# 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?

-

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.

-
For the first question, if the temperature is slowly raised, do the energy levels gradually get smeared out? For the second question, since experiments are only pseudo-2D, how can we similarly argue (like in the first question) that anyons exist? –  leongz Mar 22 '12 at 18:36
First question: If you do not touch the Fermi level of the semiconductors via the voltage gates, I think yes. However, the interest is to have this 2D electron gases, isn't it?. –  DaniH Mar 22 '12 at 21:29
Second question: it is very difficult to prove that anyons exist! I only know this result about an interferometer with edge states [Laughlin quasiparticles], see link at arxiv.org/abs/cond-mat/0502406. Appart from the excitations in the FQHE for Laughlin fractions, [nonabelian] anyons are also believed to be related to certain fractions in the quantum Hall regime, see this Physics Focus, physics.aps.org/articles/v3/93. –  DaniH Mar 22 '12 at 21:34
I asked a question here, inspired by this question –  hlew May 13 '13 at 18:13

Leongz, DaniH has already given you an excellent discussion of the issues and parameters involved, so I'm not going to attempt to make any additions to that. What I will mention instead is just a heuristic that can be helpful in understanding pseudo-2D and pseudo-1D systems.

The heuristic is this: If for the effect you are looking at there are one or two dimensions of the apparatus that are smaller than the smallest possible wavelength of the effect, then a pseudo-2D or pseudo-1D analysis has a good chance of working well, possibly very well indeed.

This is principle behind single-mode optical fiber, which is the foundation of modern ground-based and intercontinental data communications. By restricting two dimension of the fiber so much that light (analogy!) cannot "bounce back and forth" between the sides of the fiber, the result is a simplified waveform that is vastly superior for carrying data.

Finally, applying that to your specific question: Pseudo-2D and pseudo-1D effects do not in general have smooth transitions into 3D, for the same reason as with the fiber example I just gave. What happens instead is that you just start getting 3D wave effects that add noise and uncertainty to whatever effect you were really interested in.

-