Correlation-Hole Induced Paired Quantum Hall States in the Lowest Landau Level

Abstract

A theory is developed for the paired even-denominator fractional quantum Hall states in the lowest Landau level. We show that electrons bind to quantized vortices to form composite fermions, interacting through an exact instantaneous interaction that favors chiral p-wave pairing. There are two canonically dual pairing gap functions related by the bosonic Laughlin wavefunction (Jastrow factor) due to the correlation holes. We find that the ground state is the Moore-Read pfaffian in the long wavelength limit for weak Coulomb interactions, a new pfaffian with an oscillatory pairing function for intermediate interactions, and a Read-Rezayi composite Fermi liquid beyond a critical interaction strength. Our findings are consistent with recent experimental observations of the 1/2 and 1/4 fractional quantum Hall effects in asymmetric wide quantum wells. [5][6][7] indeed observed FQHE at ν = 1/2 and 1/4, suggesting that these, too, may be in the MRP phase. Although the abelian two-component Halperin (331) and (553) states [8] can be strong contenders for these FQHE [9], fresh experiments and numerical studies found strong evidence for the one-component FQHE at ν = 1/2 and 1/4 in asymmetric wide quantum wells [7,10]. Whether the observed FQHE can be understood as pfaffians in the LLL is the focus of this work.The MRP is a chiral p-wave paired quantum Hall state [11]. In principle, it can emerge as a p-wave pairing instability of the composite Fermi liquid (CFL), a gapless state of electrons attached to flux tubes [13]. The leading-order statistical interaction mediated by the Chern-Simons (CS) gauge field fluctuations can produce a p-wave pairing potential for the composite fermion [14] (CF). However, since the coupling between the CF and the CS gauge field is not small, diagrammatic perturbation theory is not controllable. Within the random-phase approximation, the gauge fluctuations are in fact singular and pair-breaking [15]. Therefore, the ground states at filling fractions 1/2 and 1/4 remained enigmatic [16].The key to solve this problem is to properly account for the effects of the correlation hole, i.e. the local charge depletion caused by attaching flux to an electron. A CF feels the correlation hole of the other CFs, which is captured by the Jastrow factor in the Laughlin wavefunction. In the unitary CF theory [12,13], only an infinitely thin flux tube associated with a U(1) phase is attached to each electron without accounting for the Jastrow factor, i.e. the correlation hole. Read improved the concept of CF by attaching finite size vortices to electrons [17,18] In this paper, we show that paired quantum Hall states emerge in the LLL using the NUCF field theory where a vortex with vorticity-φ (φ = even integer) is attached to an electron at filling fraction ν = 1/φ. An important feature of attaching vortices to electrons is that the diamagnetic coupling, quadratic in the gauge field, to the CF density is canceled by its dual contribution from the correlation hole associated with the vortex. As a result, we sho...

Fractional quantum Hall states versus Wigner crystals in wide quantum wells in the half-filled lowest and second Landau levels

“…The well width can even lead to the formation of novel FQHSs, such as the ones observed at ν = 1/2 and 1/4 in the lowest LL. [18][19][20][21] The reminiscence with even-denominator states in bilayer quantum Hall systems 22 hints at a multi-component origin later confirmed theoretically both for symmetric [23][24][25][26][27] and asymmetric 28,29 quantum wells in terms of a (331) Halperin state 30 that competes with the compressible composite-fermion Fermi liquid (CFFL) 31 and, under certain circumstances, 24,25,28,32 with a Pfaffian. 5,6 Furthermore, these liquid phases compete with an insulating phase in the limit of very large well widths.…”

Dynamical mass generation of composite Dirac fermions and fractional quantum Hall effects near charge neutrality in graphene

“…Thus, the large distance behaviors of the pairing wavefunction are g 1 (r) ∝ 1/z + c √ 2/π cos(κr − π/4)/z √ r and g 2 (r) ∝ −c √ 2/π cos(κr − π/4)/z √ r, where c and κ are numerical constants. The oscillatory terms in the above equations are due to the k 2 terms in the expansions, originating from the k dependence of the mass, which should be distinguished from the oscillatory wavefunction in [40]. The many-body real space wavefunction for CDFs can be obtained by [39] CDF…”

“…The oscillatory terms in the above equations are due to the k 2 terms in the expansions, originating from the k dependence of the mass, which should be distinguished from the oscillatory wave function in Ref. 40 . The many-body real space wave func-tion for CDFs can be obtained by…”

Identification of topological order in the fractional quantum Hall state at ν=1/4

“…Similar to the case of ν = 5/2 FQHE, different topological orders have been proposed to describe the ν = 1/4 FQH state [36,[38][39][40][41][42][43]. The original experiment by Luhman et al [31] reported that the FQH state was strengthened by tilting the sample in a magnetic field.…”