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PROFESSOR: OK, let's start.
00:00:25.710 --> 00:00:30.460
So we are going to change
perspective again and think
00:00:30.460 --> 00:00:32.070
in terms of lattice models.
00:00:38.750 --> 00:00:42.810
So for the first
part of this course,
00:00:42.810 --> 00:00:45.780
I was trying to change
your perspective
00:00:45.780 --> 00:00:49.400
from thinking in terms of
microscopic degrees of freedom
00:00:49.400 --> 00:00:51.640
to a statistical field.
00:00:51.640 --> 00:00:55.930
Now we are going to go back
and try to build pictures
00:00:55.930 --> 00:00:59.310
around things that
look more microscopic.
00:00:59.310 --> 00:01:03.860
Typically, in many solid
state configurations,
00:01:03.860 --> 00:01:09.860
we are dealing with transitions
that take place on a lattice.
00:01:09.860 --> 00:01:14.440
For example, imagine a square
lattice, which is easy to draw,
00:01:14.440 --> 00:01:19.570
but there could be all kinds
of cubic and more complex
00:01:19.570 --> 00:01:20.970
lattices.
00:01:20.970 --> 00:01:24.730
And then, at each
side of this lattice,
00:01:24.730 --> 00:01:28.380
you may have one microscopic
degrees of freedom
00:01:28.380 --> 00:01:32.640
that is ultimately participating
in the ordering and the phase
00:01:32.640 --> 00:01:34.930
transition that
you have in mind.
00:01:34.930 --> 00:01:37.130
Could, for example,
be a spin, or it
00:01:37.130 --> 00:01:42.670
could be one atom
in a binary mixture.
00:01:42.670 --> 00:01:48.010
And what you would like to do
is to construct a partition
00:01:48.010 --> 00:01:54.330
function, again, by summing
over all degrees of freedom.
00:01:54.330 --> 00:01:58.220
And we need some
kind of Hamiltonian.
00:01:58.220 --> 00:02:01.510
And what we are going
to assume governs
00:02:01.510 --> 00:02:06.950
that Hamiltonian is the analog
of this locality assumption
00:02:06.950 --> 00:02:09.759
that we have in
statistical field theories.
00:02:09.759 --> 00:02:11.590
That is, we are going
to assume that one
00:02:11.590 --> 00:02:13.960
of our degrees of
freedom essentially
00:02:13.960 --> 00:02:18.100
talks to a small
neighborhood around it.
00:02:18.100 --> 00:02:19.900
And the simplest
neighborhood would
00:02:19.900 --> 00:02:24.970
be to basically talk to
the nearest neighbor.
00:02:24.970 --> 00:02:29.730
So if I, for example,
assign index i
00:02:29.730 --> 00:02:32.420
to each side of
the lattice, let's
00:02:32.420 --> 00:02:36.880
say I have some variable at each
side that could be something.
00:02:36.880 --> 00:02:40.430
Let's call it SI.
00:02:40.430 --> 00:02:42.360
Then my partition
function would be
00:02:42.360 --> 00:02:47.800
obtained by summing
over all configurations.
00:02:47.800 --> 00:02:53.090
And the weight I'm going to
assume in terms of this lattice
00:02:53.090 --> 00:02:58.220
picture to be a sum
over interactions
00:02:58.220 --> 00:03:01.360
that exist between
pairs of sides.
00:03:01.360 --> 00:03:05.680
So that's already an assumption
that it's a pairwise thing.
00:03:05.680 --> 00:03:11.720
And I'm going to use this
symbol ij with an angular
00:03:11.720 --> 00:03:15.390
bracket along it,
around it, to indicate
00:03:15.390 --> 00:03:19.605
the sum over nearest neighbors.
00:03:25.230 --> 00:03:30.870
And there is some
function of the variables
00:03:30.870 --> 00:03:32.518
that live on these neighbors.
00:03:36.790 --> 00:03:40.490
So basically, in the
picture that I have drawn,
00:03:40.490 --> 00:03:47.250
interactions exist only between
the places that you see lines.
00:03:47.250 --> 00:03:49.630
So that this pin
does not interact
00:03:49.630 --> 00:03:52.930
with this pin, this
pin, or this pin,
00:03:52.930 --> 00:03:55.710
but it interacts
with these four pins,
00:03:55.710 --> 00:03:57.610
to which it is near neighbor.
00:04:00.940 --> 00:04:03.120
Now clearly, the form
of the interaction
00:04:03.120 --> 00:04:07.410
has to be dictated by
your degrees of freedom.
00:04:07.410 --> 00:04:12.900
And the idea of
this representation,
00:04:12.900 --> 00:04:16.040
as opposed to the previous
statistical field theory
00:04:16.040 --> 00:04:21.519
that we had, is that in
several important instances,
00:04:21.519 --> 00:04:27.190
you may want to know not only
what these universal properties
00:04:27.190 --> 00:04:32.170
are, but also, let's say, the
explicit temperature or phase
00:04:32.170 --> 00:04:36.630
diagram of the system as a
function of external parameters
00:04:36.630 --> 00:04:38.140
as well as temperature.
00:04:38.140 --> 00:04:43.610
And if you have some idea of
how your microscopic degrees
00:04:43.610 --> 00:04:45.940
of freedom interact
with each other,
00:04:45.940 --> 00:04:50.630
you should be able to solve
this kind of partition function,
00:04:50.630 --> 00:04:52.470
get the singularities,
et cetera.
00:04:55.390 --> 00:05:01.690
So let's look at some simple
versions of this construction
00:05:01.690 --> 00:05:03.930
and gradually discuss
the kinds of things
00:05:03.930 --> 00:05:05.338
that we could do with it.
00:05:07.970 --> 00:05:09.695
So some simple models.
00:05:12.570 --> 00:05:22.680
The simplest one
is the Ising model
00:05:22.680 --> 00:05:27.090
where the variable that
you have at each side
00:05:27.090 --> 00:05:29.750
has two possibilities.
00:05:29.750 --> 00:05:37.320
So it's a binary variable in
the context of a binary alloy.
00:05:37.320 --> 00:05:40.780
It could be, let's
say, atom A or atom
00:05:40.780 --> 00:05:44.800
B that is occupying
a particular site.
00:05:44.800 --> 00:05:48.080
There are also cases
where there will be some--
00:05:48.080 --> 00:05:51.450
if this is a surface and
you're absorbing particles
00:05:51.450 --> 00:05:55.120
on top of it, there could be
a particle sitting here or not
00:05:55.120 --> 00:05:55.790
sitting here.
00:05:55.790 --> 00:05:59.690
So that would be also another
example of a binary variable.
00:05:59.690 --> 00:06:06.010
So you could indicate that by
empty or occupied, zero or one.
00:06:06.010 --> 00:06:10.470
But let's indicate it by
plus or minus one as the two
00:06:10.470 --> 00:06:11.995
possible values
that you can have.
00:06:15.640 --> 00:06:20.330
Now, if I look at the
analog of the interaction
00:06:20.330 --> 00:06:24.120
that I have between two
sites that are neighboring
00:06:24.120 --> 00:06:25.980
each other, what
can I write down?
00:06:29.684 --> 00:06:33.560
Well, the most general
thing that I can write down
00:06:33.560 --> 00:06:38.630
is, first of all, a
constant I can put,
00:06:38.630 --> 00:06:41.250
such as shift of energy.
00:06:41.250 --> 00:06:46.280
There could be a linear
term in-- let's put
00:06:46.280 --> 00:06:48.540
all of these with minus signs.
00:06:48.540 --> 00:06:51.520
Sigma i and sigma j.
00:06:51.520 --> 00:06:57.080
I assume that it is symmetric
with respect to the two sides.
00:06:57.080 --> 00:06:59.730
For reasons to become
apparent shortly,
00:06:59.730 --> 00:07:04.130
I will divide by z, which
is the coordination number.
00:07:07.350 --> 00:07:11.100
How many bonds per side?
00:07:15.920 --> 00:07:17.380
And then the next
term that I can
00:07:17.380 --> 00:07:23.480
put is something like
j sigma i sigma j.
00:07:23.480 --> 00:07:27.340
And actually, I can't
put anything else.
00:07:27.340 --> 00:07:31.270
Because if you s of this as
a power series in sigma i
00:07:31.270 --> 00:07:34.940
and sigma j, and sigma
has only two values,
00:07:34.940 --> 00:07:35.950
it will terminate here.
00:07:35.950 --> 00:07:39.890
Because any higher power of
sigma is either one or sigma
00:07:39.890 --> 00:07:40.390
itself.
00:07:43.010 --> 00:07:46.930
So another way of writing
this is that the partition
00:07:46.930 --> 00:07:53.670
function of the Ising model is
obtained by summing over all 2
00:07:53.670 --> 00:07:56.043
to the n configurations
that I have.
00:07:56.043 --> 00:07:59.230
If I have a lattice
of n sides, each side
00:07:59.230 --> 00:08:11.270
can have two possibilities of
a kind of Hamiltonian, which
00:08:11.270 --> 00:08:20.180
is basically some constant
plus h sum over i sigma i kind
00:08:20.180 --> 00:08:24.250
of field that prefers one
side or the other side.
00:08:24.250 --> 00:08:29.010
So it is an analog of a magnetic
field in this binary system.
00:08:29.010 --> 00:08:34.010
And basically, I convert it from
a description that is overall
00:08:34.010 --> 00:08:37.039
balanced to a description
overall sides.
00:08:37.039 --> 00:08:40.690
And that's why I had put the
coordination number there.
00:08:40.690 --> 00:08:44.630
It's kind of a
matter of convention.
00:08:44.630 --> 00:08:53.010
And then a term that
prefers neighboring sites
00:08:53.010 --> 00:08:55.480
to be aligned as
long as k positive.
00:09:00.560 --> 00:09:03.338
So that's one
example of a model.
00:09:06.760 --> 00:09:12.050
Another model that
we will also look at
00:09:12.050 --> 00:09:15.540
is what I started to
draw at the beginning.
00:09:15.540 --> 00:09:19.820
That is, at each site,
you have a vector.
00:09:19.820 --> 00:09:26.340
And again, going in terms of
the pictures that we had before,
00:09:26.340 --> 00:09:31.600
let's imagine that we have a
vector that has n components.
00:09:31.600 --> 00:09:39.980
So SI is something
that is in RN.
00:09:39.980 --> 00:09:45.380
And I will assume that the
magnitude of this vector
00:09:45.380 --> 00:09:47.260
is one.
00:09:47.260 --> 00:09:49.130
So essentially,
imagine that you have
00:09:49.130 --> 00:09:55.050
a unit vector, and each
site that can rotate.
00:09:55.050 --> 00:10:03.210
So if n equals to one, you
have essentially one component
00:10:03.210 --> 00:10:04.970
vector.
00:10:04.970 --> 00:10:07.610
Its square has to
be one, so the two
00:10:07.610 --> 00:10:11.170
values that it can have
are plus one and minus one.
00:10:11.170 --> 00:10:18.140
So the n equals to one case
is the Ising model again.
00:10:18.140 --> 00:10:22.200
So this ON is a
generalization of the Ising
00:10:22.200 --> 00:10:25.630
model to multiple components.
00:10:25.630 --> 00:10:30.320
n equals to two corresponds
to a unit vector
00:10:30.320 --> 00:10:34.620
that can take any angle
in two dimensions.
00:10:34.620 --> 00:10:39.130
And that is usually
given the name xy model.
00:10:42.310 --> 00:10:44.970
n equals to three is
something that maybe you
00:10:44.970 --> 00:10:52.540
want to use to describe
magnetic ions in this lattice.
00:10:52.540 --> 00:10:57.120
And classically, the direction
of this ion could be anywhere.
00:10:57.120 --> 00:11:01.130
The spin of the
ion can be anywhere
00:11:01.130 --> 00:11:03.870
on the surface of a sphere.
00:11:03.870 --> 00:11:06.470
So that's three
components, and this model
00:11:06.470 --> 00:11:08.392
is sometimes called
the Heisenberg model.
00:11:17.420 --> 00:11:19.051
Yes.
00:11:19.051 --> 00:11:21.012
AUDIENCE: In the
Ising model, what's
00:11:21.012 --> 00:11:27.520
the correspondence between g
hat h hat j hat and g, h, and k?
00:11:27.520 --> 00:11:28.580
PROFESSOR: Minus beta.
00:11:28.580 --> 00:11:29.080
Yeah.
00:11:29.080 --> 00:11:31.570
So maybe I should have written.
00:11:31.570 --> 00:11:38.910
In order to take this-- if I
think of this as energy pair
00:11:38.910 --> 00:11:44.310
bond, then in order to
get the Boltzmann weight,
00:11:44.310 --> 00:11:50.860
I have to put a minus beta.
00:11:50.860 --> 00:11:54.120
So I would have said
that this g, for example,
00:11:54.120 --> 00:11:56.850
is minus beta g hat.
00:11:56.850 --> 00:12:00.790
k is minus beta j hat.
00:12:00.790 --> 00:12:03.790
And h is minus beta h hat.
00:12:11.390 --> 00:12:13.970
So I should have
emphasized that.
00:12:13.970 --> 00:12:18.920
What I meant by b--
actually, I wrote
00:12:18.920 --> 00:12:24.680
b-- so what I should have
done here to be consistent,
00:12:24.680 --> 00:12:26.470
let's write this as minus beta.
00:12:34.340 --> 00:12:34.840
Thank you.
00:12:34.840 --> 00:12:37.930
That was important.
00:12:40.820 --> 00:12:44.690
If I described these b's
as being energies, then
00:12:44.690 --> 00:12:46.817
minus beta times
that will be what
00:12:46.817 --> 00:12:48.150
will go in the Boltzmann factor.
00:12:53.530 --> 00:12:56.160
OK
00:12:56.160 --> 00:13:01.130
So whereas these were examples
that we had more or less seen
00:13:01.130 --> 00:13:05.920
their continuum version in
the Landau-Ginzburg model,
00:13:05.920 --> 00:13:09.580
there are other symmetries
that get broken.
00:13:09.580 --> 00:13:11.900
And things for which
we didn't discuss
00:13:11.900 --> 00:13:15.630
what the corresponding
statistical field theory is.
00:13:15.630 --> 00:13:19.220
A commonly used case
pertains to something
00:13:19.220 --> 00:13:20.470
that's called a Potts model.
00:13:23.770 --> 00:13:28.470
Where at each side
you have a variable,
00:13:28.470 --> 00:13:33.200
let's call it SI,
that takes q values,
00:13:33.200 --> 00:13:35.870
one, two, three,
all the way to q.
00:13:39.580 --> 00:13:42.200
And I can write
what would appear
00:13:42.200 --> 00:13:47.080
in the exponent
minus beta h to be
00:13:47.080 --> 00:13:50.770
a sum over nearest neighbors.
00:13:53.520 --> 00:14:00.480
And I can give some kind of
an interaction parameter here,
00:14:00.480 --> 00:14:04.870
but a delta si sj.
00:14:04.870 --> 00:14:08.900
So basically, what it says is
that on each side of a lattice,
00:14:08.900 --> 00:14:10.920
you put a number.
00:14:10.920 --> 00:14:13.820
Could be one, two,
three, up to q.
00:14:13.820 --> 00:14:18.710
And if two neighbors are
the same, they like it,
00:14:18.710 --> 00:14:21.980
and they gain some
kind of a weight.
00:14:21.980 --> 00:14:26.740
That is, if k is positive,
encourages that to happen.
00:14:26.740 --> 00:14:29.660
If the two neighbors
are different,
00:14:29.660 --> 00:14:30.970
then it doesn't matter.
00:14:30.970 --> 00:14:33.050
You don't gain energy,
but you don't really
00:14:33.050 --> 00:14:36.510
care as to which one it is.
00:14:36.510 --> 00:14:42.620
The underlying symmetry that
this has is permutation.
00:14:42.620 --> 00:14:47.400
Basically, if you were to
permute all of these indices
00:14:47.400 --> 00:14:50.420
consistently across the
lattice in any particular way,
00:14:50.420 --> 00:14:53.010
the energy would not change.
00:14:53.010 --> 00:15:04.930
So permutation symmetry
is what underlies this.
00:15:04.930 --> 00:15:12.640
And again, if I look at the
case of two, then at each side,
00:15:12.640 --> 00:15:15.190
let's say I have one or two.
00:15:15.190 --> 00:15:19.390
And one one and two two are
things that gain energy.
00:15:19.390 --> 00:15:21.530
One two or two one don't.
00:15:21.530 --> 00:15:25.110
Clearly it is the same
as the Ising model.
00:15:25.110 --> 00:15:29.310
So q equals to two is another
way of writing the Ising model.
00:15:32.680 --> 00:15:36.810
Q equals to three is something
that we haven't seen.
00:15:36.810 --> 00:15:40.760
So at each site, there
are three possibilities.
00:15:40.760 --> 00:15:47.040
Actually, when I started at MIT
as a graduate student in 1979,
00:15:47.040 --> 00:15:50.770
the project that I had
to do related to the q
00:15:50.770 --> 00:15:53.660
equals to three Potts model.
00:15:53.660 --> 00:15:55.280
Where did it come from?
00:15:55.280 --> 00:16:00.620
Well, at that time,
people were looking
00:16:00.620 --> 00:16:03.846
at surface of graphite,
which as you know,
00:16:03.846 --> 00:16:08.910
has this hexagonal structure.
00:16:08.910 --> 00:16:12.230
And then you can
absorb molecules
00:16:12.230 --> 00:16:15.910
on top of that, such as,
for example, krypton.
00:16:15.910 --> 00:16:19.190
And krypton would
want to come and sit
00:16:19.190 --> 00:16:22.620
in the center of one
of these hexagons.
00:16:22.620 --> 00:16:26.320
But its size was such
that once it sat there,
00:16:26.320 --> 00:16:29.800
you couldn't occupy
any of these sides.
00:16:29.800 --> 00:16:35.340
So the next one would have
to go, let's say, over here.
00:16:35.340 --> 00:16:41.600
Now, it is possible to
subdivide this set of hexagons
00:16:41.600 --> 00:16:43.820
into three sub lattices.
00:16:43.820 --> 00:16:49.300
One, two, three, one,
two, three, et cetera.
00:16:49.300 --> 00:16:51.750
Actually, I drew
this one incorrectly.
00:16:51.750 --> 00:16:54.690
It would be sitting here.
00:16:54.690 --> 00:17:01.230
And what happens is that
basically the agile particles
00:17:01.230 --> 00:17:06.550
would order by occupying one of
three equivalent sublattices.
00:17:06.550 --> 00:17:10.630
So the way that that order
got destroyed was then
00:17:10.630 --> 00:17:17.020
described by the q equals to
three Potts universality class.
00:17:17.020 --> 00:17:18.890
You can think of
something like q
00:17:18.890 --> 00:17:26.339
equals to four that would
have a symmetry of a tetragon.
00:17:26.339 --> 00:17:30.820
And so some structure that
is like a tetragon getting,
00:17:30.820 --> 00:17:34.250
let's say, distorted in
some particular direction
00:17:34.250 --> 00:17:38.250
would then have four equivalent
directions, et cetera.
00:17:38.250 --> 00:17:42.510
So there's a whole set of other
types of universality classes
00:17:42.510 --> 00:17:46.460
and symmetry breakings that
we did not discuss before.
00:17:46.460 --> 00:17:49.110
And I just want
to emphasize that
00:17:49.110 --> 00:17:53.420
what we discussed before does
not cover all possible symmetry
00:17:53.420 --> 00:17:54.510
breakings.
00:17:54.510 --> 00:17:58.100
It was just supposed to
show you an important class
00:17:58.100 --> 00:18:01.090
and the technology
to deal with that.
00:18:01.090 --> 00:18:03.570
But again, in this
particular system,
00:18:03.570 --> 00:18:07.460
let's say you really wanted
to know at what temperature
00:18:07.460 --> 00:18:11.890
the phase transition occurs,
as well as what potential phase
00:18:11.890 --> 00:18:14.340
diagrams and
critical behavior is.
00:18:14.340 --> 00:18:16.570
And then you would
say, well, even
00:18:16.570 --> 00:18:19.490
if I could construct a
statistical field theory
00:18:19.490 --> 00:18:21.460
and analyze it in
two dimensions,
00:18:21.460 --> 00:18:24.170
and we've seen how
hard it is to go
00:18:24.170 --> 00:18:27.000
below some other
critical dimension,
00:18:27.000 --> 00:18:30.810
it doesn't tell me things about
phase diagrams, et cetera.
00:18:30.810 --> 00:18:33.960
So maybe trying to understand
and deal with this lattice
00:18:33.960 --> 00:18:38.120
model itself would tell
us more information.
00:18:38.120 --> 00:18:43.280
Although about quantities that
are not necessarily inverse.
00:18:43.280 --> 00:18:46.770
Depending on your
microscopic model,
00:18:46.770 --> 00:18:49.840
you may try to introduce
more complicated systems,
00:18:49.840 --> 00:18:53.870
such as inspired by quantum
mechanics, you can think
00:18:53.870 --> 00:18:56.730
of something that
I'll call a spin S
00:18:56.730 --> 00:19:04.980
model in which your SI
takes values from minus s,
00:19:04.980 --> 00:19:11.260
minus s plus 1, all
the way to plus s.
00:19:11.260 --> 00:19:14.330
There's 2s plus 1 possibilities.
00:19:14.330 --> 00:19:18.110
And you can think of this
as components of, say,
00:19:18.110 --> 00:19:23.640
a quantum spin of s
along the zed axis.
00:19:23.640 --> 00:19:28.410
Write some kind of
Hamiltonian for this.
00:19:28.410 --> 00:19:31.235
But as long as you deal
with things classically,
00:19:31.235 --> 00:19:33.980
it turns out that
this kind of system
00:19:33.980 --> 00:19:37.942
will not really have different
universality from the Ising
00:19:37.942 --> 00:19:38.442
model.
00:19:42.110 --> 00:19:46.830
So let's say we have
this lattice model.
00:19:46.830 --> 00:19:48.420
Then what can we do?
00:19:52.360 --> 00:19:56.280
So in the next
set of lectures, I
00:19:56.280 --> 00:20:02.380
will describe some tools for
dealing with these models.
00:20:02.380 --> 00:20:08.110
One set of approaches, the
one that we will start today,
00:20:08.110 --> 00:20:20.080
has to do with the position
space renormalization group.
00:20:20.080 --> 00:20:22.340
That is the approach
that we were following
00:20:22.340 --> 00:20:24.900
for renormalization previously.
00:20:24.900 --> 00:20:27.500
Dealt with going
to Fourier space.
00:20:27.500 --> 00:20:30.110
We had this sphere,
hyper sphere.
00:20:30.110 --> 00:20:33.010
And then we were basically
eliminating modes
00:20:33.010 --> 00:20:37.340
at the edge of this
sphere in Fourier space.
00:20:37.340 --> 00:20:40.100
We started actually by
describing the process
00:20:40.100 --> 00:20:41.130
in real space.
00:20:41.130 --> 00:20:44.040
So we will see
that in some cases,
00:20:44.040 --> 00:20:48.520
it is possible to do a
renormalization group directly
00:20:48.520 --> 00:20:49.610
on these lattice models.
00:20:53.760 --> 00:21:00.860
Second thing is, it turns out
that as combinatorial problems,
00:21:00.860 --> 00:21:05.600
some, but a very small
subset of these models,
00:21:05.600 --> 00:21:07.730
are susceptible to
exact solutions.
00:21:10.460 --> 00:21:14.020
Turns out that practically
all models in one dimension,
00:21:14.020 --> 00:21:18.990
as we will start today,
one can solve exactly.
00:21:18.990 --> 00:21:21.810
But there's one
prominent case, which
00:21:21.810 --> 00:21:25.450
is the two dimensionalizing
model that one can also
00:21:25.450 --> 00:21:26.680
solve exactly.
00:21:26.680 --> 00:21:29.450
And it's a very
interesting solution
00:21:29.450 --> 00:21:36.030
that we will also examine in, I
don't know, a couple of weeks.
00:21:36.030 --> 00:21:38.310
Finally, there are
approximate schemes
00:21:38.310 --> 00:21:44.680
that people have evolved for
studying these problems, where
00:21:44.680 --> 00:21:52.980
you have series expansions
starting from limits, where
00:21:52.980 --> 00:21:55.320
you know what is happening.
00:21:55.320 --> 00:21:58.570
And one simple
example would be to go
00:21:58.570 --> 00:22:01.320
to very high temperatures.
00:22:01.320 --> 00:22:04.910
And at high temperatures,
essentially every degree
00:22:04.910 --> 00:22:07.290
of freedom does what it wants.
00:22:07.290 --> 00:22:09.810
So it's essentially a
zero dimensional problem
00:22:09.810 --> 00:22:11.080
that you can solve.
00:22:11.080 --> 00:22:14.960
And then you can start treating
interactions perturbatively.
00:22:14.960 --> 00:22:18.780
So this is kind of
similar to the expansions
00:22:18.780 --> 00:22:23.720
that we had developed in 8
333, the virial expansions,
00:22:23.720 --> 00:22:27.530
et cetera, about
the ideal gas limit.
00:22:27.530 --> 00:22:30.940
But now done on a system
that is a lattice,
00:22:30.940 --> 00:22:33.240
and going to
sufficiently high order
00:22:33.240 --> 00:22:38.070
that you can say something
about the phase transition.
00:22:38.070 --> 00:22:41.570
There is another extreme.
00:22:41.570 --> 00:22:45.740
In these systems, typically
the zero temperature state
00:22:45.740 --> 00:22:46.600
is trivial.
00:22:46.600 --> 00:22:48.050
It is perfectly ordered.
00:22:48.050 --> 00:22:50.480
Let's say all the
spins are aligned.
00:22:50.480 --> 00:22:53.760
And then you can start
expanding in excitations
00:22:53.760 --> 00:22:57.220
around that state and
see whether eventually,
00:22:57.220 --> 00:23:00.030
by including more
and more excitations,
00:23:00.030 --> 00:23:04.730
you can see the phase transition
out of the ordered state.
00:23:04.730 --> 00:23:07.790
And something that is actually
probably the most common use
00:23:07.790 --> 00:23:12.090
of these models, but I
won't cover in class,
00:23:12.090 --> 00:23:16.470
is to put them on the computer
and do some kind of a Monte
00:23:16.470 --> 00:23:23.320
Carlo simulation,
which is essentially
00:23:23.320 --> 00:23:27.270
a numerical way of trying to
generate configurations that
00:23:27.270 --> 00:23:29.260
are governed by this weight.
00:23:29.260 --> 00:23:33.570
And by changing the temperature
as it appears in that weight,
00:23:33.570 --> 00:23:36.134
whether or not one can,
in the simulation, see
00:23:36.134 --> 00:23:37.300
the phase transition happen.
00:23:41.210 --> 00:23:44.330
So that's the change
in perspective
00:23:44.330 --> 00:23:47.060
that I want you to have.
00:23:47.060 --> 00:23:49.860
So the first thing
that we're going to do
00:23:49.860 --> 00:23:56.100
is to do the number one here,
to do the position space
00:23:56.100 --> 00:24:05.290
renormalization group of
one dimensional Ising model.
00:24:05.290 --> 00:24:09.030
And the procedure that
I describe for you
00:24:09.030 --> 00:24:11.190
is sufficiently general
that in fact you
00:24:11.190 --> 00:24:15.070
can apply to any other
one dimensional model,
00:24:15.070 --> 00:24:19.660
as long as you only have these
nearest neighbor interactions.
00:24:19.660 --> 00:24:26.890
So here you have a lattice
that is one dimensional.
00:24:26.890 --> 00:24:31.720
So you have a set
of sites one, two.
00:24:31.720 --> 00:24:38.820
At some point, you have i,
i minus one, i plus one.
00:24:38.820 --> 00:24:41.200
Let's say we call
the last one n.
00:24:43.910 --> 00:24:46.060
So there are n sites.
00:24:46.060 --> 00:24:51.940
There are going to be 2 to
the n possible configurations.
00:24:51.940 --> 00:24:55.300
And your task is given
that at each site,
00:24:55.300 --> 00:24:59.440
there's a variable
that is binary.
00:24:59.440 --> 00:25:03.020
You want to calculate a
partition function, which
00:25:03.020 --> 00:25:08.700
is a sum over all these 2
to the n configurations.
00:25:08.700 --> 00:25:14.380
Of a weight that is
this e to the sum
00:25:14.380 --> 00:25:22.220
over i B, the interaction
that couples SI and SI plus 1.
00:25:22.220 --> 00:25:29.400
Maybe I should have
called this e hat.
00:25:29.400 --> 00:25:34.132
And B is the thing that has
minus beta absorbed in it.
00:25:39.130 --> 00:25:43.350
So notice that basically, the
way that I have written it,
00:25:43.350 --> 00:25:46.380
one is interacting with two.
00:25:46.380 --> 00:25:48.740
i minus one is
interacting with i.
00:25:48.740 --> 00:25:50.850
i is interacting
with i plus one.
00:25:50.850 --> 00:25:53.750
So I wrote the nearest
neighbor interaction
00:25:53.750 --> 00:25:57.170
in this particular fashion.
00:25:57.170 --> 00:26:00.890
We may or may not worry
about the last spin,
00:26:00.890 --> 00:26:04.980
whether I want to
finish the series here,
00:26:04.980 --> 00:26:08.810
or sometimes I will use
periodic boundary condition
00:26:08.810 --> 00:26:12.280
and bring it back and couple
it to the first one, where
00:26:12.280 --> 00:26:13.300
I have a ring.
00:26:13.300 --> 00:26:14.550
So that's another possibility.
00:26:17.320 --> 00:26:21.880
Doesn't really matter all
that much at this stage.
00:26:21.880 --> 00:26:30.330
So this runs for i
going from one to n.
00:26:30.330 --> 00:26:34.320
There are n degrees of freedom.
00:26:34.320 --> 00:26:37.830
Now, renormalization
group is a procedure
00:26:37.830 --> 00:26:43.150
by which I get rid of
some degrees of freedom.
00:26:43.150 --> 00:26:47.270
So previously, I have
emphasized that what we did
00:26:47.270 --> 00:26:50.200
was some kind of an averaging.
00:26:50.200 --> 00:26:51.900
So we said that
let's say I could
00:26:51.900 --> 00:26:55.060
do some averaging of
three sites and call
00:26:55.060 --> 00:26:58.930
some kind of a representative
of those three.
00:26:58.930 --> 00:27:04.210
Let's say that we
want to do a RG
00:27:04.210 --> 00:27:06.780
by a factor of b equals to two.
00:27:09.790 --> 00:27:17.230
So then maybe you say that
I will pick sigma i prime
00:27:17.230 --> 00:27:24.330
and u sigma i to be sigma i
plus sigma i plus 1 over 2,
00:27:24.330 --> 00:27:27.230
doing some kind of an average.
00:27:27.230 --> 00:27:31.120
The problem with this choice
is that if the two spins are
00:27:31.120 --> 00:27:33.290
both pluses, I will get plus.
00:27:33.290 --> 00:27:35.375
If they're both minuses,
I will get minus.
00:27:35.375 --> 00:27:39.680
If there is one plus and
one minus, I will get zero.
00:27:39.680 --> 00:27:42.860
Why that is not
nice is that that
00:27:42.860 --> 00:27:46.370
changes the structure
of the theory.
00:27:46.370 --> 00:27:48.810
So I started with
binary variables.
00:27:48.810 --> 00:27:50.110
I do this rescaling.
00:27:50.110 --> 00:27:56.390
If I choose this scheme, I will
have three variables per site.
00:27:56.390 --> 00:28:01.340
But I can insist upon keeping
two variables per site, as long
00:28:01.340 --> 00:28:05.460
as I do everything
consistently and precisely.
00:28:05.460 --> 00:28:11.140
So maybe I can say that
when this occurs, where
00:28:11.140 --> 00:28:17.400
the two sites are different,
and the average would be zero,
00:28:17.400 --> 00:28:20.340
I choose as tiebreaker
the left one.
00:28:29.240 --> 00:28:32.940
So then I will
have plus or minus.
00:28:32.940 --> 00:28:35.770
Now you can convince
yourself that if I do this,
00:28:35.770 --> 00:28:39.400
and I choose always the
left one as tiebreaker,
00:28:39.400 --> 00:28:43.130
the story is the same as
just keeping the left one.
00:28:46.610 --> 00:28:52.500
So essentially, this kind of
averaging with a tiebreaker
00:28:52.500 --> 00:28:58.555
is equivalent to getting
rid of every other spin.
00:29:01.190 --> 00:29:06.460
And so essentially
what I can do is
00:29:06.460 --> 00:29:11.640
to say that I call
a sigma i prime.
00:29:11.640 --> 00:29:16.989
Now, in the new listing that I
have, this thing is no longer,
00:29:16.989 --> 00:29:18.030
let's say, the tenth one.
00:29:18.030 --> 00:29:21.980
It becomes the fifth one because
I removed half of things.
00:29:21.980 --> 00:29:28.200
So sigma i prime is
really sigma 2i minus 1.
00:29:28.200 --> 00:29:39.590
So basically, all the odd ones
I will call to be my new spins.
00:29:39.590 --> 00:29:43.475
All the even ones I want to
get rid of, I'll call them SI.
00:29:47.200 --> 00:29:50.740
So this is just a
renaming of the variables.
00:29:50.740 --> 00:29:53.540
I did some handwaving
to justify.
00:29:53.540 --> 00:29:57.970
Effectivity, all I did
was I broke this sum
00:29:57.970 --> 00:30:04.050
into two sets of sums, but
I call sigma i prime and SI.
00:30:04.050 --> 00:30:06.740
And each one of
them the index i,
00:30:06.740 --> 00:30:10.500
rather than running from
1 to n in this new set,
00:30:10.500 --> 00:30:15.691
the index i runs
from 1 to n over 2.
00:30:21.350 --> 00:30:24.450
So what I have said,
agian, is very trivial.
00:30:24.450 --> 00:30:27.610
I've said that the
original sum, I
00:30:27.610 --> 00:30:31.790
bring over as a sum over
the odd spin, whose names
00:30:31.790 --> 00:30:36.530
I have changed, and a
sum over even spins,
00:30:36.530 --> 00:30:40.180
whose names I have called SI.
00:30:40.180 --> 00:30:48.240
And I have an interaction,
which I can write as sum over i,
00:30:48.240 --> 00:30:53.750
essentially running
from 1 to n over 2.
00:30:53.750 --> 00:30:58.310
I start with the interaction
that involves sigma i
00:30:58.310 --> 00:31:03.840
prime with si because
now each sigma i
00:31:03.840 --> 00:31:09.280
prime is acting with
an s on one side.
00:31:09.280 --> 00:31:11.160
And then there's
another interaction,
00:31:11.160 --> 00:31:13.650
which is SI, and the next.
00:31:19.290 --> 00:31:24.810
So essentially, I rename
things, and I regrouped bonds.
00:31:24.810 --> 00:31:31.590
And the sum that was n terms
now n over 2 pairs of terms.
00:31:31.590 --> 00:31:34.530
Nothing significant.
00:31:34.530 --> 00:31:37.920
But the point is
that over here, I
00:31:37.920 --> 00:31:43.400
can rewrite this as a
sum over sigma i prime.
00:31:43.400 --> 00:31:54.430
And this is a product over
terms where within each term,
00:31:54.430 --> 00:32:02.260
I can sum over the spin that
is sitting between two spins
00:32:02.260 --> 00:32:03.140
that I'm keeping.
00:32:03.140 --> 00:32:12.190
So I'm getting rid of this spin
that sits between spin sigma
00:32:12.190 --> 00:32:16.840
i prime and sigma
i plus 1 prime.
00:32:27.090 --> 00:32:33.800
Now, once I perform
this sum over SI here,
00:32:33.800 --> 00:32:36.790
then what I will
get is some function
00:32:36.790 --> 00:32:42.470
that depends on sigma i prime
and sigma i plus 1 prime.
00:32:42.470 --> 00:32:46.360
And I can choose to
write that function as e
00:32:46.360 --> 00:32:52.840
to the b prime sigma i
prime sigma i plus 1 prime.
00:32:52.840 --> 00:32:56.850
And hence, the
partition function
00:32:56.850 --> 00:33:02.210
after removing every
other spin is the same
00:33:02.210 --> 00:33:07.970
as the partition function that
I have for the remaining spins
00:33:07.970 --> 00:33:11.815
weighted with this b prime.
00:33:20.110 --> 00:33:25.030
So you can see that I took the
original partition function
00:33:25.030 --> 00:33:29.340
and recast it in
precisely the same form
00:33:29.340 --> 00:33:32.800
after removing half of
the degrees of freedom.
00:33:32.800 --> 00:33:36.340
Now, the original b
for the Ising model
00:33:36.340 --> 00:33:40.980
is going to be parameterized
by g, h, and k.
00:33:40.980 --> 00:33:46.540
So the b prime I
did parameterize.
00:33:46.540 --> 00:33:52.750
So this, let's say, emphasizes
parameterized by g, h, k.
00:33:52.750 --> 00:33:56.110
I can similarly
parameterize this
00:33:56.110 --> 00:34:00.140
by g prime, h prime, k prime.
00:34:00.140 --> 00:34:02.980
And how do I know that?
00:34:02.980 --> 00:34:04.890
Because when I was
writing this, I
00:34:04.890 --> 00:34:07.510
emphasized that this is
the most general form
00:34:07.510 --> 00:34:10.860
that I can write down.
00:34:10.860 --> 00:34:14.800
There is nothing else
other than this form
00:34:14.800 --> 00:34:17.719
that I can write down for this.
00:34:17.719 --> 00:34:24.850
So what I have essentially is
that this e to the b prime,
00:34:24.850 --> 00:34:30.454
which is e to the g prime plus
h prime sigma i prime plus sigma
00:34:30.454 --> 00:34:36.460
i plus 1 prime plus k
prime sigma i prime sigma
00:34:36.460 --> 00:34:42.210
i plus 1 prime involves
these three parameters,
00:34:42.210 --> 00:34:46.190
is obtained by summing over SI.
00:34:46.190 --> 00:34:54.659
Let me just call it s being
minus plus 1 of e to the g plus
00:34:54.659 --> 00:35:06.630
h sigma 1 sigma i prime plus
SI plus k sigma i prime SI
00:35:06.630 --> 00:35:20.610
plus g plus kh sigma SI plus
sigma i plus 1 prime plus k SI
00:35:20.610 --> 00:35:21.653
sigma i plus 1.
00:35:26.970 --> 00:35:30.270
So it's an implicit
equation that
00:35:30.270 --> 00:35:33.740
relates g prime, h prime,
k prime, to g, h, and k.
00:35:36.440 --> 00:35:40.620
And in particular, just
to make the writing
00:35:40.620 --> 00:35:46.170
of this thing explicit more
clearly, I will give names.
00:35:46.170 --> 00:35:53.990
I will call e to the k to
the x, e to the h to by,
00:35:53.990 --> 00:35:57.430
e to the g to bz.
00:35:57.430 --> 00:36:01.720
And here, similarly,
I will write x prime e
00:36:01.720 --> 00:36:05.396
to the k prime, y
prime e to the h prime,
00:36:05.396 --> 00:36:08.800
and z prime is e to the g prime.
00:36:15.230 --> 00:36:19.590
So now I just have
to make a table.
00:36:19.590 --> 00:36:23.830
I have sigma i prime
sigma i plus 1 prime.
00:36:26.860 --> 00:36:34.400
And here also I can
have values of s.
00:36:34.400 --> 00:36:43.430
And the simplest possibility
here is I have plus plus.
00:36:43.430 --> 00:36:48.580
Actually, let's put
this number further out.
00:36:48.580 --> 00:36:52.390
So if both the sigma
primes are plus,
00:36:52.390 --> 00:36:54.800
what do I have on
the left hand side?
00:36:54.800 --> 00:37:00.430
I have e to the g
prime, which is z prime.
00:37:00.430 --> 00:37:06.150
e to the 2h prime, which
is y prime squared.
00:37:06.150 --> 00:37:08.500
e to the k prime,
which is x prime.
00:37:11.300 --> 00:37:14.120
What do I have on
the left hand side?
00:37:14.120 --> 00:37:16.780
On the left hand side, I
have two possible things
00:37:16.780 --> 00:37:18.190
that I can put.
00:37:18.190 --> 00:37:21.420
I can put s to be either
plus or s to be minus,
00:37:21.420 --> 00:37:25.900
and I have to sum over
those two possibilities.
00:37:25.900 --> 00:37:30.380
You can see that in all
cases, I have e to the 2g.
00:37:30.380 --> 00:37:32.060
That's a trivial thing.
00:37:32.060 --> 00:37:35.650
So I will always
have a z squared.
00:37:35.650 --> 00:37:38.780
Irrespective of s, I
have two factors of e
00:37:38.780 --> 00:37:40.710
to the h sigma prime.
00:37:49.080 --> 00:37:52.260
OK, you know what happened?
00:37:52.260 --> 00:37:55.080
I should've used--
since I was using
00:37:55.080 --> 00:37:58.600
b, this factor of h over 2.
00:37:58.600 --> 00:38:06.420
So I really should have
put here an h prime over 2,
00:38:06.420 --> 00:38:09.940
and I should have
put here an h over 2
00:38:09.940 --> 00:38:17.100
and h over 2 because the field
that was residing on the sites,
00:38:17.100 --> 00:38:21.170
I am dividing half of it
to the right and half of it
00:38:21.170 --> 00:38:23.120
to the left bond.
00:38:23.120 --> 00:38:25.620
And since I'm writing things
in terms of the bonds,
00:38:25.620 --> 00:38:27.940
that's how I should go.
00:38:27.940 --> 00:38:35.644
So what I have here actually
is now one factor of i prime.
00:38:35.644 --> 00:38:37.880
That's better.
00:38:37.880 --> 00:38:42.400
Now, what I have on
the right is similarly
00:38:42.400 --> 00:38:46.810
one factor of y
from the h's that
00:38:46.810 --> 00:38:50.390
go with sigma 1
prime, sigma 2 prime.
00:38:50.390 --> 00:38:56.460
And then if s is
plus, then you can
00:38:56.460 --> 00:39:02.510
see that I will get two
factors of e to the k
00:39:02.510 --> 00:39:04.830
because both bonds
will be satisfied.
00:39:04.830 --> 00:39:07.540
Both bonds will be plus plus.
00:39:07.540 --> 00:39:12.020
So I will get x squared,
and the contribution
00:39:12.020 --> 00:39:14.740
to the h of the
intermediate bond s
00:39:14.740 --> 00:39:17.130
is going to be 1e to the h.
00:39:17.130 --> 00:39:21.310
So I will get x squared y.
00:39:21.310 --> 00:39:26.940
Whereas if I put it the
intermediate sign for s
00:39:26.940 --> 00:39:30.890
to be minus 1, then I have
two pluses at the end.
00:39:30.890 --> 00:39:32.410
The one in the middle is minus.
00:39:32.410 --> 00:39:36.020
So I would have two
dissatisfied factors of e
00:39:36.020 --> 00:39:39.270
to the k becoming
e to the minus k.
00:39:39.270 --> 00:39:41.360
So it's x to the minus 2.
00:39:41.360 --> 00:39:45.160
And the field also
will give a factor of e
00:39:45.160 --> 00:39:50.860
to the minus h or y inverse.
00:39:50.860 --> 00:39:53.130
So there are four
possibilities here.
00:39:53.130 --> 00:39:56.320
The next one is minus minus.
00:39:56.320 --> 00:40:00.070
z prime will stay the way it is.
00:40:00.070 --> 00:40:04.500
The field has switched sign.
00:40:04.500 --> 00:40:08.510
So this becomes y prime inverse.
00:40:08.510 --> 00:40:11.030
But since both of them
are minus-- minus,
00:40:11.030 --> 00:40:14.900
minus-- the k factor
is satisfied and happy.
00:40:14.900 --> 00:40:20.300
Gives me x prime because
it's e to the plus k.
00:40:20.300 --> 00:40:24.260
On the right hand side, I
will always get the z squared.
00:40:24.260 --> 00:40:29.590
The first factor
becomes the inverse.
00:40:29.590 --> 00:40:34.080
Now, s plus 1 is a
plus site, plus spin,
00:40:34.080 --> 00:40:36.830
that is sandwiched
between two minus spins.
00:40:36.830 --> 00:40:39.240
So there are two unhappy bonds.
00:40:39.240 --> 00:40:42.760
This gives me x to the minus 2.
00:40:42.760 --> 00:40:47.260
Why the spin is pointing in
the direction of the field.
00:40:47.260 --> 00:40:49.440
So there's the y here.
00:40:49.440 --> 00:40:52.670
And here I will get x
squared and y inverse
00:40:52.670 --> 00:40:56.470
because now I have
three negative signs.
00:40:56.470 --> 00:40:58.230
So all the k's are happy.
00:41:01.320 --> 00:41:03.960
There's the next one,
which is plus and minus.
00:41:03.960 --> 00:41:08.320
Plus and minus, we can see that
the contribution to the field
00:41:08.320 --> 00:41:10.540
vanishes.
00:41:10.540 --> 00:41:15.070
Because sigma i prime and plus
sigma i plus 1 prime are zero.
00:41:15.070 --> 00:41:19.170
I will still get z,
the contribution,
00:41:19.170 --> 00:41:22.180
because plus and minus,
the bond between them
00:41:22.180 --> 00:41:25.740
gives me e to the minus k prime.
00:41:25.740 --> 00:41:27.345
I have here z squared.
00:41:30.070 --> 00:41:34.910
There is no overall contribution
of y for the same reason
00:41:34.910 --> 00:41:37.640
that there was nothing here.
00:41:37.640 --> 00:41:47.160
But from s, I will get a
factor of y plus y inverse.
00:41:47.160 --> 00:41:49.560
And there's no
contribution for x
00:41:49.560 --> 00:41:52.100
because I have a
plus and a minus.
00:41:52.100 --> 00:41:56.230
And if the spin is either
plus or minus in the middle,
00:41:56.230 --> 00:41:59.820
there will be one satisfied
and one dissatisfied bond.
00:41:59.820 --> 00:42:03.740
Again, by symmetry,
the other configuration
00:42:03.740 --> 00:42:04.810
is exactly the same.
00:42:09.230 --> 00:42:14.650
So while I had four
configuration, and hence
00:42:14.650 --> 00:42:22.360
four things to match, the
last two are the same.
00:42:22.360 --> 00:42:25.280
And that's consistent
with my having
00:42:25.280 --> 00:42:29.020
three variables, x prime, y
prime, and z prime, to solve.
00:42:29.020 --> 00:42:31.700
So there are three equations
and three variables.
00:42:36.140 --> 00:42:41.620
Now, to solve these equations,
we can do several things.
00:42:46.660 --> 00:42:53.980
Let's, for example, multiply
all four equations together.
00:42:53.980 --> 00:42:56.420
What do we get on
the left hand side?
00:42:56.420 --> 00:43:01.110
There are two x's, two
inverse x's, y, and inverse y,
00:43:01.110 --> 00:43:02.930
but four factors of z.
00:43:02.930 --> 00:43:07.130
So I get z prime to
the fourth factor.
00:43:07.130 --> 00:43:10.610
On the other side, I
will get z to the eight,
00:43:10.610 --> 00:43:12.775
and then the product
of those four factors.
00:43:33.110 --> 00:43:39.820
I can divide one, equation
one, by equation two.
00:43:39.820 --> 00:43:41.580
What do I get?
00:43:41.580 --> 00:43:43.240
The x's cancel.
00:43:43.240 --> 00:43:44.350
The x's cancel.
00:43:44.350 --> 00:43:46.440
So I will get y prime squared.
00:43:49.300 --> 00:43:52.110
On the other side,
divide these two.
00:43:52.110 --> 00:43:53.350
I will get y squared.
00:43:56.910 --> 00:44:01.580
x squared y plus x
squared y inverse,
00:44:01.580 --> 00:44:11.768
x minus 2y plus x2 minus--
yeah, x2 y inverse.
00:44:15.600 --> 00:44:21.040
And finally, to get the equation
for x prime, what I can do
00:44:21.040 --> 00:44:29.002
is I can multiply 1 and
2, divide by 3 and 4.
00:44:29.002 --> 00:44:32.740
And if I do that, on
the left hand side
00:44:32.740 --> 00:44:38.075
I will get x prime
to the fourth.
00:44:38.075 --> 00:44:41.410
And on the right
hand side, I will
00:44:41.410 --> 00:44:49.450
get x squared y plus x
to the minus 2 y inverse
00:44:49.450 --> 00:44:57.660
x minus 2y x squared y inverse
divided by y plus y inverse
00:44:57.660 --> 00:44:58.160
squared.
00:45:07.070 --> 00:45:09.840
So I can take the log of
these equations, if you like,
00:45:09.840 --> 00:45:13.950
to get the recurrence
relations for the parameters.
00:45:13.950 --> 00:45:16.610
For example, taking the
log of that equation,
00:45:16.610 --> 00:45:19.540
you can see that I
will get g prime.
00:45:19.540 --> 00:45:26.990
From here, I would get
2g plus some function.
00:45:26.990 --> 00:45:35.420
There's some delta g
that depends on k and h.
00:45:35.420 --> 00:45:37.440
If I do the same
thing here, you can
00:45:37.440 --> 00:45:39.540
see that I will
get an h prime that
00:45:39.540 --> 00:45:43.910
is h plus some function
from the log of this
00:45:43.910 --> 00:45:46.400
that depends on k and h.
00:45:46.400 --> 00:45:51.231
And finally, I will get k
prime some function of k and h.
00:45:58.570 --> 00:46:02.840
This parameter g is
not that important.
00:46:02.840 --> 00:46:05.150
It's basically an
overall constant.
00:46:05.150 --> 00:46:07.370
The way that we started
it, we certainly
00:46:07.370 --> 00:46:11.810
don't expect it to modify
phase diagrams, et cetera.
00:46:11.810 --> 00:46:16.380
And you can see that it never
affects the equations that
00:46:16.380 --> 00:46:20.410
govern h and k, the
two parameters that
00:46:20.410 --> 00:46:23.820
give the relative weight of
different configurations.
00:46:23.820 --> 00:46:27.340
Whether you are in a ferromagnet
or a disordered state
00:46:27.340 --> 00:46:30.030
is governed by these
two parameters.
00:46:30.030 --> 00:46:32.510
And indeed, we can ignore this.
00:46:32.510 --> 00:46:37.120
Essentially, what it amounts to
is as follows, that every time
00:46:37.120 --> 00:46:41.230
I remove some of the
spins, I gradually
00:46:41.230 --> 00:46:44.410
build a contribution
to my free energy.
00:46:44.410 --> 00:46:47.070
Because clearly, once
I have integrated out
00:46:47.070 --> 00:46:51.070
over all of the energies, what I
will have will be the partition
00:46:51.070 --> 00:46:51.700
function.
00:46:51.700 --> 00:46:54.230
Its log would be
the free energy.
00:46:54.230 --> 00:46:56.720
And actually, we encountered
exactly the same thing
00:46:56.720 --> 00:46:59.530
when we were doing
momentum space RG.
00:46:59.530 --> 00:47:02.460
There was always, as a
result of the integration,
00:47:02.460 --> 00:47:07.020
some contribution that I
call delta f that I never
00:47:07.020 --> 00:47:10.230
looked at because, OK, it's
part of the free energy
00:47:10.230 --> 00:47:13.130
but does not govern
the relative weights
00:47:13.130 --> 00:47:15.440
of the different configurations.
00:47:15.440 --> 00:47:17.440
So these are really
the two things
00:47:17.440 --> 00:47:20.980
that we need to focus on.
00:47:20.980 --> 00:47:27.220
Now, if we did things
correctly and these
00:47:27.220 --> 00:47:32.190
are correct equations, had
I started in the subspace
00:47:32.190 --> 00:47:37.040
where h equals to zero,
which had up-down symmetries,
00:47:37.040 --> 00:47:40.700
I could have changed all of
my sigmas to minus sigma,
00:47:40.700 --> 00:47:44.350
and for h equals to zero, the
energy would not have changed.
00:47:44.350 --> 00:47:47.070
So a check that we
did things correctly
00:47:47.070 --> 00:47:51.390
is that if h equals to zero,
then h prime has to be zero.
00:47:51.390 --> 00:47:52.960
So let's see.
00:47:52.960 --> 00:47:59.530
If h equals to zero
or y is equal to one,
00:47:59.530 --> 00:48:01.690
well if y is equal
to one, you can
00:48:01.690 --> 00:48:05.660
see that those two factors in
the numerator and denominator
00:48:05.660 --> 00:48:07.250
are exactly the same.
00:48:07.250 --> 00:48:11.630
And so y prime stays to one.
00:48:11.630 --> 00:48:17.060
So this check has
been performed.
00:48:17.060 --> 00:48:22.000
So if I am on h equals to
zero on the subspace that
00:48:22.000 --> 00:48:28.030
has symmetry, then I have only
one parameter, this k or x.
00:48:28.030 --> 00:48:32.810
And so on this space, the
recursion relation that I have
00:48:32.810 --> 00:48:36.060
is that x prime
to the fourth is x
00:48:36.060 --> 00:48:39.280
squared plus x to
the minus 2 squared.
00:48:44.520 --> 00:48:46.110
I think I made a mistake here.
00:49:09.400 --> 00:49:12.390
No, it's correct.
00:49:12.390 --> 00:49:14.850
y plus y inverse.
00:49:14.850 --> 00:49:16.970
OK, but y is one.
00:49:16.970 --> 00:49:23.490
So this is divided by
2 squared, which is 4.
00:49:23.490 --> 00:49:40.090
But let me double check
so that I don't go-- yeah.
00:49:42.750 --> 00:49:45.690
So this is correct.
00:49:45.690 --> 00:49:54.480
So I can write it in terms
of x being e to the k.
00:49:54.480 --> 00:49:58.610
So what I have here
is e to the 4k prime.
00:49:58.610 --> 00:50:03.810
I have here e to the 2k plus e
to the minus 2k divided by 2,
00:50:03.810 --> 00:50:09.310
which is hyperbolic
cosine of 2k, squared.
00:50:09.310 --> 00:50:18.650
So what I will get is k
prime is 1/2 log hyperbolic
00:50:18.650 --> 00:50:19.890
cosine of 2k.
00:50:30.030 --> 00:50:39.490
Now, k is a parameter
that we are changing.
00:50:39.490 --> 00:50:43.980
As we make k stronger,
presumably things
00:50:43.980 --> 00:50:46.540
are more coupled to each
other and should go more
00:50:46.540 --> 00:50:48.110
towards order.
00:50:48.110 --> 00:50:52.900
As k goes towards zero,
basically the spins
00:50:52.900 --> 00:50:53.730
are independent.
00:50:53.730 --> 00:50:56.650
They can do whatever they like.
00:50:56.650 --> 00:50:59.530
So it kind of makes
sense that there
00:50:59.530 --> 00:51:06.180
should be a fixed point at
zero corresponding to zero
00:51:06.180 --> 00:51:09.110
correlation length.
00:51:09.110 --> 00:51:10.910
And let's check that.
00:51:10.910 --> 00:51:19.240
So if k is going to zero--
it's very small-- then
00:51:19.240 --> 00:51:28.400
k prime is 1/2 log hyperbolic
cosine of a small number,
00:51:28.400 --> 00:51:34.330
is roughly 1 plus that
small number squared over 2.
00:51:37.170 --> 00:51:39.670
There's a series like that.
00:51:39.670 --> 00:51:43.890
Log of 1 plus a small
number is a small number.
00:51:43.890 --> 00:51:49.130
So this becomes 4k
squared over 2 over 2.
00:51:49.130 --> 00:51:49.970
It's k squared.
00:51:52.880 --> 00:51:58.200
So it says that if I, let's say,
start with a k that is 1/10,
00:51:58.200 --> 00:52:02.650
my k prime would be
1/100 and then 1/10,000.
00:52:02.650 --> 00:52:07.600
So basically, this
is a fixed point
00:52:07.600 --> 00:52:11.220
that attracts everything to it.
00:52:11.220 --> 00:52:13.670
Essentially, what
it says is you may
00:52:13.670 --> 00:52:17.130
have some weak
amount of coupling.
00:52:17.130 --> 00:52:19.930
As you go and look at the spins
that are further and further
00:52:19.930 --> 00:52:24.080
apart, the effective coupling
between them is going to zero.
00:52:24.080 --> 00:52:25.810
And the spins that
are further apart
00:52:25.810 --> 00:52:27.190
don't care about each other.
00:52:27.190 --> 00:52:29.820
They do whatever they like.
00:52:29.820 --> 00:52:33.129
So that all makes
physical sense.
00:52:33.129 --> 00:52:35.170
Well, we are really
interested in this other end.
00:52:35.170 --> 00:52:37.750
What happens at k
going to infinity?
00:52:43.820 --> 00:52:47.850
We can kind of look
at this equation.
00:52:47.850 --> 00:52:50.690
Or actually, look at this
equation, also doesn't matter.
00:52:50.690 --> 00:52:52.630
But here, maybe it's clearer.
00:52:52.630 --> 00:52:56.390
So I have e to the 4k prime.
00:52:56.390 --> 00:52:58.755
And this is going to
be dominated by this.
00:52:58.755 --> 00:53:00.440
This is e to the 2k.
00:53:00.440 --> 00:53:02.840
e to the minus 2k is very small.
00:53:02.840 --> 00:53:04.910
So it's going to
be approximately
00:53:04.910 --> 00:53:09.110
e to the 4k divided by 4.
00:53:13.790 --> 00:53:16.710
So what I have out here
is that k is very large.
00:53:16.710 --> 00:53:20.790
k prime is roughly
k, but then smaller
00:53:20.790 --> 00:53:25.300
by a factor that
is 1/2 of log 2.
00:53:29.560 --> 00:53:34.090
So we try to take two spins that
are next to each other, couple
00:53:34.090 --> 00:53:35.810
them with a very strong k.
00:53:35.810 --> 00:53:39.550
Let's say a million,
10 million, whatever.
00:53:39.550 --> 00:53:43.880
And then I look at spins
that are too further apart.
00:53:43.880 --> 00:53:47.710
They're still very strongly
coupled, but slightly less.
00:53:47.710 --> 00:53:49.690
It's a million minus log 2.
00:53:49.690 --> 00:53:56.150
So it very, very gradually
starts to move away from here.
00:53:56.150 --> 00:54:01.850
But then as it kind of goes
further, it kind of speeds up.
00:54:01.850 --> 00:54:04.550
What it says is that it is true.
00:54:04.550 --> 00:54:08.100
At infinity, you
have a fixed point.
00:54:08.100 --> 00:54:11.320
But that fixed
point is unstable.
00:54:11.320 --> 00:54:17.210
And even if you have very
strong but finite coupling,
00:54:17.210 --> 00:54:21.970
because things are finite, as
you go further and further out,
00:54:21.970 --> 00:54:24.810
the spins become less
and less correlated.
00:54:24.810 --> 00:54:28.540
This, again, is another
indication of the statement
00:54:28.540 --> 00:54:31.140
that a one dimensional
system will not order.
00:54:35.030 --> 00:54:39.030
So one thing that you
can do to sort of make
00:54:39.030 --> 00:54:43.160
this look slightly better, since
we have the interval from zero
00:54:43.160 --> 00:54:47.040
to infinity, is to
change variables.
00:54:47.040 --> 00:54:52.380
I can ask what does
tang of k prime do?
00:54:52.380 --> 00:54:56.790
So tang k prime is e to the
2k prime minus 1 divided by
00:54:56.790 --> 00:55:00.160
e to the 2k prime plus 1.
00:55:00.160 --> 00:55:07.750
You write e to the 2k's
in terms of this variable.
00:55:07.750 --> 00:55:11.750
At the end of the day,
a little bit of algebra,
00:55:11.750 --> 00:55:15.320
you can convince yourself
that the recursion relation
00:55:15.320 --> 00:55:18.650
for tang k has a
very simple form.
00:55:18.650 --> 00:55:21.660
The new tang k is simply the
square of the old tang k.
00:55:28.870 --> 00:55:34.380
If I plot things as
a function of tang k
00:55:34.380 --> 00:55:41.890
that runs between zero
and one, that fixed point
00:55:41.890 --> 00:55:46.200
that was at infinity
gets mapped into one,
00:55:46.200 --> 00:55:50.970
and the flow is
always towards here.
00:55:50.970 --> 00:55:55.080
There is no other
fixed point in between.
00:55:55.080 --> 00:55:57.670
Previously, it wasn't
clear from the way
00:55:57.670 --> 00:56:00.790
that I had written whether
potentially there's
00:56:00.790 --> 00:56:03.380
other fixed points
along the k-axis.
00:56:03.380 --> 00:56:08.280
If you write it as t
prime tang is t squared,
00:56:08.280 --> 00:56:11.380
where t stands for tang,
clearly the only fixed points
00:56:11.380 --> 00:56:14.750
are at zero and one.
00:56:14.750 --> 00:56:16.860
But now this also
allows us to ask
00:56:16.860 --> 00:56:19.590
what happens if
you also introduce
00:56:19.590 --> 00:56:25.410
the field direction,
h, in this space.
00:56:28.070 --> 00:56:33.710
Now, one thing to
check is that if you
00:56:33.710 --> 00:56:39.225
start with k equals to zero,
that is independent spins,
00:56:39.225 --> 00:56:42.700
you look at the equation here.
00:56:42.700 --> 00:56:47.450
k equals to 0 corresponds
to x equals to one.
00:56:47.450 --> 00:56:48.870
This factor drops out.
00:56:48.870 --> 00:56:52.010
You see y prime is equal to y.
00:56:52.010 --> 00:56:54.950
So if you have no
coupling, there
00:56:54.950 --> 00:56:58.700
is no reason why the
magnetic field should change.
00:56:58.700 --> 00:57:05.230
So essentially, this entire line
corresponds to fixed points.
00:57:05.230 --> 00:57:10.386
Every point here
is a fixed point.
00:57:10.386 --> 00:57:12.760
And we can show
that, essentially,
00:57:12.760 --> 00:57:16.460
if you start along
field zero, you go here.
00:57:16.460 --> 00:57:21.210
If you start along different
values of the field,
00:57:21.210 --> 00:57:25.690
you basically go to
someplace along this line.
00:57:25.690 --> 00:57:30.700
And all of these flows
originate, if you go and plot
00:57:30.700 --> 00:57:34.450
them, from back here
at this fixed point
00:57:34.450 --> 00:57:38.800
that we identified before
at h equals to zero.
00:57:38.800 --> 00:57:43.500
So in order to find out what is
happening along that direction,
00:57:43.500 --> 00:57:49.730
all we need to do is to go and
look at x going to infinity.
00:57:49.730 --> 00:57:51.650
With x going to
infinity, you can
00:57:51.650 --> 00:57:55.730
see that y prime squared, the
equation that we have for y,
00:57:55.730 --> 00:57:58.760
y prime squared is y squared.
00:57:58.760 --> 00:58:01.020
And then there's this fraction.
00:58:01.020 --> 00:58:04.050
Clearly, the terms that are
proportional to x squared
00:58:04.050 --> 00:58:06.170
will overwhelm
those proportional
00:58:06.170 --> 00:58:08.350
to x to the minus 2.
00:58:08.350 --> 00:58:10.910
So I will get a
y and a y inverse
00:58:10.910 --> 00:58:13.280
from the numerator
and denominator.
00:58:13.280 --> 00:58:18.270
And so this goes
like y to the fourth.
00:58:18.270 --> 00:58:22.990
Which means that in the vicinity
of this point, what you have is
00:58:22.990 --> 00:58:24.775
that h prime is 2h.
00:58:28.770 --> 00:58:32.490
So this, again, is
irrelevant direction.
00:58:32.490 --> 00:58:38.420
And here, you are flowing
in this direction.
00:58:38.420 --> 00:58:40.650
And the combination
of these two really
00:58:40.650 --> 00:58:44.930
justifies the kind of flows
that I had shown you before.
00:58:48.790 --> 00:58:52.050
So essentially, in the
one dimensional model,
00:58:52.050 --> 00:58:55.560
you can start with any
microscopic set of k and h's
00:58:55.560 --> 00:58:57.000
that you want.
00:58:57.000 --> 00:59:00.895
As you rescale, you essentially
go to independent spins
00:59:00.895 --> 00:59:02.270
with an effective
magnetic field.
00:59:06.690 --> 00:59:13.200
So let's say we start with
very close to this fixed point.
00:59:13.200 --> 00:59:16.510
So we had a very
large value of k.
00:59:16.510 --> 00:59:19.510
I expect if I have
a large value of k,
00:59:19.510 --> 00:59:23.860
neighboring spins are
really close to each other.
00:59:23.860 --> 00:59:29.290
I have to go very far before,
if I have a patch of pluses,
00:59:29.290 --> 00:59:32.080
I can go over to a
patch of minuses.
00:59:32.080 --> 00:59:34.550
So there's a large
correlation length
00:59:34.550 --> 00:59:36.290
in the vicinity of this point.
00:59:39.160 --> 00:59:43.620
And that correlation
length is a function
00:59:43.620 --> 00:59:48.020
of our parameters k and h.
00:59:48.020 --> 00:59:51.240
Now, the point is that
the recursion relation
00:59:51.240 --> 00:59:57.150
that I have for k does not look
like the type of recursions
00:59:57.150 --> 00:59:59.020
that I had before.
00:59:59.020 --> 01:00:03.780
This one is fine because I can
think of this as my usual h
01:00:03.780 --> 01:00:08.830
prime is b to the yh h, where
here I'm clearly dealing
01:00:08.830 --> 01:00:15.890
with a b equals to two, and so
I can read off my yh to be one.
01:00:15.890 --> 01:00:19.640
But the recursion
relation that I have for k
01:00:19.640 --> 01:00:22.730
is not of that form.
01:00:22.730 --> 01:00:26.700
But who said that I
choose k as the variable
01:00:26.700 --> 01:00:28.470
that I put over here?
01:00:28.470 --> 01:00:31.290
I have been doing all of
these manipulations going
01:00:31.290 --> 01:00:34.380
from k to x, et cetera.
01:00:34.380 --> 01:00:36.870
One thing that behaves
nicely, you can see,
01:00:36.870 --> 01:00:40.700
is a variable if I
call e to the minus k
01:00:40.700 --> 01:00:48.660
prime is square root
of 2 e to the minus k.
01:00:52.460 --> 01:00:55.080
At k equals to infinity,
this is something
01:00:55.080 --> 01:00:57.450
that goes to zero on both ends.
01:00:57.450 --> 01:01:00.350
But if its k is
large but not finite,
01:01:00.350 --> 01:01:03.070
it says that on
the rescaling, it
01:01:03.070 --> 01:01:06.770
changes by a factor of root 2.
01:01:06.770 --> 01:01:10.150
So if I say, well,
what is c as a function
01:01:10.150 --> 01:01:12.930
of the magnetic field,
rather than writing k,
01:01:12.930 --> 01:01:16.010
I will write it as
e to the minus k.
01:01:16.010 --> 01:01:20.080
It's just another way
of writing things.
01:01:20.080 --> 01:01:24.450
Well, I know that under
one step of RG, all
01:01:24.450 --> 01:01:27.730
my length scales have
shrunk by a factor of two.
01:01:27.730 --> 01:01:34.920
So this is twice the c that
I should write down with 2h
01:01:34.920 --> 01:01:37.180
and root 2 e to the minus k.
01:01:41.050 --> 01:01:44.950
So I just related the
correlation length
01:01:44.950 --> 01:01:49.760
before and after one
step of RG, starting
01:01:49.760 --> 01:01:51.935
from very close to this point.
01:01:54.830 --> 01:01:58.250
And for these two
factors, 2 and root 2,
01:01:58.250 --> 01:02:02.320
I use the results that
I have over there.
01:02:02.320 --> 01:02:05.130
And this I can keep doing.
01:02:05.130 --> 01:02:06.900
I can keep iterating this.
01:02:06.900 --> 01:02:09.720
After l steps of
RG, this becomes 2
01:02:09.720 --> 01:02:15.630
to the l, c of 2
to the lh, and 2
01:02:15.630 --> 01:02:18.070
to the l over 2
e to the minus k.
01:02:25.530 --> 01:02:28.770
I iterate this sufficiently
so that I have moved away
01:02:28.770 --> 01:02:33.280
from this fixed point, where
everything is very strongly
01:02:33.280 --> 01:02:35.110
correlated.
01:02:35.110 --> 01:02:37.730
And so that means
I move, let's say,
01:02:37.730 --> 01:02:41.110
until this number is something
that is of the order of one.
01:02:41.110 --> 01:02:41.830
Could be 2.
01:02:41.830 --> 01:02:42.840
Could be 1/2.
01:02:42.840 --> 01:02:45.090
I really don't care.
01:02:45.090 --> 01:02:47.230
But the number of
iterations, the number
01:02:47.230 --> 01:02:49.340
of rescalings by a
factor of two that I
01:02:49.340 --> 01:02:53.085
have to do in order to achieve
this, is when 2 to the l
01:02:53.085 --> 01:02:55.498
is of the order of e to the k.
01:02:58.920 --> 01:03:00.200
e to the 2k, sorry.
01:03:11.770 --> 01:03:15.320
And if I do that, I
find that c should
01:03:15.320 --> 01:03:25.600
be e to the 2k some
function of h e to the 2k.
01:03:37.660 --> 01:03:42.590
So let's say I am
at h equals to zero.
01:03:42.590 --> 01:03:46.530
And I make the strength of the
interaction between neighbors
01:03:46.530 --> 01:03:48.430
stronger and stronger.
01:03:48.430 --> 01:03:51.140
If I asked you, how does
the correlation length,
01:03:51.140 --> 01:03:55.350
how does the size of the patches
that are all plus and minus
01:03:55.350 --> 01:03:58.440
change as k becomes
stronger and stronger?
01:03:58.440 --> 01:04:02.740
Well, RG tells you that
it goes in this fashion.
01:04:02.740 --> 01:04:04.620
In the problem
set that you have,
01:04:04.620 --> 01:04:08.357
you will solve the problem
exactly by a different method
01:04:08.357 --> 01:04:09.440
and get exactly this form.
01:04:15.610 --> 01:04:16.155
What else?
01:04:16.155 --> 01:04:21.370
Well, we said that one of the
characteristics of a system
01:04:21.370 --> 01:04:23.620
is that the free
energy has a singular
01:04:23.620 --> 01:04:27.010
part as a function
of the variables
01:04:27.010 --> 01:04:31.910
that we have that is related
to the correlation length
01:04:31.910 --> 01:04:33.690
to the power d.
01:04:33.690 --> 01:04:37.100
In this case, we
have d equals to one.
01:04:37.100 --> 01:04:40.020
So the statement is
that the singular part
01:04:40.020 --> 01:04:44.170
of the free energy, as you
approach infinite coupling
01:04:44.170 --> 01:04:47.580
strength, behaves
as e to the minus 2k
01:04:47.580 --> 01:04:52.404
some other function
of h, e to the 2k.
01:04:58.020 --> 01:05:01.440
Once you have a
function such as this,
01:05:01.440 --> 01:05:03.930
you can take two
derivatives with respect
01:05:03.930 --> 01:05:07.330
to the field to get
the susceptibility.
01:05:07.330 --> 01:05:11.530
So the susceptibility would
go like two derivatives
01:05:11.530 --> 01:05:14.850
of the free energy with
respect to the field.
01:05:14.850 --> 01:05:18.410
You can see that each derivative
brings forth a factor of e
01:05:18.410 --> 01:05:19.930
to the 2k.
01:05:19.930 --> 01:05:23.850
So two derivatives will bring
a factor of e to the 2k,
01:05:23.850 --> 01:05:28.790
if I evaluate it at
h equals to zero.
01:05:28.790 --> 01:05:35.660
So the statement is that
if I'm at zero field,
01:05:35.660 --> 01:05:41.640
and I put on a small
infinitesimal magnetic field,
01:05:41.640 --> 01:05:44.080
it tends to overturn
all of the spins
01:05:44.080 --> 01:05:46.380
in the direction of the field.
01:05:46.380 --> 01:05:50.010
And the further you are
close to k to infinity,
01:05:50.010 --> 01:05:53.270
there are larger responses
that you would see.
01:05:53.270 --> 01:05:55.130
The susceptibility
of the response
01:05:55.130 --> 01:05:58.070
diverges as k goes to infinity.
01:05:58.070 --> 01:06:02.780
So in some sense, this model
does not have phase transition,
01:06:02.780 --> 01:06:05.970
but it demonstrates some
signatures of the phase
01:06:05.970 --> 01:06:08.910
transition, such as
diverging correlation length
01:06:08.910 --> 01:06:11.530
and diverging
susceptibility if you
01:06:11.530 --> 01:06:14.340
go to very, very strong
nearest neighbor coupling.
01:06:19.840 --> 01:06:23.170
There is one other
relationship that we had.
01:06:23.170 --> 01:06:28.410
That is, the susceptibility
is an integral
01:06:28.410 --> 01:06:37.120
in d dimension over
the correlation length,
01:06:37.120 --> 01:06:40.120
which we could say
in one dimension--
01:06:40.120 --> 01:06:42.710
well, actually, let's
write it in d dimension.
01:06:42.710 --> 01:06:47.550
e to the minus x
over c divided by x.
01:06:47.550 --> 01:06:52.940
We introduce an exponent,
eta, to describe
01:06:52.940 --> 01:06:55.990
the decay of correlations.
01:06:55.990 --> 01:07:00.050
So this would generally behave
like c to the 2 minus eta.
01:07:05.680 --> 01:07:11.590
Now, in this case, we see that
both the correlation length
01:07:11.590 --> 01:07:17.840
and susceptibility diverge with
the same behavior, e to the 2k.
01:07:17.840 --> 01:07:20.050
They're proportional
to each other.
01:07:20.050 --> 01:07:23.250
So that immediately
tells me that for the one
01:07:23.250 --> 01:07:26.070
dimensional system
that we are looking at,
01:07:26.070 --> 01:07:29.700
eta is equal to one.
01:07:29.700 --> 01:07:34.170
And if I substitute back
here, so eta is one,
01:07:34.170 --> 01:07:37.180
the dimension is one,
and the two cancels.
01:07:37.180 --> 01:07:39.890
Essentially, it says that
the correlation length
01:07:39.890 --> 01:07:45.490
in one dimension have a
pure exponential decay.
01:07:45.490 --> 01:07:48.010
They don't have this
sub leading power load
01:07:48.010 --> 01:07:51.310
that you would have
in higher dimensions.
01:07:51.310 --> 01:07:53.640
So when you do
things exactly, you
01:07:53.640 --> 01:07:57.510
will also be able
to verify that.
01:07:57.510 --> 01:08:03.230
So all of the predictions
of this position space
01:08:03.230 --> 01:08:09.390
RG method that we can carry out
in this one dimensional example
01:08:09.390 --> 01:08:14.920
very easily, you can
also calculate and obtain
01:08:14.920 --> 01:08:18.229
through the method that
is called transfer matrix,
01:08:18.229 --> 01:08:20.319
and is the subject
of the problem set
01:08:20.319 --> 01:08:21.487
that was posted yesterday.
01:08:27.220 --> 01:08:32.090
Also, you can see that
this approach will
01:08:32.090 --> 01:08:35.890
work for any system
in one dimension.
01:08:35.890 --> 01:08:44.040
All I really need to ensure is
that the b that I write down
01:08:44.040 --> 01:08:47.229
is sufficiently
general to include
01:08:47.229 --> 01:08:51.720
all possible interactions that
I can write between two spins.
01:08:51.720 --> 01:08:55.200
Because if I have some
subset of those interactions,
01:08:55.200 --> 01:08:58.350
and then I do this procedure
that I have over here
01:08:58.350 --> 01:09:02.189
and then take the
log, there's no reason
01:09:02.189 --> 01:09:05.160
why that would not
generate all things that
01:09:05.160 --> 01:09:07.130
are consistent with symmetry.
01:09:07.130 --> 01:09:10.300
So you really have to
put all possible terms,
01:09:10.300 --> 01:09:13.300
and then you will get
all possible terms here,
01:09:13.300 --> 01:09:16.960
and there would be a recursion
relation that would relate.
01:09:16.960 --> 01:09:20.720
You can do this very easily, for
example, for the Potts model.
01:09:20.720 --> 01:09:23.759
For the continuous spin systems,
it becomes more difficult.
01:09:33.839 --> 01:09:42.380
Now let's say briefly as to why
we can solve things exactly,
01:09:42.380 --> 01:09:45.210
let's say, for the one
dimensionalizing model
01:09:45.210 --> 01:09:46.859
by this procedure.
01:09:46.859 --> 01:09:51.540
But this procedure we cannot
do in higher dimensions.
01:09:51.540 --> 01:09:56.640
So let's, for example, think
that we have a square lattice.
01:09:56.640 --> 01:09:59.045
Just generalize what we
had to two dimensions.
01:10:03.170 --> 01:10:10.940
And let's say that we want
to eliminate this spin
01:10:10.940 --> 01:10:14.180
and-- well, let's see,
what's the best way?
01:10:14.180 --> 01:10:18.730
Yeah, we want to eliminate
a checkerboard of spins.
01:10:18.730 --> 01:10:22.750
So we want to eliminate
half of the spins.
01:10:22.750 --> 01:10:26.255
Let's say the white
squares on a checkerboard.
01:10:29.750 --> 01:10:34.090
And if I were to eliminate these
spins, like I did over here,
01:10:34.090 --> 01:10:37.520
I should be able to
generate interactions
01:10:37.520 --> 01:10:41.640
among spins that are left over.
01:10:41.640 --> 01:10:42.870
So you say, fine.
01:10:42.870 --> 01:10:45.500
Let's pick these four spins.
01:10:45.500 --> 01:10:52.030
Sigma 1, sigma 2,
sigma 3, sigma 4,
01:10:52.030 --> 01:10:55.070
that are connected
to this spin, s,
01:10:55.070 --> 01:10:59.670
that is sitting in the middle,
that I have to eliminate.
01:10:59.670 --> 01:11:03.930
So let's also stay
in the space where
01:11:03.930 --> 01:11:07.900
h equals to zero,
just for simplicity.
01:11:07.900 --> 01:11:10.760
So the result of
eliminating that is
01:11:10.760 --> 01:11:14.080
I have to do a sum over s.
01:11:14.080 --> 01:11:19.500
I have e to the-- let's say
the original interaction is k,
01:11:19.500 --> 01:11:27.611
and s is coupled by these bonds
to sigma 1, sigma 2, sigma 3,
01:11:27.611 --> 01:11:28.110
sigma 4.
01:11:33.070 --> 01:11:35.940
Now, summing over the
two possible values of s
01:11:35.940 --> 01:11:36.920
is very simple.
01:11:36.920 --> 01:11:40.950
It just gives me e to
the k times the sum plus
01:11:40.950 --> 01:11:42.850
e to the minus k, that sum.
01:11:42.850 --> 01:11:48.110
So it's the same thing as 2
hyperbolic cosine of k, sigma
01:11:48.110 --> 01:11:51.220
1 plus sigma 2 plus
sigma 3 plus sigma 4.
01:11:55.960 --> 01:11:57.660
We say good.
01:11:57.660 --> 01:12:00.370
I had something like
that, and I took a log
01:12:00.370 --> 01:12:02.530
so that I got the k prime.
01:12:02.530 --> 01:12:07.030
So maybe I'll do something
like a recasting of this,
01:12:07.030 --> 01:12:10.760
and maybe a recasting of this
will give me some constant,
01:12:10.760 --> 01:12:13.450
will give me some
kind of a k prime,
01:12:13.450 --> 01:12:19.270
and then I will have sigma
1, sigma 2, sigma 2, sigma 3,
01:12:19.270 --> 01:12:22.890
sigma 3, sigma 4,
sigma 4, sigma 1.
01:12:26.240 --> 01:12:28.870
But you immediately see that
that cannot be the entire
01:12:28.870 --> 01:12:30.590
story.
01:12:30.590 --> 01:12:35.220
Because there is really
no ordering among sigma 1,
01:12:35.220 --> 01:12:38.140
sigma 2, sigma 3, sigma 4.
01:12:38.140 --> 01:12:40.760
So clearly, because of
the symmetries of this,
01:12:40.760 --> 01:12:45.530
you will generate also sigma 1
sigma 3 plus sigma 2 sigma 4.
01:12:51.260 --> 01:12:56.610
That is, eliminating this
spin will generate for you
01:12:56.610 --> 01:12:59.800
interactions among
these, but also
01:12:59.800 --> 01:13:03.640
interactions that go like this.
01:13:03.640 --> 01:13:06.960
And in fact, if you
do it carefully,
01:13:06.960 --> 01:13:10.780
you'll find that you will also
generate an interaction that
01:13:10.780 --> 01:13:14.465
is product of all four spins.
01:13:14.465 --> 01:13:18.350
You will generate something
that involves all of the force.
01:13:18.350 --> 01:13:21.550
So basically, because
of the way of geometry,
01:13:21.550 --> 01:13:25.110
et cetera, that you have
in higher dimensions,
01:13:25.110 --> 01:13:28.340
there is no trick that
is analogous to what
01:13:28.340 --> 01:13:31.980
we could do in one dimension
where you would eliminate
01:13:31.980 --> 01:13:37.140
some subset of spins and not
generate longer and longer
01:13:37.140 --> 01:13:40.920
range interactions, interactions
that you did not have.
01:13:40.920 --> 01:13:43.840
You could say, OK, I
will start including
01:13:43.840 --> 01:13:46.580
all of these
interactions and then
01:13:46.580 --> 01:13:50.020
have a higher, larger
parameter space.
01:13:50.020 --> 01:13:51.880
But then you do
something else, you'll
01:13:51.880 --> 01:13:54.790
find that you need to
include further and further
01:13:54.790 --> 01:13:57.360
neighboring interactions.
01:13:57.360 --> 01:14:04.000
So unless you do some kind of
termination or approximation,
01:14:04.000 --> 01:14:07.140
which we will do
next time, then there
01:14:07.140 --> 01:14:13.008
is no way to do this exactly
in higher dimensions.
01:14:13.008 --> 01:14:14.111
AUDIENCE: Question.
01:14:14.111 --> 01:14:14.735
PROFESSOR: Yes.
01:14:17.450 --> 01:14:20.730
AUDIENCE: I mean, are you
putting any sort of weight
01:14:20.730 --> 01:14:26.610
on the fact that, for
example, sigma 1 and sigma 3
01:14:26.610 --> 01:14:29.010
are farther apart than
sigma 1 and sigma 2,
01:14:29.010 --> 01:14:34.680
or are we using taxi cab
distances on this lattice?
01:14:34.680 --> 01:14:40.220
PROFESSOR: Well, we are
thinking of an original model
01:14:40.220 --> 01:14:42.660
that we would like
to solve, in which
01:14:42.660 --> 01:14:45.670
I have specified that
things are coupled only
01:14:45.670 --> 01:14:47.900
to nearest neighbors.
01:14:47.900 --> 01:14:50.910
So the ones that correspond
to sigma 1, sigma 3,
01:14:50.910 --> 01:14:52.520
are next nearest neighbors.
01:14:52.520 --> 01:14:55.720
They're certainly farther
apart on the lattice.
01:14:55.720 --> 01:14:59.430
You could say, well,
there's some justification,
01:14:59.430 --> 01:15:03.180
if these are ions
and they have spins,
01:15:03.180 --> 01:15:06.260
to have some weaker interaction
that goes across here.
01:15:06.260 --> 01:15:09.350
There has to be some
notion of space.
01:15:09.350 --> 01:15:13.360
I don't want to couple everybody
to everybody else equivalently.
01:15:13.360 --> 01:15:17.520
But if I include this,
then I have further more
01:15:17.520 --> 01:15:19.670
complicated Hamiltonian.
01:15:19.670 --> 01:15:22.180
And when I do RG,
I will generate
01:15:22.180 --> 01:15:23.922
an even more
complicated Hamiltonian.
01:15:27.300 --> 01:15:28.164
Yes.
01:15:28.164 --> 01:15:29.080
AUDIENCE: [INAUDIBLE].
01:15:35.330 --> 01:15:39.039
PROFESSOR: Question is, suppose
I have a square lattice.
01:15:39.039 --> 01:15:39.957
Let's go here.
01:15:43.170 --> 01:15:46.640
And the suggestion
is, why don't I
01:15:46.640 --> 01:15:50.730
eliminate all of
the spins over here,
01:15:50.730 --> 01:15:53.740
maybe all of the
spins over here?
01:15:53.740 --> 01:15:56.620
So the problem is that I
will generate interactions
01:15:56.620 --> 01:16:00.212
that not only go like this, but
interactions that go like this.
01:16:03.030 --> 01:16:09.040
So the idea of what happens is
that imagine that there's these
01:16:09.040 --> 01:16:11.240
spins that you're eliminating.
01:16:11.240 --> 01:16:14.790
As long as there's paths that
connect the spins that you're
01:16:14.790 --> 01:16:18.620
eliminating to any
other spin, you
01:16:18.620 --> 01:16:20.220
will generate that
kind of couple.
01:16:28.650 --> 01:16:31.860
Again, the reason that the
one dimensional model works
01:16:31.860 --> 01:16:35.920
is also related to
its exact solvability
01:16:35.920 --> 01:16:38.700
by this transfer matrix method.
01:16:38.700 --> 01:16:43.070
So I will briefly mention
that in the last five minutes.
01:16:47.780 --> 01:16:53.300
So for one dimensional
models, the partition function
01:16:53.300 --> 01:16:57.730
is a sum over whatever
degree of freedom you have.
01:16:57.730 --> 01:17:01.410
Could be Ising, Potts,
xy, doesn't matter.
01:17:01.410 --> 01:17:04.480
But the point is
that the interaction
01:17:04.480 --> 01:17:10.820
is a sum of bonds that connect
one site to the next site.
01:17:10.820 --> 01:17:18.566
I can write this as a product of
e to the b of SI and SI plus 1.
01:17:23.920 --> 01:17:27.320
Now, I can regard
this entity-- let's
01:17:27.320 --> 01:17:31.710
say I have the Potts
model q values.
01:17:31.710 --> 01:17:33.490
This is q possible values.
01:17:33.490 --> 01:17:34.940
This is q possible values.
01:17:34.940 --> 01:17:38.530
So there are q squared possible
values of the interaction.
01:17:38.530 --> 01:17:40.790
And there is q squared
possible values
01:17:40.790 --> 01:17:42.930
of this Boltzmann weight.
01:17:42.930 --> 01:17:49.170
I can regard that
as a matrix, but I
01:17:49.170 --> 01:17:50.520
can write in this fashion.
01:17:53.690 --> 01:17:56.020
And so what you
have over there, you
01:17:56.020 --> 01:18:07.240
can see is effectively you have
s1, ts2, s2, ts3, and so forth.
01:18:07.240 --> 01:18:10.090
And if I use periodic boundary
condition like the one
01:18:10.090 --> 01:18:14.070
that I indicated there so
that the last one is connected
01:18:14.070 --> 01:18:22.190
to the first one, and then I
do a sum over all of these s's,
01:18:22.190 --> 01:18:28.130
this is just the
product of two matrices.
01:18:28.130 --> 01:18:33.800
So this is going to become, when
I sum over s2, s1 t squared s3
01:18:33.800 --> 01:18:35.055
and so forth.
01:18:35.055 --> 01:18:37.410
You can see that
the end result is
01:18:37.410 --> 01:18:41.840
trace of the matrix t
raised to the power of n.
01:18:45.820 --> 01:18:50.220
Now, the trace you can
calculate in any representation
01:18:50.220 --> 01:18:51.980
of the matrix.
01:18:51.980 --> 01:18:54.350
If you manage to find
the representation where
01:18:54.350 --> 01:18:57.700
t is diagonal, then
the trace would
01:18:57.700 --> 01:19:02.280
be sum over alpha
lambda alpha to the n,
01:19:02.280 --> 01:19:07.240
where these lambdas are the
eigenvalues of this matrix.
01:19:07.240 --> 01:19:09.690
And if n is very large,
the thermodynamic
01:19:09.690 --> 01:19:12.630
limit that we are
interested, it will
01:19:12.630 --> 01:19:14.460
be dominated by the
largest eigenvalue.
01:19:21.460 --> 01:19:25.010
Now, if I write this for
something like Potts model
01:19:25.010 --> 01:19:29.180
or any of the spin models that
I had indicated over there,
01:19:29.180 --> 01:19:32.070
you can see that all
elements of this matrix
01:19:32.070 --> 01:19:37.440
being these Boltzmann
weights are plus, positive.
01:19:37.440 --> 01:19:47.550
Now, there's a theorem called
Frobenius's theorem, which
01:19:47.550 --> 01:19:53.120
states that if you have a
matrix, all of its eigenvalues
01:19:53.120 --> 01:20:03.880
are positive, then the largest
eigenvalues is non-degenerate.
01:20:06.810 --> 01:20:12.170
So what that means is that if
this matrix is characterized
01:20:12.170 --> 01:20:17.630
by a set of parameters, like
our k's, et cetera, and I
01:20:17.630 --> 01:20:24.120
change that parameter,
k, well the eigenvalue
01:20:24.120 --> 01:20:26.490
is obtained by looking
at a single matrix.
01:20:26.490 --> 01:20:29.330
It doesn't know
anything about that.
01:20:29.330 --> 01:20:32.520
The only way that the
eigenvalue can become singular
01:20:32.520 --> 01:20:36.330
is if two eigenvalues
cross each other.
01:20:36.330 --> 01:20:41.070
And since Frobenius's
theorem does not allow that,
01:20:41.070 --> 01:20:45.290
you conclude that this
largest eigenvalue
01:20:45.290 --> 01:20:49.330
has to be a perfectly nice
analytical function of all
01:20:49.330 --> 01:20:53.790
of the parameters that go into
constructing this Hamiltonian.
01:20:53.790 --> 01:20:58.190
And that's a mathematical way
of saying that there is no phase
01:20:58.190 --> 01:21:05.850
transition for one dimensional
model because you cannot have
01:21:05.850 --> 01:21:10.760
a crossing of eigenvalues, and
there is no singularity that
01:21:10.760 --> 01:21:11.600
can take place.
01:21:15.630 --> 01:21:19.150
Now, an interesting then
question or caveat to that
01:21:19.150 --> 01:21:24.690
comes from the very question
that was asked over here.
01:21:24.690 --> 01:21:31.380
What if I have, let's say,
a two dimensional model,
01:21:31.380 --> 01:21:35.470
and I regard it essentially
as a complicated one
01:21:35.470 --> 01:21:41.160
dimensional model in
which I have a complicated
01:21:41.160 --> 01:21:46.240
multi-variable
thing over one side,
01:21:46.240 --> 01:21:49.650
and then I can go through this
exact same procedure over here
01:21:49.650 --> 01:21:52.090
also?
01:21:52.090 --> 01:21:56.860
And then I would have to
diagonalize this huge matrix.
01:21:56.860 --> 01:22:01.596
So if this is l, it would be a
2 to the l by 2 to the l matrix.
01:22:04.270 --> 01:22:08.340
And you may naively
think that, again,
01:22:08.340 --> 01:22:09.910
according to
Frobenius's theorem,
01:22:09.910 --> 01:22:13.020
there should be no
phase transition.
01:22:13.020 --> 01:22:17.480
Now, this is exactly what
Lars Onsager did in order
01:22:17.480 --> 01:22:19.390
to solve the two
dimensionalizing model.
01:22:19.390 --> 01:22:22.320
He constructed this matrix
and was clever enough
01:22:22.320 --> 01:22:27.060
to diagonalize it and show
that in the limit of l going
01:22:27.060 --> 01:22:30.010
to infinity, then the
Frobenius's theorem
01:22:30.010 --> 01:22:33.120
can and will be violated.
01:22:33.120 --> 01:22:37.070
And so that's something
that we will also
01:22:37.070 --> 01:22:41.570
discuss in some of
our future lectures.
01:22:41.570 --> 01:22:43.265
Yes.
01:22:43.265 --> 01:22:45.140
AUDIENCE: But it won't
be violated in the one
01:22:45.140 --> 01:22:48.450
dimensional case, even
if n goes to infinity?
01:22:48.450 --> 01:22:51.850
PROFESSOR: Yeah, because
n only appears over here.
01:22:51.850 --> 01:22:55.770
Lambda max is a perfectly
analytic function.