# Question on universailty classes

Hi

When a particular model (say Ising) is solved on a particular lattice (say 2D triangular), do the critical exponents of the same model fall within the same universality class (have same critical exponents) as when solved on a different lattice (say 3D cubic)?

Thanks,
Mavi

I found out one part of the answer: changing the dimensionality of the lattice will change the critical exponents for the same model (say Ising). So the modified question would be whether the exponents change while changing the lattice structure (say square to triangle) for the same model in a given dimension?

Mavi

Guessing another part of the answer: it seems that since the continuum limit or the lattice limit must yield the same critical exponents (due to scaling invariance after graining), the lattice structure itself must not matter for the exponents. I wonder if this reasoning is correct.

Mavi

For the ferromagnetic Ising model case, outside of the dimensionality, it would be fair to assume that the lattice does not matter. There is no guarantee on this though. For a dramatic example, it should be fairly clear that the physical properties of an antiferromagnetic Ising model with be strongly dependent on the lattice geometry.

The problem with your argument is that you are assuming that given a model on an arbitrary lattice (with fixed dimensionality) you'll always arrive at the same long distance limit. It is difficult to show such a thing directly from a given model, mostly since the coarse graining procedure must done within some approximation (which may or may not be justified).