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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.

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The best way to numerically work with continuous phase transitions is to study observables that have a vanishing length dimension (or mass dimension in the language of QFT). Take for example the Binder's cumulant ($\langle m^4\rangle/\langle m^2\rangle^2$ modulo factors of 3 and constants, where $m$ is the order parameter) or the correlation length scaled by the system size ($\xi/L$) and vary these for different temperatures and system sizes. The intersection of these plots will give you the point of scale invariance, which is where the system achieves criticality. Having found the critical point, you now sit at that point and perform finite size scaling to extract the exponents. The uncertainty in the value of $T_c$ will lead to uncertainties in the values of the exponents (only two of which you need to estimate in equilibrium systems as all the others can be found through scaling relations).

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