This seminar will summarize insights that emerge from theoretically treating a finite number of (3-8) electrons confined to a two-dimensional plane with a magnetic field oriented transversely. This is the prototype system of strongly-correlated electrons in condensed matter physics, which gives rise to phenomena such as the fractional quantum Hall effect, where certain combinations of electron density and magnetic field strength (filling fractions) exhibit intriguing features in their resistivity. Instead of formulating this theoretical problem in the usual framework using independent electron coordinates, we treat it in collective hyperspherical coordinates, which are particularly well-suited to describing nonperturbative behavior of the electron conglomerate as a whole. A number of insights have emerged from this way of treating the problem, such as an observation that one coordinate of the system, the hyperradius, is quite accurately separable from the other degrees of freedom. Another observation from our analysis is that the degree of exceptional degeneracy in the non-interacting electron system often correlates with the filling fractions where the famous Laughlin-type or composite-fermion-type states occur experimentally.
Wild, unrestrained speculations on future directions for this line of research will be offered.
 K.M. Daily, R.E. Wooten, and Chris H. Greene, Hyperspherical theory of the quantum Hall effect: The role of exceptional degeneracy, Phys. Rev. B 92, 125427-1 to -16 (2015).