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In condensed matter literature, at many places, the phrase 'deep lattice limit' is used. Please tell what is the deep lattice limit and the shallow lattice limit?

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Deep lattice limit corresponds to the tight binding limit where the physical entities (bosons or fermions) are localized and there is very less hopping allowed and no long range hopping is considered.

While shallow lattice limit is the one in which the wavefunction for an entity on one of the lattice sites has significant amplitude on other near lattice sites and thus there is a non-trivial long range hopping involved.

The scale for these limits is the depth and half-width of the lattice potential.

Higher depth and lower width indicate deep lattice limit and vice-versa.

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