What is the Hamiltonian for a 2D Ising model on a square lattice?

In summary, the conversation discusses the construction of a mean field equation for the magnetization in a 2 dimensional three state Ising model with ferromagnetic coupling on a triangular lattice and in a magnetic field. The individuals in the conversation are unsure how to start and what the Hamiltonian would look like for a square lattice.
  • #1
WarnK
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Homework Statement


Consider a 2 dimensional three state Ising model with ferromagnetic coupling J, where si = −1, 0, 1
in a magnetic field h on a triangular lattice. Construct a mean field equation for the magnetization m.

The Attempt at a Solution


I don't know how to start. 2d ising model on a triangular lattice? I need some hamiltionian to start with, I'm not sure what that would look like, and then continue with makeing a mean field theory approximation, getting a partition function and so on
 
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  • #2
2d lattice ising model

I have the same problem, but for a square lattice. (For a take home final due monday 9am. :frown: ) Do you know what the Hamiltonian for that would look like?
 

1. What is the 2D Ising model?

The 2D Ising model is a mathematical model used to study the behavior of interacting spins in a two-dimensional lattice. It is commonly used in statistical mechanics to understand phase transitions and critical phenomena.

2. What does MFT stand for in the 2D Ising model?

MFT stands for mean field theory, which is a simplification of the 2D Ising model that assumes all spins are independent and interact with an average field from their nearest neighbors.

3. How is the 2D Ising model solved using MFT?

In MFT, the spins are treated as independent and the energy of the system is minimized by solving a set of self-consistent equations. These equations determine the average spin orientation and the critical temperature at which a phase transition occurs.

4. What are the main assumptions of MFT in the 2D Ising model?

The main assumptions of MFT in the 2D Ising model are that the spins are independent, the interactions between spins are only through a mean field, and the lattice is infinite.

5. How does the 2D Ising model in MFT differ from other theories?

MFT simplifies the 2D Ising model by assuming independent spins and interactions through a mean field, while other theories may consider more complex interactions or finite size effects. However, MFT is often a good approximation for understanding the behavior of the 2D Ising model and can provide valuable insights into phase transitions and critical phenomena.

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