## Bose-Hubbard Model Single Particle Excitation Gap

How do I use this algorithm? What does that parameter do?
quarkonia
Posts: 10
Joined: 27 Jul 2022, 15:51

### Bose-Hubbard Model Single Particle Excitation Gap

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.
Posts: 6
Joined: 09 Apr 2021, 06:06

### 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 $$\rho$$.
Johannes