> 00:55:47.460
So that when I'm dealing
with sigma sigma being plus,
00:55:47.460 --> 00:55:49.100
I add those two factors.
00:55:49.100 --> 00:55:50.730
e to the minus K's disappear.
00:55:50.730 --> 00:55:54.000
I will get e to the
K. When I'm dealing
00:55:54.000 --> 00:55:57.190
with this thing to the minus,
the e to the K's disappear,
00:55:57.190 --> 00:56:00.470
and I will get e to
the minus K. So it's
00:56:00.470 --> 00:56:03.000
correct rewriting
of that factor.
00:56:03.000 --> 00:56:04.885
The first term you,
of course, recognize
00:56:04.885 --> 00:56:08.640
as the hyperbolic cosine
of K, the second one
00:56:08.640 --> 00:56:12.740
as the hyperbolic
sine of K. And so I
00:56:12.740 --> 00:56:16.930
can write the whole thing
as hyperbolic cosine
00:56:16.930 --> 00:56:24.345
1 plus hyperbolic tanh
of K sigma i sigma j.
00:56:24.345 --> 00:56:24.845
OK?
00:56:29.740 --> 00:56:37.270
So this tanh is really
same thing as here.
00:56:37.270 --> 00:56:40.390
It's the high-temperature
expansion variable.
00:56:40.390 --> 00:56:43.620
As K goes to 0 at
high temperature,
00:56:43.620 --> 00:56:46.150
tanh K also goes to 0.
00:56:46.150 --> 00:56:50.620
And it turns out that a much
nicer variable to expand
00:56:50.620 --> 00:56:55.830
is this quantity tanh K. And so
that I don't have to repeat it
00:56:55.830 --> 00:57:00.140
throughout, I will
give it the symbol t.
00:57:00.140 --> 00:57:04.570
So small t stands not for
reduced temperature anymore,
00:57:04.570 --> 00:57:07.980
but for hyperbolic tanh of K.
00:57:07.980 --> 00:57:11.115
So my partition function
now, Z-- maybe I'll
00:57:11.115 --> 00:57:13.760
go to another page.
00:57:49.920 --> 00:58:00.580
So my partition function is a
sum over the 2 to the N binary
00:58:00.580 --> 00:58:10.630
variables e to the K sigma i
sigma j sum over all bonds.
00:58:10.630 --> 00:58:16.150
I can write that as a product of
these exponential factors over
00:58:16.150 --> 00:58:17.910
[INAUDIBLE].
00:58:17.910 --> 00:58:28.910
Each of these exponential
factors I can write as cosh K 1
00:58:28.910 --> 00:58:31.150
plus t sigma i sigma j.
00:58:35.270 --> 00:58:39.570
All the factors of cosh K
I will take to the outside.
00:58:39.570 --> 00:58:43.610
So I will get cosh K
raised to the power
00:58:43.610 --> 00:58:47.030
of the number of bonds
that I have in my lattice
00:58:47.030 --> 00:58:49.330
because each bond
will contribute
00:58:49.330 --> 00:58:52.440
one of these factors.
00:58:52.440 --> 00:59:02.045
And then I have this sum over
sigma i product over bonds.
00:59:07.980 --> 00:59:12.980
So this is the product
of 1 plus t factors.
00:59:12.980 --> 00:59:18.485
So for each-- maybe
I'll do it over here.
00:59:28.170 --> 00:59:33.810
So for each i, j, I have to
pick one of these factors.
00:59:33.810 --> 00:59:37.780
I can either pick
1, nothing, or I
00:59:37.780 --> 00:59:43.715
can pick a factor of
t sigma i sigma j.
00:59:43.715 --> 00:59:44.215
OK?
00:59:47.430 --> 00:59:49.390
So the first term
in this series--
00:59:49.390 --> 00:59:54.160
since it's a series in powers of
t, the first term in the series
00:59:54.160 --> 00:59:56.800
is to pick 1 everywhere.
00:59:56.800 --> 01:00:01.230
The next term is to pick
one factor at some point.
01:00:01.230 --> 01:00:04.320
But then when I
pick that factor,
01:00:04.320 --> 01:00:08.460
that term in the series, I
have to sum over sigma i.
01:00:08.460 --> 01:00:11.870
And when I sum over sigma
i, since this sigma i can
01:00:11.870 --> 01:00:16.414
be plus or minus with equal
probability, it will give me 0.
01:00:16.414 --> 01:00:17.670
OK?
01:00:17.670 --> 01:00:22.590
So I cannot leave this
sigma i by itself.
01:00:22.590 --> 01:00:26.545
So maybe I will pick
another higher-order term
01:00:26.545 --> 01:00:31.490
in the series that has a t, a
sigma i that would make this
01:00:31.490 --> 01:00:36.734
into a sigma i squared, and
then I will have a sigma K here.
01:00:36.734 --> 01:00:38.090
OK?
01:00:38.090 --> 01:00:42.540
Now, note it was kind of similar
to what I was doing here.
01:00:42.540 --> 01:00:45.660
But here I could
pick as many bonds
01:00:45.660 --> 01:00:52.860
as I like on as many factors of
K. Now what has happened here
01:00:52.860 --> 01:00:57.310
is, effectively, I
have only two choices.
01:00:57.310 --> 01:01:00.330
One choice is having
gone many, many times,
01:01:00.330 --> 01:01:04.830
so summing all of the terms
that had 2, 4, et cetera.
01:01:04.830 --> 01:01:08.420
That's what gives you the
cosh K. Or including something
01:01:08.420 --> 01:01:12.150
like this, sum of
1, 3, 5, et cetera.
01:01:12.150 --> 01:01:15.410
That's what gives you
tanh K. But the good thing
01:01:15.410 --> 01:01:17.590
is that it's really
now a binary choice.
01:01:17.590 --> 01:01:21.550
You either draw one line,
or you don't draw anything.
01:01:21.550 --> 01:01:23.030
OK?
01:01:23.030 --> 01:01:30.530
So again, your first choice is
to somehow complete the series
01:01:30.530 --> 01:01:33.970
by drawing something like this.
01:01:33.970 --> 01:01:40.430
And quite generically-- OK,
so after that has happened,
01:01:40.430 --> 01:01:42.280
then this is sigma i squared.
01:01:42.280 --> 01:01:43.300
This is sigma j squared.
01:01:43.300 --> 01:01:47.320
These are all--
they have gone to 1.
01:01:47.320 --> 01:01:49.420
And then you do the
sum over sigma i,
01:01:49.420 --> 01:01:52.240
you will get a factor of 2.
01:01:52.240 --> 01:01:57.800
So the answer is going to
be 2 to the number of sites,
01:01:57.800 --> 01:02:03.005
N, cosh K to the power
of the number of bonds.
01:02:07.610 --> 01:02:10.500
And then I would
have a series, which
01:02:10.500 --> 01:02:22.810
is the sum over all graphs
with even number of bonds
01:02:22.810 --> 01:02:26.320
per site like here.
01:02:26.320 --> 01:02:31.710
So I either have 0 bond going
here, or I can have two bonds.
01:02:31.710 --> 01:02:34.520
I could very well have
something like this, four bonds.
01:02:34.520 --> 01:02:36.790
That doesn't violate anything.
01:02:36.790 --> 01:02:40.740
So all I need to ensure
in order that sigma i does
01:02:40.740 --> 01:02:45.430
not average to 0 is that I
have an even number per site.
01:02:45.430 --> 01:02:47.800
And then the
contribution of the graph
01:02:47.800 --> 01:02:51.550
is t to the number of
bonds in the graph.
01:02:55.600 --> 01:02:58.100
And at this stage when I'm
calculating a partition
01:02:58.100 --> 01:03:02.000
function, there is no
reason why I could not
01:03:02.000 --> 01:03:05.060
have disconnected graphs.
01:03:05.060 --> 01:03:07.190
For the partition function,
there is no problem.
01:03:07.190 --> 01:03:10.305
Presumably, when I take the
log, the disconnected pieces
01:03:10.305 --> 01:03:12.560
will go away.
01:03:12.560 --> 01:03:14.010
OK?
01:03:14.010 --> 01:03:14.579
Yes?
01:03:14.579 --> 01:03:16.745
AUDIENCE: Where does the 2
to the N come from again?
01:03:16.745 --> 01:03:17.680
PROFESSOR: OK.
01:03:17.680 --> 01:03:25.170
So at each site, I have
to sum over sigma i.
01:03:25.170 --> 01:03:28.130
So sigma i is either
minus 1 or plus 1.
01:03:28.130 --> 01:03:33.400
What I'm doing is sum over
sigma i sigma i to some power.
01:03:33.400 --> 01:03:36.530
And this is either
going to give me 2 or 0
01:03:36.530 --> 01:03:42.520
depending on whether
P is even or P is odd.
01:03:42.520 --> 01:03:45.415
All right?
01:03:45.415 --> 01:03:45.915
OK?
01:03:49.810 --> 01:03:58.640
So you can try to calculate
general terms for this series.
01:03:58.640 --> 01:04:04.540
Let's say we go to
hypercubic lattice, which
01:04:04.540 --> 01:04:07.250
is what we were doing before.
01:04:07.250 --> 01:04:12.940
The number of bonds
per site is d.
01:04:12.940 --> 01:04:16.060
So this, for the hypercubic
lattice, the number of bonds
01:04:16.060 --> 01:04:17.930
will be dN.
01:04:17.930 --> 01:04:20.620
You could do this calculation
for a triangular lattice.
01:04:20.620 --> 01:04:23.740
You don't have to
stick with FCC lattice.
01:04:23.740 --> 01:04:27.780
You don't have to stick with
these hypercubic lattices.
01:04:27.780 --> 01:04:35.770
The first diagram that you can
create is always the square.
01:04:35.770 --> 01:04:37.130
OK?
01:04:37.130 --> 01:04:42.370
And in d dimensions, one leg
has a choice of d direction.
01:04:42.370 --> 01:04:44.740
The next one would be d minus 1.
01:04:44.740 --> 01:04:51.320
So this would be d d minus
1 over 2 t to the fourth.
01:04:51.320 --> 01:04:54.190
But you could start it from
any site on the lattice
01:04:54.190 --> 01:04:57.030
so you would have
something like this.
01:04:57.030 --> 01:05:00.110
The next term that you
would have in the series
01:05:00.110 --> 01:05:06.800
is something that involves,
let's say, six bonds.
01:05:06.800 --> 01:05:11.420
So the next term
will be N t to the 6.
01:05:11.420 --> 01:05:15.470
And I think I sometimes
convince myself
01:05:15.470 --> 01:05:18.285
that the numerical factor
was something like this,
01:05:18.285 --> 01:05:19.347
but doesn't matter.
01:05:19.347 --> 01:05:20.680
You could calculate out of this.
01:05:20.680 --> 01:05:20.935
Yes?
01:05:20.935 --> 01:05:22.351
AUDIENCE: What if
we have diagrams
01:05:22.351 --> 01:05:25.402
of order of t squared, just
[INAUDIBLE] there and back?
01:05:25.402 --> 01:05:26.810
PROFESSOR: OK.
01:05:26.810 --> 01:05:29.502
Where would I get the
t squared from here?
01:05:33.310 --> 01:05:34.740
OK?
01:05:34.740 --> 01:05:37.900
So from this bond,
I have this factor,
01:05:37.900 --> 01:05:41.570
1 plus t sigma i sigma j.
01:05:41.570 --> 01:05:44.090
There is no t squared.
01:05:44.090 --> 01:05:48.620
I would have had K squared,
K to the fourth, et cetera.
01:05:48.620 --> 01:05:52.280
But I re-summed all of
them into hyperbolic cosine
01:05:52.280 --> 01:05:54.450
and the hyperbolic sine.
01:05:54.450 --> 01:05:55.050
So this--
01:05:55.050 --> 01:05:57.740
AUDIENCE: So [INAUDIBLE]
taking this product
01:05:57.740 --> 01:06:01.730
along all the bonds, you can
kind of go along the same bond.
01:06:01.730 --> 01:06:04.580
PROFESSOR: We already
summed all of those things
01:06:04.580 --> 01:06:07.809
together into this factor t.
01:06:07.809 --> 01:06:08.350
AUDIENCE: OK.
01:06:08.350 --> 01:06:09.320
PROFESSOR: Yeah?
01:06:09.320 --> 01:06:10.290
OK?
01:06:10.290 --> 01:06:11.760
Yeah, it's good.
01:06:11.760 --> 01:06:18.360
And that's why this tanh
is such a nice variable.
01:06:18.360 --> 01:06:20.480
OK?
01:06:20.480 --> 01:06:24.130
So there is actually
the nicer series
01:06:24.130 --> 01:06:27.810
to work with in terms of
trying to extract exponent
01:06:27.810 --> 01:06:30.120
is this high-temperature
series in terms
01:06:30.120 --> 01:06:32.810
of these new
diagrams, et cetera.
01:06:32.810 --> 01:06:37.680
But I'm not going to
be doing diagrammatics.
01:06:37.680 --> 01:06:44.770
What I will be using this
high-temperature series
01:06:44.770 --> 01:06:47.510
is the following.
01:06:47.510 --> 01:06:52.270
One, to show that
in a few minutes
01:06:52.270 --> 01:07:02.910
we can use it to exactly solve
the one-dimensional Ising model
01:07:02.910 --> 01:07:05.870
and gain a physical
understanding of what's
01:07:05.870 --> 01:07:15.482
happening, and 2, to
re-derive Gaussian model.
01:07:19.730 --> 01:07:23.470
Turns out that there is a
close connection between all
01:07:23.470 --> 01:07:26.500
of these loops that you
can draw on a lattice
01:07:26.500 --> 01:07:29.220
through some kind of a path
integral way of thinking
01:07:29.220 --> 01:07:31.660
about it with the
Gaussian model.
01:07:31.660 --> 01:07:35.650
And that we actually will use as
a stepping stone towards where
01:07:35.650 --> 01:07:42.724
we are really headed,
which is the exact solution
01:07:42.724 --> 01:07:44.544
of the 2D Ising model.
01:08:11.930 --> 01:08:13.960
OK?
01:08:13.960 --> 01:08:20.069
So the 1D Ising model.
01:08:23.779 --> 01:08:28.870
And actually, the method
is sufficiently powerful
01:08:28.870 --> 01:08:34.380
that we can compare and
contrast two cases, one
01:08:34.380 --> 01:08:40.100
when you have open chain.
01:08:40.100 --> 01:08:50.240
So this is a system that is
composed of sites 1, 2, 3, 4,
01:08:50.240 --> 01:08:55.710
N minus 1, N. On each one of
them I have an Ising variable.
01:08:58.689 --> 01:09:11.270
And if I follow my nose, it's
a Z is 2 to the number of sites
01:09:11.270 --> 01:09:17.010
cosh K to the power of
the number of bonds.
01:09:17.010 --> 01:09:21.420
Actually, clearly with open
systems, the number of bonds
01:09:21.420 --> 01:09:24.380
is 1 less than the
number of sites.
01:09:24.380 --> 01:09:26.890
So I can be extremely precise.
01:09:26.890 --> 01:09:30.240
It is N minus 1.
01:09:30.240 --> 01:09:34.560
And then I have
to draw all graphs
01:09:34.560 --> 01:09:39.710
that I can on this lattice that
have an even number of bonds
01:09:39.710 --> 01:09:43.540
emanating from each site.
01:09:43.540 --> 01:09:44.547
Find one.
01:09:44.547 --> 01:09:46.937
[LAUGHTER]
01:09:46.937 --> 01:09:47.899
OK.
01:09:47.899 --> 01:09:51.560
Since you won't have
one, that stands.
01:09:51.560 --> 01:09:53.630
So you can take the log of that.
01:09:53.630 --> 01:09:57.455
You have this free
energy, whatever you like.
01:09:57.455 --> 01:09:59.650
We can't.
01:09:59.650 --> 01:10:06.650
1 is essentially not the zeroth
order term in this series.
01:10:06.650 --> 01:10:07.792
Yes?
01:10:07.792 --> 01:10:08.960
That was the question.
01:10:08.960 --> 01:10:10.881
OK.
01:10:10.881 --> 01:10:11.380
All right?
01:10:14.080 --> 01:10:17.310
You can use the same
thing, same technology,
01:10:17.310 --> 01:10:19.640
to calculate
spin-spin correlation.
01:10:19.640 --> 01:10:23.055
So I pick spins m
and n on this chain.
01:10:23.055 --> 01:10:28.000
Let's say this is spin m
here, and somewhere here I
01:10:28.000 --> 01:10:29.180
put spin n.
01:10:29.180 --> 01:10:33.090
And I want to know the
average of that quantity.
01:10:33.090 --> 01:10:35.240
What am I supposed to do?
01:10:35.240 --> 01:10:41.730
I'm supposed to sum over all
configurations with the weight
01:10:41.730 --> 01:10:48.216
sigma i sigma i plus 1 product
over all-- well, actually,
01:10:48.216 --> 01:10:49.720
we can be general with this.
01:10:49.720 --> 01:10:54.980
Let's call it product over all
bonds, which, in this case,
01:10:54.980 --> 01:10:57.930
are near neighbors,
sigma i sigma j.
01:10:57.930 --> 01:11:03.580
That weight I have to
multiply by sigma m sigma n.
01:11:03.580 --> 01:11:09.450
And then I have to divide
by the partition function
01:11:09.450 --> 01:11:11.892
so that this is
appropriately weighted.
01:11:11.892 --> 01:11:12.391
OK?
01:11:15.160 --> 01:11:20.960
So I can do precisely the
same decomposition over here.
01:11:20.960 --> 01:11:27.730
So I will have 2 to the N
cosh K to the number of bonds.
01:11:27.730 --> 01:11:33.380
In fact, this I can
do in any dimensions.
01:11:33.380 --> 01:11:44.280
It's not really what I would
have only in one dimension.
01:11:44.280 --> 01:11:47.010
And the partition
function, you have seen,
01:11:47.010 --> 01:11:51.190
is the sum over
all graphs, where
01:11:51.190 --> 01:11:58.590
t to the number of bonds
in graph is called g.
01:12:01.270 --> 01:12:06.700
Now I can do the same kind of
expansion that I did over here.
01:12:06.700 --> 01:12:13.050
If I multiply with an
additional sigma m sigma n,
01:12:13.050 --> 01:12:15.980
it is just like I
have already a sigma m
01:12:15.980 --> 01:12:17.300
and a sigma n somewhere.
01:12:20.470 --> 01:12:22.360
And when I sum
over sigmas, I have
01:12:22.360 --> 01:12:29.260
to make sure that these
things don't average to 0.
01:12:29.260 --> 01:12:33.050
So what I need to
do is to draw graphs
01:12:33.050 --> 01:12:35.540
that have an even
number at all sites
01:12:35.540 --> 01:12:39.220
and an odd number
at these two sites.
01:12:39.220 --> 01:12:39.720
All right?
01:12:39.720 --> 01:12:52.230
So this is sum over g with
even number except on m and n,
01:12:52.230 --> 01:12:54.880
where you have to
have an odd number,
01:12:54.880 --> 01:12:57.821
and t is subset of graphs.
01:13:01.120 --> 01:13:01.830
OK?
01:13:01.830 --> 01:13:08.110
So if I do this
for the 1D model,
01:13:08.110 --> 01:13:20.480
sigma m sigma n, I have to draw
graphs that have, essentially,
01:13:20.480 --> 01:13:24.830
an odd number.
01:13:24.830 --> 01:13:27.210
Essentially, sigma
m and sigma n should
01:13:27.210 --> 01:13:32.330
be the origins or ends of lines.
01:13:32.330 --> 01:13:38.350
And clearly, I can draw a
graph that connects these two.
01:13:42.520 --> 01:13:47.210
And so what I will get is
t to the number of steps
01:13:47.210 --> 01:13:50.250
that I have to make
between the two of them.
01:13:50.250 --> 01:13:53.720
The rest of the it is going to
be the same, 2 to the N cosh K
01:13:53.720 --> 01:13:56.700
to the N minus 1 in the
numerator and denominator,
01:13:56.700 --> 01:13:58.633
they cancel each other.
01:13:58.633 --> 01:14:00.480
OK?
01:14:00.480 --> 01:14:04.740
So you can see
explicitly that this
01:14:04.740 --> 01:14:11.980
is a function that decays
since t is less than 1
01:14:11.980 --> 01:14:15.440
as I go further and further out.
01:14:15.440 --> 01:14:17.620
And that it is a
pure exponential.
01:14:17.620 --> 01:14:20.630
So you remember that we said in
general you would have a power
01:14:20.630 --> 01:14:23.730
law line in front that would
have an exponent [? theta. ?]
01:14:23.730 --> 01:14:25.950
And when we did r of
g, I told you, well,
01:14:25.950 --> 01:14:29.070
[? theta ?] came out to be
1 such that you have pure
01:14:29.070 --> 01:14:29.900
exponential.
01:14:29.900 --> 01:14:32.280
Well, here is the proof.
01:14:32.280 --> 01:14:37.910
And furthermore, from this
we see that c is minus 1
01:14:37.910 --> 01:14:41.470
over log of the
hyperbolic tanh of K.
01:14:41.470 --> 01:14:43.390
And if you expand
that, you will find
01:14:43.390 --> 01:14:46.420
that as K goes to
infinity, it has
01:14:46.420 --> 01:14:50.250
precisely that e to
the 2K divergence
01:14:50.250 --> 01:14:52.710
that we had calculated.
01:14:52.710 --> 01:14:56.570
So you can see that
calculating things
01:14:56.570 --> 01:15:01.080
using this graphical
method is very simple.
01:15:01.080 --> 01:15:03.830
And essentially, the
interpretation of t
01:15:03.830 --> 01:15:07.590
is that it is the fidelity
with which information
01:15:07.590 --> 01:15:10.300
goes from one site
to the next site.
01:15:10.300 --> 01:15:13.790
And so the further
away you go every time,
01:15:13.790 --> 01:15:16.440
you lose a factor
of t in how sure
01:15:16.440 --> 01:15:20.310
you are about the nature
of where you started with.
01:15:20.310 --> 01:15:22.900
And so as you go further, you
have this exponential decay.
01:15:26.360 --> 01:15:28.370
OK?
01:15:28.370 --> 01:15:32.816
And the other thing that
we can do at no cost
01:15:32.816 --> 01:15:34.585
is periodic boundary conditions.
01:15:39.680 --> 01:15:45.680
So we take, again,
our spins 1, 2, 3,
01:15:45.680 --> 01:15:53.450
except that we then bend it
such that the last one comes
01:15:53.450 --> 01:15:56.755
and gets connected
to the first one.
01:15:56.755 --> 01:15:59.030
OK?
01:15:59.030 --> 01:16:05.590
So what's the partition
function in this case?
01:16:05.590 --> 01:16:09.910
It is 2 to the N.
01:16:09.910 --> 01:16:11.810
The number of
bonds, in this case,
01:16:11.810 --> 01:16:13.960
is exactly the same as
the number of sites.
01:16:13.960 --> 01:16:16.040
It's one more than
before, so I get
01:16:16.040 --> 01:16:26.870
to cosh K raised to the power
of N. And then is it just one?
01:16:26.870 --> 01:16:30.190
There is one thing that
goes all the way around,
01:16:30.190 --> 01:16:36.220
so I have 1 plus t to the N. So
this is an exponentially small
01:16:36.220 --> 01:16:40.740
correction as we go
further and further out.
01:16:40.740 --> 01:16:45.700
You can kind of regard that as
some finite-size interaction.
01:16:45.700 --> 01:16:53.841
I can similarly calculate sigma
m sigma n, the expectation
01:16:53.841 --> 01:16:54.340
value.
01:16:59.576 --> 01:17:01.970
OK?
01:17:01.970 --> 01:17:05.400
And in the denominator
from the partition
01:17:05.400 --> 01:17:12.700
function, I have this factor
of 1 to the t to the N.
01:17:12.700 --> 01:17:14.820
In the numerator,
again, you should
01:17:14.820 --> 01:17:16.615
be able to see two graphs.
01:17:16.615 --> 01:17:20.760
We can either connect this way
or we can connect that way.
01:17:20.760 --> 01:17:25.410
So you'll have t to
the power of n minus m,
01:17:25.410 --> 01:17:28.710
but you don't know which angle
is the smaller one, so you'll
01:17:28.710 --> 01:17:31.867
have to also include
the other one.
01:17:35.220 --> 01:17:37.620
OK?
01:17:37.620 --> 01:17:40.920
So again, if we take N
to infinity and these two
01:17:40.920 --> 01:17:43.080
sufficiently close,
you can see that all
01:17:43.080 --> 01:17:47.120
of these finite-size
effects, boundary effects,
01:17:47.120 --> 01:17:49.090
et cetera disappear.
01:17:49.090 --> 01:17:51.520
But this is, again,
a toy model in which
01:17:51.520 --> 01:17:55.330
to think about what the effects
of boundaries is, et cetera.
01:17:55.330 --> 01:17:59.090
You can see how nicely
this graphical method
01:17:59.090 --> 01:18:03.790
can enable you to calculate
things very rapidly.
01:18:03.790 --> 01:18:08.600
We'll see that, again, it
provides the right tools
01:18:08.600 --> 01:18:14.010
conceptually to think about what
happens in higher dimensions.