This is how critical systems are universal.... Here comes the amazing fact about critical exponents. Until now we have been concentrating on the nearest neighbor Ising model on the lattice Zd. Now consider the nearest neighbor Ising model on another lattice in the same number of dimensions. For example, instead of the square lattice Z2 we could use the hexagonal or triangular lattices. In general the critical β for the model will depend on the lattice. However, the critical exponents do not. They are believed to be exactly the same for all lattices in the same number of dimensions. This phenomenon is called “ universality”. But wait, there is more. We can also change the Hamiltonian. For example, for Z2 we can include terms −σ σ in the Hamiltonian when the distance between i and j √ij is either 1 or 2. The new terms in the Hamiltonian favor the spins lining up even more than in the original Ising model, so we might expect that the critical temperature of this model is higher than that of the original Ising model. However, the critical exponents of the two models are again believed to be exactly the same. To spell out a few more examples, we could include all terms −σiσj with |i−j| ≤ L, and we should get the same critical exponents regardless of the choice of the cutoff distance l. We could include interactions between four spins at a time. In general a wide class of models will all have the same critical exponents. These critical exponents will depend on the number of dimensions, but not on the details of the microscopic interaction or the particular lattice.
Posted on: Wed, 15 Oct 2014 02:44:49 +0000