How close to the critical point is sufficient close for measuring critical exponents?

Question

I am learning Monte Carlo and just manage to simulate a phase transition by computing the heat capacity or the susceptibility. I wish I can also compute critical exponents.To this purpose, I have read some references, especially about the finite size scaling. As far as my understanding, one of the key difficulty is that one has to be very close to the critical point to make sure he/she is in the critical region. However, I don't know how close is close enough. For example, the critical temperature of the $2d$ Ising model is known as $T_c = 2.27$. If I want compute critical exponents, is $T = 2.26$ or $T=2.28$ close enough? Put in another way, is $T = 0.99$ $T_c$ or $T = 0.999$ $T_c$ sufficiently close?

I would be very appreciate for any hints or references.

Answer 1

Typically, one considers a 5% relative deviation from $T_c$ as close to the critical point, and a deviation of 0.2% as very close. But this does not mean that a fit will then give accurate values for the critical exponent.

For the 2D Ising model, exact results are available, and there close enough just means that you neglect all nonleading order terms. For numerical fits to simulation data, everything depends on the accuracy you are aiming at (and also on which data you are fitting). The bigger the neighborhood, the more a fitted exponent is contaminated by the errors made in neglecting terms not present in the fit.

Just do the fits and you'll find out. But be warned that it is a nontrivial task to devise a numerical setting that will give very accurate critical exponents; you need to include enough terms accounting for finite scale effects and for corrections to scaling.

There is also an article called "How close is close to the critical point" by Levelt Sengers and Sengers (pp. 239-271 in: Perspectives in Statistical Physics, North-Holland, New York 1981).