Section \[sec:driven\], where it becomes physically relevant. In the deep lattice limit the eigenstates become M-fold degenerate as individual lattice sites effectively decouple. To avoid numerical issues and have a clearly-defined ground-state Slater determinant, we avoid this limit and stay within a range of lattice depths (0 to 20 $E_r$) for which the first $M$ states are not yet completely degenerate.

![Lowest lying 150 single-particle eigenenergies as a function of lattice depth for $M=50$.[]{data-label="fig:SPspectrum"}](singleparticlespectrum.pdf){width="0.7\linewidth"}

Statics of the TG gas {#sec:statics}
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In the following we will first discuss the ground state properties of the TG gas in the presence of a lattice.

The pinned and the superfluid phase
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The Hamiltonian is known to possesses two distinct ground-state phases in the TG limit, which appear as a function of the ratio of number of bosons to lattice sites $$F=\frac{N}{M}\;.$$ For incommensurate fillings ($F \neq \mathcal{N}$, with $\mathcal{N}$ a positive integer) in shallow lattices the system has superfluid-like characteristics with long-range coherence and good conductivity due to the delocalisation of the wavefunction over many lattice sites. However for commensurate filling ($F= \mathcal{N}$), the bosons becomes localised at individual lattice sites and the total system becomes *pinned* to the lattice. This pinned 