Duality in Quantum Hall Compressible States*

It has long been known (at least since 1997) empirically that
quantum Hall plateau transitions enjoy an apparent self-dualty symmetry. The
emaning of this statement is that the current-voltage (I-V) curves are nonlinear
and are symmetric around their liner behavior at the quantum phase
transition. This behavior was first observed in dirtier samples and is now
know to occur in the compressible sattes at filling fractions 1/2 and 1/4. This
empirical result was interpreted by Shimshoni, Sondhi and Shahar (PRB 55,
13730 (1997)) as a manifestation of a particle-vortex duality.Yet, the origin of
this symmetry has been misterious and has remained unexplained until quite
recently. In a recent paper (Hart Goldman and Eduardo Fradkin, PRB
98, 165137 (2018)) we used the recently developed
dualities of Dirac fermions (D. T. Son Phys. Rev X. 5, 031027 (2015); N.
Seiberg, T. Senthil, C. Wang and E. Witten, Ann. Phys. 374, 3095 (2016)) to
show that the mirror symmetry of the Jain sequences converging on 1/2n
compressible states explain the observed self-duality as a property of the
compressible states. I will also discuss the role of duality in nonabelian fractional quantum Hall states.