Comp. GS of hubbard hamiltonian

[babylonia]

Hi,

I'm trying to repeat the numerical calculation of D Jaksch's article PRL 81,3108.
It is about using variational method for the ground state of bose hubbard hamiltonian:
H=-J\sum{a⁺_i+1a_i}+U\sum{n_in_i},where i denotes the lattice index
the trial function is based on Gutzwiller ansatz:
G=\prod_i{\sum_m=0toInffⁱ_m|m>_i, where m denotes the number of atoms in a certain lattice, fⁱ_m is the variational parameter,
What should be done is to minimize
<G|H|G>-mu <G|\sum {n_i}|G>, where mu is the given chemical potential.
As I see it, this is done by inserting G, and should lead to a set of nonlinear equations, solve it will give the solution of variational parameter.
However, I am having trouble how to solve it. since fⁱ_mis complex, and they must satisfy normalization condition, the resulting nonlinear equations seem difficult.

This is really a big problem for me. if any tips on such problem, I'd appreciated it

 

[eljose79]

.

Hi there,

Thank you for reaching out with your question. I understand that you are trying to replicate the numerical calculation of D Jaksch's article PRL 81, 3108, which involves using the variational method for the ground state of the Bose-Hubbard Hamiltonian. It seems like you are having trouble solving the resulting set of nonlinear equations due to the complexity and normalization conditions of the variational parameter.

Firstly, I would recommend reviewing the methodology and equations used in the article to ensure that you have a clear understanding of the problem and its solution. It may also be helpful to consult with a colleague or supervisor for their insights and suggestions. Additionally, there are many resources available online and in textbooks that provide step-by-step guides for solving nonlinear equations, such as using numerical methods or approximations.

Furthermore, it may be beneficial to break down the problem into smaller, more manageable steps. For example, you could start by solving the equations without the normalization condition and then incorporating it into your solution afterwards. This may help simplify the problem and make it easier to solve.

I also recommend considering different software or programming languages that may aid in solving these types of equations. There are many numerical computation tools available that can handle complex equations and constraints, such as MATLAB, Mathematica, or Python.

Lastly, don't be afraid to reach out to the author of the article or other experts in the field for their insights. They may have valuable tips or suggestions that could help you in your calculations.

I hope this helps and good luck with your research!