WEBVTT
00:00:00.060 --> 00:00:01.780
The following
content is provided
00:00:01.780 --> 00:00:04.019
under a Creative
Commons license.
00:00:04.019 --> 00:00:06.870
Your support will help MIT
OpenCourseWare continue
00:00:06.870 --> 00:00:10.730
to offer high quality
educational resources for free.
00:00:10.730 --> 00:00:13.340
To make a donation or
view additional materials
00:00:13.340 --> 00:00:17.217
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.217 --> 00:00:17.842
at ocw.mit.edu.
00:00:21.650 --> 00:00:25.210
OK, let's start.
00:00:25.210 --> 00:00:28.600
So let's go back to
our starting point
00:00:28.600 --> 00:00:30.720
for the past couple of weeks.
00:00:33.440 --> 00:00:39.130
The square lattice, let's
say, where each side
00:00:39.130 --> 00:00:43.260
we assign a binary variable
sigma, which is minus 1.
00:00:46.550 --> 00:00:52.420
And a weight that tends
to make subsequent nearest
00:00:52.420 --> 00:00:55.750
neighbors [? things ?] to
be parallel to each other.
00:00:55.750 --> 00:00:59.620
So into the plus k,
they are parallel.
00:00:59.620 --> 00:01:03.400
Penalized into the minus k
if they are anti-parallel.
00:01:03.400 --> 00:01:08.810
And of course, over all
pairs of nearest neighbors.
00:01:08.810 --> 00:01:12.361
And the partition function that
is obtained by summing over 2
00:01:12.361 --> 00:01:14.600
to the n configurations.
00:01:14.600 --> 00:01:17.560
Let's make this up
to give us a function
00:01:17.560 --> 00:01:19.660
of the rate of this
coupling, which
00:01:19.660 --> 00:01:23.850
is some energy provided
by temperature.
00:01:23.850 --> 00:01:29.075
And we expect this-- at least,
in two and higher dimensions--
00:01:29.075 --> 00:01:32.600
to capture a phase transition.
00:01:32.600 --> 00:01:36.640
And the way that we have
been proceeding with this
00:01:36.640 --> 00:01:41.840
to derive this factor as
the hyperbolic cosine of k.
00:01:41.840 --> 00:01:49.440
1 plus r variable t, which is
the hyperbolic [? sine ?] of k.
00:01:52.950 --> 00:01:55.710
Sigma i, sigma j.
00:01:55.710 --> 00:02:02.640
And then this becomes
a cos k to the number
00:02:02.640 --> 00:02:07.760
of bonds, which is 2n
on the square lattice.
00:02:07.760 --> 00:02:11.900
And then expanding
these factors,
00:02:11.900 --> 00:02:15.270
we saw that we could get
things that are either
00:02:15.270 --> 00:02:18.940
1 from each bond
or a factor that
00:02:18.940 --> 00:02:22.990
was something like
t sigma sigma.
00:02:22.990 --> 00:02:25.830
And then summing over
the two values of sigma
00:02:25.830 --> 00:02:31.550
would give us 0 unless we
added another factor of sigma
00:02:31.550 --> 00:02:33.520
through another bond.
00:02:33.520 --> 00:02:39.340
And going forth, we had to
draw these kinds of diagrams
00:02:39.340 --> 00:02:44.060
where at each site, I have an
even number of 1's selected.
00:02:44.060 --> 00:02:47.156
Then summing over the sigmas
would give me a factor of 2
00:02:47.156 --> 00:02:49.200
to the n.
00:02:49.200 --> 00:02:55.881
And so then I have a sum over a
whole bunch of configurations.
00:02:55.881 --> 00:02:57.380
[? There are ?]
[? certainly ?] one.
00:02:57.380 --> 00:02:59.830
There are configurations
that are composed
00:02:59.830 --> 00:03:04.860
of one way of drawing
an object on the lattice
00:03:04.860 --> 00:03:08.060
such as this one,
or objects that
00:03:08.060 --> 00:03:13.005
correspond to doing two of
these loops, and so forth.
00:03:17.620 --> 00:03:23.150
So that's the expression
for the partition function.
00:03:23.150 --> 00:03:27.140
And what we are
really interested
00:03:27.140 --> 00:03:31.440
is the log of the partition
function, which gives us
00:03:31.440 --> 00:03:33.960
the free energy in terms
of thermodynamic quantities
00:03:33.960 --> 00:03:38.150
that potentially will tell
us about phase transition.
00:03:38.150 --> 00:03:42.460
So here we will get
a log 2 to the n.
00:03:42.460 --> 00:03:46.930
Well actually, you want
to divide everything by n
00:03:46.930 --> 00:03:48.930
so that we get the
intensive part.
00:03:48.930 --> 00:03:54.120
So here we get log 2
hyperbolic cosine squared of k.
00:03:56.680 --> 00:04:03.050
And then I have to take the
log of this expression that
00:04:03.050 --> 00:04:05.540
includes things
that are one loop,
00:04:05.540 --> 00:04:07.682
disjointed loop, et cetera.
00:04:07.682 --> 00:04:10.230
And we've seen that
the particular loop
00:04:10.230 --> 00:04:14.640
I can slide all over the place--
so if you have a factor of n--
00:04:14.640 --> 00:04:16.380
whereas things that
are multiple loops
00:04:16.380 --> 00:04:19.420
have factors of n
squared, et cetera,
00:04:19.420 --> 00:04:22.660
which are incompatible
with this.
00:04:22.660 --> 00:04:29.970
So it was very tempting for
us to do the usual thing
00:04:29.970 --> 00:04:32.980
and say that the
log of a sum that
00:04:32.980 --> 00:04:37.540
includes these multiple
occurrences of the loops
00:04:37.540 --> 00:04:43.620
is the same thing as sum
over the configurations
00:04:43.620 --> 00:04:46.560
that involve a single loop.
00:04:46.560 --> 00:04:50.760
And then we have to sum over
all shapes of these loops.
00:04:50.760 --> 00:04:54.530
And each loop will
get a factor of t
00:04:54.530 --> 00:04:56.961
per the number of bonds
that are occurring in that.
00:05:00.050 --> 00:05:04.160
Of course, what we said was
that this equality does not
00:05:04.160 --> 00:05:11.110
hold because if I
exponentiate this term,
00:05:11.110 --> 00:05:14.800
I will generate things where
the different loops will
00:05:14.800 --> 00:05:17.540
coincide with each
other, and therefore
00:05:17.540 --> 00:05:22.110
create types of terms that are
not created in the original sum
00:05:22.110 --> 00:05:24.780
that we had over there.
00:05:24.780 --> 00:05:30.560
So this sum over phantom
loops neglected the condition
00:05:30.560 --> 00:05:34.640
that these loops, in some sense,
have some material to them
00:05:34.640 --> 00:05:39.330
and don't want to
intersect with each other.
00:05:39.330 --> 00:05:43.050
Nonetheless, it was
useful, and we followed up
00:05:43.050 --> 00:05:44.520
this calculation.
00:05:44.520 --> 00:05:48.075
So that's repeat what the result
of this incorrect calculation
00:05:48.075 --> 00:05:49.260
is.
00:05:49.260 --> 00:05:54.997
So we have log of 2 hyperbolic
cosine squared k, 1 over n.
00:05:57.710 --> 00:06:02.280
Then we said the
particular way to organize
00:06:02.280 --> 00:06:06.430
the sum over the
loops is to sum over
00:06:06.430 --> 00:06:09.190
there the length of the loop.
00:06:09.190 --> 00:06:12.400
So I sum over the
length of the loop
00:06:12.400 --> 00:06:16.022
and count the number of
loops that have length l.
00:06:16.022 --> 00:06:18.319
All of them will be
giving me a contribution
00:06:18.319 --> 00:06:19.110
that is t to the l.
00:06:22.620 --> 00:06:26.890
So then I said, well,
let's, for example,
00:06:26.890 --> 00:06:33.700
pick a particular
point on the lattice.
00:06:33.700 --> 00:06:36.490
Let's call it r.
00:06:36.490 --> 00:06:44.380
And I count the number of
ways that I can start at r,
00:06:44.380 --> 00:06:50.870
do a walk of l steps,
and end at r again.
00:06:55.760 --> 00:06:59.780
We saw that for
these phantom loops,
00:06:59.780 --> 00:07:03.110
this w had a very
nice structure.
00:07:03.110 --> 00:07:05.680
It was simply what
was telling me
00:07:05.680 --> 00:07:12.230
about one step raised to the l.
00:07:12.230 --> 00:07:16.450
This was the Markovian property.
00:07:16.450 --> 00:07:19.830
There was, of course,
an important thing here
00:07:19.830 --> 00:07:24.510
which said that I could have
set the origin of this loop
00:07:24.510 --> 00:07:28.870
at any point along the loop.
00:07:28.870 --> 00:07:31.747
So there is an over-counting
by a factor of l
00:07:31.747 --> 00:07:35.730
because the same loop
would have been constructed
00:07:35.730 --> 00:07:40.240
with different points
indicated as the origin.
00:07:40.240 --> 00:07:44.640
And actually, I can go the loop
clockwise or anti-clockwise,
00:07:44.640 --> 00:07:48.620
so there was a factor of 2
because of this degeneracy
00:07:48.620 --> 00:07:51.630
of going clockwise or
anti-clockwise when
00:07:51.630 --> 00:07:52.460
I perform a walk.
00:07:55.040 --> 00:08:01.870
And then over here, there's
also an implicit sum
00:08:01.870 --> 00:08:06.320
over this starting
point and endpoint.
00:08:06.320 --> 00:08:09.250
If I always start and
end at the origin,
00:08:09.250 --> 00:08:13.140
then I will get rid
of the factor of n.
00:08:13.140 --> 00:08:17.780
But it is useful to
explicitly include
00:08:17.780 --> 00:08:22.120
this sum over r because
then you can explicitly
00:08:22.120 --> 00:08:26.010
see that sum over
r of this object
00:08:26.010 --> 00:08:29.360
is the trace of that matrix.
00:08:29.360 --> 00:08:32.010
And I can actually
[INAUDIBLE] the order,
00:08:32.010 --> 00:08:35.760
the trace, and the
summation over l.
00:08:35.760 --> 00:08:40.909
And when that happens, I get
log 2 cos squared k exactly
00:08:40.909 --> 00:08:43.080
as before.
00:08:43.080 --> 00:08:48.650
And then I have 1 over n.
00:08:48.650 --> 00:08:54.995
I have sum over r replaced
by the trace operation.
00:08:57.510 --> 00:09:04.750
And then sum over l-- t, T
raised to the l divided by
00:09:04.750 --> 00:09:17.470
l-- is the expansion for
minus log of 1 minus t T.
00:09:17.470 --> 00:09:21.510
And there's the
factor of 2 over there
00:09:21.510 --> 00:09:24.590
that I have to put over here.
00:09:24.590 --> 00:09:28.220
So note that this
plus became minus
00:09:28.220 --> 00:09:32.290
because of the expansion
for log of 1 minus
00:09:32.290 --> 00:09:34.640
x is minus x minus
x squared over 2
00:09:34.640 --> 00:09:36.550
minus x cubed over 3, et cetera.
00:09:43.200 --> 00:09:52.690
And the final step
that we did was
00:09:52.690 --> 00:10:00.380
to note that the trace I
can calculate in any basis.
00:10:00.380 --> 00:10:04.500
And in particular, this
matrix t is diagonalized
00:10:04.500 --> 00:10:07.920
by going to Fourier
representation.
00:10:07.920 --> 00:10:13.580
In the Fourier representation,
the trace operation
00:10:13.580 --> 00:10:16.750
becomes sum over all q values.
00:10:16.750 --> 00:10:19.800
Sum over all q values,
I go to the continuum
00:10:19.800 --> 00:10:25.950
and write as n integral
over q, so the n's cancel.
00:10:25.950 --> 00:10:29.470
I will get 2 integration over q.
00:10:29.470 --> 00:10:33.900
These are essentially each
one of them, qz and qz,
00:10:33.900 --> 00:10:34.780
in the interval.
00:10:34.780 --> 00:10:38.300
Let's say 0 to 2 pi or minus
pi to 2 pi, doesn't matter.
00:10:38.300 --> 00:10:42.300
Interval of size of 2pi.
00:10:42.300 --> 00:10:44.120
So that's the trace operation.
00:10:44.120 --> 00:10:49.950
Log of 1 minus t.
00:10:49.950 --> 00:10:55.540
Then the matrix that represents
walking along the lattice
00:10:55.540 --> 00:10:57.620
represented in Fourier.
00:10:57.620 --> 00:11:00.510
And so basically at
the particular site,
00:11:00.510 --> 00:11:03.210
we can step to the
right or to the left.
00:11:03.210 --> 00:11:07.695
So that's e to the i qx, e to
the minus i qx, e to the i qy,
00:11:07.695 --> 00:11:09.770
e to the minus i qy.
00:11:09.770 --> 00:11:13.890
Adding all of those up,
you get 2 cosine of qz
00:11:13.890 --> 00:11:15.814
plus cosine of qy.
00:11:21.590 --> 00:11:25.160
So that was our expression.
00:11:25.160 --> 00:11:29.300
And then we realized,
interestingly,
00:11:29.300 --> 00:11:31.710
that whereas this final
expression certainly
00:11:31.710 --> 00:11:37.090
was not the Ising partition
function that we were after,
00:11:37.090 --> 00:11:39.980
that it was, in
fact, the partition
00:11:39.980 --> 00:11:46.630
function of a Gaussian
model where at each site
00:11:46.630 --> 00:11:50.060
I had a variable
whose variance was 1,
00:11:50.060 --> 00:11:52.300
and then I had this
kind of coupling
00:11:52.300 --> 00:11:54.900
rather than with
the sigma variable,
00:11:54.900 --> 00:11:57.100
with these Gaussian
variables that
00:11:57.100 --> 00:11:58.420
will go from minus to infinity.
00:12:05.190 --> 00:12:09.240
But then we said, OK, we
can do better than that.
00:12:09.240 --> 00:12:15.950
And we said that log
z over n actually
00:12:15.950 --> 00:12:19.880
does equal a very similar sum.
00:12:19.880 --> 00:12:24.310
It is log 2 hyperbolic
cosine squared k,
00:12:24.310 --> 00:12:26.650
and then I have 1 over n.
00:12:31.740 --> 00:12:39.570
Sum over all kinds
of loops where
00:12:39.570 --> 00:12:46.430
I have a similar diagram that
I draw, but I put a star.
00:12:49.230 --> 00:12:54.080
And this star
implied two things--
00:12:54.080 --> 00:13:00.270
that just like before, I draw
all kinds of individual loops,
00:13:00.270 --> 00:13:04.560
but I make sure
that my loops never
00:13:04.560 --> 00:13:07.720
have a step that goes
forward and backward.
00:13:07.720 --> 00:13:10.930
So there was no U-turn.
00:13:10.930 --> 00:13:15.030
And importantly, there
was a factor of minus 1
00:13:15.030 --> 00:13:18.625
to the number of times that
the walk crossed itself.
00:13:21.480 --> 00:13:25.490
And we showed that
when we incorporate
00:13:25.490 --> 00:13:29.890
both of these conditions,
the can indeed
00:13:29.890 --> 00:13:37.270
exponentiate this expression and
get exactly the same diagrams
00:13:37.270 --> 00:13:40.100
as we had on the
first line, all coming
00:13:40.100 --> 00:13:42.240
with the correct
weights, and none
00:13:42.240 --> 00:13:45.660
of the diagrams that had
multiple occurrences of a bond
00:13:45.660 --> 00:13:46.160
would occur.
00:13:49.410 --> 00:13:52.220
So then the question
was, how do you
00:13:52.220 --> 00:13:55.590
calculate this
given that we have
00:13:55.590 --> 00:14:01.860
this dependence on the number
of crossings, which offhand may
00:14:01.860 --> 00:14:05.910
look as if it is something
that requires memory?
00:14:05.910 --> 00:14:13.930
And then we saw that, indeed,
just like the previous case,
00:14:13.930 --> 00:14:23.630
we could write the
result as a sum over
00:14:23.630 --> 00:14:29.015
walks that have a
particular length l.
00:14:29.015 --> 00:14:34.430
Right here, we have the
factor of t to the l.
00:14:34.430 --> 00:14:42.520
Those walks could start and
end at the particular point r.
00:14:42.520 --> 00:14:47.240
But we also specified
the direction mu
00:14:47.240 --> 00:14:50.890
along which you started.
00:14:50.890 --> 00:14:56.200
So previously I only
specified the origin.
00:14:56.200 --> 00:15:00.460
Now I have to specify the
starting point as well
00:15:00.460 --> 00:15:02.670
as the direction.
00:15:02.670 --> 00:15:08.660
I have to end at the same point
and along the same direction
00:15:08.660 --> 00:15:11.430
to complete the loop.
00:15:11.430 --> 00:15:18.490
And these were accomplished by
having these factors of walks
00:15:18.490 --> 00:15:21.990
that are length l.
00:15:21.990 --> 00:15:27.635
So to do that, we can certainly
incorporate this condition
00:15:27.635 --> 00:15:34.780
of no U-turn in the description
of the steps that I take.
00:15:34.780 --> 00:15:38.020
So for each step, I
know where I came from.
00:15:38.020 --> 00:15:42.100
I just make sure I don't
step back, so that's easy.
00:15:42.100 --> 00:15:46.970
And we found that this
minus 1 to the power of nc
00:15:46.970 --> 00:15:50.880
can be incorporated
through a factor of e
00:15:50.880 --> 00:15:57.920
to the i sum over the changes of
the orientation of the walker--
00:15:57.920 --> 00:16:02.610
as I step through the lattice--
provided that I included also
00:16:02.610 --> 00:16:06.090
an additional factor of minus.
00:16:06.090 --> 00:16:09.315
So that factor of minus I
could actually put out front.
00:16:09.315 --> 00:16:10.770
It's an important factor.
00:16:13.770 --> 00:16:16.990
And then there's
the over-counting,
00:16:16.990 --> 00:16:21.950
but as before, the walk can go
in either one of two directions
00:16:21.950 --> 00:16:25.930
and can have l starting points.
00:16:25.930 --> 00:16:31.500
OK, so now we can
proceed just as before.
00:16:31.500 --> 00:16:35.680
log 2 hyperbolic
cosine squared k,
00:16:35.680 --> 00:16:44.010
and then we have
a sum over l here,
00:16:44.010 --> 00:16:49.380
which we can, again, represent
as the log of 1 minus
00:16:49.380 --> 00:16:53.065
t-- this 4 by 4 matrix t star.
00:16:55.590 --> 00:16:59.130
Going through the log
operation will change this sign
00:16:59.130 --> 00:17:01.670
from minus to plus.
00:17:01.670 --> 00:17:06.380
I have the 1 over 2n as before.
00:17:06.380 --> 00:17:16.290
And I have to sum over r and
mu, which are the elements that
00:17:16.290 --> 00:17:19.829
characterize this
4n by 4n matrix.
00:17:19.829 --> 00:17:22.369
So this amounts to doing
the trace log operation.
00:17:27.750 --> 00:17:37.350
And then taking advantage of
the fact that, just as before,
00:17:37.350 --> 00:17:41.710
Fourier transforms can
at least partially block
00:17:41.710 --> 00:17:46.030
diagonalize this
4n by 4n matrix.
00:17:46.030 --> 00:17:50.260
I go to that basis,
and the trace
00:17:50.260 --> 00:17:52.200
becomes an integral over q.
00:17:55.890 --> 00:17:59.400
And then I would have to do
the trace of a log of a 4
00:17:59.400 --> 00:18:01.580
by 4 matrix.
00:18:01.580 --> 00:18:05.240
And for that, I use the
identity that a trace
00:18:05.240 --> 00:18:09.060
of log of any matrix is
the log of the determinant
00:18:09.060 --> 00:18:12.060
of that same matrix.
00:18:12.060 --> 00:18:15.390
And so the thing that
I have to integrate
00:18:15.390 --> 00:18:22.100
is the log of the
determinant of a 4
00:18:22.100 --> 00:18:26.060
by 4 matrix that
captures these steps
00:18:26.060 --> 00:18:28.680
that I take on the
square lattice.
00:18:28.680 --> 00:18:32.800
And we saw that,
for example, going
00:18:32.800 --> 00:18:37.305
in the horizontal direction
would give me a factor of t
00:18:37.305 --> 00:18:39.040
e to the minus i qx.
00:18:41.670 --> 00:18:50.140
Going in the vertical
direction-- up-- y qy.
00:18:50.140 --> 00:18:53.085
Going in the horizontal
direction-- down.
00:19:00.880 --> 00:19:04.170
These are the diagonal elements.
00:19:04.170 --> 00:19:08.520
And then there were
off-diagonal elements,
00:19:08.520 --> 00:19:14.240
so the next turn here was
to go and then bend upward.
00:19:14.240 --> 00:19:19.600
So that gave me in addition
to e to the minus i qx,
00:19:19.600 --> 00:19:22.520
which is the same
forward step here.
00:19:22.520 --> 00:19:25.900
A factor of, let's
call it omega so I
00:19:25.900 --> 00:19:28.335
don't have to write
it all over the place.
00:19:28.335 --> 00:19:33.880
Omega is e to the pi pi over 4.
00:19:33.880 --> 00:19:38.080
And the next one was a U-turn.
00:19:38.080 --> 00:19:45.130
The next one was minus t
e minus i qx omega star.
00:19:45.130 --> 00:19:50.840
And we could fill out similarly
all of the other places
00:19:50.840 --> 00:19:53.560
in this 4 by 4 matrix.
00:19:53.560 --> 00:19:59.470
And then the whole problem comes
down to having to evaluate a 4
00:19:59.470 --> 00:20:01.560
by 4 determinant,
which you can do
00:20:01.560 --> 00:20:07.620
by hand with a couple of sheets
of paper to do your algebra.
00:20:07.620 --> 00:20:11.520
And I wrote for you
the final answer,
00:20:11.520 --> 00:20:16.660
is 1/2 2 integrals
from minus pi 2 pi
00:20:16.660 --> 00:20:22.540
over qx and qy divided
by 2 pi squared.
00:20:22.540 --> 00:20:29.030
And then the result of
this, which is log of 1
00:20:29.030 --> 00:20:37.180
plus t squared squared minus 2t
1 minus t squared cosine of qx
00:20:37.180 --> 00:20:41.740
plus cosine of qy.
00:20:41.740 --> 00:20:47.410
This was the expression
for the partition function.
00:20:57.460 --> 00:21:03.560
OK, so this is where
we ended up last time.
00:21:03.560 --> 00:21:08.740
Now the question is we
have here on the board
00:21:08.740 --> 00:21:19.300
two expressions, the project
one and the incorrect one--
00:21:19.300 --> 00:21:25.470
the Gaussian model and the
2-dimensionalizing one.
00:21:28.330 --> 00:21:33.780
They look surprisingly
similar, and that
00:21:33.780 --> 00:21:39.500
should start to worry us
potentially because we expect
00:21:39.500 --> 00:21:41.710
that many when we
have some functional
00:21:41.710 --> 00:21:45.230
form, that functional
form carries
00:21:45.230 --> 00:21:48.990
within it certain singularities.
00:21:48.990 --> 00:21:52.210
And you say, well, these
two functions, both of them
00:21:52.210 --> 00:21:56.220
are a double integral
of log of something
00:21:56.220 --> 00:21:58.080
minus something
cosine plus cosine.
00:22:00.690 --> 00:22:04.870
So after all of
this work, did we
00:22:04.870 --> 00:22:08.450
end up with an expression
that has the same singular
00:22:08.450 --> 00:22:12.500
behavior as the Gaussian model?
00:22:12.500 --> 00:22:18.160
OK, so let's go and look
at things more carefully.
00:22:18.160 --> 00:22:25.120
So in both cases, what I need
to do is to integrate A function
00:22:25.120 --> 00:22:30.600
a that appears inside the log.
00:22:30.600 --> 00:22:32.905
There is an A for the
Gaussian, and then there
00:22:32.905 --> 00:22:38.000
is this object--
let's call it A star--
00:22:38.000 --> 00:22:42.090
for the correct solution.
00:22:42.090 --> 00:22:47.710
So the thing that I have to
integrate is, of course, q.
00:22:47.710 --> 00:22:51.160
So this is a function
of the vector q,
00:22:51.160 --> 00:22:54.730
as well as the parameter that
is a function of which I expect
00:22:54.730 --> 00:22:57.220
to have a phase
transition, which is t.
00:23:01.930 --> 00:23:08.580
Now where could I potentially
get some kind of a singularity?
00:23:08.580 --> 00:23:12.610
The only place that I
can get a singularity
00:23:12.610 --> 00:23:19.400
is if the argument of the log
goes to 0 because log of 0
00:23:19.400 --> 00:23:23.380
is something that's derivatives
of singularities, et cetera.
00:23:23.380 --> 00:23:28.180
So you may say, OK, that's
where I should be looking at.
00:23:28.180 --> 00:23:33.770
So where is it this
most likely to happen
00:23:33.770 --> 00:23:36.080
when I'm integrating over q?
00:23:36.080 --> 00:23:40.450
So basically, I'm
integrating over qx and qy
00:23:40.450 --> 00:23:41.980
over a [INAUDIBLE]
[? zone ?] that
00:23:41.980 --> 00:23:48.110
goes from minus pi to
pi in both directions.
00:23:48.110 --> 00:23:53.490
And potentially somewhere in
this I encounter a singularity.
00:23:53.490 --> 00:23:57.030
Let's come from the site
of high temperatures
00:23:57.030 --> 00:23:59.940
where t is close to 0.
00:23:59.940 --> 00:24:02.490
Then I have log
of 1, no problem.
00:24:02.490 --> 00:24:06.410
As I go to lower and
lower temperatures,
00:24:06.410 --> 00:24:08.320
the t becomes larger.
00:24:08.320 --> 00:24:12.100
Then from the 1, I start
to subtract more and more
00:24:12.100 --> 00:24:14.050
with these cosines.
00:24:14.050 --> 00:24:16.300
And clearly the place
that I'm subtracting
00:24:16.300 --> 00:24:20.110
most is right at the
center of q equals to 0.
00:24:23.050 --> 00:24:27.120
So let's expand this
in the vicinity of q
00:24:27.120 --> 00:24:32.220
goes to 0 in the
vicinity of this place.
00:24:32.220 --> 00:24:37.200
And there what I see
is it is 1 minus.
00:24:37.200 --> 00:24:41.470
Cosines are approximately
1 minus q squared over 2.
00:24:41.470 --> 00:24:45.910
So I have 1 minus
4t, and then I have
00:24:45.910 --> 00:24:51.070
plus t qx squared plus qy
squared, which is essentially
00:24:51.070 --> 00:24:52.590
the net q squared.
00:24:52.590 --> 00:24:56.510
And then I have order of
higher power z, qx and qy.
00:25:00.160 --> 00:25:01.700
Fine.
00:25:01.700 --> 00:25:06.360
So this part is
positive, no problem.
00:25:06.360 --> 00:25:11.970
I see that this
part goes through 0
00:25:11.970 --> 00:25:15.660
when I hit tc, which is 1/4.
00:25:15.660 --> 00:25:18.300
And this we had already
seen, that basically this
00:25:18.300 --> 00:25:22.930
is the place where the
exponentially increasing
00:25:22.930 --> 00:25:26.780
number of walks-- as 4
to the number of steps--
00:25:26.780 --> 00:25:29.990
overcomes the
exponentially decreasing
00:25:29.990 --> 00:25:33.510
fidelity of information carried
through each walk, which
00:25:33.510 --> 00:25:35.460
was t to the l.
00:25:35.460 --> 00:25:40.470
So 4dt being 1, tc
is of the order 1/4.
00:25:40.470 --> 00:25:42.600
We are interested
in the singularities
00:25:42.600 --> 00:25:45.790
in the vicinity of
this phase transition,
00:25:45.790 --> 00:25:50.780
so we additionally
go and look at what
00:25:50.780 --> 00:25:57.160
happens when t approaches
tc, but from this side above,
00:25:57.160 --> 00:26:01.440
because clearly if I go to
t that is larger than 1/4,
00:26:01.440 --> 00:26:03.620
it doesn't make any sense.
00:26:03.620 --> 00:26:08.290
So t has to be less than 1/4.
00:26:08.290 --> 00:26:15.795
And so then what I have here is
that I can write this as 4tc,
00:26:15.795 --> 00:26:21.490
so this is 4 times tc minus t.
00:26:21.490 --> 00:26:25.543
And this to the lowest order
I can replace as tc-- q
00:26:25.543 --> 00:26:28.500
squared plus higher orders.
00:26:28.500 --> 00:26:32.650
This 4 I can write as 1 over tc.
00:26:32.650 --> 00:26:36.790
So the whole thing
I can write as q
00:26:36.790 --> 00:26:48.852
squared plus delta t divided by
tc, and there's a factor of 4.
00:26:53.190 --> 00:27:01.200
And delta t I have
defined to be tc minus 10.
00:27:01.200 --> 00:27:05.650
How close I am to the location
where this singularity takes
00:27:05.650 --> 00:27:08.030
places.
00:27:08.030 --> 00:27:12.830
So what I'm interested
is not in the whole form
00:27:12.830 --> 00:27:15.400
of this function, but
only the singularities
00:27:15.400 --> 00:27:17.150
that it expresses.
00:27:17.150 --> 00:27:27.230
So I focus on the singular part
of this Gaussian expression.
00:27:27.230 --> 00:27:30.230
I don't have to worry
about that term.
00:27:30.230 --> 00:27:31.670
So I have minus 1/2.
00:27:34.370 --> 00:27:37.180
I have double integral.
00:27:37.180 --> 00:27:41.760
The argument of
the log, I expanded
00:27:41.760 --> 00:27:43.900
in the vicinity of
the point that I
00:27:43.900 --> 00:27:47.320
see a singularity to take place.
00:27:47.320 --> 00:27:54.320
And I'll write the answer as q
squared plus 4 delta t over tc.
00:27:54.320 --> 00:27:58.540
If I am sufficiently close in
my integration to the origin
00:27:58.540 --> 00:28:04.430
so that the expansion
in q is acceptable,
00:28:04.430 --> 00:28:08.140
there is an additional
factor of tc.
00:28:08.140 --> 00:28:11.390
But if I take a log of
tc, it's just a constant.
00:28:11.390 --> 00:28:13.660
I can integrate that out.
00:28:13.660 --> 00:28:17.376
It's going to not contribute
to the singular part, just
00:28:17.376 --> 00:28:21.630
an additional regular component.
00:28:21.630 --> 00:28:25.060
Now if I am in the
vicinity of q equals to 0
00:28:25.060 --> 00:28:30.440
where all of the action is,
at this order, the thing
00:28:30.440 --> 00:28:34.050
that I'm integrating
has circular symmetry so
00:28:34.050 --> 00:28:36.050
that 2-dimensional integration.
00:28:36.050 --> 00:28:39.250
I can write whether
or not it's d qx, d qy
00:28:39.250 --> 00:28:46.325
as 2 pi q dq divided
by 2 pi squared,
00:28:46.325 --> 00:28:47.740
which was the density of state.
00:28:50.960 --> 00:28:57.472
And this approximation
of a thing being
00:28:57.472 --> 00:29:01.245
isotropic and
circular only holds
00:29:01.245 --> 00:29:04.020
when I'm sufficiently
close to the origin,
00:29:04.020 --> 00:29:08.730
let's say up to some value
that I will call lambda.
00:29:08.730 --> 00:29:11.450
So I will impose
some cut-off here,
00:29:11.450 --> 00:29:15.210
lambda, which is
certainly less than one
00:29:15.210 --> 00:29:17.795
of the order of pi--
let's say pi over 10.
00:29:17.795 --> 00:29:19.550
It doesn't matter what it is.
00:29:19.550 --> 00:29:21.285
As we will see,
the singular part,
00:29:21.285 --> 00:29:25.000
it ultimately does not
matter what I put for lambda.
00:29:25.000 --> 00:29:28.820
But the rest of the integration
that I haven't exclusively
00:29:28.820 --> 00:29:31.890
written down from
all of this-- again,
00:29:31.890 --> 00:29:37.980
in analogy to what we had seen
before for the Landau-Ginzburg
00:29:37.980 --> 00:29:39.880
calculation will
give me something
00:29:39.880 --> 00:29:42.100
that is perfectly analytic.
00:29:42.100 --> 00:29:47.860
So I have extracted from this
expression the singular part.
00:29:47.860 --> 00:29:52.590
OK, now let's do this
integral carefully.
00:29:52.590 --> 00:29:54.914
So what do I have?
00:29:54.914 --> 00:29:59.440
I have 1 over 2 pi with
minus 1/2, so I have minus 1
00:29:59.440 --> 00:30:02.690
over 4 pi.
00:30:02.690 --> 00:30:09.360
If I call this whole
object here x-- so x is q
00:30:09.360 --> 00:30:11.760
squared plus this
something-- then
00:30:11.760 --> 00:30:16.210
we can see that dx is 2 q dq.
00:30:19.280 --> 00:30:23.420
So what I have to do is
the integral of dx log x,
00:30:23.420 --> 00:30:26.460
which is x log x minus x.
00:30:26.460 --> 00:30:35.870
So essentially, let's keep the
2 up there and make this 8 pi.
00:30:35.870 --> 00:30:50.570
And then what I have is
x log x minus x itself,
00:30:50.570 --> 00:30:55.570
which I can write as log of e.
00:30:55.570 --> 00:30:59.416
And then this whole thing has
to be evaluated between the two
00:30:59.416 --> 00:31:01.773
limits of integration,
lambda and [INAUDIBLE].
00:31:05.370 --> 00:31:09.795
Now you explicitly see
that if I substitute
00:31:09.795 --> 00:31:14.900
in this expression for q
the upper part of lambda,
00:31:14.900 --> 00:31:17.270
it will give me
something like lambda
00:31:17.270 --> 00:31:20.480
squared plus delta
t-- an expandable
00:31:20.480 --> 00:31:22.140
and analytical function.
00:31:22.140 --> 00:31:26.330
Log of a constant plus delta t
that I can start analytically
00:31:26.330 --> 00:31:27.550
expanding.
00:31:27.550 --> 00:31:29.880
So anything that I get
from the other cut-off
00:31:29.880 --> 00:31:33.660
of the integration is
perfectly analytic.
00:31:33.660 --> 00:31:35.410
I don't have to worry about it.
00:31:35.410 --> 00:31:39.980
If I'm interested in
the singular part,
00:31:39.980 --> 00:31:44.760
I basically need to evaluate
this as it's lower cut-off.
00:31:44.760 --> 00:31:47.040
So I evaluate it
at q equals to 0.
00:31:47.040 --> 00:31:50.010
Well first off all, I
will get a sign change
00:31:50.010 --> 00:31:53.560
because I'm at
the lower cut-off.
00:31:53.560 --> 00:31:59.420
I will get from here
4 delta t over tc.
00:31:59.420 --> 00:32:04.155
And from here, I will get
log of a bunch of constants.
00:32:04.155 --> 00:32:05.510
It doesn't really matter.
00:32:05.510 --> 00:32:08.438
4 over e delta t over tc.
00:32:14.800 --> 00:32:18.720
What is the leading singularity?
00:32:18.720 --> 00:32:24.010
Is delta t log t-- log delta t?
00:32:24.010 --> 00:32:27.146
And so again, the
leading singularity
00:32:27.146 --> 00:32:30.830
is delta t log of delta t.
00:32:30.830 --> 00:32:35.685
There's an overall factor of 1
over pi. [? Doesn't match. ?]
00:32:35.685 --> 00:32:37.660
You take two derivatives.
00:32:37.660 --> 00:32:39.940
You find that the
heat capacity, let's
00:32:39.940 --> 00:32:43.655
say that is proportion
to two derivatives of log
00:32:43.655 --> 00:32:47.340
z by delta t squared.
00:32:47.340 --> 00:32:50.340
You take one derivative,
it goes like the log.
00:32:50.340 --> 00:32:52.406
You take another
derivative of the log,
00:32:52.406 --> 00:32:57.600
you find that the singularity
is 1 over delta t.
00:32:57.600 --> 00:33:01.290
That corresponds to a
heat capacity divergence
00:33:01.290 --> 00:33:04.460
with an exponent
of unity, which is
00:33:04.460 --> 00:33:07.580
quite consistent with the
generally Gaussian formula
00:33:07.580 --> 00:33:12.085
that we had in d dimensions,
which was 2 minus t over 2.
00:33:17.690 --> 00:33:20.340
So that's the Gaussian.
00:33:20.340 --> 00:33:30.860
And of course, this
whole theory breaks down
00:33:30.860 --> 00:33:35.430
for t that is greater than tc.
00:33:35.430 --> 00:33:38.000
Once I go beyond
tc, my expressions
00:33:38.000 --> 00:33:39.310
just don't make sense.
00:33:39.310 --> 00:33:43.480
I can't integrate a lot
of a negative number.
00:33:43.480 --> 00:33:46.300
And we understand why that is.
00:33:46.300 --> 00:33:49.690
That's because we are including
all of these loops that
00:33:49.690 --> 00:33:52.480
can go over each
other multiple times.
00:33:52.480 --> 00:33:55.640
The whole theory
does not make sense.
00:33:55.640 --> 00:33:57.640
So we did this one to death.
00:34:00.830 --> 00:34:05.100
Will the exact result
be any different?
00:34:05.100 --> 00:34:08.940
So let's carry out the
corresponding procedure--
00:34:08.940 --> 00:34:12.809
for A star that is a
function of q and t.
00:34:15.460 --> 00:34:20.889
And again, singularities
should come from the place
00:34:20.889 --> 00:34:23.960
where this is most
likely to go to 0.
00:34:23.960 --> 00:34:27.650
You can see it's 1
something minus something.
00:34:27.650 --> 00:34:30.190
And clearly when the
q's are close to 0
00:34:30.190 --> 00:34:32.719
is when you subtract
most, and you're
00:34:32.719 --> 00:34:34.780
likely to become negative.
00:34:34.780 --> 00:34:38.000
So let's expand it around 0.
00:34:38.000 --> 00:34:44.960
This as q goes to 0 is 1
plus t squared squared.
00:34:44.960 --> 00:34:47.730
And then I have minus.
00:34:47.730 --> 00:34:51.260
Each one of the cosines
starts at unity,
00:34:51.260 --> 00:34:57.330
so I will have 4t
1 minus t squared.
00:34:57.330 --> 00:35:02.090
And then from the qx squared
over 2 qy squared over 2,
00:35:02.090 --> 00:35:06.330
I will get a plus
t 1 minus t squared
00:35:06.330 --> 00:35:10.520
q squared-- qx squared plus
qy squared-- plus order of q
00:35:10.520 --> 00:35:11.498
to the fourth.
00:35:20.310 --> 00:35:26.560
So the way that we identified
the location of the singular
00:35:26.560 --> 00:35:32.125
part before was to focus
on exactly q equals to 0.
00:35:32.125 --> 00:35:37.610
And what we find is that
A star at q equals to 0
00:35:37.610 --> 00:35:41.660
is essentially this part.
00:35:41.660 --> 00:35:46.270
This part I'm going to rewrite
the first end slightly.
00:35:46.270 --> 00:35:48.860
1 plus t squared squared
is the same thing
00:35:48.860 --> 00:35:54.160
as 1 minus t squared
squared plus 4t squared.
00:35:54.160 --> 00:35:56.020
The difference
between the expansion
00:35:56.020 --> 00:35:57.970
of each one of
these two terms is
00:35:57.970 --> 00:36:01.940
that this has plus 2t squared,
this has minus 2t squared,
00:36:01.940 --> 00:36:04.200
which I have added here.
00:36:04.200 --> 00:36:09.600
And then minus 4t
1 minus t squared.
00:36:09.600 --> 00:36:12.240
And the reason
that I did that is
00:36:12.240 --> 00:36:16.530
that you can now see
that this term is twice
00:36:16.530 --> 00:36:20.230
this times this when
I take the square.
00:36:20.230 --> 00:36:24.710
So the whole thing is the same
thing as 1 minus t squared
00:36:24.710 --> 00:36:26.470
minus 2t squared.
00:36:31.270 --> 00:36:36.720
So the first thing that gives
us reassurance happens now,
00:36:36.720 --> 00:36:40.600
whereas previously for the
Gaussian model 1 minus 4t could
00:36:40.600 --> 00:36:44.470
be both positive and
negative, this you can see
00:36:44.470 --> 00:36:45.525
is always positive.
00:36:48.150 --> 00:36:50.540
So there is no
problem with me not
00:36:50.540 --> 00:36:53.170
being able to go from one
side of the phase transition
00:36:53.170 --> 00:36:55.840
to another side of
the phase transition.
00:36:55.840 --> 00:37:00.270
This expression will
encounter no difficulties.
00:37:00.270 --> 00:37:05.670
But there is a special
point when this thing is 0.
00:37:05.670 --> 00:37:13.200
So there is a point when 1
minus 2t c plus tc squared
00:37:13.200 --> 00:37:14.330
is equal to 0.
00:37:14.330 --> 00:37:17.210
The whole thing goes to 0.
00:37:17.210 --> 00:37:20.770
And you can figure
out where that is.
00:37:20.770 --> 00:37:22.710
tc is 1.
00:37:22.710 --> 00:37:24.070
It's a quadratic form.
00:37:24.070 --> 00:37:30.560
It has two solutions-- 1
minus plus square root of 2.
00:37:30.560 --> 00:37:32.365
A negative solution
is not acceptable.
00:37:41.131 --> 00:37:41.630
Minus.
00:37:45.244 --> 00:37:50.780
I have to recast this slightly.
00:37:50.780 --> 00:37:54.150
Knowing the answer sometimes
makes you go too fast.
00:37:54.150 --> 00:38:00.210
tc squared plus 2tc
minus 1 equals to 0.
00:38:00.210 --> 00:38:05.862
tc is minus 1 plus or
minus square root of 2.
00:38:05.862 --> 00:38:09.110
The minus solution
is not acceptable.
00:38:09.110 --> 00:38:12.630
The plus solution would
give me root 2 minus 1.
00:38:15.860 --> 00:38:20.280
Just to remind you,
we calculated a value
00:38:20.280 --> 00:38:23.900
for the critical point
based on duality.
00:38:23.900 --> 00:38:28.520
So let's just recap
that duality argument.
00:38:28.520 --> 00:38:32.250
We saw that the series
that we had calculated,
00:38:32.250 --> 00:38:38.510
which was an expansion in
high temperatures times k,
00:38:38.510 --> 00:38:42.980
reproduced the expansion that
we had for 0 temperature,
00:38:42.980 --> 00:38:46.500
including islands of
minus in a sea of plus,
00:38:46.500 --> 00:38:49.690
where the contribution of each
bond was going from e to the k
00:38:49.690 --> 00:38:51.710
to e to the minus k.
00:38:51.710 --> 00:38:59.530
So there was a correspondence
between a dual coupling.
00:38:59.530 --> 00:39:04.600
And the actual coupling,
that was like this.
00:39:04.600 --> 00:39:08.750
At a critical point,
we said the two of them
00:39:08.750 --> 00:39:11.470
have to be the same.
00:39:11.470 --> 00:39:14.900
And what we had
calculated based on that,
00:39:14.900 --> 00:39:18.260
since the hyperbolic
tang I can write in terms
00:39:18.260 --> 00:39:23.290
of the exponentials was that
the value of e to the minus
00:39:23.290 --> 00:39:28.760
e to the plus 2kc was, in
fact, square root of 2 plus 1.
00:39:31.630 --> 00:39:37.730
And the inverse of this will
be square root of 2 minus 1,
00:39:37.730 --> 00:39:39.360
and that's the same as the tang.
00:39:39.360 --> 00:39:40.837
That's the same
thing as the things
00:39:40.837 --> 00:39:43.380
that we have already written.
00:39:43.380 --> 00:39:47.450
So the calculation
that we had done
00:39:47.450 --> 00:39:52.740
before-- obtain this
critical temperature based
00:39:52.740 --> 00:39:56.390
on this Kramers-Wannier
duality--
00:39:56.390 --> 00:40:00.890
gave us a critical point here,
which is precisely the place
00:40:00.890 --> 00:40:04.705
that we can identify as the
origin of the singularity
00:40:04.705 --> 00:40:06.020
in this expression.
00:40:10.070 --> 00:40:11.475
So what did we do next?
00:40:11.475 --> 00:40:16.340
The next thing that we did
was having identified up there
00:40:16.340 --> 00:40:19.530
where the critical point
was-- which was at 14--
00:40:19.530 --> 00:40:23.250
was to expand our
A, the integrand,
00:40:23.250 --> 00:40:27.670
inside the log in the
vicinity of that point.
00:40:27.670 --> 00:40:30.480
So what I want to do
is similarly expand
00:40:30.480 --> 00:40:34.650
a star of q for t that
goes into vicinity of tc.
00:40:38.410 --> 00:40:41.950
And what do I have to do?
00:40:41.950 --> 00:40:59.010
So what I can do, let's write
it as t is tc plus delta t
00:40:59.010 --> 00:41:02.520
and make an expansion
for delta t small.
00:41:07.760 --> 00:41:13.060
So I have to make an expansion
or delta t small, first
00:41:13.060 --> 00:41:14.380
of all, of this quantity.
00:41:18.290 --> 00:41:23.916
So if I make a
small change in t,
00:41:23.916 --> 00:41:26.650
I'll have to take a
derivative inside here.
00:41:26.650 --> 00:41:31.780
So I have minus 2t minus
2, evaluate it at tc,
00:41:31.780 --> 00:41:33.810
times the change in delta t.
00:41:36.610 --> 00:41:42.260
So that's the change
of this expression
00:41:42.260 --> 00:41:46.640
if I go slightly away from
the point where it is 0.
00:41:46.640 --> 00:41:47.314
Yes?
00:41:47.314 --> 00:41:48.480
AUDIENCE: I have a question.
00:41:48.480 --> 00:41:53.300
T here is [? tangent ?]
of k where we calculated
00:41:53.300 --> 00:41:59.020
the [INAUDIBLE] to
[? be as ?] temperature.
00:41:59.020 --> 00:42:01.850
PROFESSOR: Now all
of these things
00:42:01.850 --> 00:42:05.000
are analytical
functions of each other.
00:42:05.000 --> 00:42:12.790
So the k is, in fact, some
unit of energy divided by kt.
00:42:12.790 --> 00:42:15.610
So Really, the
temperature is here.
00:42:15.610 --> 00:42:19.800
And my t is tang of
the above objects,
00:42:19.800 --> 00:42:24.260
so it's tang of j divided by kt.
00:42:24.260 --> 00:42:26.980
So the point is
that whenever I look
00:42:26.980 --> 00:42:31.020
at the delta in
temperature, I can translate
00:42:31.020 --> 00:42:35.440
that delta in temperature to
a delta int times the value
00:42:35.440 --> 00:42:38.370
of the derivative at
the location of this,
00:42:38.370 --> 00:42:40.330
which is some finite number.
00:42:40.330 --> 00:42:44.160
Of basically up
to some constant,
00:42:44.160 --> 00:42:46.430
taking derivatives with
respect to temperature,
00:42:46.430 --> 00:42:49.130
with respect to k,
with respect to tang k,
00:42:49.130 --> 00:42:51.420
with respect to beta.
00:42:51.420 --> 00:42:55.190
Evaluate it at the
finite temperature,
00:42:55.190 --> 00:42:57.500
which is the location
of the critical point.
00:42:57.500 --> 00:43:00.980
They're all the same up to
proportionality constants,
00:43:00.980 --> 00:43:04.300
and that's why I wrote
"proportionality" there.
00:43:04.300 --> 00:43:07.000
One thing that you
have to make sure--
00:43:07.000 --> 00:43:10.930
and I spent actually half an
hour this morning checking--
00:43:10.930 --> 00:43:14.060
is that the signs work out fine.
00:43:14.060 --> 00:43:17.120
So I didn't want to write
an expression for the heat
00:43:17.120 --> 00:43:19.350
capacity that was negative.
00:43:19.350 --> 00:43:21.710
So proportionalities
aside, that's
00:43:21.710 --> 00:43:23.530
the one thing that
I better have,
00:43:23.530 --> 00:43:26.174
is that the sign of the heat
capacity, that is positive.
00:43:34.100 --> 00:43:39.750
So that's the expansion of
the term that corresponds to q
00:43:39.750 --> 00:43:41.990
equals to 0.
00:43:41.990 --> 00:43:44.360
The term that was
proportional to q squared,
00:43:44.360 --> 00:43:46.740
look at what we did in
the above expression
00:43:46.740 --> 00:43:48.660
for the Gaussian model.
00:43:48.660 --> 00:43:52.980
Since it was lower
order, it was already
00:43:52.980 --> 00:43:55.100
proportional to q squared.
00:43:55.100 --> 00:43:58.420
We evaluated it at
exactly t equals to tc.
00:43:58.420 --> 00:44:04.750
So I will put here tc 1
minus tc squared q squared,
00:44:04.750 --> 00:44:06.250
and then I will
have high orders.
00:44:10.920 --> 00:44:14.580
Now fortunately, we
have a value for tc
00:44:14.580 --> 00:44:18.800
that we can substitute in
a couple of places here.
00:44:18.800 --> 00:44:24.420
You can see that this is,
in fact, twice tc plus 1,
00:44:24.420 --> 00:44:27.000
and tc plus 1 is root 2.
00:44:27.000 --> 00:44:29.650
So this is minus 2 root 2.
00:44:29.650 --> 00:44:32.320
I square that, and
this whole thing
00:44:32.320 --> 00:44:35.425
becomes 8 delta t squared.
00:44:39.740 --> 00:44:49.790
This object here, 1
minus tc squared--
00:44:49.790 --> 00:44:53.690
you can see if I put the 2 tc
on the other side-- 1 minus tc
00:44:53.690 --> 00:44:55.620
squared' is the
same thing as 2 tc.
00:45:00.460 --> 00:45:06.400
So I can write the whole thing
here as 2 tc squared q squared.
00:45:09.530 --> 00:45:12.720
And the reason I
do that is, like
00:45:12.720 --> 00:45:14.620
before, there's
an overall factor
00:45:14.620 --> 00:45:18.690
that I can take out
of this parentheses.
00:45:18.690 --> 00:45:22.760
And the answer will
be q squared now
00:45:22.760 --> 00:45:31.210
plus 4 delta t over tc squared.
00:45:31.210 --> 00:45:34.130
It's very similar to
what we had before,
00:45:34.130 --> 00:45:37.640
except that when we had q
squared plus 4 delta t over tc,
00:45:37.640 --> 00:45:42.480
we have q squared plus 4
delta t over tc squared.
00:45:42.480 --> 00:45:45.700
Now this square was very
important for allowing
00:45:45.700 --> 00:45:48.320
us to go for both
positive and negative,
00:45:48.320 --> 00:45:52.720
but let's see its consequence
on the singularity.
00:45:52.720 --> 00:45:59.190
So now log z of the correct
form divided by n-- the singular
00:45:59.190 --> 00:46:04.520
part-- and we calculate
it just as before.
00:46:04.520 --> 00:46:08.100
First of all, rather than
minus 1/2 I have a plus 1/2.
00:46:11.020 --> 00:46:16.360
I have the same integral, which
in the vicinity of the origin
00:46:16.360 --> 00:46:21.700
is symmetric, so I will
write it as 2 pi qd
00:46:21.700 --> 00:46:27.110
q divided by 4 pi squared.
00:46:27.110 --> 00:46:29.300
And then I have log.
00:46:29.300 --> 00:46:33.160
I will forget about this
factor for consideration
00:46:33.160 --> 00:46:38.245
of singularities 4
delta t over tc squared.
00:46:42.430 --> 00:46:50.010
And now again, it is
exactly the same structure
00:46:50.010 --> 00:46:55.410
as x dx that I had before.
00:46:55.410 --> 00:46:57.500
So it's the same integral.
00:46:57.500 --> 00:47:11.470
And what you will find is that I
evaluate it as 1/8 pi integral,
00:47:11.470 --> 00:47:19.820
essentially q squared plus
4 delta t over tc squared,
00:47:19.820 --> 00:47:26.700
log of q squared plus 4
delta t over tc squared
00:47:26.700 --> 00:47:32.140
over e evaluated
between 0 and lambda.
00:47:32.140 --> 00:47:37.760
And the only singularity
comes from the evaluation
00:47:37.760 --> 00:47:38.890
that we have at the origin.
00:47:43.330 --> 00:47:45.455
And so that I will
get a factor of minus.
00:47:45.455 --> 00:47:48.980
So I will get 1/8 pi.
00:47:48.980 --> 00:47:52.620
Actually then, I
substitute this factor.
00:47:52.620 --> 00:47:56.060
The 4 and the 8 will give me 2.
00:47:56.060 --> 00:47:59.510
And then I evaluate the
log of delta t squared,
00:47:59.510 --> 00:48:01.820
so that's another factor of 2.
00:48:01.820 --> 00:48:05.760
So actually, only one
factor of pi survives.
00:48:05.760 --> 00:48:12.710
I will have delta t over tc
squared log of, let's say,
00:48:12.710 --> 00:48:15.033
absolute value of
delta t over tc.
00:48:20.220 --> 00:48:24.420
So the only thing that
changed was that whereas I
00:48:24.420 --> 00:48:29.160
had the linear term sitting
in front of the log, now
00:48:29.160 --> 00:48:32.000
have a quadratic term.
00:48:32.000 --> 00:48:35.780
But now when I take two
derivatives, and now
00:48:35.780 --> 00:48:39.900
we are sure that taking
derivatives does not really
00:48:39.900 --> 00:48:44.230
matter whether I'm doing it with
respect to temperature or delta
00:48:44.230 --> 00:48:45.301
t or any other variable.
00:48:48.250 --> 00:48:50.250
You can see that
the leading behavior
00:48:50.250 --> 00:48:53.040
will come taking two
derivatives out here
00:48:53.040 --> 00:48:56.170
and will be
proportional to the log.
00:48:56.170 --> 00:49:03.410
So I will get minus 1 over
pi log of delta t over tc.
00:49:07.640 --> 00:49:15.790
So that if I were to plug the
heat capacity of the system,
00:49:15.790 --> 00:49:18.040
as a function of, let's
say, this parameter t--
00:49:18.040 --> 00:49:23.550
which is also something that
stands for temperature-- but t
00:49:23.550 --> 00:49:26.890
goes between, say, 0 and 1.
00:49:26.890 --> 00:49:28.900
It's a hyperbolic tangent.
00:49:28.900 --> 00:49:33.830
There's a location which is this
tc, which is root 2 minus 1.
00:49:36.590 --> 00:49:40.290
And the singular part
of the heat capacity--
00:49:40.290 --> 00:49:42.780
there will be some other
part of the heat capacity
00:49:42.780 --> 00:49:46.160
that is regular-- but
the singular part we see
00:49:46.160 --> 00:49:47.690
has a [INAUDIBLE] divergence.
00:49:55.800 --> 00:50:00.990
And furthermore, you can
see that the amplitudes-- so
00:50:00.990 --> 00:50:03.460
essentially this
goes approaching
00:50:03.460 --> 00:50:08.040
from different sides of the
transition, A plus or A minus
00:50:08.040 --> 00:50:12.260
log of absolute
value of delta a.
00:50:12.260 --> 00:50:15.280
And the ratio of the
amplitudes, which we have also
00:50:15.280 --> 00:50:18.350
said is universal,
is equal to 1.
00:50:18.350 --> 00:50:20.172
And you had
anticipated that based
00:50:20.172 --> 00:50:21.380
on [? duality ?] [INAUDIBLE].
00:50:37.600 --> 00:50:45.130
All right, so indeed, there
is a different behavior
00:50:45.130 --> 00:50:46.785
between the two models.
00:50:46.785 --> 00:50:49.730
The exact solution allows
us to go both above
00:50:49.730 --> 00:50:53.700
and below, has this
logarithmic singularity.
00:50:53.700 --> 00:51:00.010
And this expression was
first written down-- well,
00:51:00.010 --> 00:51:07.350
first published Onsager in 1944.
00:51:07.350 --> 00:51:10.870
Even couple of
years before that,
00:51:10.870 --> 00:51:14.830
he had written the expression on
boards of various conferences,
00:51:14.830 --> 00:51:17.270
saying that this is
the answer, but he
00:51:17.270 --> 00:51:19.900
didn't publish the paper.
00:51:19.900 --> 00:51:24.960
The way that he did it is based
on the transfer matrix network,
00:51:24.960 --> 00:51:26.680
as we said.
00:51:26.680 --> 00:51:32.590
Basically, we can
imagine that we
00:51:32.590 --> 00:51:37.190
have a lattice that is,
let's say, I parallel
00:51:37.190 --> 00:51:39.250
in one direction,
l perpendicular
00:51:39.250 --> 00:51:41.390
in the other direction.
00:51:41.390 --> 00:51:46.460
And then for the problems
that you had to do,
00:51:46.460 --> 00:51:51.850
the transform matrix for
one dimensional model,
00:51:51.850 --> 00:51:54.340
we can easy to do it
for a [? ladder ?].
00:51:54.340 --> 00:51:56.820
It's a 4 by 4.
00:51:56.820 --> 00:51:59.860
For this, it becomes a 2 to
the l by 2 to the l matrix.
00:52:05.850 --> 00:52:08.140
And of course, you are
interested in the limit where
00:52:08.140 --> 00:52:13.000
l goes to infinity so that
you can come 2-dimensional.
00:52:13.000 --> 00:52:18.840
And so he was able to sort
of look at this structure
00:52:18.840 --> 00:52:23.580
of this matrix, recognize that
the elements of this matrix
00:52:23.580 --> 00:52:26.980
could be represented in
terms of other matrices
00:52:26.980 --> 00:52:29.310
that had some
interesting algebra.
00:52:29.310 --> 00:52:31.970
And then he arguably
could figure out
00:52:31.970 --> 00:52:38.202
what the diagonalization looked
like in general for arbitrary
00:52:38.202 --> 00:52:42.880
l, and then calculate
log z in terms
00:52:42.880 --> 00:52:46.896
of the log of the
largest eigenvalue.
00:52:46.896 --> 00:52:52.910
I guess we have to
multiply by l parallel.
00:52:52.910 --> 00:52:58.210
And showed that, indeed,
it corresponds to this
00:52:58.210 --> 00:53:01.190
and has this phase transition.
00:53:01.190 --> 00:53:05.310
And before this solution,
people were not even sure
00:53:05.310 --> 00:53:08.440
that when you sum an expression
such as that for partition
00:53:08.440 --> 00:53:12.770
function if you ever get
a singularity because,
00:53:12.770 --> 00:53:14.950
again, on the face
of it, it's basically
00:53:14.950 --> 00:53:16.780
a sum of exponential functions.
00:53:16.780 --> 00:53:19.020
Each one of them is
perfectly analytic,
00:53:19.020 --> 00:53:20.670
sums of analytical functions.
00:53:20.670 --> 00:53:22.700
It's supposed to be analytical.
00:53:22.700 --> 00:53:29.160
The whole key lies in the limit
of taking, say, l to infinity
00:53:29.160 --> 00:53:31.430
and n to infinity,
and then you'll
00:53:31.430 --> 00:53:34.960
be able to see these
kinds of singularities.
00:53:34.960 --> 00:53:37.520
And then again,
some people thought
00:53:37.520 --> 00:53:39.320
that the only type
of singularities
00:53:39.320 --> 00:53:42.890
that you will be able to
get are the kinds of things
00:53:42.890 --> 00:53:45.200
that we saw [INAUDIBLE] point.
00:53:45.200 --> 00:53:48.210
So to see a different
type of singularity
00:53:48.210 --> 00:53:53.000
with a heat capacity that was
actually divergent and could
00:53:53.000 --> 00:53:55.240
explicitly be shown
through mathematics
00:53:55.240 --> 00:53:59.650
was quite an
interesting revelation.
00:53:59.650 --> 00:54:02.940
So the kind of
relative importance
00:54:02.940 --> 00:54:07.170
of that is that
after the war-- you
00:54:07.170 --> 00:54:12.430
can see this is all around the
time of World War II-- Casimir
00:54:12.430 --> 00:54:18.950
wrote to Pauli saying that I
have been away from thinking
00:54:18.950 --> 00:54:22.230
about physics the past few years
with the war and all of that.
00:54:22.230 --> 00:54:24.890
Anything interesting happening
in theoretical physics?
00:54:24.890 --> 00:54:28.290
And Pauli responded, well,
not much except that Onsager
00:54:28.290 --> 00:54:32.040
solved the
2-dimensionalizing model.
00:54:32.040 --> 00:54:37.200
Of course, the solution that
he has is quite obscure.
00:54:37.200 --> 00:54:43.140
And I don't think many
people understand that.
00:54:43.140 --> 00:54:52.590
Then before, the form that
people refer to was presented
00:54:52.590 --> 00:54:56.000
is actually kind of interesting,
because this paper has
00:54:56.000 --> 00:54:58.950
something about
crystal statistics,
00:54:58.950 --> 00:55:01.880
and then there's the following
paper by a different author--
00:55:01.880 --> 00:55:06.580
Bruria Kaufman, 1949--
has this same title
00:55:06.580 --> 00:55:10.130
except it goes from
number two or something.
00:55:10.130 --> 00:55:13.500
So they were clearly
talking to each other,
00:55:13.500 --> 00:55:19.950
but what she was able
to show, Bruria Kaufman,
00:55:19.950 --> 00:55:22.650
was that the structure
of these matrices
00:55:22.650 --> 00:55:25.490
can be simplified
much further and can
00:55:25.490 --> 00:55:28.160
be made to look
like spinners that
00:55:28.160 --> 00:55:30.930
are familiar from other
branches of physics.
00:55:30.930 --> 00:55:33.550
And so this kind
of 50-page paper
00:55:33.550 --> 00:55:37.510
was kind of reduced to
something like a 20-page paper.
00:55:37.510 --> 00:55:45.240
And that's the solution that
is reproduced in Wong's book.
00:55:45.240 --> 00:55:48.710
Chapter 15 of Wong has
essentially a reproduction
00:55:48.710 --> 00:55:50.540
of this.
00:55:50.540 --> 00:55:53.430
I was looking at this
because there aren't really
00:55:53.430 --> 00:55:58.100
that many women
mathematical physicists,
00:55:58.100 --> 00:56:00.870
so I was kind of
looking at her history,
00:56:00.870 --> 00:56:04.870
and she's quite
an unusual person.
00:56:04.870 --> 00:56:07.200
So it turns out
that for a while,
00:56:07.200 --> 00:56:11.100
she was mathematical
assistant to Albert Einstein.
00:56:11.100 --> 00:56:17.200
She was first married to one of
the most well known linguists
00:56:17.200 --> 00:56:18.970
of the 20th century.
00:56:18.970 --> 00:56:24.050
And for a while, they were
both in Israel in a kibbutz
00:56:24.050 --> 00:56:27.890
where this important linguist
was acting as a chauffeur
00:56:27.890 --> 00:56:29.970
and driving people around.
00:56:29.970 --> 00:56:35.400
And then later in life,
she married briefly
00:56:35.400 --> 00:56:37.300
Willis Lamb, of Lamb shift.
00:56:37.300 --> 00:56:39.920
He turned out the term, 1949.
00:56:39.920 --> 00:56:44.680
She had done some calculation
that if Lamb had paid attention
00:56:44.680 --> 00:56:47.500
to, he would have
also potentially won
00:56:47.500 --> 00:56:50.080
a Nobel Prize for
Mossbauer effect,
00:56:50.080 --> 00:56:52.210
but at that time
didn't pay attention
00:56:52.210 --> 00:56:56.900
to it so somebody
else got there first.
00:56:56.900 --> 00:56:58.850
So very interesting person.
00:56:58.850 --> 00:57:02.040
So to my mind, a good
project for somebody
00:57:02.040 --> 00:57:04.650
is to write a biography
for this person.
00:57:04.650 --> 00:57:08.010
It doesn't seem to exist.
00:57:08.010 --> 00:57:14.700
OK, so then both of these are
based on this transfer matrix
00:57:14.700 --> 00:57:16.230
method.
00:57:16.230 --> 00:57:18.330
The method that
I have given you,
00:57:18.330 --> 00:57:24.900
which is the graphical
solution, was first
00:57:24.900 --> 00:57:31.320
presented by Kac
and Ward in 1952.
00:57:31.320 --> 00:57:35.550
And it is reproduced
in Feynman's book.
00:57:35.550 --> 00:57:42.260
So Feynman apparently also
had one of these crucial steps
00:57:42.260 --> 00:57:44.830
of the conjecture with
his factor of minus 1
00:57:44.830 --> 00:57:47.750
to the power of the
number of crossings,
00:57:47.750 --> 00:57:52.860
giving you the correct
factor to do the counting.
00:57:56.770 --> 00:58:04.360
Now it turns out that I also
did not prove that statement,
00:58:04.360 --> 00:58:07.110
so there is a missing
mathematical link
00:58:07.110 --> 00:58:10.450
to make my proof of this
expression complete.
00:58:10.450 --> 00:58:14.050
And that was provided by a
mathematician called Sherman,
00:58:14.050 --> 00:58:19.080
1960, that essentially
shows very rigorously
00:58:19.080 --> 00:58:22.320
that these factors of minus
1 to the number of crossings
00:58:22.320 --> 00:58:28.680
will work out and magically
make everything happen.
00:58:28.680 --> 00:58:30.840
Now the question to
ask is the following.
00:58:34.090 --> 00:58:39.270
We expect things to
be exactly solvable
00:58:39.270 --> 00:58:42.030
when they are trivial
in some sense.
00:58:42.030 --> 00:58:44.310
Gaussian model is
exactly solvable
00:58:44.310 --> 00:58:48.420
because there is no
interaction among the modes.
00:58:48.420 --> 00:58:52.190
So why is it that the
2-dimensionalizing model
00:58:52.190 --> 00:58:54.311
is solvable?
00:58:54.311 --> 00:58:58.970
And one of the keys to
that is a realization
00:58:58.970 --> 00:59:01.770
of another way of looking
at the problem that
00:59:01.770 --> 00:59:14.870
appeared by Lieb, Mattis,
and Schultz in 1964.
00:59:14.870 --> 00:59:20.310
And so basically what
they said is let's
00:59:20.310 --> 00:59:23.090
take a look at these
pictures that I
00:59:23.090 --> 00:59:26.880
have been drawing
for the graphs.
00:59:26.880 --> 00:59:30.630
And so I have graphs
that basically
00:59:30.630 --> 00:59:34.010
on a kind of coarse level, they
look something like this maybe.
00:59:40.160 --> 00:59:46.950
And what they said was that
if we look at the transfer
00:59:46.950 --> 00:59:50.630
matrix-- the one that
Onsager and Bruria
00:59:50.630 --> 00:59:52.220
Kaufman were looking at.
00:59:52.220 --> 00:59:56.560
And the reason it was
solvable, it looked very much
00:59:56.560 --> 01:00:00.060
like you had a
system of fermions.
01:00:00.060 --> 01:00:05.420
And then the insight is that
if we look at these pictures,
01:00:05.420 --> 01:00:09.870
you can regard this as
a 1-dimensional system
01:00:09.870 --> 01:00:13.760
of fermions that is
evolving in time.
01:00:13.760 --> 01:00:17.320
And what you are looking at
these borderline histories
01:00:17.320 --> 01:00:20.740
of two particles
that are propagating.
01:00:20.740 --> 01:00:23.160
Here they annihilate each other.
01:00:23.160 --> 01:00:25.140
Here they annihilate each other.
01:00:25.140 --> 01:00:26.861
Another pair gets
created, et cetera.
01:00:30.090 --> 01:00:35.400
But in one dimensions, fermions
you can regard two ways--
01:00:35.400 --> 01:00:41.350
either they cannot occupy the
same site or you can say, well,
01:00:41.350 --> 01:00:46.360
let them occupy in same site,
but then I introduce these
01:00:46.360 --> 01:00:50.400
factors of minus 1 for
the exchange of fermions.
01:00:50.400 --> 01:00:54.060
So when two fermions cross
each other in one dimension,
01:00:54.060 --> 01:00:56.610
their position
has been exchanged
01:00:56.610 --> 01:00:59.830
so you have to put a
minus 1 for crossing.
01:00:59.830 --> 01:01:03.180
And then when you sum over all
histories for every crossing,
01:01:03.180 --> 01:01:05.680
there will be one that
touches and goes away.
01:01:05.680 --> 01:01:09.860
And the sum total of
the two of them is 0.
01:01:09.860 --> 01:01:14.310
So the point is that
at the end of the day,
01:01:14.310 --> 01:01:18.450
this theory is a theory
of three fermions.
01:01:23.050 --> 01:01:27.430
So we have not solved,
in fact, an interacting
01:01:27.430 --> 01:01:28.700
complicated problem.
01:01:28.700 --> 01:01:30.100
It is.
01:01:30.100 --> 01:01:32.470
But in the right
perspective, it looks
01:01:32.470 --> 01:01:37.240
like a bunch of fermions that
completely non-interacting pass
01:01:37.240 --> 01:01:38.760
through each other
as long as we're
01:01:38.760 --> 01:01:43.115
willing to put the minus 1 phase
that you have for crossings.
01:01:49.090 --> 01:01:53.637
One last aspect to
think about that you
01:01:53.637 --> 01:01:56.460
learn from this
fermionic perspective.
01:02:03.900 --> 01:02:05.720
Look at this expression.
01:02:05.720 --> 01:02:09.960
Why did I say that this
is a Gaussian model?
01:02:09.960 --> 01:02:15.140
We saw that we could
get this, z Gaussian,
01:02:15.140 --> 01:02:24.020
by doing an integral
essentially over weights
01:02:24.020 --> 01:02:27.470
that were continuous by i.
01:02:27.470 --> 01:02:36.300
And the weight I could click
here as kij phi i phi j.
01:02:36.300 --> 01:02:41.490
Because in principle, if I
only count an interaction once,
01:02:41.490 --> 01:02:43.260
let's say I count it twice.
01:02:43.260 --> 01:02:48.370
I could put a factor of 1/2
if I allow i and j to be some.
01:02:48.370 --> 01:02:51.280
Especially I said the weight
that I have to put here
01:02:51.280 --> 01:02:53.955
is something like
phi i squared over 2.
01:02:53.955 --> 01:03:00.610
So this essentially you
can see z of the Gaussian
01:03:00.610 --> 01:03:07.470
ultimately always
becomes something like z
01:03:07.470 --> 01:03:14.520
of the Gaussian is going to
be 1 over the square root
01:03:14.520 --> 01:03:19.010
of a determinant of whatever
the quadratic form is up there.
01:03:21.840 --> 01:03:25.870
And when you take the log
of the partition function,
01:03:25.870 --> 01:03:31.010
the square root of the
determinant becomes minus 1/2
01:03:31.010 --> 01:03:32.679
of the log of the
determinant, which
01:03:32.679 --> 01:03:33.970
is what we've been calculating.
01:03:36.560 --> 01:03:39.700
So that's obvious.
01:03:39.700 --> 01:03:42.220
You say this object
over here that you
01:03:42.220 --> 01:03:48.215
write as an answer is also
log of some determinant.
01:03:51.150 --> 01:03:56.460
So can I think of these as
some kind of a rock that
01:03:56.460 --> 01:04:01.170
is prescribed according to
these rules on a lattice,
01:04:01.170 --> 01:04:04.980
give weights to jump
according to what I have,
01:04:04.980 --> 01:04:07.081
then do a kind of
Gaussian integration
01:04:07.081 --> 01:04:08.080
and get the same answer?
01:04:10.620 --> 01:04:14.420
Well, the difficulty
is precisely
01:04:14.420 --> 01:04:19.310
this 1/2 versus minus
1/2, because when
01:04:19.310 --> 01:04:21.210
you do the Gaussian
integration, you
01:04:21.210 --> 01:04:22.885
get the determinant
in the denominator.
01:04:26.050 --> 01:04:37.230
So is there a trick to get the
determinant in the numerator?
01:04:37.230 --> 01:04:40.390
And the answer is
that people who
01:04:40.390 --> 01:04:49.450
do have integral formulations
for fermionic systems
01:04:49.450 --> 01:04:52.530
rely on coherent
states that involve
01:04:52.530 --> 01:04:57.560
anti-commuting variables,
called Grassmann variables.
01:04:57.560 --> 01:05:00.970
And the very interesting thing
about Grassmann variables
01:05:00.970 --> 01:05:06.810
is that if I do the analog
of the Gaussian integration,
01:05:06.810 --> 01:05:08.910
the answer-- rather than
being in the determinant,
01:05:08.910 --> 01:05:12.750
in the denominator--
goes into the numerator.
01:05:12.750 --> 01:05:17.930
And so one can, in fact,
rewrite this partition function,
01:05:17.930 --> 01:05:22.290
sort of working
backward, in terms
01:05:22.290 --> 01:05:28.230
of an integration over
Gaussian distributed
01:05:28.230 --> 01:05:31.910
Grassmannian variables
on the lattice,
01:05:31.910 --> 01:05:34.776
which is also equivalent
to another way of thinking
01:05:34.776 --> 01:05:35.400
about fermions.
01:05:41.550 --> 01:05:43.680
Let's see.
01:05:43.680 --> 01:05:45.660
What else is known
about this model?
01:05:48.270 --> 01:05:54.560
So I said that the specific
heat singularity is known,
01:05:54.560 --> 01:05:57.992
so we have this
alpha which is 0 log.
01:06:01.940 --> 01:06:05.280
Given that the
structure that we have
01:06:05.280 --> 01:06:11.970
involves inside the log
something that is like this,
01:06:11.970 --> 01:06:18.070
you won't be surprised that if I
think in terms of a correlation
01:06:18.070 --> 01:06:21.445
length-- so typically q's
would be an inverse correlation
01:06:21.445 --> 01:06:26.850
length, some kind of a q
at which I will be reaching
01:06:26.850 --> 01:06:30.920
appropriate saturation for
a delta t-- I will arrive
01:06:30.920 --> 01:06:35.710
at a correlation range
that diverge as delta t
01:06:35.710 --> 01:06:38.130
to the minus 1.
01:06:38.130 --> 01:06:42.210
Again, I can write it
more precisely as b plus b
01:06:42.210 --> 01:06:46.410
minus to the minus mu minus.
01:06:46.410 --> 01:06:52.840
The ratio of the b's
this is 1, and the mus
01:06:52.840 --> 01:06:55.560
are the same and equal to 1.
01:06:55.560 --> 01:07:00.845
So the correlation length
diverges with an exponent 1,
01:07:00.845 --> 01:07:04.590
and one can [INAUDIBLE]
this exactly.
01:07:04.590 --> 01:07:10.820
One can then also calculate
actual correlations
01:07:10.820 --> 01:07:18.450
at criticality and show that
that criticality correlations
01:07:18.450 --> 01:07:21.760
decay with separation
between the points
01:07:21.760 --> 01:07:27.270
that you are looking at
as 1 over r to the 1/4.
01:07:27.270 --> 01:07:32.760
So the exponent that we call
mu is 1/4 in 2 dimensions.
01:07:35.730 --> 01:07:38.240
Once you have
correlations, you certainly
01:07:38.240 --> 01:07:41.145
know that you can calculate
the susceptibility
01:07:41.145 --> 01:07:46.260
as an integral of the
correlation function.
01:07:46.260 --> 01:07:52.390
And so it's going to be an
integral b 2r over r to the 1/4
01:07:52.390 --> 01:07:54.710
that is cut off at the
correlation length.
01:07:54.710 --> 01:07:59.370
So that's going to give me c
to the power of 2 minus 1/4,
01:07:59.370 --> 01:08:01.370
which is 7/4.
01:08:01.370 --> 01:08:06.940
So that's going to diverge as
delta t 2 to the minus gamma.
01:08:06.940 --> 01:08:08.140
Gamma is, again, 7/4.
01:08:13.940 --> 01:08:20.069
So these things--
correlations-- can
01:08:20.069 --> 01:08:23.720
be calculated like
you saw already
01:08:23.720 --> 01:08:26.060
for the case of
the Gaussian model
01:08:26.060 --> 01:08:27.720
with a appropriate modification.
01:08:27.720 --> 01:08:30.470
That is, I have to
look at walks that
01:08:30.470 --> 01:08:33.470
don't come back and
close on themselves,
01:08:33.470 --> 01:08:36.939
but walks that go from one
point to another point.
01:08:36.939 --> 01:08:40.413
So the same types of techniques
that is described here
01:08:40.413 --> 01:08:44.082
will allow you to get to
all of these other results.
01:08:44.082 --> 01:08:44.910
AUDIENCE: Question?
01:08:44.910 --> 01:08:46.119
PROFESSOR: Yes?
01:08:46.119 --> 01:08:49.505
AUDIENCE: Do people try to
experiment on the builds
01:08:49.505 --> 01:08:52.170
as things that should be
[INAUDIBLE] to the Ising model?
01:08:52.170 --> 01:08:55.500
PROFESSOR: There are by now
many experimental realizations
01:08:55.500 --> 01:08:59.220
of [INAUDIBLE] Ising model
in which these exponents--
01:08:59.220 --> 01:09:01.700
actually, the next
one that I will
01:09:01.700 --> 01:09:05.060
tell you have been
confirmed very nicely.
01:09:05.060 --> 01:09:08.710
So there are a number of
2-dimensional absorb, systems,
01:09:08.710 --> 01:09:12.910
a number of systems of
mixtures in 2 dimensions
01:09:12.910 --> 01:09:14.090
that phase separate.
01:09:14.090 --> 01:09:17.662
So there's a huge number of
experimental realizations.
01:09:22.090 --> 01:09:25.250
At that time, no, because
we're talking about 70 years.
01:09:30.800 --> 01:09:32.740
So the last thing
that I want to mention
01:09:32.740 --> 01:09:39.879
is, of course, when
you go below tc,
01:09:39.879 --> 01:09:44.240
we expect that-- let's
call it temperate--
01:09:44.240 --> 01:09:48.540
when I go below temperature,
there will be a magnetization.
01:09:48.540 --> 01:09:54.530
That has always been our
signature of symmetry breaking.
01:09:54.530 --> 01:10:00.050
And so the question is,
what is the magnetization?
01:10:00.050 --> 01:10:04.830
And then this is another
interesting story,
01:10:04.830 --> 01:10:10.400
that around 1950s in a
couple of conferences,
01:10:10.400 --> 01:10:14.310
at the end of somebody's talk,
Onsager went to the board
01:10:14.310 --> 01:10:17.870
and said that he
and Bruria Kaufman
01:10:17.870 --> 01:10:28.495
have found this expression for
the magnetization of the system
01:10:28.495 --> 01:10:30.860
at low temperature as a
function of temperature
01:10:30.860 --> 01:10:33.010
or the coupling constant.
01:10:33.010 --> 01:10:39.440
But they never wrote the
solution down until in 1952,
01:10:39.440 --> 01:10:43.560
actually CN Yang
published a paper
01:10:43.560 --> 01:10:45.730
that has derived this result.
01:10:45.730 --> 01:10:50.330
And since this goes to 1
at the critical point-- why
01:10:50.330 --> 01:10:53.760
duality [INAUDIBLE]
2k [INAUDIBLE] 2k--
01:10:53.760 --> 01:10:56.000
[? dual ?] plus 1.
01:10:56.000 --> 01:10:58.430
This vanishes with
an exponent beta,
01:10:58.430 --> 01:11:02.112
which is 1/8, which is the
other exponent in this series.
01:11:10.890 --> 01:11:14.206
So as I said, there
are many people
01:11:14.206 --> 01:11:18.640
who since the '50s and
'60s devoted their life
01:11:18.640 --> 01:11:22.790
to looking at various
generalizations, extensions
01:11:22.790 --> 01:11:24.880
of the Ising model.
01:11:24.880 --> 01:11:26.860
There are many people
who try to solve it
01:11:26.860 --> 01:11:30.300
with a finite in 2 dimensions,
a finite magnetic field.
01:11:30.300 --> 01:11:31.690
You can't do that.
01:11:31.690 --> 01:11:33.610
This magnetization
is obtained only
01:11:33.610 --> 01:11:36.970
at the limit of
[INAUDIBLE] going to 0.
01:11:36.970 --> 01:11:40.150
And clearly, people
have thought a lot
01:11:40.150 --> 01:11:44.780
about doing things in 3
dimensions or higher dimensions
01:11:44.780 --> 01:11:48.000
without much success.
01:11:48.000 --> 01:11:51.680
So this is basically
the end of the portion
01:11:51.680 --> 01:11:56.690
that I had to give with
discrete models and lattices.
01:11:56.690 --> 01:12:00.400
And as of next lecture, we
will change our perspective
01:12:00.400 --> 01:12:02.160
one more time.
01:12:02.160 --> 01:12:04.720
We'll go back to
the continuum and we
01:12:04.720 --> 01:12:10.110
will look at n component
models in the low temperature
01:12:10.110 --> 01:12:14.370
expansion and see what
happens over there.
01:12:16.984 --> 01:12:17.983
Are there any questions?
01:12:20.970 --> 01:12:23.860
OK, I will give you
a preview of what
01:12:23.860 --> 01:12:28.880
I will be doing in
the next few minutes.
01:12:28.880 --> 01:12:36.100
So we have been looking at these
lattice models in the high t
01:12:36.100 --> 01:12:39.770
limit where expansion
was graphical,
01:12:39.770 --> 01:12:42.015
such as the one that
I have over there.
01:12:42.015 --> 01:12:44.430
But the partition
function turned out
01:12:44.430 --> 01:12:47.960
to be something of
two constants, as 1
01:12:47.960 --> 01:12:51.720
plus something that involves
loops and things that
01:12:51.720 --> 01:12:55.210
involve multiple
loops, et cetera.
01:12:55.210 --> 01:12:58.445
This was for the Ising model.
01:12:58.445 --> 01:13:02.165
It turns out that if I
go from the Ising model
01:13:02.165 --> 01:13:04.190
to n component space.
01:13:04.190 --> 01:13:07.280
So at each side
of the lattice, I
01:13:07.280 --> 01:13:11.430
put something that
has n components
01:13:11.430 --> 01:13:20.998
subject to the condition
that it's a unit vector.
01:13:20.998 --> 01:13:25.280
And I try to calculate
the partition function
01:13:25.280 --> 01:13:30.350
by integrating over all
of these unit vectors
01:13:30.350 --> 01:13:34.643
in n dimension of a
weight that is just
01:13:34.643 --> 01:13:40.330
a generalization of the Ising,
except that I would have
01:13:40.330 --> 01:13:44.090
the dot product of these things.
01:13:44.090 --> 01:13:49.000
And I start making appropriate
high temperature expansions
01:13:49.000 --> 01:13:50.200
for these models.
01:13:50.200 --> 01:13:53.590
I will generate a
very similar series,
01:13:53.590 --> 01:13:56.615
except that whenever
I see a loop,
01:13:56.615 --> 01:13:59.280
I have to put a
factor of n where
01:13:59.280 --> 01:14:02.240
n is the number of components.
01:14:02.240 --> 01:14:04.250
And this we already
saw when we were
01:14:04.250 --> 01:14:06.320
doing Landau-Ginzburg
expansions.
01:14:06.320 --> 01:14:10.650
We saw that the expansions
that we had over here
01:14:10.650 --> 01:14:14.770
could be graphically
interpreted as representations
01:14:14.770 --> 01:14:17.550
of the various terms that we
had in the Landau-Ginzburg
01:14:17.550 --> 01:14:18.840
expansion.
01:14:18.840 --> 01:14:25.240
And essentially, this factor
of n is the one difficulty.
01:14:25.240 --> 01:14:30.010
You can use the same methods
for numerical expansions
01:14:30.010 --> 01:14:34.080
for all n models that are
these n component models.
01:14:34.080 --> 01:14:38.170
You can't do anything
exactly with them.
01:14:38.170 --> 01:14:42.640
Now the low temperature
expansion for Ising
01:14:42.640 --> 01:14:49.650
like models we saw involved
starting with some ground
01:14:49.650 --> 01:14:57.270
state-- for example, all up--
and then including excitations
01:14:57.270 --> 01:15:04.400
that were islands of
minus in a sea of plus.
01:15:04.400 --> 01:15:09.570
And so then there higher
order in this series.
01:15:09.570 --> 01:15:13.980
Now for other discrete models,
such as the Potts model,
01:15:13.980 --> 01:15:15.680
you can use the same procedure.
01:15:15.680 --> 01:15:18.600
And again, you see that
in some of the problems
01:15:18.600 --> 01:15:21.250
that you've had to solve.
01:15:21.250 --> 01:15:26.190
But this will not work when we
come to these continuous spin
01:15:26.190 --> 01:15:33.590
markets, because for
continuous spin models,
01:15:33.590 --> 01:15:38.515
the ground state would be when
everybody is pointing in one
01:15:38.515 --> 01:15:45.280
direction, but the excitations
on this ground state
01:15:45.280 --> 01:15:49.070
are not islands that
are flipped over.
01:15:49.070 --> 01:15:58.770
They are these long
wavelength Goldstone modes,
01:15:58.770 --> 01:16:02.580
which we described
earlier in class.
01:16:02.580 --> 01:16:07.226
So if we want to
make an expansion
01:16:07.226 --> 01:16:12.840
to look at the low temperature
[INAUDIBLE] for systems with n
01:16:12.840 --> 01:16:16.670
greater than 1, we have to
make an expansion involving
01:16:16.670 --> 01:16:20.010
Goldstone modes
and, as we will see,
01:16:20.010 --> 01:16:22.110
interactions among
Goldstone modes.
01:16:22.110 --> 01:16:25.730
So you can no longer regard
at that appropriate level
01:16:25.730 --> 01:16:29.590
of sophistication as the
Goldstone modes maintain
01:16:29.590 --> 01:16:31.770
independence from each other.
01:16:31.770 --> 01:16:36.160
And very roughly, the main
difference between discrete
01:16:36.160 --> 01:16:40.500
and these continuous symmetry
models is captured as follows.
01:16:40.500 --> 01:16:46.510
Suppose I have a huge
system that is L by L
01:16:46.510 --> 01:16:51.654
and I impose boundary conditions
on one side where all of this
01:16:51.654 --> 01:16:56.850
spins in one
direction, and ask what
01:16:56.850 --> 01:17:02.960
happens if I impose a different
condition on the other side.
01:17:02.960 --> 01:17:08.010
Well, what would happen is you
would have a domain boundary.
01:17:08.010 --> 01:17:10.370
And the cost of
that domain boundary
01:17:10.370 --> 01:17:13.950
will be proportional to
the area of the boundary
01:17:13.950 --> 01:17:16.000
in d dimensions.
01:17:16.000 --> 01:17:23.420
So the cost is
proportional to some energy
01:17:23.420 --> 01:17:26.580
per bond times L
to the d minus 1.
01:17:30.710 --> 01:17:34.930
Whereas if I try to do the same
thing for a continuous spin
01:17:34.930 --> 01:17:43.360
system, if I align one side like
this, the other side like this,
01:17:43.360 --> 01:17:48.940
but in-between I can gradually
change from one direction
01:17:48.940 --> 01:17:51.740
to another direction.
01:17:51.740 --> 01:17:54.050
And the cost would
be the gradient,
01:17:54.050 --> 01:17:59.830
which is 1 over L
squared times integrated
01:17:59.830 --> 01:18:01.900
over the entire system.
01:18:01.900 --> 01:18:05.570
So the energy cost
of this excitation
01:18:05.570 --> 01:18:13.390
will be some parameter
j, 1 over L-- which
01:18:13.390 --> 01:18:18.120
is the shift-- squared-- which
is the [? strain ?] squared--
01:18:18.120 --> 01:18:19.845
integrated over
the entire volume,
01:18:19.845 --> 01:18:24.760
so it goes L to the d minus 2.
01:18:24.760 --> 01:18:31.000
So we can see that these
systems are much softer
01:18:31.000 --> 01:18:33.990
than discrete systems.
01:18:33.990 --> 01:18:36.930
For discrete systems,
thermal fluctuations
01:18:36.930 --> 01:18:41.862
are sufficient to destroy
order at low temperature
01:18:41.862 --> 01:18:47.660
as soon as this cost is of the
order of kt, which for large L
01:18:47.660 --> 01:18:51.850
can only happen in d
[INAUDIBLE] 1 and lower.
01:18:51.850 --> 01:18:56.930
Whereas for these systems, it
happens in d of 2 and lower.
01:18:56.930 --> 01:18:59.940
So the lower critical
dimension for these models.
01:18:59.940 --> 01:19:03.370
With continuous symmetry,
we already saw it's 2.
01:19:03.370 --> 01:19:06.715
For these models, it is 1.
01:19:06.715 --> 01:19:09.520
Now we are going to be
interested in this class
01:19:09.520 --> 01:19:11.070
of models.
01:19:11.070 --> 01:19:14.380
I have told you that
in 2 dimensions,
01:19:14.380 --> 01:19:17.400
they should not order.
01:19:17.400 --> 01:19:22.210
So presumably, the critical
temperature-- if I continuously
01:19:22.210 --> 01:19:24.700
regard it as a
function of dimension,
01:19:24.700 --> 01:19:30.600
will go to 0 as a function
of dimension as d minus 2.
01:19:30.600 --> 01:19:34.610
So the insight of
Polyakov was that maybe we
01:19:34.610 --> 01:19:38.570
can look at the interactions
of these old Goldstone modes
01:19:38.570 --> 01:19:41.830
and do a systematic low
temperature expansion that
01:19:41.830 --> 01:19:47.056
reaches the phase transition of
a critical point systematically
01:19:47.056 --> 01:19:48.830
ind minus 2.
01:19:48.830 --> 01:19:51.970
And that's what we will
attempt in the future.