Pfaffian paired states for half-integer fractional quantum Hall effect

Abstract

<jats:p> In this review, the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ([Formula: see text]-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the particle–hole (PH) Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern–Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states. </jats:p>

Approximate two-body generating Hamiltonian for the PH-Pfaffian wavefunction

“…For example, the pseudopotentials that perturbatively account for the Landau level mixing and finite width [10,13,55] to lowest order do not to our knowledge produce the suitable two-body corrections. It has been argued that 3-body pseudopotentials may be required to stabilize PH-Pfaffian [56][57][58] and it would be interesting to include 3-body and higher-order pseudopotentials into the variational Hamiltonian ansatz. The 2-body Hamiltonians presented here will be a valuable starting point for such a study.…”

Microscopic derivation of Dirac composite fermion theory: Aspects of noncommutativity and pairing instabilities

“…We may associate the plus combination with SU L c (N ), and the minus combination with SU L s (N ) -"spin" transformations -which are inverse in the d sector with respect to the ones in the c sector. Together, (12) and (13) with the plus sign lead to conclusion that |phy states are spin-singlet(s) under SU L c (N ). On the other hand, in the physical states, the generators of SU L s (N ) transformations, = N/2, furnished two adjoint representations of SU (N ) group.…”

“…On the other hand, in the physical states, the generators of SU L s (N ) transformations, = N/2, furnished two adjoint representations of SU (N ) group. However, with the hard-core constraint (12), we have only one non-trivial representation of SU (N ) group, SU L s (N ). The physical states are invariant under global U s (1) transformation because…”

“…the constraint that eliminates the hole degrees of freedom as additional degrees of freedom, by precluding the double occupancy as in (12) in the special bilayer problem. In the following paragraph we will recapitulate the necessary constraints in the formulation of the half-filled LL problem.…”

“…The classical action S (0) and the Chern-Simons term in S (1) give the description of a version of the Son's theory [5] of the Dirac composite fermion [9,10] that assumes the Pauli-Villars type of regularization [11,12] because of the presence of the Chern-Simons term for field a µ . Note that CS term has the correct coefficient, 1 8π .…”