How to simulate a lattice with boundary conditions?

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Discussion Overview

The discussion revolves around simulating a lattice with boundary conditions, specifically in the context of the Hubbard model. Participants explore various aspects of lattice simulation, including the implementation of boundary conditions, the construction of Hamiltonian matrices, and the computation of physical properties such as eigenvalues and density of states.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant seeks guidance on simulating a lattice with boundary conditions, specifically for the Hubbard model.
  • Another participant suggests using periodic boundary conditions but expresses uncertainty about their applicability to the Hubbard model in two dimensions.
  • A participant clarifies their question by asking how to model a simple system of two sites with two electrons, indicating a desire to understand the numerical linking of sites.
  • There is a suggestion to explore quantum Monte Carlo simulations related to the Hubbard model.
  • A participant questions whether mean field approximations are included in quantum Monte Carlo simulations, noting the differences between analytical and numerical solutions.
  • One participant outlines a detailed plan for their simulation, including creating a lattice, constructing a Hamiltonian matrix, diagonalizing it, and computing the density of states.
  • A participant shares a resource (PDF) that may assist with quantum Monte Carlo simulations.
  • Another participant acknowledges the complexity of the shared resource but expresses intent to revisit it later.

Areas of Agreement / Disagreement

Participants express varying levels of uncertainty regarding the implementation of boundary conditions and the relationship between quantum Monte Carlo simulations and mean field approximations. No consensus is reached on the best approach to simulate the lattice or the specifics of the methods discussed.

Contextual Notes

Participants mention the need for clarity on boundary conditions and the numerical implementation of the Hubbard model, indicating potential gaps in understanding the foundational aspects of the simulation process.

Amentia
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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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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.
 
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 parameter, 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.
 
You should search for quantum monte carlo simulations regarding the Hubbard model.
 
Ok thank you. By the way, is the mean field approximation included into the quantum monte carlo simulations?
 
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.
 
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...
 
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.
 

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