How to simulate a lattice with boundary conditions?

In summary, you would need to create a lattice with sites, link them with a hopping parameter, diagonalize the matrix, and get the eigenvalues. You would then use a mean field approximation for the second part of the Hamiltonian and plot a curve to see how the material will behave.
  • #1
Amentia
110
5
Hello,

My question is very simple but I do not have a lot of experience with simulation. I want to write some code to simulate a lattice with boundary conditions and then I will perform calculations with the Hubbard model to find different kinds of properties of interest. I would like to know how to do this first step, for any lattice (you can choose a simple one like a square lattice with few atoms to give an example).

Thank you!
 
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  • #2
Maybe, you might use "periodic boundary" conditions, but I am not really sure whether it works when simulating a Hubbard model on a two dimensional lattice. I am sure, however, that there is many stuff around if you search in this direction.
 
  • #3
Hello,

Thank you for your answer. In fact, I did not find anything, maybe because it is too simple. My question is even more simple than that: assume no boundary conditions, what should I write to model a system like two sites with two electrons that can be on either site with a hopping parameter t=1? Because I think once I am sure how to link the sites numerically, I should just have to link bottom atomic sites with top atomic sites with the correct t or 0 hopping paramater, in order to simulate the periodic boundary conditions. I know it is possible for a Hubbard model on a 2D lattice because there is already literature on that. But they give their results, not their codes.
 
  • #4
You should search for quantum monte carlo simulations regarding the Hubbard model.
 
  • #5
Ok thank you. By the way, is the mean field approximation included into the quantum monte carlo simulations?
 
  • #6
Amentia said:
Ok thank you. By the way, is the mean field approximation included into the quantum monte carlo simulations?

Mean field approximations provide analytical, approximative solutions. Monte Carlo simulations provide numerical solutions.
 
  • #7
I am still confused. Maybe I will detail more what I want to do.

1) Create a lattice with sites that can have up to two electrons and link them with a hopping parameter t (main question of this thread)
2) Use this lattice to make a hamiltonian matrix for the kinetic part of the Hubbard model
3) Diagonalize and get the eigenvalues
4) Compute density of states
5) Knowing the number of electrons and the number of sites, find the chemical potential for finite temperature
6) Use a mean field approximation for the second part of the Hamiltonian
7) Plot a curve to see how the material will behave, for example if it is supposed to be ferromagnetic, if there are phase transitions...
 
  • #9
Thank you for the book. I think it is much more involved than what I want to do now but I will keep it for later. It could be of great help.
 

1. What is a lattice in scientific simulations?

A lattice is a regular arrangement of points or sites in space that is used to model and simulate physical systems, such as crystals, polymers, or fluids. It is often represented as a grid or network of interconnected nodes.

2. How are boundary conditions incorporated in lattice simulations?

Boundary conditions are rules or constraints that define the behavior of the system at the edges or boundaries of the lattice. They are typically implemented by assigning specific values or interactions to the nodes or points at the edges of the lattice.

3. What are the different types of boundary conditions used in lattice simulations?

The most commonly used boundary conditions in lattice simulations are periodic, reflecting, and open. Periodic boundary conditions wrap the lattice around itself, reflecting conditions mirror the system at the boundary, and open conditions allow for the exchange of particles or energy with the surroundings.

4. How do boundary conditions affect the results of lattice simulations?

Boundary conditions can significantly influence the behavior and properties of the simulated system. They can introduce artifacts or bias in the results and may need to be carefully chosen and tested to ensure accurate and meaningful simulations.

5. Are there any tools or software available for simulating lattices with boundary conditions?

Yes, there are various software packages and libraries, such as LAMMPS, GROMACS, and HOOMD-blue, that include features for simulating lattices with different types of boundary conditions. These tools also often have user-friendly interfaces and tutorials for setting up and running lattice simulations.

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