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PROFESSOR: OK.
Let's start.
So if we have been thinking about critical points.
And these arise in many phased diagrams
such as that we have for the liquid gas system where there's
a coexistence line, let's say, between the gas and the liquid
that terminate, or we looked in the case of a magnet
where as a function of [INAUDIBLE] temperature
there was in some sense coexistence
between magnetizations in different directions
terminating at the critical point.
So why is it interesting to take it a whole phase
diagram that we have over here?
For example, for this system, we can also have solid, et cetera.
AUDIENCE: So isn't this [INAUDIBLE] [INAUDIBLE]
PROFESSOR: [INAUDIBLE].
Yes.
Thank you.
And focus on just the one point.
In the vicinity of this one point.
And the reason for that was this idea of universality.
There many things that are happening
in the vicinity of this point as far
as singularitities, correlations, et cetera,
are concerned that are independent of whatever
the consequence of the system are.
And these singularities, we try to capture through some scaling
laws for the singularities.
And I've been kind of constructing
a table of your singularities.
Let's do it one more time here.
So we could look at system such as the liquid gas--
so let's have here system-- and then we
could look at the liquid gas.
And for that, we can look at a variety of exponents.
We have alpha, beta, gamma, delta, mu, theta.
And for the liquid gas, I write you some numbers.
The heat capacity diverges with an exponent
that is 0.11-- slightly more accurate
than I had given you before.
The case for beta is 0.33 gamma.
OK.
I will give you a little bit more digits just
to indicate the accuracy of experiments.
This is 1.238 minus plus 0.012.
So these exponents are obtained by looking at the fluid system
with light scattering-- doing this critical opalescence
that we were talking about in more detail
and accurately, Delta is 4.8.
The mu is, again from light scattering 0.629 minus plus
0.003.
Theta is 0.032 0 minus plus 0.013.
And essentially these three are [INAUDIBLE] light scattered.
Sorry.
Another case that I mentioned is that of the super fluid.
And in this general construction of the lambda
gives [INAUDIBLE] theories that we had,
liquid gas would be in question one.
Superfluid would be in question two.
And I just want to mention that actually the most
experimentally accurate exponent that has been determined
is the heat capacity for dry superfluid helium transition.
I had said that it kind of looks like a logarithmic divergence.
You look at it very closely.
And it is in fact a cusp, and does not diverge all the way
to infinity, so it corresponds to a slightly negative value
of alpha, which is the 0.0127 minus plus 0.0003.
And the way that this has been data-mined
is they took superfluid helium to the space shuttle,
and this experiments were done away
from the gravity of the earth in order
to not to have to worry about the density difference
that we would have across the system.
Other exponents that you have for this system--
let me write down-- beta is around 0.35.
Gamma is 1.32.
Delta is 4.79.
Mu is is 0.67.
Theta is 0.04.
And we don't need this for system.
Any questions we could do players kind of
add the exponents here I've booked usability
even if it's minus 1 is a research data
0.7 down all those are long this is more to say about new ideas
and so on I think is that these numbers aren't you
think that is simplest way for us is net
my position and the question is why these numbers are all
of the same as all the systems is therefore profound.
These are dimensionless numbers.
So in some sense, it is a little bit of mathematics.
It's not like you calculate the charge of the electron
and you get a number.
These don't depend on a specific material.
Therefore, what is important about them
is that they must somehow be capturing
some aspect of the collective behavior of all
of these degrees of freedom, in which the details of what
the degrees of freedom are is not that important.
Maybe the type of synergy rating is important.
So unless we understand and derive these numbers,
there is something important about the collective behavior
of many degrees of freedom that we have not understood.
And it is somehow a different question
if you are thinking about phase transitions.
So let's say you're thinking about superconductors.
There's a lot of interest in making high temperature
superconductor pushing TC further and further up.
So that's certainly a material problem.
We are asking a different problem.
Why is it, whether you have a high temperature superconductor
or any other type of system, the collective behavior
is captured by the same set of exponents.
So in an attempt to try to answer that,
we did this Landau-Ginzburg and try to calculate its singular
behavior using this other point of approximation.
And the numbers that we got, alpha
was 0, meaning that there was discontinuity.
Beta was 1/2, gamma was 1, delta was 3, my was 1/2,
theta was 0, which don't quite match with these numbers
that we have up there.
So question is, what should you do?
We've made an attempt and that attempt was not successful.
So we are going to completely for a while forget about that
and try to approach the problem from a different perspective
and see how far we can go, whether we
can gain any new insights.
So that new approach I put on there
the name of the scaling hypothesis.
And the reason for that will become apparent shortly.
So what we have in common in both of these examples
is that there is a line where there
are discontinuities in calculating
some thermodynamic function that terminates
at a particular point.
And in the case of the magnetic system,
we can look at the singularities approaching that point either
along the direction that corresponds
to change in temperature and parametrize that through heat,
or we can change the magnetic field
and approach the problem from this other direction.
And we saw that there were analogs for doing
so in the liquid gas system also.
And in particular, let's say we calculated a magnetization,
we found that there was one form of singularity coming this way,
one form of singularity coming that way.
We look at the picture for the liquid gas system
that I have up there, and it's not necessarily clear
which direction would correspond to this nice symmetry
breaking or non-symmetry breaking
that you have for the magnetic system.
So you may well ask, suppose I approach the critical point
along some other direction.
Maybe I come in along the path such as this.
I still go to the critical point.
We can imagine that for the liquid gas system.
And what's the structure of the singularities?
I know that there are different singularities in the t and h
direction.
What is it if I come and approach
the system along a different direction,
which we may well do for a liquid gas system?
Well, we could actually answer that if we go back
to our graph saddlepoint approximation.
In the saddlepoint approximation,
we said that ultimately, the singularities in terms
of these two parameters t and h--
so this is in the saddlepoint.
Part obtained by minimizing this function that
was appearing in the expansion in the exponent.
There was a t over 2m squared.
There was a mu n to the 4th.
And there is an hm.
So we had to minimize this with respect to m.
And clearly, what that gives us is m.
If I really solve the equation, that
corresponds to this minimization,
which is a function of t and h.
And in particular, approaching two directions that's
indicated, if I'm along the direction where h equals 0,
I essentially balance these two terms.
Let's just write this as a proportionality.
I don't really care about the numbers.
Along the direction where h equals 0,
I have to balance m to the 4th and tm squared.
So m squared will scale like e.
m will scale like square root of t.
And more precisely, we calculated
this formula for t negative and h equals to 0.
If I, on the other hand, come along the direction
that corresponds to t equals to 0, along that direction
I don't have a first term.
I have to balance um the 4th and hm.
So we immediately see that m will scale like h over u.
In fact, more correctly h over 4u to the power of one third.
You substitute this in the free energy
and you find that the singular part of the free energy
as a function of t and h in this saddlepoint approximation
has the [INAUDIBLE] to the form of proportionality.
If I substitute this in the formula for t negative,
I will get something like minus t squared
over 4 we have-- forget about the number t squared over u.
If I go along the t equals to 0 direction,
substitute that over there, I will get n to the 4th.
I will get h to the 4 thirds divided by mu to the one third.
Even the mu dependence I'm not interested.
I'm really interested in the behavior close to t and h
as a function of t and h.
Mu is basically some non-universal number
that doesn't go to 0.
I could in some sense capture these two expressions
by a form that is t squared and then
some function-- let's call it g sub
f which is a function of-- let's see
how I define the delta h over t to the delta.
So my claim is that I toyed with the behavior coming
across these two different special direction.
In general, anywhere else where t and h are both nonzero,
the answer for m will be some solution of a cubic equation,
but we can arrange it to only be a function of h
over [INAUDIBLE] and have this form.
Now I could maybe rather than explicitly
show you how that arises, which is not
difficult-- you can do that-- since there's something
that we need to do later on, I'll
show it in the following manner.
I have not specified what this function g sub f is.
But I know its behavior along h equals to 0 here.
And so if I put h equals to 0, the argument of the function
goes to 0.
So if I say that the argument of the function
is a constant-- the constant let's say is minus 1
over u on one side, 0 on the other side,
then everything's fine.
So I have is that the limit as its argument goes to 0
should be some constant.
Well, what about the other direction?
How can I reproduce from a form such as this
the behavior when t equals to 0?
Because I see that when t equals to 0,
the answer of course cannot depend on t itself,
but as a power law as a function of h.
Is it consistent with this form?
Well, as t goes to 0 in this form,
the numerator here goes to 0, the argument of the function
goes to infinity, I need to know something
about the behavior of the function of infinity.
So let's say that the limiting behavior as the argument
of the function goes to infinity of gf
of x is proportional to the argument to some other peak.
And I don't know where that power is.
Then if I look at this function, the whole function
in this limit where t goes to 0 will behave.
There's a t squared out front the goes to 0,
the argument of the function goes to infinity.
So the function will go like the argument to some power.
So I go like h t to the delta to some other peak.
So what do I know?
I know that the answer should really
be proportional to h to the four thirds.
So I immediately know that my t should be four thirds.
But what about this delta?
I never told you what delta was.
Now I can figure out what delta is, because the answer should
not depend on t. t has gone to 0.
And so what power of t do I have?
I have 2 minus delta p should be 0.
So my delta should be 2 over p, 2 over four thirds,
so it should be three halves.
Why is this exponent relevant to the question that I had before?
You can see that the function that describes the free energy
as a function of these two coordinates.
If I look at the combination where h and t are non-zero,
is very much dependent on this h divided by t to the delta,
and that delta is three halves.
So, for example, if I were to draw here
curves where h goes like 3 to the three halves--
it's some coefficient, I don't know what that coefficient is--
then essentially, everything that is on the side
that hogs the vertical axis behaves like the h singularity.
Everything that is over here depends like a t singularity.
So a path that, for example, comes along a straight line,
if I, let's say, call the distance that I have
to the critical point s, then t is something
like s cosine of theta. h is something like s sine theta.
You can see however that the information
h over t to the delta as s goes to 0 will diverge,
because I have other three halves down here
for s that will overcome the linear cover I have over there.
So for any linear path that goes through the critical point,
eventually for small s I will see the type of singularity
that is characteristic of the magnetic field
if the exponents are according to this other point.
We have this assumption, of course.
But if I therefore knew the correct delta
for all of those systems, I would be also able to answer,
let's say for the liquid gas, whether if I
take a linear path that goes through the critical point
I would see one set of singularities or deltas
that have singularities.
So this delta which is called a gap exponent,
gives you the answer to that.
But of course I don't know the other exponents.
There is no reason for me to trust the gap exponent
that I obtained in this fashion.
So what I say is let's assume that for any critical point,
the singular part of the free energy
on approaching the critical point which
depends on this pair of coordinates
has a singular behavior that is similar to what
we had over here, except that I don't know the exponent.
So rather than putting 2 t squared,
I write t to the 2 minus alpha for reason
that will become apparent shortly,
and some function of h t to the delta and for some
alpha and delta.
So this is certainly already an assumption.
This mathematically corresponds to having
homogeneous functions.
Because if I have a function of x and y,
I can certainly write lots of functions such as x squared
plus y squared plus a constant plus x cubed y cubed that I
cannot rearrange into this form.
But there are certain functions of x and y
that I can rearrange so that I can pull out
some factor of let's say x squared out front,
and everything that is then in a series
is a function of let's say y over x cubed.
Something like that.
So there's some class of functions
of two arguments that have this homogeneity.
So we are going to assume that the singular behavior close
to the critical point is described by such a function.
That's an assumption.
But having made that assumption, let's follow its consequence
and let's see if we learned something
about that table of exponents.
Now the first thing to note is clearly
I chose this alpha over here so that when
I take two derivatives with respect to t,
I would get something like a heat capacity,
for which I know what the divergence is.
That's a divergence called alpha.
But there's one thing that I have
to show you is that when I take a derivative of one
of these homogeneous functions, with respect
to one of its arguments, I will generate
another homogeneous function.
If I take one derivative with respect to t,
that derivative can either act on this,
leaving the function unchanged, or it
can act on the argument of the function
and give me d to the 2 minus alpha.
I will have minus h t to the power of delta plus 1.
There will be a factor of delta and then
I will have the derivative function ht to the delta.
So I just took derivatives.
I can certainly pull out a factor of t
to the 1 minus alpha.
Then the first term is just 2 minus alpha times
the original function.
The second term is minus delta h divided by t to the delta.
Because I pulled out the 1 minus alpha,
this t gets rid of the factor of 1 there.
And I have the derivative.
So this is completely different function.
It's not the derivative of the original function.
But whatever it is it is still only a function
of the combination h over t to the delta.
So the derivative of a homogeneous function
is some other homogeneous function.
Let's call it g2.
It doesn't matter.
Let's call it g1 ht to the delta.
And this will happen if I take a second derivative.
So I know that if I take two derivatives,
I will get t to the minus alpha.
I will basically drop two factors over there.
And then some other function, ht to the delta.
Clearly again, if I say that I'm looking at the line
where h equals to zero for a magnet,
then the argument of the function goes to 0.
If I say that the function of the argument goes to 0
is a constant, like we had over here,
then I will have the singularity t to the minus alpha.
So I've clearly engineered whatever the value of alpha
is in this table, I can put over here
and I have the right singularity for the heat capacity.
Essentially I've put it there by hand.
Let me comment on one other thing, which
is when we are looking at just the temperature,
let's say we are looking at something
like a superfluid, the only parameter
that we have at our disposal is temperature and tens of ITC.
Let's say we plug the heat capacity
and then we see divergence of the heat capacity on the two
sides.
Who said that I should have the same exponent on this side
and on this side?
So we said that generally, in principle,
I could say I would do that.
And in principle, there is no problem with that.
If there is function that has one behavior here,
another behavior there, who says that two exponents
have to be the same?
But I have said something more.
I have said that in all of the cases that I'm looking at,
I know that there is some other axis.
And for example, if I am in the liquid gas system,
I can start from down here, go all the way around
back here without encountering a singularity.
I can go from the liquid all the way to gas
without encountering a singularity.
So that says that the system is different from a system
that, let's say, has a line of singularities.
So if I now take the functions that in principle
have two different singularities,
t to the minus alpha minus t to the minus alpha plus on the h
equals to 0 axis and try to elevate them
into the entire space by putting this homogeneous functions
in front of them, there is one and only one way
in which the two functions can match exactly on this t
equals to 0 line, and that's if the two exponents are the same
and you are dealing with the same function.
So that we put in a bit of physics.
So in principle, mathematically if you don't have the h axis
and you look at the one line and there's a singularity,
there's no reason why the two singularities
should be the same.
But we know that we are looking at the class
of physical systems where there is the possibility
to analytically go from one side to the other side.
And that immediately imposes this constraint
that alpha plus should be alpha minus, and one alpha
is in fact sufficient.
And I gave you the correct answer for why that is.
If you want to see the precise mathematical details
step by step, then that's in the notes.
So fine.
So far we haven't learned much.
We've justified why the two alphas should
be the same above and below, but we
put the alpha, the one alpha, then by hand.
And then we have this unknown delta also.
But let's proceed.
Let's see what other consequence emerge, because now we
have a function of two variables.
I took derivatives in respect to t.
I can take derivatives with respect to m.
And in particular, the magnetization
m as a function of t and h is obtained
from a derivative of the free energy with respect to h.
There's potential.
It's the response to adding a field could
be some factor of beta c or whatever.
It's not important.
The singular part will come from this.
And so taking a derivative of this function
I will get t this to the 2 minus alpha.
The derivative of a can be respect to h,
but h comes in the combination h over t
to the delta will bring down a factor of minus delta up front.
Then the derivative function-- let's call it gf1, for example.
So now I can look at this function in the limit
where h goes to 0, climb along the coexistence line,
h2 goes to 0.
The argument of the function has gone to 0.
Makes sense that the function should be constant
when its argument goes to 0.
So the answer is going to be proportional to t
to the 2 minus alpha minus delta.
But that's how beta was defined.
So if I know my beta and alpha, then I
can calculate my delta from this exponent identity.
Again, so far you haven't done much.
You have translated two unknown exponents, this singular form,
this gap exponent that we don't know.
I can also look at the other limit
where t goes to 0 that is calculating
the magnetization along the critical isotherm.
So then the argument of the function has gone to infinity.
And whatever the answer is should not depend on t,
because I have said t goes to 0.
So I apply the same trick that I did over here.
I say that when the argument goes to infinity,
the function goes like some power of its argument.
And clearly I have to choose that power such that the t
dependence, since t is going to 0, I have to get rid of it.
The only way that I can do that is if p is 2 minus alpha
minus delta divided by that.
So having done that, the whole thing will then
be a function of h to the p.
But the shape of the magnetization
along the critical isotherm, which was also
the shape of the isotherm of the liquid gas system,
we were characterizing by an exponent
that we were calling 1 over delta.
So we have now a formula that says
my delta shouldn't in fact be the inverse of p.
It should be the delta 2 minus alpha minus delta.
Yes?
AUDIENCE: Why isn't the exponent t minus 1
after you've differentiated [INAUDIBLE]?
Because g originally was defined as [INAUDIBLE].
PROFESSOR: Let's call it pr.
Because actually, you're right.
If this is the same g and this has particular singularity
[INAUDIBLE].
But at the end of the day, it doesn't matter.
So now I have gained something that I didn't have before.
That is, in principle I hit alpha
and beta, my two exponents, I'm able to figure out
what delta is.
And actually I can also figure out what gamma is ,
because gamma describes the divergence
of the susceptibility.
[INAUDIBLE] which is the derivative of magnetization
with respect to field, I have to take
another derivative of this function.
Taking another derivative with respect to h
will bring down another factor of delta.
So this becomes minus 2 delta.
Some other double derivative function h 2 to the delta.
And susceptibilities, we are typically
interested in the limit where the field goes to 0.
And we define them to diverge with exponent gamma.
So we have identified gamma to be 2 delta plus alpha minus 2.
So we have learned something.
Let's summarize it.
So the consequences-- one is we established
that same critical exponents above and below.
Now since various quantities of interest
are obtained by taking derivatives
of our homogeneous function and they
turn into homogeneous functions, we
conclude that all quantities are homogeneous functions
of the same combination ht to the delta.
Same delta governs it.
And thirdly, once we make this answer our assumption
for the free energy, we can calculate the other exponents
on the table.
So all, of almost all other exponents related to 2,
in this case alpha and delta.
Which means that if you have a number of different exponents
that all depend on 2, there should
be some identities, exponent identities.
It's these numbers in the table, we predict if all of this
is varied have some relationships with t.
So let's show a couple of these relationships.
So let's look at the combination alpha plus 2 beta plus gamma.
Measurement of heat capacity, magnetization, susceptibility.
Three different things.
So alpha is alpha 2.
My beta up there is 2 minus alpha minus delta.
My gamma is 2 delta plus alpha minus 2.
We got algebra.
There's one alpha minus 2 alpha plus alpha.
Alpha is cancelled.
Minus 2 deltas plus 2 deltas then it does cancel.
I have 2 times 2 minus 2, so that 2.
So the prediction is that you take some line on the table,
add alpha, beta, 2 beta plus gamma, they
should add up to one.
So let's pick something.
Let's pick a first-- actually, let's
pick the last line that has a negative alpha.
So let's do n equals to 3.
For n equals to 3 I have alpha which is minus .12.
I have twice beta, that is .37, so that becomes 74.
And then I have gamma, which is 1.39.
So this is 74.
I have 9 plus 413 minus 2, which is 1.
I have 3 plus 7, which is 10, minus 1, which is 9.
But then I had a 1 that was carried over, so I will have 0.
So then I have 1, so 201.
Not bad.
Now this goes by the name of the Rushbrooke identity.
The Rushbrooke made a simple manipulation
based on thermodynamics and you have a relationship with these.
Let's do another one.
Let's do delta and subtract 1 from it.
What is my delta?
I have delta to the delta 2 plus alpha minus delta.
This is small delta versus big delta.
And then I have minus 1.
Taking that into the numerator with the common denominator
of 2 plus alpha minus delta, this minus delta
becomes plus delta, which this becomes 2 delta minus
alpha minus 2.
2 delta
AUDIENCE: Should that be a minus alpha in the denominator?
PROFESSOR: It better be.
Yes.
2 delta plus alpha minus 2.
Then we can read off the gamma.
So this is gamma over beta.
And let's check this, let's say for m equals to 2.
No, let's check it for m plus 21, for the following
reason, that for n equals to 1, what we have for delta is
4.8 minus 1, which would be 3.8.
And on the other side, we have gamma over beta.
Gamma is 1.24, roughly, divided by beta .33,
which is roughly one third.
So I multiply this by 3.
And that becomes 3.72.
This one is known after another famous physicist, Ben Widom,
as the Widom identity.
So that's nice.
We can start learning that although we don't know anything
about this table, these are not independent numbers.
There's relationship between them.
And they're named after famous physicists.
Yes?
AUDIENCE: Can we briefly go over again what extra assumption
we had put in to get these in and these out?
Is it just that we have this homogeneous function
[INAUDIBLE]?
PROFESSOR: That's right.
So you assume that the singularity
in the vicinity of the critical point as a function
of deviations from that critical point
can be expressed as a homogeneous function.
The homogeneous function, you can rearrange any way you like.
One nice way to rearrange it is in this fashion.
It will depend, the homogeneous function on two exponents.
I chose to write it as 2 minus alpha
so that one of the exponents would immediately be alpha.
The other one I couldn't immediately
write in terms of beta or gamma.
I had to do these manipulations to find out
what the relationship [INAUDIBLE].
But the physics of it is simple.
That is, once you know the singularity of a free energy,
various other quantities you obtain
by taking derivatives of the free energy.
That's [INAUDIBLE] And so then you
would have the singular behavior of [INAUDIBLE].
So I started by saying that all other exponents,
but then I realized we have nothing so far that tells us
anything about mu and eta.
Because mu and eta relate to correlations.
They are in microscopic quantities.
Alpha, beta, gamma depend on macroscopic thermodynamic
quantities, magnetization susceptibility.
So there's no way that I will be able to get information,
almost.
No easy way or no direct way to get information
about mu and eta.
So I will go to assumption 2.0.
Go to the next version of the homogeneity assumption, which
is to emphasize that we certainly
know, again from physics and the relationship
between susceptibility and correlations,
that the reason for the divergence
of the susceptibility is that the correlations become large.
So we'll emphasize that.
So let's write our ansatz not about the free energy,
but about the correlation length.
So let's replace that ansatz with homogeneity
of correlation length.
So once more, we have a structure
where is a line that is terminate
when two parameters, t and h go to 0.
And we know that on approaching this point,
the system will become cloudy.
There's a correlation length that
diverges on approaching that point a function of these two
arguments.
I'm going to make the same homogeneity assumption
for the correlation length.
And again, this is an assumption.
I say that this is a to the minus mu.
The exponent mu was a divergence of the correlation length.
Some other function, it's not that first g that we wrote.
Let's call it g psi of ht to the delta.
So we never discussed it, but this function immediately
also tells me if you approach the critical point
along the criticalizer term, how does the correlation length
diverge through the various tricks that we have discussed?
But this is going to be telling me something more
if from here, I can reproduce my scaling assumption 1.0.
So there is one other step that I can make.
Assume divergence of c is responsible--
let's call it even solely responsible-- for singular
behavior.
And you say, what does all of this mean?
So let's say that I have a system could be my magnet,
could be my liquid gas that has size l on each search.
And I calculate the partition function log z.
Log z will certainly have the part that is regular.
Well-- log z will have a part that is certainly-- let's
say the contribution phonons, all kinds
of other regular things that don't have anything
to do with singularity of the system.
Those things will give you some regular function.
But one thing that I know for sure
is that the answer is going to be extensive.
If I have any nice thermodynamic system
and I am in v dimensions, then it
will be proportional to the volume of that system
that I have.
Now the way that I have written it is not entirely nice,
because log z is-- a log is a dimensionless quantity.
Maybe I measured my length in meters or centimeters
or whatever, so I have dimensions here.
So it makes sense to pick some landscape to dimensionalize it
before multiplying it by some kind of irregular function
of whatever I have, t and h, for example.
But what about the singular part?
For the singular part, the statement
was that somehow it was a connective behavior.
It involved many, many degrees of freedom.
We saw for the heat capacity of the solid at low temperatures,
it came from long wavelength degrees of freedom.
So no lattice parameter is going to be important.
So one thing that I could do, maintaining extensivity,
is to divide by l over c times something.
So that's the only thing that I did
to ensure that extensivity is maintained when
I have kind of benign landscape, but in addition a landscape
that is divergent.
Now you can see that immediately that says that log
z singular as a function of t and h,
will be proportional to c to the minus t.
And using that formula, it will be
proportional to t to the du, some other scaling function.
And it's go back to gf ht to the delta.
Physically, what it's saying is that when
I am very close, but not quite at the critical point,
I have a long correlation length, much larger
than microscopic length scale of my system.
So what I can say is that within a correlation length,
my degrees of freedom for magentization or whatever it is
are very much coupled to each other.
So maybe what I can do is I can regard
this as an independent lock.
And how many independent locks do I have?
It is l over c to the d.
So the statement roughly is a part
of the assumption is that this correlation length that is
getting bigger and bigger.
Because things are correlated, the number
of independent degrees of freedom that you
are having gets smaller and smaller.
And that's changing the number of degrees of freedom
is responsible for the singular behavior of the free energy.
If I make this assumption about this correlation then diverges,
then I will get this form.
So now my ansatz 2.0 matches my ansatz 1.0
provided du is 2 minus alpha.
So I have du2 plus 2 minus alpha which
is known after Brian Josephson, so this
is the Josephson relation.
And it is different from the other exponent identities
that we have because it explicitly
depends on the dimensionality of space.
d appears in the problem.
It's called hyperscale for that reason.
Yes?
AUDIENCE: So does the assumption that the divergence in c
is solely responsible for the singular behavior, what
are we excluding when we assume that?
What else could happen that would make that not true?
PROFESSOR: Well, what is appearing here maybe
will have some singular function of t and h.
AUDIENCE: So this similar to what
we were assuming before when we said that our free energy could
have some regular part that depends
on [INAUDIBLE] the part that [INAUDIBLE].
PROFESSOR: Yes, exactly.
But once again, the truth is really whether or not
this matches up with experiments.
So let's, for example, pick anything in that
table, v equals to t.
Let's pick n goes to 2, which we haven't done so far.
And so what the formula would say is 3 times mu.
Mu for the superfluid is 67 is 2 minus-- well,
alpha is almost 0 but slightly negative.
So it is 0.01.
And what do we have?
3 times 67 is 2.01.
So it matches.
Actually, we say, well, why do you emphasize
that it's the function of dimension?
Well, a little bit later on in the course,
we will do an exact solution of the so-called 2D Ising model.
So this is a system that first wants to be close to 2,
n equals to 1.
And it was an important thing that people could actually
solve an interacting problem, not in three dimensions
but in two.
And the exponents for that, alpha is 0,
but it really is a logarithmic divergence.
Beta is 1/8.
Gamma is 7/4, delta is 15, mu is 1, and eta is 1/4.
And we can check now for this v equals to 2 n
equals to 1 that we have two times
our mu, which exactly is known to be
1 is 2 minus logarithmic divergence corresponding to 0.
So again, there's something that works.
One thing that you may want to see and look at
is that the ansatz that we made first also
works for the result of saddlepoint,
not surprisingly because again in the saddlepoint
we start with a singular free energy and go through all this.
But it does not work for this type of scaling,
because 2 minus alpha would be 0 is not
equal to d times one half, except in the case of four
dimensions.
So somehow, this ansatz and this picture
breaks down within the saddlepoint approximation.
If you remember what we did when we calculated fluctuation
corrections for the saddlepoint, you
got actually an exponent alpha that was 2 minus mu over 2.
So the fluctuating part that we get around the saddlepoint
does satisfy this.
But on top of that there's another part that
is doe to the saddlepoint value itself
that violates this hyperscaling solution.
Yes?
AUDIENCE: Empirically, how well can we probe the dependence
on dimensionality that we're finding in these expressions?
PROFESSOR: Experimentally, we can
do d equals to 2 d equals to 3.
And computer simulations we can also do d equals to 2 d
equals to 3.
Very soon, we will do analytical expressions
where we will be in 3.99 dimensions.
So we will be coming down conservatively around 4.
So mathematically, we can play tricks such as that.
But certainly empirically, in the sense of experimentally
we are at a disadvantage in those languages.
OK?
So we are making progress.
We have made our way across this table.
We have also an identity that involves mu.
But so far I haven't said anything about eta.
I can say something about the eta reasonably simply,
but then you try to build something profound
based on that.
So let's look at exactly at tc, at the critical point.
So let's say you are sitting at t and h equals to 0.
You have to prepare your system at that point.
There's nothing physically that says you can't.
At that point, you can look at correlations.
And the exponent eta for example is a characteristic
of those correlations.
And one of the things that we have is that m of x m of 0,
the connected parts-- well, actually at the critical point
we don't even have to put the connected part
because the average of n is going to be 0.
But this is a quantity that behaves
as 1 over the separation that's actually
include two possible points, x minus y.
When we did the case of the fluctuations
at the critical point within the saddlepoint method,
we found that the behavior was like the Coulomb law.
It was falling off as 1x to the d minus 2.
But we said that experiment indicated
that there is a small correction for this
that we indicate with exponent eta.
So that was how the exponent eta was defined.
So can we have an identity that involves the exponent eta?
We actually have seen how to do this already.
Because we know that in general, the susceptibilities
are related to integrals of the correlation functions.
Now if I put this power law over here,
you can see that the answer is like trying
to be integrate x squared all the way to infinity down.
So it will be divergent and that's no problem.
At the critical point we know that the susceptibility
is divergent.
But you say, OK, if I'm away from the critical point,
then I will use this formula, but only
up to the correlation length.
And I say that beyond the correlation length,
then the correlations will decay exponentially.
That's too rapid a falloff, and essentially the only part
that's contributing is because what
was happening at the critical point.
Once I do that, I have to integrate ddx over x to the d
minus 2 plus eta up to the correlation length.
The answer will be proportional to the correlation length
to the power of 2 minus eta.
And this will be proportional to p to the power of c goes
[INAUDIBLE] to the minus mu.
2 minus eta times mu.
But we know that the susceptibilities
diverge as t to the minus gamma.
So we have established an exponent identity
that tells us that gamma is 2 minus eta times mu.
And this is known as the Fisher identity, after Michael Fisher.
Again, you can see that in all of the cases in three
dimensions that we are dealing with,
exponent eta is roughly 0.
It's 0.04 And all of our gammas are
roughly twice what our mus are in that table.
It's time we get that table checked.
The one case that I have on that table where eta is not 0
is when I'm looking at v positive 2 where eta is 1/4.
So I take 2 minus 1/4, multiply it
by the mu that is one in two dimension,
and the answer is the 7/4, which we
have for the exponent gamma over there.
So we have now the identity that is
applicable to the last exponents.
So all of this works.
Let's now take the conceptual leap
that then allows us to do what we
will do later on to get the exponents.
Basically, you can see that what we have imposed
here conceptually is the following.
That when I'm away from the critical point,
I look at the correlations of this important statistical
field.
And I find that they fall off with separation,
according to some power.
And the reason is that at the critical point,
the correlation length has gone to infinity.
That's not the length scale that you have to play with.
You can divide x minus y divided by c, which is what we do away
from the critical point. c has gone to infinity.
The other length scale that we are worried about
are things that go into the microscopics.
but we are assuming that microscopics is irrelevant.
It has been washed out.
So if we don't have a large length scale,
if we don't have a short length scale, some function
of distance, how can it decay?
The only way it can decay is [INAUDIBLE].
So this statement is that when we are at a critical point,
I look at some correlation.
And this was the magnetization correlation.
But I can look at correlation of anything else
as a function of separation.
And this will only fall off as some power of separation.
Another way of writing it is that if I
were to multiply this by some length scale,
so rather than looking at things that are some distance apart
at twice that distance apart or hundred times
that distance apart, I will reproduce the correlation
that I have up to some other of the scale factor.
So the scale factor here we can read off
has to be related to t minus 2 plus eta.
But essentially, this is a statement again
about homogeneity of correlation functions
when you are at a critical point.
So this is a symmetry here.
It says you take your statistical correlations
and you look at them at the larger scale
or at the shorter scale.
And up to some overall scale factor,
you reproduce what you had before.
So this is something to do with invariance on the scale.
This scaling variance is some property
that was popular a while ago as being associated
with the kind of geometrical objects
that you would call fractals.
So the statement is that if I go across my system
and there is some pattern of magnetization fluctuations,
let's say I look at it.
I'm going along this direction x.
And I plot at some particular configuration
that is dominant and is contributing
to my free energy, the magnetization,
that it has a shape that has this characteristic self
similarity kind of maybe looking like a mountain landscape.
And the statement is that if I were
to take a part of that landscape and then blow it up,
I will generate a pattern that is of course not the same
as the first one.
It is not exactly scale invariant.
But it has the same kind of statistics as the one
that I had originally after I multiplied this axis
by some factor lambda.
Yes?
AUDIENCE: Under what length scales are those subsimilarity
properties evident and how do they compare to the length
scale over which you're doing your course
grading for this field?
PROFESSOR: OK, so basically we expect this to be applicable
presumably at length scales that are
less than the size of your system
because once I get to the size of the system
I can't blow it up further or whatever.
It has to certainly be larger than whatever
the coarse-graining length is, or the length scale at which
I have confidence that I have washed out the microscoping
details.
Now that depends on the system in question,
so I can't really give you an answer for that.
The answer will depend on the system.
But the point is that I'm looking
in the vicinity of a point where mathematically I'm
assured that there's a correlation length that
goes to infinity.
So maybe there is some system number 1 that average out
very easily, and after a distance of 10
I can start applying this.
But maybe there's some other system
where the microscopic degrees of freedom are very problematic
and I have to go further and further out before they average
out.
But in principle, since my c has gone to infinity,
I can just pick a bigger and bigger piece of my system
until that has happened.
So I can't tell you what the short distance length
scale is in the same sense that when [INAUDIBLE] says
that coast of Britain is fractal, well,
I can't tell you whether the short distance is
the size of a sand particle, or is it
the size of, I don't know, a tree or something like that.
I don't know.
So we started thinking about our original problem.
And constructing this Landau-Ginzburg,
[INAUDIBLE] that we worked with on the basis of symmetries
such as invariance on the rotation, et cetera.
Somehow we've discovered that the point that we
are interested has an additional symmetry that maybe we
didn't anticipate, which is this self-similarity and scale
invariance.
So you say, OK, that's the solution to the problem.
Let's go back to our construction
of the Landau-Ginsburg theory and add
to the list of symmetries that have
to be obeyed, this additional self-similarity of scaling.
And that will put us at t equals to 0, h equals to 0.
And for example, we should be able to calculate
this correlation.
Let me expand a little bit on that because we
will need one other correlation.
Because we've said that essentially,
all of the properties of the system
I can get from two independent exponents.
So suppose I constructed this scale invariant theory
and I calculated this.
That would be on exponent.
I need another one.
Well, we had here a statement about alpha.
We made the statement that heat capacity diverges.
Now in the same sense that the susceptibility is a response--
it came from two derivatives of the free energy with respect
to the field.
The derivative of magnetization with respect
to field magnetization is one derivative [INAUDIBLE].
The heat capacity is also two derivatives
of free energy with respect to some other variable.
So in the same sense that there is
a relationship between the susceptibility
and an integrated correlation function,
there is a relationship that says that the heat capacity is
related to an integrated correlation function.
So c as a function of say t and h, let's say the singular part,
is going to be related to an integral of something.
And again, we've already seen this.
Essentially, you take one derivative of the free energy
let's say with respect the beta or temperature,
you get the energy.
And you take another derivative of the energy
you will get the heat capacity.
And then that derivative, if we write
in terms of the first derivative of the partition function
becomes converted to the variance in energy.
So in the same way that the susceptibility was
the variance of the net magnetization,
the heat capacity is related to the variance
of the net energy of the system at an even temperature.
The net energy of the system we can write this
as an integral of an energy density,
just as we wrote the magnetization
as an integral of magnetization density.
And then the heat capacity will be related to the correlation
functions of the energy density.
Now once more, you say that I'm at the critical point.
At the critical point there is no length scale.
So any correlation function, not only that of the magnetization,
should fall off as some power of separation.
And you can call that exponent whatever you like.
There is no definition for it in the literature.
Let me write it in the same way as magnetization
as d minus 2 plus eta prime.
So then when I go and say let's terminate it at the correlation
length, the answer is going to be
proportional to c to the 2 minus eta prime,
which would be t to the minus mu.
2 minus eta prime.
So then I would have alpha being mu 2 minus eta.
So all I need to do in principle is
to construct a theory, which in addition
to rotational invariance or there's
whatever is appropriate to the system in question,
has this statistical scale invariance.
Within that theory, calculate the correlation functions
of two quantities, such as magnetization and energy.
Extract two exponents.
Once we have two exponents, then we
know why your manipulations will be
able to calculate all the exponents.
So why doesn't this solve the problem?
The answer is that whereas I can write immediately for you
a term such as m squared, that is rotational invariant,
I don't know how to write down a theory that is scale invariant.
The one case where people have succeeded to do that
is actually two dimensions.
So in two dimensions, one can show
that this kind of scale invariance
is related to conformal invariance
and that one can explicitly write down conformal invariant
theories, extract exponents et cetera out of those.
But say in three dimensions, we don't know how to do that.
So we will still, with that concept
in the back of our mind, approach it
slightly differently by looking at the effects of the scale
transformation on the system.
And that's the beginning of this concept of normalization.