Basically, I am trying to find the boundaries delimiting the first two Mott lobes by calculating the single-particle excitation gap. I do not really get how I should fix the lattice size and vary the particle numbers (N,N+1,N-1) such that I can calculate the quantity,\[ \Delta = \mu^+ - \mu^-, \quad\mu^+ = E_0(L,N+1) - E_0(L,N), \quad \mu^- = E_0(L,N) - E_0(L,N-1) \]

Then I can obtain the phase boundaries in \( (\mu,t) \) plane.

## Bose-Hubbard Model Single Particle Excitation Gap

### Re: Bose-Hubbard Model Single Particle Excitation Gap

One suggestion I can give is that you first keep the density fixed( say \(\rho=1\) ). Start with some system size(say L=40). Then find \( E_0 (N+1),E_0 (N),E_0(N-1)\). \(N=\rho \times L \). Find \(\mu^{+}\),\( \mu^{-}\). Now keep increasing the system size and correspondingly find \(\mu^{+}\),\( \mu^{-}\). Extrapolate for \(\mu^{+}\) ,\(\mu^{-}\) up to the thermodynamic limit. It should be a 1/L fit. Do this for different values of t. Plot \(\mu^{+}\) ,\(\mu^{-}\) vs t. The point where the two curves meet is the critical point. The region encompassed by the two curves is the mott lobe. Again repeat this procedure for different values of \(\rho\).

### Re: Bose-Hubbard Model Single Particle Excitation Gap

Thanks for the suggestion, this is indeed the way to go here. See e.g. arXiv:cond-mat/9906019 for details.