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PROFESSOR: OK, let's start.
So we have been looking at the problem of phase transitions
from the perspective of a simple system which
is a piece of magnet.
And we find that if we change the temperature of the system,
there is a critical temperature, Tc,
that separates paramagnetic behavior
on the high temperature side and ferromagnetic behavior
on the low temperature side.
Clearly in the vicinity of this point,
whether you're on one side or the other,
the magnetization is small and we
are relying on that to make us a good parameter
to expand things in.
The other thing is that we anticipated
that over here there are long wavelength
fluctuations of the magnetization field
and so we did averaging and defined a statistical field,
this magnetization as a function of position.
Then we said, OK, if I'm just changing temperature,
what potential is the behavior of the probability
that I will see in my sample some particular configuration
of this magnetization field?
So there is a functional that governs that.
And the statement that we made was
that whatever this functional is,
I can write it as the exponential of something
if I want.
This probability is positive.
I will assume that it is-- locally there is a probability
density that I will integrate across the system.
Probability density is a function
of whatever magnetization I have around point x.
And so then when we expanded this, what did we have?
We said that the terms that are consistent with rotational
symmetry have to be things like m squared, m to the fourth,
and so forth.
In principal, there is a long series
but hopefully since m is small, I
don't have to include that many terms in the series.
And additionally, I can have an expansion in gradient
and the lowest order term in that series
was the gradient of m squared and potentially higher
order terms.
OK?
We said that we could, if we relied
on looking at the most probable configuration of this weight,
make a connection between what is going
on here and the experimental observation.
And essentially the only thing that we needed to do
was to basically start T from here,
so T was made to be proportional to T minus Tc.
And then we could explain these kinds of phenomena
by looking at the behavior of the most
probable magnetization.
Now I kind of said that we are going
to have long wavelength fluctuations.
There was one case where we actually
saw a video of those long wavelength fluctuations
and that was for the case of critical opalescence
taking place at the liquid gas mixture at its critical point.
Can we try to quantify that a little bit better?
The answer is yes, we can do so through scattering experiments.
And looking at the sample was an example
of a scattering experiment, which
if you want to do more quantitatively,
we can do the following- we can say that there
is some incoming field, electromagnetic wave that
is impingent on the system.
It's a pass on incoming wave vector k.
It sort of goes through the sample
and then when it comes out, it gets scattered and so what
I will see is some k [INAUDIBLE] that
comes from the other part of the system.
In principle, I guess I can put a probe here and measure
what is coming out.
And essentially it will depend on the angle
towards which this has rotated.
If I asked well, how much has been scattered?
We'd say, well it's a complicated problem
in quantum mechanics.
Let's say this is a quantum mechanical procedure-
you would say that there's an amplitude that you have
the scattering that is proportional to some overlap
between the kind of state that you started with,
what we started is an incoming wave with k initial,
presumably there is the initial state of my sample
before the wave hits it, and then I end up
with the final configuration which is k f, whatever
the final version of my system is.
Now between these two, I have to put whatever
is responsible for scattering this wave so there
is in some sense some overall potential
that I have to put over here.
Now let's think about the case of this thing being,
say, a mixture of gas and liquid,
well what is scattering light?
Well, it is the individual atoms that are scattering light
and there are lots of them.
So basically I have to sum over all of the scattering elements
that I have my system.
Let's say I have a u for a scattering element, i that
is located at position-- maybe bad choice,
let's call it sum over alpha.
X alpha is the position of let's say
the atom that is scattered here.
OK?
So now since I'm dealing with, say, linear order,
not multiple scattering, what I can do
is I can basically take this sum outside.
So this thing is related to a sum
over alpha of the scattering I would have
for individual elements that are scattering.
And then roughly each individual element
will scatter an amount that I will call sigma q.
If you have elastic scattering what happens
is that essentially your initial k simply gets rotated
without changing its magnitude.
So what happens is that essentially everything
will end up being a function of this momentum transfer which
is proportional to q k f minus k i
whose magnitude would be twice the magnitude of your k
sine of the half of the angle if you just
do the simple geometry over there.
So this is for elastic scattering
which is what we will be thinking about.
Now the amount that each individual element scatters
like each atom is indeed a function
of your momentum transferred from the scattering probe.
But the thing that you're scattering from
is something that is very small like an atom,
so it turns out that the resulting q will give
significant-- well, the resulting sigma will vary all
the over scales where q is related
to the inverse of whatever is scattering
which is something that is very large.
So most of this stuff that is happening at small q, most
of the variation that is observed,
comes from summing over the contributions
of the different elements.
So going to the continuum limit, this becomes an integral
across your system of whatever the density of the thing that
is scattering is.
Indeed if I'm thinking about the light scattering experiment
that we saw with critical opalescence, what you would
be looking at is this density of liquid
versus gas, which if I want to convert to q,
I have to do a Fourier transform here.
And so this is the amplitude of scattering that I expect and we
can see that it is directly probing
the fluctuations of the system, Fourier
transform [INAUDIBLE] number q.
Eventually of course this is the amplitude
what you will be seeing is the amount that is scattered,
s of q will be proportional to this amplitude squared.
We'll have a part that at small q is roughly constant,
so basically at small q I can regard this as a constant.
So at small q all of the variations
is going to come from this Roth q squared.
Of course again thinking about the case of the liquid gas
system, where we were seeing the picture, there were variations,
so there's essentially lots and lots
of these Roth q's depending on which instant of time
you're looking at.
And then it would be useful to do some kind of a time average
and hope that the time average comes
from the result of a probability measure such as this.
OK?
So that's the procedure that we'll follow.
We're going to go slightly beyond what we did before.
What we did before was we started with the probability
distribution, such as this that we
posed on the basis of symmetry and then calculated singular
behavior of various thermodynamic functions
such as heat capacity, susceptibility, magnetization,
et cetera, all of them at macroscopic quantities.
But this is a probability that also
works at the level of microscopics.
It's really a probability as a function of our configurations
and the way that that is probed is
through scattering experiments.
So scattering experiments really probe the Fourier
transform of this probability that we have posed over here.
OK?
Now again the full probability that I have written down there
is rather difficult.
Let's say this in the case of the liquid gas system,
this row would be explicitly the magnetization.
It would be the fluctuations of the magnetization
around the mean.
I should note that in the case of the magnet,
you can say, well how do you probe things?
In that case you need something, some probe
that scatters from magnetization at each point.
And the appropriate probe for magnetization is neutrons.
So you basically hit the system with a beam of neutrons
that may be polarizing them in this particular direction,
their spins-- and they hit the spins of whatever
is in your sample and they get scattered
according to this mechanism.
And what you will be seeing at small q
is related to fluctuations of this magnetization field.
Think he was looking for that.
Now I realize-- I'm not going to run after him.
OK.
So teaches them to leave the room earlier.
OK.
So let's see what we have to-- we can
do for the case of calculating this quantity.
Now I'm not going to calculate it for the nonlinear form,
it's rather difficult.
What I'm going to do is to sort of expand on the trick
that we were using last time which
led to this other point which is to look
at the most probable state.
So basically we looked at that function
where we were calculating the subtle point
integration and the first thing that we
did was to find the configuration
of the magnetization field that was the most likely.
And the answer was that because k is positive,
your magnetization that extremizes that probability
is something that is uniform, does not depend on x,
and is pointing all in one direction.
Let's call it e hat 1.
That is indicating this one thing
is symmetry breaking in the zero field limit.
Of course that spontaneous symmetry breaking
only occurs when t is negative and for t positive m bar is 0.
For t negative just minimizing the expression
t m squared plus u m to the fourth
gives you minus t over 4u.
OK?
So that's the most probable configuration.
What this thing is probing is fluctuations so
let's expand around the most probable configuration.
So let's say that we-- say that I have thermally excited and m
of x which is m bar plus a little bit that varies
from each location to another location,
like if I'm looking at this critical opalescence,
it's the variation in density from one
location to another location.
But that is true as long as I'm looking
at the case of something that has only one component.
If I have multiple components, I can also
have fluctuations in the remaining m minus 1 directions,
and this is n of let's say alpha go from 2 of phi t of e alpha.
So I have broken the fluctuations into two types.
I've said that let's say if you were n equals to 2
your m bar would be pointing in some particular direction.
And so phi l's correspond to increasing the length
or decreasing the length whereas phi t corresponds
to going in the orthogonal direction which
in general would be n minus 1, different components.
OK?
So I want to ask, what's the probability
of this set of fluctuations?
All I need to do and again this is x-dependent
is to substitute into my general expression for the probability
so for that I need a few things.
One of the things that I need is the gradient of m squared.
The uniform part has no gradient so I will either
get the gradient from phi l squared or I will get gradient
of the n minus 1 component vector field phi t
that I will simply write as gradient of phi t squared.
So phi t is an n minus 1 component.
I can ask, what is m squared?
Basically the first term I need to put over there,
but m squared, I have to square this expression.
I will get m bar squared to m bar phi l plus phi l squared
that comes from the component that is along e 1.
All the other components here will add up
to give me the magnitude of this transverse field that
exists in the other m minus 1 directions.
The other term that I need is m to the fourth
and in particular we saw that it is absolutely necessary
if t is negative to include the m to the fourth term
because otherwise the probability that we were
writing just didn't make sense and we need
to write expressions for probability that are physically
sensible.
So I just take the line above and square it.
But I want to only keep terms to quadratic order.
So to zero order I have m bar to the fourth to third order
I have-- to first order I have 4 m bar cubed phi l
then there are a bunch of terms that are order of phi squared.
Squaring this will give me 4 m bar squared phi l
squared, but the dot product of these two terms
will also gives me two m bar squared phi l
squared for a total of six m bar squared phi l squared.
And then the phi t squared comes simply
from twice m bar squared phi t squared.
And then there's higher order terms cubic and fourth order m
phi t and phi l that I don't write
assuming that the fluctuations that I'm looking at our small
around the most probable state.
OK?
So if I stick with this quadratics
then the probability of fluctuations
across my system characterized by phi l of x and phi t
of x is proportional to exponential
of minus integral dd x.
I have an overall factor-- well I have a factor of K over 2
for the first term.
I have a gradient of phi l squared
and then you can see that I have a bunch of them.
Let's put the K over 2, let's put the one-half here.
I have K phi l gradient of phi l squared.
I'm going to put everything that has phi l squared in it.
I have here t over 2 phi l squared from t over 2m squared.
So I have phi l squared.
I have t over 2.
Actually I have taken-- I'm going to make mistakes
unless I put the one-half over here, too.
I have another phi l squared from over here.
That gets multiplied by u so I will get here
plus 12 u m bar squared, 12 rather than 6
because I divided by 2.
And then I have a term that is K over 2 gradient of the vector
phi t squared.
And then I have t over 2 phi t squared
and then 2 m bar squared multiplied by u
becomes 4u m bar squared.
And there are higher order terms that I will not write down.
Yes?
Did I--
AUDIENCE: You said phi l [INAUDIBLE].
PROFESSOR: Good.
Yes, so the question is I immediately
jumped to second order, so what happened to the linear term?
There's a linear term here and there's a linear term here.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Let's write them down.
So the coefficient of phi l would
be t over 2 m bar plus 4 u m bar cubed and minimizing
that expression is setting this first derivative to zero
so if you are expanding around an extreme on the most probable
state then by construction you're
not going to get any terms that are linear either m phi l
or m phi t.
OK?
Yes?
AUDIENCE: Is there-- what's the reason for not including
a term in our general probability,
a term like [INAUDIBLE] of m squared?
PROFESSOR: We could.
He said that essentially that amount
over here to an expansion in powers of the gradient, which
if I go to Fourier space this would become q
squared plus n squared would be q to the fourth, et cetera.
If you are looking at more q or large wavelengths and so
we are going to focus on the first few terms.
But they exist just as they existed
for the phonon spectrum.
We looked at the linear portion and then realized
that going away from q equals to 0
generates all kinds of other steps.
OK?
I mean, these are very important things to ask again and again
to ultimately convince yourself, because in reality
that expansion that I have written
has an infinity of terms in it.
You have to always convince yourself
that close enough to the critical point all
of those other terms that I don't write down
are not going to make any difference.
OK?
All right.
So that's the weight.
What I'm going to do is the same thing that we did
last time for the case of Goldstone modes,
et cetera, which is to go to a Fourier presentation
so any one of the components, be the longitudinal or transverse,
I will write as a sum over q e to the i q dot, x some Fourier
component and then get a square root of v
just so that the normalization would look simple.
And if I substitute for phi of x in terms of phi of q,
just as we saw last time the probability
will decompose into independent contribution for each q
because once you substitute it here,
every quadratic term will have both a sum over x and-- sorry,
and integral over x and sums over q and q
prime, e to the i q plus q prime x the integration over x forces
q and q prime to be the same up to a sign.
So then we find that the probability distribution
as a function of these Fourier amplitudes
phi alpha and phi t decomposes into a product.
Basically each q mode is acting independently
of all the others.
And also at the quadratic level we
see that there is no crosstalk between transverse
and longitudinal so we will have one weight for the transverse,
one weight for the longitudinal.
And what's it actually going to look like?
Essentially it's going to look like something
that is proportional, let's say, for phi l,
it is proportional to phi l of q squared.
I have here something and then I have k q squared over 2
so then a Fourier transform that I will get k q squared over 2.
So let's write it in this fashion, q over 2 q squared
and then that's something that's not q dependent.
And by convention I will write it as x e l minus 2.
It has dimensions of inverse length scale
because q has dimensions of inverse length scale
so I will shortly define a length so that these two
terms would have the same dimensional,
so c l is defined in that fashion.
And similarly I have an exponential
that governs k over 2 q squared plus c t to the minus 2
phi tilde of t of q squared and this is a vector
so there are n minus 1 components there.
OK?
And you can see that basically potentially these two terms,
c l and c t are different.
In fact, let's just write down what they are.
So this coefficient K over c l squared
is defined to be t plus 12 u m bar squared.
OK?
Question?
AUDIENCE: [INAUDIBLE]
PROFESSOR: OK.
Now this depends on whether you are
for t positive or t negative-- better be positive
since I have written it as one over some positive quantity--
for t positive, m bar is 0 so this is just t.
For t negative, then m bar squared is minus t over 4 u
so this becomes minus 3t, so this becomes minus 2t.
OK?
And k over c t squared is t plus 4u m bar squared.
It is t for t positive.
For t negative, substitute it for m bar
squared, it will give me 0.
OK?
Actually the top one hopefully you recognize
or you remember from last time.
We had exactly this expression t and minus
2t when we calculated the susceptibility.
This was the inverse susceptibility.
In fact, I can be now more precise
and call that the inverse of the longitudinal susceptibility.
And what we have here is the inverse
of the transverse susceptibility.
What does that mean?
Let me remind you what susceptibility is.
Susceptibility is you have a system
and you put on a little bit of the field
and then see how the magnetization responds.
If you are in the ordered phase so that your system is
spontaneously pointing in one direction,
then if you put the field in this direction,
you have to climb this Mexican hat potential
and you have to pay a cost to do so.
Whereas if I put it the field perpendicular, all that happens
is that the magnetization rotates,
so it can respond without any cost and that's what this is.
These are really the Goldstone modes
that we were discussing are the transverse fluctuations that I
have written before.
So again, we discussed last time you
break a continual symmetry you will have Goldstone modes
and these Goldstone modes are the ones that
are perpendicular to the average magnetization, if you like.
OK?
So now we have a prediction.
We say that if I look at these phi phi fluctuations,
I can pick a particular q, let's say,
and here I have to put q star in order
to get something that is non-zero.
Actually that's put here q prime and then pick two different
and this is alpha and beta.
Well, if I look at this average since the weight is
the product of contributions from different q's the answer
will be 0 unless q and q prime add up to 0.
And if I'm looking at the same q,
I better make sure that I'm looking at the same component
because the longitudinal and transverse component or any
of the n minus 1 transverse components among each other
have completely independent Gaussian rates.
If I'm now looking at the same Gaussian, for the same Gaussian
I can just immediately read off it's variance
which is K q squared plus whatever the appropriate c is
for that direction whether it's c l
or c t potentially would make a difference.
OK?
Right.
So now we have a prediction for our experimentals.
I said that these guys can go and measure
the scattering as a function of angle at small angle
and they can fit how much is scattered
as you go as a function of angle and fit it as a function of q.
So we predict that if they look at something
that's say like phi l squared, if you're thinking
about the liquid gas system that's really
the only thing that you have because there
is no transverse component if you have a Scalar variable.
We claim that if you go and look at those critical opalescent
pictures that we saw, and do it more precisely
and see what happens as a function of the scattered wave
number q, that you will get a shape that
is 1 over q squared plus c to the minus 2.
This kind of shape that is called a Lorentzian
is indeed what you commonly see for all kinds of scattering
line shapes.
OK?
So we have a prediction.
Of course, the reason that it works
is because in principle we know that this series will have
higher order terms as we discussed
like q to the fourth, q to the sixth, et cetera,
but they fall way down here where you're not
going to be seeing all that much anyway.
Now the place where this curve turns around
from being something that is dominated by 1 over K c
to the minus 2 to something that falls off as 1
over K q squared or maybe even faster,
the borderline is this inverse length scale that we indicated,
c l minus 1.
So now what happens if I go closer to the phase transition
point?
As I go closer to the phase transition point,
t goes to zero, this c inverse goes towards zero.
So if this is for some temperature above Tc
and I go to some lower temperature, then what happens
is that the curve will start higher and then [INAUDIBLE].
Yeah?
Actually it doesn't cross the other curve
which is what I wrote down.
Just because it starts higher it can bend and go
and join this curve at a further point.
Eventually when you go through exactly the critical point then
you get the union of all of these curves, which is a 1
over q squared type of curve.
So right at the point where t equals to 0
the prediction is that the Lorentzian shape,
the coefficient that appears in front of q squared
vanishes you will see one over q squared.
OK?
Now the results of experiments in reality that are very
happily fitted, the Lorentzian, when you're away
from the critical point they claim
that when you are exactly at the critical point,
it's not quite 1 over q squared.
Seems to be slightly different.
At Tc, the scattering appears to be
more similar to 1 over q to 2 minus a small amount.
That's where another critical exponent theta
is introduced so that's another thing that ultimately you
have to try to figure out and understand.
OK?
Of course I drew the curves for the longitudinal component.
If I look at the curves for the transverse components,
and again, by appropriate choice of spin polarized neutrons
you can decompose different components
of scattering from the magnetization
field of a piece of iron, for example.
If you are above Tc, there is really
no difference between longitudinal and transverse
because there is no direction that is selected.
And you can see that the forms that you will get above Tc
would be exactly the same.
When you go below Tc, that's where the difference appears
because the length scale that would appear
for the longitudinal parameters would be finite
and it corresponds to having to push
the magnetization above this bottom of the Mexican hat
potential whereas there is no cost in the other direction.
So if you can probe in the fluctuations that
would correspond to these Goldstone modes,
you would see the 1 over q squared type of behavior.
OK?
So the story that we were talking about last time around
about the Goldstone modes and they're
fluctuating a lot because of their low cost also certainly
remains in this case.
We have now explicitly separated out
the longitudinal fluctuations that
are finite because they are controlled
by this stiffness of going up the bottom of the potential
whereas there is no stiffness associated
with the transverse ones.
OK?
All right, so good.
So we've talked about some of the things that
are experimentally observed.
Any questions?
Now we looked at things here in the Fourier space corresponding
to this momentum transfer in the scattering experiment,
but we can also ask about what is happening in physical space.
That is if I have a fluctuation at one point,
how much does the influence of that fluctuation
propagate in space?
So for that I need to calculate things like phi-- let's say,
do we need to put an invert, why not?--
phi l of x phi l of x prime.
Let's say we want to calculate this quantity.
OK?
Now I can certainly decompose phi l
of x in terms of these Fourier components.
And so what do I get?
I will get a sum over q-- maybe I should write it in one case
explicitly, q and q prime e to i cube of x e
to the i q prime the x prime.
Two factors of root 3 giving me the V
and then I have phi l of q phi l of q prime
and the expectation value will go over here.
Now we said that the different q's and q primes
are uncorrelated.
So here I immediately will have a delta function
mq plus q prime but if I'm looking at the same q and q
prime, I have this factor of 1 over K q squared
plus c to the minus 2-- c l to the minus 2.
OK?
So then the whole thing becomes due to the delta function
the sum over 1 q of 1 over V each with i q
dot x minus x prime because q prime was 2 minus q.
And then I have K 2 squared plus c l to the minus.
OK?
So then I go to the continuum limit of a large size,
the sum over q gets replaced with integral
over q d times the density of state.
The V's disappear, I will have a factor
of 2 pi to the d and what I have is the Fourier transform of k q
squared plus c to the minus-- l to the minus 2.
And I will write this as minus 1 over K
a function I that depends on d dimension
and clearly depends on the separation
x minus x prime at the correlation length c l.
Why I said correlation length shortly becomes apparent.
So I introduce a function I d which depends on x and c
to be minus integral over q 2 pi to the d Fourier transform of q
squared plus c to the minus 2.
If that c was not there that's the integral that we
did last time and it was the Coulomb
potential in d dimension.
So this presumably is related to that.
And we can use the same trick that we employed
to make it explicit last time around.
We can take the Laplacian of this potential i d and what
happens is that I will bring two factors of i
q squared so the minus goes away.
I will have the integral d d cubed 2 pi to the d.
I will have a q squared, denominator
is q squared plus c to the minus 2.
I add and subtract the c to the minus 2 to the numerator
and I have to Fourier transform.
OK?
The first part if I divide by the denominator is simply 1.
Integral of Fourier transform of 1
will give me a delta function.
And then what I have is minus c squared, the same integral
that I used to define i d.
So this becomes plus i d of x divided by c squared.
OK?
So whereas in the absence of c you have
the potential do to a charge, the presence of c
adds this additional term that corresponds
to some kind of a damping.
So this equation you probably have
seen in the context of screened Coulomb interaction
and giving rise to the [INAUDIBLE] potential
in three dimensions.
We would like to look at it in d dimension
so that we know what the behavior is in general.
OK?
So again what I'm looking at is the potential
that is due to a charge at the origin
so this idea of x in principle only
depends on the magnitude of x and not on the direction of it.
It is something that has general spherical symmetry
in d dimensions.
So I use that fact of spherical symmetry
to write down what the expression for the Laplacian
is.
OK?
We can again use Gauss's law if you've forgotten
or whatever but in the presence of spherical symmetry
the general expression for Laplacian in d dimension
is this.
OK?
So if this d was equal to 1 it would
be a simple second derivative.
And in higher dimensions you would have additional factors
if you basically apply Gauss's law to shares around here.
You can very easily convince yourself of that.
This is some kind of an aerial term
that comes in d minus 1 dimension.
And then I can write this as either the second derivative
if d by the x acts on this, the x to the minus 1
disappears or if it acts on this one,
it will gives me d minus 1 x to the d
minus 2, x to the d minus 1 gives me an x, the i by d x.
So the equation that I have to solve
is this object equals to i over c squared plus a delta function
[INAUDIBLE].
OK?
Now if you vary one dimension you wouldn't
have this term at all and you would have-- except that x
equals to 0, the second derivative proportional
to the function divided by c squared.
So you will immediately write a way
x equals to 0 that the answer is e to the minus x over c.
OK?
Actually proportional because you
have to fit out with the amplitude, et cetera.
Now in higher dimensions what happens
is that this solution gets modified,
falls off with some additional x to the p
but we have to be somewhat careful with this.
So let's look at this a little bit more closely.
If I were to substitute this ansatz into this expression,
what would happen?
What I need to do is to take the first and the second
derivative.
Now if I take the first derivative,
the derivative either acts on this factor,
gives me a factor of minus 1 over c and then
the exponential back, so I can get the i back.
If I had an exponential I take a derivative,
I will get just minus 1 over psi exponential.
If I act on x to the minus b I will get minus p
x to the p minus 1, which is different from the original
solution by a factor of p over x.
OK?
If I now take two derivatives I can take the second derivative
on I itself and then d I by d x will give me I
back with this factor.
So I will get 1 over c squared plus 2 P c x plus P squared
over x squared with I but that's not
the whole story because the derivative can also leave I
aside and act on P over x, which if it does so,
it will get P over x squared so that
will be an additional term here.
So that's the second derivative.
So now what I have done is I have evaluated
with this ansatz the terms that should appear
in that equation of a from x equals to 0, so let's
substitute it.
Everything now I have is proportional to I so I just
forget about the I.
I have the second derivative 1 over c
squared plus 2 P divided by x c plus p p
plus 1 divided by x squared.
And then I have d minus 1, the first derivative,
so I have minus d minus 1 over c minus d minus 1 p over x.
Both of these terms get an additional factor
of x because of here so I will get x c and x squared
and what I have on the right hand side from the origin
is I over c squared.
Divide by the I, I have 1 over c squared.
Now if I'm moving away from x I can organize
things in powers of 1 over x.
The most important term is the constant
and clearly you can see that I chose the decay
constant of the exponential correctly as
evidenced by the absence or removal of 1
over the constant term on the two sides.
But now I have two terms, two types of terms.
Terms that are proportional to x squared and terms
that are proportional to x psi and there's
no way that I can simultaneously satisfy both of these.
So the assumption that the solution of this equation
is a single exponential divided by a power law
is in fact not correct.
But it can be correct in two regions.
So for x that is much less than psi then
the more important term is the 1 over x squared part.
For x but going towards 0 the 1 over x squared
is more important than 1 over x.
OK so then what I do is I will match these two
terms and those two terms that are 1 over x squared tell me
that P p plus 1 should be P d minus 1.
OK?
And that immediately getting rid of the p's tells me
that the P in this regime is d minus 2.
OK?
Now the d minus 2 you recall is what
we had for the Coulomb potential.
Right?
So basically at short distances you
are still not screened by this additional term.
You don't see its effect and you get essentially
the standard Coulomb potential whereas if you are away
what you get is that you have to match the terms that
are proportional to x c because they're more important than 1
over x squared.
And there you get that 2 P should be d minus 1
or P should be d minus 1 over 2.
OK?
So let's just plot that function over here.
So if I plot this function as a function of the separation x
and it only depends on the magnitude,
in fact, what I should plot is minus i
d because it's the minus i d that
depends on the fluctuations once I divide
by K. I find that it has two regimes.
Let's say above two dimensions you have one regime that
is a simple Coulomb type of potential
and the Coulomb potential last time
actually we normalized properly.
We saw that it is x to the 2 minus d S d d minus 2.
The e to x into the minus x over c I
can in fact ignore in this regime
because I'm at distance x that is much less than c
so the exponential term has not kicked in yet
whereas I go at large distances and the exponential term does
kick in.
So the overall behavior is e to the minus x over c.
That's the most dominant behavior that you have.
On top of that, we have a power log
which is x to the power of d minus 1 over 2.
Now those of you who know what the screened Coulomb
potential is know that the screened Coulomb potential
in three dimensions is the 1 over r, the Coulomb potential,
and you put an exponential on top of that.
There is no difference in the powers
that you have whether or not you are
smaller than this correlation length or larger.
You can check here, if I put d equals to 3,
this becomes a 1 over x and this becomes a 1 over x.
So it's just an accident of three dimensions
that the screened Coulomb potential
is the 1 over r with an exponential on top.
In general dimensions you have different powers.
But having different powers also means
that somehow the amplitude that goes over
here has to carry dimensions so that it can be matched to what
we have here at this distance of C.
And so if I try to match those terms, roughly
when you're at order of c, what I would do is I would put
s d d minus 2 and c to the 3 minus d over 2.
And now you can check that the two expressions
will have the right dimension and will match roughly
at order of c.
OK?
So essentially what it says is that if I ask in my system what
is the nature of these fluctuations,
how correlated they are, they would
know to be more or less the same although falling off as if you
were at the critical point because we said
that the critical point or when you have Goldstone modes,
you have just this term.
But then they know that you are not exactly
sitting at the critical point and then
they are no longer correlated.
So basically there is this length scale
that we also saw when we were looking
at these critical opalescence and we
were seeing things that were moving together.
That length scale where things are moving together
is this parameter c that we have defined over here.
So what we have is-- where do we want to put it?
Let's put it here.
A correlation length which measures
the extent to which things are fluctuating together,
although when I'm saying fluctuating together,
they are still correlations that are falling off
but they're not falling off exponentially.
They start to fall off exponentially
when you are beyond this length scale c.
And we have the formula for c.
So what we find is that if I were
to invert that, for example, what I find for c l
as a function of t is that it is simply square root of k over t
when I am on the t positive side.
When I go to the t negative side,
it just becomes square root of K minus 2 t.
OK?
So this correlation length I indicated we could state
has behavior close to a transition,
there's a divergence.
We can parametrize those divergences through something
like t minus Tc to exponent u, potentially different
on the two sides of the transition.
But this t is simply proportional to the real t
minus Tc so we conclude that u plus is
the same as u minus we've just indicated by u,
should be one-half.
The amplitudes themselves depend on all kinds of things.
We don't know much about them.
But we can see that the amplitude ratio B
plus over B minus, if I were to divide those two
the ratio of those two is universal,
it gives me a factor of square root of 2.
OK?
If I were to plot c t, for example,
on the high temperature side c t and c l are of course the same.
On the low temperature side, we said that the Goldstone modes
have these long range correlations.
They fall off or grow according to the Coulomb potential
but there is no length scale so in some sense
the correlation length for the transverse modes
is always infinity.
OK.
Now actually in the second lecture what I said
was that the fact that the response
function such as susceptibility diverges
immediately tells you that there have to be long
range correlations, so we had predicted
before that c has to diverge.
But we were not sufficiently precise about the way
that it does, so let's try to do that.
Let's see.
A relationship more precisely with susceptibility
and these correlation lengths, so what we said more
generally was that the susceptibilities up
to various factors of data, et cetera, that are not
that important are related to the integrated magnetization
to magnetization connected correlation.
So basically, what I have to do is
to look at m minus its average at x,
m minus its average at some other point, which
means that what I'm really looking at
is the phi phi averages.
OK?
Now what we have shown right now is
that these averages are significant.
These phi phi correlations are significant
only over a distance that is this correlation length
and then they die off.
So we could basically as far as scaling and things
like that is concerned terminate this integration at c.
And that when we are looking at distances that are below that,
you don't see the effect of the exponential,
you just see the Coulomb power law
so you would see here fluctuations
that decay as x to the 2 minus d.
Right?
So essentially what you're doing is integrating x to the 2 minus
d in d dimension of space.
So you can see that immediately gets related
to the square of the correlation length.
X to the minus d n d d x, the d part vanishes,
there's a 2 that remains and gives you c squared.
If you like you can write it in spherical coordinates,
et cetera, but dimensions have to work out
to be something like this.
So now we can-- yes?
AUDIENCE: Just to clarify, when you sat phi of x and phi of 0,
are those both longitudinal or both transverse?
PROFESSOR: I wasn't precise so if I, thinking about chi l,
these will be both longitudinal.
OK?
And then we have this expression.
If I'm talking about chi t and I'm above the transition
temperature, there's no problem.
If I'm below the transition temperature,
I can use the same thing but have
to set c to infinity so I have to integrate
all the way to infinity.
OK?
But now you can see that the divergence of susceptibility
is very much related to the divergence of correlations,
in some sense very precisely in that
if this goes like t to the minus gamma
and the correlation length diverges as t to the minus u,
then gamma should be 2 nu.
And indeed, our ne is one-half.
We had seen previously that gamma was 2 nu.
Secondly that the amplitude ratio for susceptibility
should be the square of the amplitude ratio
for the correlation length and again this
is something that we have seen before.
The amplitude ratio for susceptibility
was the square root of 2.
Now it turns out that all of this
is a gain within this [INAUDIBLE] point approximation
looking at the most probable state, et cetera.
Because what we find in reality is that at the critical point,
the correlations don't decay simply according to the Coulomb
law but there is this additional eta
which is the same eta that we had over here.
OK?
And that because of that eta, here what you would have
is 2 minus eta and you would get an example
of a number of things that we will see a lot later on.
That is there even if you don't know what the exponents are,
you know that there are relationships
among the exponents.
This is an example of an exponent identity called
a Fisher exponent and there are several
of these exponents identities.
OK?
But that also brings us to the following-
that we did all of this work and we came up
with answers for the singular behaviors at critical points
and why they are universal.
And actually as far as the thermodynamic quantities
were concerned, all we ended up doing
was to write some expression that was analytical
and then find its minimum.
And we found that the minimum of an analytical expression
always has the same type of singularities, which
we can characterize by these exponents.
So maybe it's now a good time to check
how these match with the experiment.
So let's look at the various types of phase transition,
an example of the material that undergoes that phase
transition, and what the exponents alpha, beta, gamma,
and u are that are experimentally obtained.
AUDIENCE: What is this again?
PROFESSOR: The material that undergoes a transition,
so for example when we are talking about the ferromagnet
to paramagnet transition, you could look at material
such as iron or nickel and if we ask
in the context of this systematics
that we were developing for the Landau-Ginzburg what they
correspond to, they are things that have three components
or fields so they correspond to n equals to 3.
Of course everything that I will be talking to in this column
will correspond to 3-dimensional systems.
Later we'll talk also about 2-dimensional and other
systems, but let's stick with real 3-dimensional world.
So that would be one set.
We will look at super fluidity.
Let's say in helium, which we discussed last semester,
that corresponds to n equals to 2.
We will talk about various examples of liquid gas
transition which correspond to a scalar density difference.
And this could be anything from say carbon dioxide, neon,
argon, whatever gas we like.
And also talk about superconductors
which to all intents and purposes
should have the same type of symmetries as super fluids.
An example of a quantum system should be n equals to 2,
gained lots of different cases such as aluminum, copper,
whatever.
So what do we find for the exponent?
Actually for ferromagnetic system
the heat capacity does not diverge.
It has a discontinuous derivative at the transition
and kind of goes in a manner that if you take its derivative
then the derivative appears to be singular and corresponds
to an alpha.
If you try to fit it to it's slightly negative.
The superfluid has this famous lambda shape for its heat
capacity and a lambda shape is very well fitted
to a logarithm type of function.
The logarithm is the limit of a power law
as the exponent goes to 0 so we can more or less indicate that
by an alpha of 0 or really it's a divergent log.
These objects, the liquid gas transition
does have weakly divergent heat capacity
so the alpha is around 0.1.
The values of betas are all less than one-half,
for ferromagnet system is of the order of 0.4.
It is almost one-third, slightly less for superfluid helium
and less for the liquid gas system.
Gammas something like 1.4.
We don't have a gamma for superfluid,
you can't put a magnetic field on the superfluid.
There's nothing that is conjugate to the quantum phase.
Here it is more like 1.3.
Mu is 0.-- it's not-- [INAUDIBLE].
OK.
So what I have here it is more like 1.3, 1.24, 0.7, 0.67,
0.63.
OK?
Now they are different from the predictions that we had.
Predictions that we had where alpha was 0 discontinuous.
Beta goes to one-half.
Gamma was 1, mu equals to one-half.
And actually these predictions that we just made
happen to match extremely well with all kinds
of super conducting systems that you look at.
So again it is important to state
that within a particular class like liquid gas
you can do a lot of different systems.
We saw that curve in the second lecture.
They all correspond to this same set of exponents,
singularly for a different magnet and so forth.
So there is something that is universal
but our Landau-Ginzburg approach with this looking
at the most probable state and fluctuations around
it has not captured it for most cases
but for some reason has captured it
for the case of superconductors.
So we have that puzzle and starting from next lecture
we'll start to unravel that.