Does the Area Law Always Hold True in Lattice Gauge Models?

[jfy4]

While reading through Kogut's lattice gauge theory introduction he goes through the area and perimeter laws for lattice gauge models. The result is something like this
$$\left\langle \prod _{l\in C}\sigma_{3}(l) \right\rangle \sim \exp(-P)$$
for low temperature, and
$$\left\langle \prod _{l\in C}\sigma_{3}(l) \right\rangle \sim \exp(-A)$$
for high temperature. He then says

So, at high T the correlation function falls very quickly as the loop is taken larger and larger, while at low T it falls off at a qualitatively slower rate.

However, I can certainly conceive of a couple different closed loops which have more perimeter units than area units. Is this relationship between area size and perimeter size more subtle, or have I missed something. Thanks.

 

[fzero]

However, I can certainly conceive of a couple different closed loops which have more perimeter units than area units. Is this relationship between area size and perimeter size more subtle, or have I missed something. Thanks.

You probably have in mind some degenerate loops which have one dimension parametrically small compared to the other. These are not loops that contribute to the discussion of confinement. Recall that one piece of evidence of confinement is that the quark-antiquark potential increases with separation. For your degenerate loops, the small dimension must always be the spatial one, since we must take the time cutoff $T\gg L$ by the physical situation. Then, for fixed $L$, we are not separating the quarks at all, so we expect no significant difference between the confining and deconfining phases. It is only in the limit that $L\rightarrow \infty$ that we should expect to see a difference.

If you have something else in mind, please explain.

 

[jfy4]

I was thinking more like this, I'm going to explain this in terms of coordinates, (x,y).

starting at (0,0), connect the dots: (1,0),(1,1),(2,1),(2,2),(1,2),(1,3),(0,3),(0,2),(-1,2),(-1,1),(0,1),(0,0).
In words, a cross. This seems to have perimeter 12, but area 5. This seems to be a possbile loop, and these crosses can be made arbitrarily large I think.

 

[fzero]

I was thinking more like this, I'm going to explain this in terms of coordinates, (x,y).

starting at (0,0), connect the dots: (1,0),(1,1),(2,1),(2,2),(1,2),(1,3),(0,3),(0,2),(-1,2),(-1,1),(0,1),(0,0).
In words, a cross. This seems to have perimeter 12, but area 5. This seems to be a possbile loop, and these crosses can be made arbitrarily large I think.

OK, suppose we scale the grid by a factor $L$, so we have points $(0,0), (L,0), (L,L)$ and so on. Now the perimeter is $P=12 L$, but the area is $A=5L^2$. In the limit $L\rightarrow \infty$, these have the behavior that was claimed. For any loop, the perimeter always scales linearly with the length of the edges, while the area is quadratic.

 

[jfy4]

When you say

. . . we scale the grid . . .

should I interpret that as

. . . under a RG transformation . . .

?

 

[fzero]

When you say

should I interpret that as

?

It's not directly related to an RG transformation, or at least, we are not using that directly. I just wanted to introduce a scale corresponding to the size of the loop that we can vary, since Kogut is discussing the behavior for large loops. Of course the RG behavior is intricately tied in anyway. For example, the fact that the coupling constant of QCD grows in the IR is also an indication of confinement. But we are not directly using RG arguments to compare the perimeter with the area.

 

[jfy4]

I see what you mean. I had in mind though loops of the form:
(0,0),(1,0)(1,n),(n+1,n),(n+1,n+1),(1,n+1)...
and n can get as large as it wants. I guess I realize that there probably are large loops which don't follow these `laws' but they contribute little I suppose.