Doubts on 2D and 3D Ising Model

[Tilde90]

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?

 

[Tilde90]

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 $h$?

 

[Tilde90]

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 $T<T_c$, there is a first order transition, i.e. a jump in the magnetization when varying the external field $h$?

 

[Jolb]

If we define Tc as the zero-field critical temperature for the ferromagnetic phase transition, then for T<Tc, varying h does not cause any phase change. There can be a discontinuous change in the magnetization (if we start in a ferromagnetic phase pointing along h and then we gradually make h go to some big value pointing opposite to the magnetization, then the magnetization can flip) but this is still a ferromagnetic phase. (Think hysteresis: Hysteresis - Wikipedia, the free encyclopedia.) You don't want to define a phase in terms of the magnetization vector because turning a magnet should not change its phase. Instead you should look at the order parameter.

 

[Tilde90]

If we define Tc as the zero-field critical temperature for the ferromagnetic phase transition, then for T<Tc, varying h does not cause any phase change. There can be a discontinuous change in the magnetization (if we start in a ferromagnetic phase pointing along h and then we gradually make h go to some big value pointing opposite to the magnetization, then the magnetization can flip) but this is still a ferromagnetic phase. (Think hysteresis: Hysteresis - Wikipedia, the free encyclopedia.) You don't want to define a phase in terms of the magnetization vector because turning a magnet should not change its phase. Instead you should look at the order parameter.

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).