# Doubts on 2D and 3D Ising Model

by Tilde90
Tags: doubts, ising, model
 P: 20 Considering $d=2$ or $d=3$, the Ising model exhibits a second order phase transition at the critical temperature $T_c$, where the system goes from an ordered phase (spins preferably aligned in a certain direction) to a disordered one. This is reflected by the behaviour of the susceptibility, similar to a Dirac delta in $T_c$; and, being the susceptibility a second order derivative of the free energy, we talk about second order phase transition. Let's pass to the specific heat. Experimental results show that in $T_c$ also the specific heat has a Dirac delta beaviour, for both $d=2$ and $d=3$; the literature usually says that $C(T) \sim |T_c-T|^{-\alpha}$, with $\alpha=0$ for $d=2$ and $\alpha\sim 0.11$ for $d=3$. Now, my questions are: - Why $\alpha=0$ for $d=2$, if numerical results show a Dirac delta behaviour? And, if this divergence really exists, I guess that we can't talk of first order transition, being the specific heat a first derivative of energy (and not of free energy). Am I right? - When does the Ising model exhibits a first order phase transition? I've read that in presence of an external magnetic field the magnetization can show a "jump", and hence a first order transition. Is this true?
 P: 20 I didn't consider that heat capacity is actually a second derivative of Helmholtz free energy $F$. Anyway, my question remains: has the 2D/3D Ising model a second order phase transition for both the susceptibility and the specific heat? And is there a first order phase transition when considering to vary the external field $h$?
 P: 20 Apparently, with $\alpha=0$ it is implied for the specific heat to diverge logarithmically, i.e. $\sim -\log(1-T/T_c)$. Hence, I guess that we can consider the heat capacity as another expression of the second phase transition, being the specific heat a second derivative of the free energy. Now, just one question remains: is it true that, for a fixed [itex]T
 P: 419 Doubts on 2D and 3D Ising Model If we define Tc as the zero-field critical temperature for the ferromagnetic phase transition, then for T
P: 20
 Quote by Jolb If we define Tc as the zero-field critical temperature for the ferromagnetic phase transition, then for T
Thank you! This makes perfectly sense.

Just a question: with order parameter you mean the temperature, right?

EDIT: The order parameter in the Ising model is the magnetization itself, which is different from zero in the ordered phase (and viceversa).

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