condensed matter - Transition between 2D and 3D quantum systems

Quantum Hall effect and anyonic particles are examples that occur in a two-dimensional system. However, experiments for such systems can only be realized in a pseudo-2D environment, where the third spatial dimension is much smaller than the other two dimensions. How do we expect the results from such experiments to differ from a true 2D system? In particular, how/when does an anyon begin or cease to exist when we transit between a 2D and 3D system?

Answer

The results from the experiment does not differ significantly from the "true 2D" system, in fact, this is why experiments and theory agree so well!

Consider a semiconductor heterostructure GaAs/GaAsAl. At the interface, alignement of the Fermi level at both sides of the semiconductor crystals creates a triangular potential well at the GaAs side of the interface. This potential well is quite narrow so that quantization in one direction can be effectively assumed. In fact, putting numbers for GaAs, $m^{\ast}=0.067m_e$, $n_{2\text{D}} \simeq 10^{15}m^{-2}$, $\epsilon^\ast=13\epsilon_0$ you get that the typical quantization energy is $\Delta E \simeq 20$meV.

Due to the triangular well, the $3\text{D}$ electron wavefunction [consider nearly-free electrons in the effective mass approximation] is modified as

$\Psi_{k_x,k_y,n\sigma}(\mathbf{r})=\dfrac{1}{A^{1/2}}{\rm e}^{i k_x x}{\rm e}^{i k_y y}\zeta_{n}(z)\chi_\sigma$

with $\zeta_n(z)$ the $n$-th eigenfunction of the triangular well with energy $\varepsilon_n^z$. THe total energy can be written as

$\varepsilon_{k_x,k_y,n} = \dfrac{\hbar^2}{2m^\ast}(k_x^2+k_y^2) + \varepsilon_n^z$

The Fermi energy can be obtained using that $k_F^2=2 \pi n$, $\varepsilon_F\simeq 10$ meV.

The difference between the highest occupied energy is $\Delta E -\varepsilon_F \simeq 10$ meV. This yields $T \simeq 100$ K.

Conclusion A quick conclusion of this calculation is the following: at temperatures $T \ll 100$K all the occupied electron states have the same orbital in the $z$ direction and promotion to other orbital requires an excitation energy of at least $10$ meV. If this is not provided, the system has indeed lost one degree of freedom and it is dynamically a true $2\text{D}$ system. Thus $2\text{D}$ systems can exist in Nature!

Concerning the second question, anyon statistics exists only in two-dimensional systems. It cannot exist in $3\text{D}$, and the reason is topological: in two dimensions the configuration space of $N$ particles is multiple connected and closed path of a particle which encloses another particle cannot be "shrinked" to a point [mathematically this is called "compatification"]. On the other hand, for higher dimensions, the configuration space is simply connected and we lose the possibility of distinguish between the interior and the exterior of a closed path.Hence, in the transition from $2\text{D}$ to $3\text{D}$ you simply lose the possibility of interpolate between Bose and Fermi statistics.