Superconducting Order Parameter for the Even-denominator Fractional Quantum Hall Effect

TL;DR

This paper develops a numerical method to directly compute the superconducting order parameter for composite fermions in the fractional quantum Hall effect at even-denominator filling factors, providing evidence for superconductivity in this system.

Contribution

It introduces a numerically exact approach to create Cooper pairs of composite fermions and measures the superconducting order parameter, advancing understanding of the 5/2 FQHE.

Findings

Direct evidence of superconductivity in composite fermion systems

Quantitative predictions of superconducting coherence length

Validation of the composite fermion pairing mechanism

Abstract

One of the most intriguing phenomena in nature is the fractional quantum Hall effect (FQHE) observed in the half-filled second Landau level which, arising in even-denominator filling factors, $\nu=5/2$ and $7/2$, is completely different from other FQHEs in its origin, all of which, except for those two filling factors, occur in odd-denominator fractions. Usually formulated in terms of a trial wave function called the Moore-Read Pfaffian wave function, current leading theories attribute the origin of the 5/2 FQHE to the formation of Cooper pairs, not of electron, but of the true quasi-particle of the system known as composite fermion. The nature of superconductivity resulting from such Cooper pairing is particularly puzzling in the sense that it apparently coexists with strong magnetic fields, which poses an interesting dilemma since the Meissner effect is {\it the} most important defining property of superconductivity. This apparent dilemma is resolved by the fact that composite fermions do not respond to external magnetic field at even-denominator filling factors. To provide direct evidence that it is composite fermions that actually form the superconducting condensate, here, we develop a numerically exact method of creating a Cooper pair of composite fermions and explicitly compute the superconducting order parameter as a function of real space coordinates. As results, in addition to direct evidence for superconductivity, we obtain quantitative predictions for superconducting coherence length. Obtaining such theoretical predictions can serve as an important step toward fault-tolerant topological quantum computation.