Pairing Effects at the Edge of Paired Quantum Hall States

Abstract

We study pairing effects in the edge states of paired fractional quantum Hall states by using persistent edge currents as a probe. We give the grand partition functions for edge excitations of paired states (Haldane-Rezayi, Pfaffian, 331) coupling to an Aharanov-Bohm flux and derive the exact formulas of the persistent edge current. We show that the currents are flux periodic with the unit flux φ0 = hc/e. At low temperatures, they exhibit anomalous oscillations in their flux dependence. The shapes of the functions depend on the bulk topological order. They converge to the sawtooth function with period φ0/2 at zero temperature, which indicates pair condensation. This phenomenon provides an interesting bridge between superconductivity in 2+1 dimensions and superconductivity in 1+1 dimensions. We propose experiments of measuring the persistent current at even denominator plateau in single or double layer systems to test our predictions. PACS: 73.40Hm, One of the surprising aspects of the fractional quantum Hall effect is that the edge state forms a new kind of state of matter beyond Fermi liquid, called the chiral Tomonaga-Luttinger liquid [1]. Some experiments have already demonstrated the characteristic behavior of chiral Tomonaga-Luttinger liquid [2,3]. Recently the Aharanov-Bohm effect (AB effect) in such systems were studied by Geller et al [4,5] and Chamon et al [6]. Especially, the former authors predict new edge-current oscillations in the persistent current at the edge of the ν = 1 q Laughlin state, which has no amplitude reduction from disorder and thus results in a universal non-Fermi-liquid temperature dependence [5]. Also the persistent current for the annulus Laughlin state was recently investigated by Kettemann [7]. These studies show the current is periodic with a unit flux quanta φ 0 = hc/e in agreement with the theorem of Byers and Young [8].Motivated by these recent studies, we investigate the persistent currents in paired fractional Hall states in this paper. The states we will consider are the 331 state [9], the Haldane-Rezayi state [10] and the Pfaffian state [11]. They are quantum Hall analogs of the BCS superconductor. The pairing symmetry is p-wave with S z = 1, 0 for the Pfaffian and the 331 states respectively, and d-wave for the Haldane-Rezayi states. The Haldane-Rezayi state and the Pfaffian state are new kinds of quantum Hall state which recently attracted considerable attentions because they are supposed to exhibit some novel features beyond ordinary quantum Hall state in their topological ordering, such as nonabelian statistics and specific degeneracy on a surface with nontrivial topology. On the other hand, the 331 state is a part of generalized hierarchy [12], but can be interpreted as a paired state. These states are proposed as a candidate for the ν = 5 2 plateau in single layer systems [13] and the ν = 1 2 plateau in double layer systems [14]. Recent numerical study by Morf suggests that the ν = 5 2 state is Pfaffian-like [15]. As in the Laughlin state, these states are incompre...

The ν=5/2 quantum Hall state revisited: spontaneous breaking of the chiral fermion parity and phase transition between Abelian and non-Abelian statistics

“…The persistent currents, flowing along the edge of a mesoscopic disk FQH sample, can give important information about the low-temperature behaviour of the FQH states [37,38]. Here we stress again the advantage of the CFT approach in the computation of the oscillating persistent currents as compared to the chiral Luttinger liquid description of the edge states which is rather unsatisfactory for general filling factors [12].…”

“…1, indicate that both currents are periodic in φ with period exactly 1. We do not see any anomalous oscillations, such as half- flux periodicity of the persistent currents 4 [38,40], which is characteristic for the BCS paired condensates, or more generally for some broken symmetries, neither in the MR nor in the EPf states for all temperatures 0.05 ≤ T /T 0 ≤ 9 that we could access numerically. This is exactly the content of the Bloch theorem, which is also known as the Byers-Yang theorem [22] in the context of superconductors.…”

Bits and Pieces in Logarithmic Conformal Field Theory

“…the enigmatic degeneracy of the ground state in the Haldane-Rezayi fractional quantum Hall effect with filling factor ν = 5/2, where conformal field theory descriptions of the bulk theory proved difficult [11,49,102], multi-fractality in disordered Dirac fermions, where the spectra did not add up correctly as long as logarithmic fields in internal channels were neglected [17], or two-dimensional conformal turbulence, where Polyakov's proposal of a conformal field theory solution did contradict phenomenological expectations on the energy spectrum [35,98,109]. Other applications worth mentioning are gravitational dressing [8], polymers and Abelian sandpiles [13,56,84,106], the (fractional) quantum Hall effect [34,53,74], and -perhaps most importantly -disorder [5,6,14,15,50,51,68,83,101]. Finally, there are even applications in string theory [67], especially in D-brane recoil [10,24,26,47,69,77,79,87], AdS/CFT correspondence [44,60,65,72,73,93,94,107], and also in Seiberg-Witten solutions to supersymmetric Yang-Mills theories, e.g.…”

A universal conformal field theory approach to the chiral persistent currents in the mesoscopic fractional quantum Hall states

“…The edge states effective CFT is relevant only for the computation of the much smaller oscillating part of the persistent currents which could nevertheless be tested directly in SQUID detectors [1] using the Josephson effect. While the persistent currents naturally express the electric properties of the system, it turns out that they could also give important information about the neutral structure of the effective field theory for the edge states [3,10]. As we shall demonstrate in Sect.…”

“…(68) for −1/2 ≤ φ ≤ 1/2 at temperature T/T 0 = 0.1, given on Fig. 2, indicate that all currents are periodic in φ with period [10,31,32] which are characteristic for the BCS paired condensates, or more generally for some broken symmetries for all numerically accessible temperatures 0.03 ≤ T/T 0 ≤ 14. This is a confirmation of the Bloch-Byers-Yang theorem [22] formulated in the context of superconductors.…”