The sixteenfold way and the quantum Hall effect at half-integer filling factors

Abstract

Fractional quantum Hall states at half-integer filling factors have been observed in many systems beyond the 5/2 and 7/2 plateaus in GaAs quantum wells. This includes bilayer states in GaAs, several half-integer plateaus in ZnO-based heterostructures, and quantum Hall liquids in graphene. In all cases, Cooper pairing of composite fermions is believed to explain the plateaus. The nature of Cooper pairing and the topological order on those plateaus are hotly debated. Different orders are believed to be present in different systems. This makes it important to understand experimental signatures of all proposed orders. We review the expected experimental signatures for all possible composite-fermion states at half-integer filling. We address Mach-Zehnder interferometry, thermal transport, tunneling experiments, and Fabry-Pérot interferometry. For this end, we introduce a uniform description of the topological orders of Kitaev's sixteenfold way in terms of their wave functions, effective Hamiltonians, and edge theories.

Model interactions for Pfaffian paired states based on Chern-Simons field theory description

“…Still, after so many years of intensive experimental and theoretical research, we are not sure which paired state can be associated with the observed even-denominator FQHE. A paired state, more precisely the theoretical concept of so-called PH Pfaffian, seems according to the recent experiment [3] and other experimental data and theory [4], and theoretical proposals [4][5][6][7], closest to the solution of the puzzle at 5/2.…”

“…Still, after so many years of intensive experimental and theoretical research, we are not sure which paired state can be associated with the observed even-denominator FQHE. A paired state, more precisely the theoretical concept of so-called PH Pfaffian, seems according to the recent experiment [3] and other experimental data and theory [4], and theoretical proposals [4][5][6][7], closest to the solution of the puzzle at 5/2.The PH Pfaffian topological phase may be a result of a disorder dominated physics [5-8], but might be also a result of Landau level (LL) mixing in a system that may be considered uniform [8,9]. It is desirable to understand whether the PH Pfaffian (state) can be supported in a uniform system, as it was done and demonstrated in numerical experiments for Pfaffian and its particle-hole (PH) conjugated partner, anti-Pfaffian [10,11], on the basis of their model interactions.…”

Fractional charge and fractional statistics in the quantum Hall effects

“…These arguments do not require that e/q is necessarily the smallest charge for a quasiparticle in the system. For example, the various competing models [10] proposed to explain the even-denominator quantized Hall state observed at ν = 5/2 have quasiparticles with charge ±e/4. Levin and Stern have argued [11], in fact, that for any fractional quantized Hall state with even denominator q, there must exist quasiparticles with charge q m = ±e/2q.…”

“…The topological order depends only on C mod 16. Thus, there are 8 Abelian and 8 non-Abelian possibilities known together as the 16-fold way [10,98]. Orders with a large Chern number are seen as unlikely, and the bulk of research has focused on the orders listed in Table I.…”

“…We finish the discussion of the 16-fold way by describing the quasiparticle types, fusion rules, and topological spins for each order [10]. (See Table I.)…”