Fractional Quantum Hall Layer As A Magical Electromagnetic Medium

At a surface between electromagnetic media the Maxwell equations are consistent with either the usual boundary conditions, or exactly one alternative: continuity of E⊥, H⊥, D‖, B‖. These alternative, classically inexplicable conditions applied to the top and bottom surfaces of an FQH layer capture exactly its unique low-frequency properties. “Magic” in science means not illusion or trickery, but rather phenomena so surprising and counterintuitive that even when explained they inspire awe. An example in classical physics is the experiment of holding a tennis ball just above a basketball at chest height over a hard floor, releasing the balls simultaneously, and seeing the tennis ball bounce up to hit even a very high ceiling! That this follows directly from conservation of energy and momentum makes it no less amazing. Quantum physics contains much that is magic in this sense, and both the original [1] and the fractional [2] quantum Hall effects are striking illustrations. The Hall effect is a steady current traveling perpendicular to the plane defined by crossed electric (E) and magnetic (B) fields. The current may be described as due to an array of charges all traveling in the same direction with speed v = |E|/|B|. In classical physics the number of these charges and hence the magnitude of the current can be arbitrary. In the quantum Hall effect, for a given sample the magnetic field may be varied over a substantial range without changing the effective number of charges ν per quantum of magnetic flux contributing to the Hall current. The value of ν is an integer for the original quantum effect, and a rational fraction for the fractional [FQHE] version. FQHE is remarkable for many reasons, not least that in a very short time it found its ‘standard model’ in the composite-fermion picture [3], which unifies Laughlin’s original description of simple Hall fractions ν = 1 2p+1 [4] with a host of other observed phenomena in the fractional Hall domain, as well as earlier understanding [5] of the integer quantum Hall effect. Perhaps because progress in microscopic theory was so rapid, a familiar stage from previous studies of macroscopic systems – phenomenological description in terms of an electromagnetic medium – appears to have been skipped. Another reason for this omission may be that Hall samples characteristically have macroscopic (O(cm)) dimensions in the directions perpendicular to the magnetic field, but a thickness of only O(500Å) in the parallel direction, so that a macroscopic description conceivably might not even exist.