Doubts on 2D and 3D Ising Model

  1. Considering [itex]d=2[/itex] or [itex]d=3[/itex], the Ising model exhibits a second order phase transition at the critical temperature [itex]T_c[/itex], 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 [itex]T_c[/itex]; 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 [itex]T_c[/itex] also the specific heat has a Dirac delta beaviour, for both [itex]d=2[/itex] and [itex]d=3[/itex]; the literature usually says that [itex]C(T) \sim |T_c-T|^{-\alpha}[/itex], with [itex]\alpha=0[/itex] for [itex]d=2[/itex] and [itex]\alpha\sim 0.11[/itex] for [itex]d=3[/itex].

    Now, my questions are:

    - Why [itex]\alpha=0[/itex] for [itex]d=2[/itex], 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?
    Last edited: Aug 26, 2013
  2. jcsd
  3. I didn't consider that heat capacity is actually a second derivative of Helmholtz free energy [itex]F[/itex]. 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 [itex]h[/itex]?
    Last edited: Aug 26, 2013
  4. Apparently, with [itex]\alpha=0[/itex] it is implied for the specific heat to diverge logarithmically, i.e. [itex]\sim -\log(1-T/T_c)[/itex]. 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<T_c[/itex], there is a first order transition, i.e. a jump in the magnetization when varying the external field [itex]h[/itex]?
  5. 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.
    Last edited: Aug 26, 2013
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  6. 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).
    Last edited: Aug 27, 2013
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