# A How far can one change a model before it leaves the universality class

#### thephystudent

Usually, critical phenomena can be categorized in some kind of universality class which determines the critical exponent.
A typical example is the class of the Ising model; adding a next-nearest-neighbour hopping term does not change the critical behavior. The typical explanation is that the 'physics on scales much longer than the lattice spacing does not depend on these interactions'.

My question is: are there any rules of thumb that determine how far a model must be changed before it changes the universality class? And when it does so, will there be a continuous crossover?

A possibility would probably be adding some different arbitrary-range coupling terms in the Hamiltonian. But on the other hand, the Ising model as I described does not contain any explicit long-range terms, although this does not say that every local model is in the quantum-ising universality class...

Note: I know my question is also valid with classical PT, but I'm posting it in this subforum since the QPT is of most interest to me at the moment.

• Demystifier
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#### A. Neumaier

Reducing the symmetry typically changes the universality class.

• thephystudent

#### atyy

In classical statistical mechanics, dimensionality is another factor, in addition to symmetry that @A. Neumaier mentioned.

• thephystudent

#### A. Neumaier

In classical statistical mechanics, dimensionality is another factor, in addition to symmetry that @A. Neumaier mentioned.
But unlike symmetry, the dimension cannot be changed by adding interactions.

#### thephystudent

Thanks for your responses. I didn't know nLab before.

Regarding symmetry, what is precisely meant by it? According to nLab, the classical 2D ising has e.g. U-symmetry, whereas I thought is was Z_2 symmetry (which gets spontaneously broken in the ferromagnetic phase). But in fact, also the dimension corresponds to some symmetry (translational symmetry), right?

To answer A. Neumaiers point, what if you have for example a 2D lattice where the coupling is much stronger in the vertical direction than the horizontal direction... does this become effectively 1D? How much should this difference be, and is there some kind of intermediate window between the two classes than (with exponents that lie between the ones of 2D and 1D)?

Related to this it is very surprising that for many models, the quantum version in D dimensions is in the same class as the classical version in D+1 dimensions. But increasing temperature would make the quantum version also more classical, so how would the crossover look?

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