Doubts on 2D and 3D Ising Model

In summary, the Ising model exhibits a second order phase transition at the critical temperature T_c, where the system goes from an ordered phase to a disordered one. This is reflected by the behaviour of the susceptibility and the specific heat, which both show a Dirac delta behaviour at T_c. The literature states that the specific heat follows a power law with exponent \alpha=0 for d=2 and \alpha\sim 0.11 for d=3. However, there is a discrepancy between this theoretical prediction and experimental results, as the specific heat actually diverges logarithmically. This can be interpreted as another expression of the second order phase transition, since the specific heat is a second derivative of the free energy. There is
  • #1
Tilde90
22
0
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:
Physics news on Phys.org
  • #2
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:
  • #3
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]?
 
  • #4
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:
  • Like
Likes 1 person
  • #5
Jolb said:
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).
 
Last edited:

1. What is the Ising Model and how does it relate to 2D and 3D systems?

The Ising Model is a mathematical model used to study magnetism in materials. It was originally developed to explain the magnetic properties of ferromagnetic materials, but has since been applied to a wide range of systems. The model assumes that each particle in a system has a magnetic moment that can be either up or down, and that the interactions between these particles have a specific range. In 2D and 3D systems, the Ising Model is used to study the behavior of magnetic materials in two and three dimensions, respectively.

2. What are the key differences between the 2D and 3D Ising Model?

The main difference between the 2D and 3D Ising Model lies in the dimensionality of the system being studied. In 2D, the particles are arranged on a flat surface and interact with their nearest neighbors, whereas in 3D, the particles are arranged in a three-dimensional lattice and interact with their nearest and next-nearest neighbors. This leads to differences in the critical temperature and phase transitions of the system.

3. How do researchers use the Ising Model to study phase transitions?

The Ising Model is a powerful tool for studying phase transitions, which are abrupt changes in the properties of a material at a critical temperature. By simulating the behavior of large systems using the Ising Model, researchers can observe how the system's properties change as it approaches the critical temperature. This allows them to make predictions about the nature of the phase transition and the behavior of the system at different temperatures.

4. Are there any limitations to the Ising Model?

Like any mathematical model, the Ising Model has its limitations. One of the main limitations is that it assumes all interactions between particles are of the same strength and range, which may not be true in real materials. Additionally, the model does not take into account quantum effects, which can be important in certain systems. However, despite these limitations, the Ising Model has been successfully applied to a wide range of systems and remains a valuable tool for studying magnetism and phase transitions.

5. What are some current research areas related to the Ising Model?

There are many ongoing research projects related to the Ising Model, including studies on phase transitions in new materials, applications in machine learning and artificial intelligence, and extensions to higher dimensions and more complex systems. Researchers are also exploring ways to incorporate quantum effects into the model to better understand the behavior of materials at the atomic level. Additionally, the Ising Model continues to be a valuable tool for studying critical phenomena in a variety of fields, including physics, chemistry, and materials science.

Similar threads

  • Atomic and Condensed Matter
Replies
8
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
1K
Replies
11
Views
1K
  • Atomic and Condensed Matter
Replies
2
Views
3K
Replies
1
Views
3K
Replies
1
Views
2K
  • Advanced Physics Homework Help
2
Replies
41
Views
4K
  • Advanced Physics Homework Help
Replies
1
Views
2K
Replies
16
Views
1K
Back
Top