renormalisation ising-model vs. phi-4

[Max]

Hello List,

I fail to understand my lecture in statistical field theory ...

First we were introduced to the ising model hamiltonian and
blockspin-transformations. We could show for example that, in the 2D
Ising model, you can find a nontrivial unstable fixed point in the space
of coupling constants when scaling.

We took the Ising hamiltonian and transformed it into the hamilton of
a continuous field. In the end it looked like a phi-4 theory with coupling
constants depending on the lattice spacing.

We need dimensionless coupling constants to do the rescaling like in
the ising model, so we multiply them, after a dimensional analysis with
the correlation-length to the power of something.
This is the first unclear point. First: Why exactly to we need
dimensionless couplings ? And why can i just multiply it with the
correlation-length ? Is is a quantity that diverges at the critical point,
but i dont want my couplings to diverge, right ?

We looked at the 1D ising model again, to find out that
you can write the exact 2-point-correlation-function like in a gaussian
Model multiplied by a factor Z1 which goes to zero at the critical point
(it goes like correlation-length to the power of minus eta=1 in 1d ising).
So we propose that if we would scale our Field phi (we are free to scale
it as we whish) with Z1, the ising model could be described by a simple
gaussian model.

We look at the 4-point-correlation-function. It always
vanishes in a gaussian model. In the ising case there is the lebowitz
disequation, stating that it should be always smaller than zero. So, the
idea to use a gaussian model does not work.

To face this problem, we defined the 4-point-vertex-function as the
difference between 4 free propagators and the real 4-point-function.
And now, we just use this vertex-function (multiplied by the correlation
length to the power of something to get it dimensionless) as our
dimensionless coupling constant. And now, it turns out that this value
does not diverge any more with the correlation length.

This is the biggest point i dont understand. Why do we do that ?
I dont see the connection between the change of our length scale and the
problem with the 4-point-function. Why should you look at the
4-point-function anyways ? Because its phi-4 ? All i know is that we
rescal our lattice, therefore changing the couplings and we want to find a
fixed point somewhere!

(After that we go on to define the beta function as the derivation of the
constant to the correlation length and so on ...)

Any help would be appreciated!

-Max

[beheiger]

Hi Max,

I share your interest in that I also do not know solid answers to
most of your questions. It is a pity that no one else seem to know either.
Let's see:

> We need dimensionless coupling constants to do the rescaling like in
> the ising model, so we multiply them, after a dimensional analysis with
> the correlation-length to the power of something.
> This is the first unclear point. First: Why exactly to we need
> dimensionless couplings ?

Probably because their scaling properties are easy/trivial?

> And why can i just multiply it with the
> correlation-length ? Is is a quantity that diverges at the critical point,
> but i dont want my couplings to diverge, right ?

Are you sure that this "power of something" is positive?
I remember that above d=4 the fourth order coupling constant scales
with a negative power, so it goes to zero on interated rescaling,
which is good news as the model becomes Gaussian.
Below d=4 this does not happen, but instead the divergence can be handled
asymptotically via the epsilon-expansion, if I remember correctly.

>
> We looked at the 1D ising model again, to find out that
> you can write the exact 2-point-correlation-function like in a gaussian
> Model multiplied by a factor Z1 which goes to zero at the critical point
> (it goes like correlation-length to the power of minus eta=1 in 1d ising).
> So we propose that if we would scale our Field phi (we are free to scale
> it as we whish) with Z1, the ising model could be described by a simple
> gaussian model.
>
> We look at the 4-point-correlation-function. It always
> vanishes in a gaussian model. In the ising case there is the lebowitz
> disequation, stating that it should be always smaller than zero. So, the
> idea to use a gaussian model does not work.
>
> To face this problem, we defined the 4-point-vertex-function as the
> difference between 4 free propagators and the real 4-point-function.
> And now, we just use this vertex-function (multiplied by the correlation
> length to the power of something to get it dimensionless) as our
> dimensionless coupling constant. And now, it turns out that this value
> does not diverge any more with the correlation length.
>
> This is the biggest point i dont understand. Why do we do that ?
> I dont see the connection between the change of our length scale and the
> problem with the 4-point-function. Why should you look at the
> 4-point-function anyways ? Because its phi-4 ? All i know is that we
> rescal our lattice, therefore changing the couplings and we want to find a
> fixed point somewhere!

Good question. The following is just my own theory from what I still
remotely remember, so do not totally rely on it but feel free to
correct/improve it:
I guess that the 4-vertex function represents a kind of
"dressed" phi4 coupling constant with the effects of interaction absorbed.
By redefining the field, the quadratic "mass" term (which is related to the
inverse correlation length) is transformed into a constant, and the effect
of
vanishing mass (criticality) transferred to the behavior of the phi4
coupling
constant. This coupling constant itself cannot be measured directly,
but through the 4point correlation function.

So have you made any progress in the meanwhile that you would like to
discuss?

Andi