Recovering 3D properties from 2D materials One of the biggest topics nowadays in Condensed Matter Physics is 2D materials. If we take graphite an peel it off we get graphene, the canonical example of 2D material which has very different properties of its 3D counterpart. This is not the only example, and heterostructures that stack different layers and still get different and nontrivial properties are everywhere.
The question is... If we start stacking layers of the same 2D material (say, graphene), when should we recover the properties of the 3D bulk material (e.g. graphite)? Do we expect a discontinuous or a continuous phase transition?
 A: Graphene
Much of the basic theory underlying graphene was known long ago - it has served as the zeroth order approximation for describing graphite, whose electronic, mechanical and other properties are well described as those of weakly coupled graphene sheets.
Heterostructures
Heterostructures are somewhat different - here the properties of a bulk region are modulated by changing composition of the material in one direction (e.g., varying $x$ in $Al_xGa_{1-x}As$). This can be interpreted as a potential superimposed on the bulk material, confining electron motion in one direction.
Transition from 2D to bulk
Heterostructures
Let us start with the heterostructures example: if we model the confining structure by an effective potential $V(z)$, our starting point for describing the properties of the material will be solutions of the non-interacting Schrödinger equation (for simplicity I ignore the underlying crystal structure and spin, although this could be generalized to Bloch functions):
$$
H\Psi_{k_x,k_y,n}(x, y,z)= \left[\frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{p_z^2}{2m} + V(z)\right]\Psi_{k_x,k_y,n}(x, y,z) = E_{k_x,k_y,n}\Psi_{k_x,k_y,n}(x, y,z),
$$
where
$$
E_{k_x,k_y,n}=\frac{\hbar^2k_x^2}{2m} + \frac{\hbar^2k_y^2}{2m} + \epsilon_n,\\
\Psi_{k_x,k_y,n}(x, y,z)\propto e^{ik_xx+ik_yy}\phi_n(z),\\
\left[\frac{p_z^2}{2m} + V(z)\right]\phi_n(z) = \epsilon_n\phi_n(z).
$$
In other words, we are dealing with 2D subbands with $\epsilon_n$ being the bottom position of the n-th subband. As long as the temperature, the energy scales of the interactions, perturbations that we want to consider in our problem are smaller than the gap between these subbands, especially
$$
k_BT\ll \epsilon_1-\epsilon_0,
$$
it is a good approximation to consider only the lowest subband - this is the sense in which heterostructures can be considered two-dimensional. When this condition breaks we would need to include excitatiosn to higher subbands, and eventually include all of them, returning to the continuous limit.
Graphene
The situation is different in graphene, in the sense that it is truly two-dimensional, and considering transition to 3D case as slowly decreasing the amplitude of the confining potential in respect to the strength of the perturbations/interactions/temperature is impractical. Instead we can think of adding layers of graphene.
One layer of graphene can be described by the same wave function as above, but with strongly confining potential - for all practical purposes we can consider only the ground state (except perhaps when treating the X-ray scattering). When joining two layers of graphene, we will have electrons tunneling back and forth between the two degenerate layers, forming bonding and antibonding subbands. When adding the their layer we get tunneling between three degenerate layers and so on.
Note how this is analogous to cimply assembling a chain of atoms, in the spirit of the tight-binding model - small hopping between degenerate states eventually transforms into a band, when the number of atoms becomes very large.
