phase has no coherence, behaves as an insulator and is the hard-core continuum analogue of the Mott-insulator phase in the Bose-Hubbard model (BHM). In the continuum TG model the pinning happens at infinitesimally small lattice depths for $F=1$. It was first theoretically proposed by Büchler *et al.* [@Zwerger2003] and has been experimentally observed in numerous cold atom experiments [@Haller2010; @Hoogerland2016].

The degree of coherence is a good property to characterize the two different phases of the system. It can be obtained from the off-diagonal elements of the reduced single-particle density matrix (RSPDM) $$\rho_1(x,y) = \langle \Psi_B | \hat{\psi}^{\dagger}(x) \hat{\psi}(y) | \Psi_B \rangle,$$ which describes the spatial auto-correlation of a single particle, giving the probability that a particle is at $y$ immediately after it has been measured at $x$. For classical particles and for non-interacting fermions, the RSPDM will always be diagonal, however for a Bose gas the off-diagonal elements play a prominent role. Efficient algorithms to numerically calculate the RSPDM in the TG regime exist [@Pezer2007], which are tractable even for large numbers of particles and which we have used in our calculations.

Diagonalising the RSPDM leads to a set of eigenstates $\phi_i(x)$, known as the natural orbitals, with the 