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

Abstract

On the basis of Chern-Simons field-theoretical description we propose a simple method for derivation of model interactions for Pfaffian paired states. We verify the method in the case of Pfaffian (i.e. Moore -Read) state, and derive a general form of the model interaction in the case of PH Pfaffian. More than one Landau level is needed to establish the correlations of the PH Pfaffian, and we present the values of relevant three-body pseudo-potentials for two Landau levels. I. INTRODUCTIONThe discovery of the fractional quantum Hall effect (FQHE) at even-denominator, 5/2, filling factor [1] initiated an intensive search for viable paired states for the explanation of the effect. A new paradigm of Pfaffian paired states (with p-wave Cooper pairs) was introduced [2] that was the most important building block for the proposals of topological superconductivity and topological quantum computing. 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. The search was proven difficult, because it was found in numerical experiments [12] that the projection of what is believed to be, an appropriate model wave-function of PH Pfaffian, to a fixed LL, represents a gapless state (not a gapped one necessary for FQHE). Therefore, for numerical experiments and in general, it is very desirable to find a model interaction for the uniform PH Pfaffian state, and in this work we will describe its general form.The proposal of the PH Pfaffian state is connected with the advance [13] in the effective (field-theoretical) description of the Fermi-liquid-like state of composite fermions (underlying quasiparticles) at compressible, even-denominator fractions. Namely, instead of classical composite fermions, we may use Dirac composite fermions to describe these fractions. The PH Pfaffian constitutes a p-wave pairing in the opposite sense of the direction dictated by external magnetic field that is materialized in the chiral motion of charge on the edge. The PH Pfaffian also makes the most natural (underlying s-wave) pairing of Dirac composite fermions. Thus the solution of the enigma of PH Pfaffian may help us to understand better pairing and superconductivity in Dirac systems raging from the FQHE ones to graphe...

Identifying the $ν=\frac{5}{2}$ topological order through charge transport measurements

“…By contrast, the quantized values for Pfaffian and anti-Pfaffian are 7 2 and 3 2 . The measured value matches theoretical expectations for the PH-Pfaffian phase, 24 which is PH invariant 25 but faces theoretical challenges: 26,27 It has never been observed numerically, and its trial states become critical upon projection into a single Landau level. [28][29][30][31] Subsequent theoretical works argued that incomplete anti-Pfaffian edge equilibration might also realize the measured value.…”

Fractional charge and fractional statistics in the quantum Hall effects

“…Some data were interpreted in terms of the Abelian 113 and 331 states [172][173][174]. Recent theoretical work suggests a complicated phase diagram in realistic disordered systems in which all topological orders with −3 C 1 are present [175][176][177] (see also [178][179][180][181] for the role of Landau level mixing). On the other hand, some theoretical proposals question [182][183][184] the existence of an energy gap at ν = 5/2.…”

PH-Pfaffian order in a translationally and rotationally invariant system

“…A possible lesson might be that the PH-Pfaffian order were impossible in uniform systems. Yet, it was argued that it might be stabilized by sufficiently strong Landau level mixing (LLM) even in uniform systems 30,31 . If so, a translationally invariant model should exist with the PH-Pfaffian ground state.…”