Is Mott insulator the same as non-compressible quantum fluid? In the field of ultracold quantum gases we study the so called Bose-Hubbard model given in second quantization:
$$\hat{\mathcal{H}} = -t\sum_{\langle i,j\rangle}\hat{a}^{\dagger}_{i}\hat{a}_{j} + \frac{U}{2}\sum\limits_{i}\hat{n}_{i}(\hat{n}_{i}-1).$$
I do not write chemical potential term because I want to work in canonical ensemble where total number of particles in fixed. When tunneling constant $t$ is much smaller than $U$ and number of particles is equal to the number of sites literature speaks about Mott insulator. Now, what really is a Mott insulator in this model because I found two different (for me) definitions:
1) We look at energies $E(N)$ and $E(N+1)$ when we have one extra particle in the system. Apart from discreteness of function $\mu(N)$ we will also have large regions where chemical potential is discontinuous. In other words, there is no chemical potential value that would correspond to some $N$ (we have a gap). So, the Mott insulator state is the state for which we see a gap - it has specific number of particles depending on $t$.
2) Second definition looks at eigenvalues of the Hamiltonian (consider e.g. exact diagonalization). If there is a gap between ground state and excited state we call the ground state a Mott insulator.
In the second case excited states have the same number of particles as the ground state, which is something different that in the first case. For me those two situations describe something different. People also consider grand canonical ensemble when you add $-\mu\hat{N}$ term to the above Hamiltonian. Then, a Mott insulator is defined as a non-compressible quantum fluid (according to S. Sachdev) because there is no change of particle density as you change chemical potential $d\langle \hat{N} \rangle/d\mu = 0$. 
 A: These two points of view are not so different in fact.
To see that, let's work in the grand-canonical ensemble (which is the most natural to talk about the chemical potential in the Mott phase, since it is not well defined in the canonical ensemble).
At a given (and small enough) $t/U$, there is a range of chemical potential $[\mu_-,\mu_+]$ where the density does not change, which defines the Mott insulating phase. Let's pick one $\mu$ in this range.
The cost to add one particle in the system is given by $\mu_+-\mu$, whereas the cost to remove one particle is $\mu-\mu_-$. Notice that the cost depends on the choice we have made for $\mu$.
On the other hand, the smallest cost for a particle-hole excitation (which conserves the number of particle in the system) corresponding to removing a particle somewhere, and putting it somewhere else in the system, very far away, will thus be  $(\mu_+-\mu)+(\mu-\mu_-)=\mu_+-\mu_-$, which is the width of the Mott lobe for the particular $t/U$ we have chosen. This cost is independent of $\mu$, and corresponds to the gap of the energy spectrum of the Hamiltonian at fixed density.
All of this is discussed in the seminal paper of Fisher et al.
