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I am doing some simulations on various systems expressed in 2nd quantization and one of the points of interest of mine was Phase transitions in the Bose-Hubbard model

$$ H = \sum_{k} \{ t_k(b^\dagger_{k} b_{k+1} + b_{k} b^\dagger_{k+1}) + U(n_{k}^2 - n_{k}) + \mu n_{k} \} $$

Assuming a linear chain where the bosons are fixed and can only hop and interact with the potential U

I set my

$$ t = -1, U = -0.13, \mu = 0 $$

for all sites

For 6 bosons on 6 sites in a linear chain the Hamiltonian looks as follows Bose-Hubbard

And we can directly diagonalize this to get the following spectrum

eigenvalues

and its corresponding eigenvectors

Now for example the Ising model that exhibits a phase transition in its magnetization which can be studied by the partition function and magnetization observable. But here i am not sure what am i looking for to determine the phase transition in the Bose-Hubbard model. For example i do notice that when varying t/U the ground state gets closer and closer to the excited states, but i am not sure how this or other factors i am unaware of lead to the phase-transition i want to observe.

Any extra educational information would be appreciated

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