Gauge Fields and Pairing in Double-Layer Composite Fermion Metals.

A symmetrically doped double layer electron system with total filling fraction ν = 1/m decouples into two even denominator (ν = 1/2m) composite fermion ‘metals’ when the layer spacing is large. Out-of-phase fluctuations of the statistical gauge fields in this system mediate a singular attractive pairing interaction between composite fermions in different layers. A strong-coupling analysis shows that for any layer spacing d this pairing interaction leads to the formation of a paired quantum Hall state with a zero-temperature gap ∆(0) ∝ 1/d2. The less singular in-phase gauge fluctuations suppress the size of the zero-temperature gap, ∆(0) ∝ 1/ ( d2(ln d)6 ) , but do not eliminate the instability. Typeset using REVTEX 1 Composite fermions were introduced by Jain in order to understand the observed hierarchy of states in the fractional quantum Hall effect (FQHE) [1]. A composite fermion is an electron confined to move in two dimensions and tied to an even number of statistical flux quanta. Jain showed that the fractional quantum Hall effect for electrons at odddenominator filling fractions can be viewed as an effective integer quantum Hall effect for composite fermions. Halperin, Lee, and Read (HLR) took Jain’s suggestion further, arguing that at Landau level filling fraction ν = 1 2 , or any even denominator filling fraction ν = 1 2m , the statistical flux attached to composite fermions can, at the Hartree level, exactly cancel the physical flux of the applied magnetic field [2]. The composite fermions then form a new type of metal, and a growing number of experiments appear to support this description [3]. HLR also showed that fluctuations of the statistical gauge field in this metal give rise to singular inelastic scattering of sufficient strength to lead to a breakdown of Landau Fermi liquid theory. Though experimental proof of the non-Fermi liquid nature of the composite fermion metal remains elusive, it has generated a great deal of excitement in the theoretical community [4]. Double-layer electron systems have been realized in both double quantum wells [5] and wide single quantum wells [6]. The (m,m, n) states at filling fraction ν = 2 m+n , proposed by Halperin [7], are double-layer generalizations of the Laughlin states. At even denominators, there are additional possibilities motivated by the composite fermion construction. As one of us has pointed out [8], in the limit where the layer spacing d is large it should be possible to view a double-layer system at ν = 2 2m as two decoupled ν = 1 2m composite fermion metals. This description will be referred to as the double-layer composite fermion metal (DLCFM) description in what follows. The main result of this Letter is that an ideal DLCFM, by which we mean a DLCFM in which the carrier densities in the two layers are precisely equal, there is no interlayer tunneling and there is no disorder, is always unstable to the formation of a paired quantum Hall state for any layer spacing d. Motivated by this result we propose the phase diagram shown in Fig. (1) for the ν = 1 double-layer system. It is unclear at present whether the line separating the paired quantum Hall state and the (1, 1, 1) state represents 2 a transition or a smooth crossover. The Lagrangian density for an ideal DLCFM as defined above is given by L(r, τ) = L1(r, τ) + L2(r, τ) (1) where L1(r, τ) = ∑