Quantum Hall effect at half-filled Landau level: Pairing of composite fermions

We discuss the possibility of the quantum Hall effect at half-filled Landau level in terms of the pairing of the composite fermions. In the absence of Coulomb energy, we show that the ground state of the system is described by the p-wave BCS pairing state of composite fermions. When the ratio α ≡ (e/ǫlB)/ǫF (lB : the magnetic length, ǫF : Fermi energy of the composite fermions) is larger than a critical value αc ∼ 8.2 the gap of the pairing state vanishes. However, α remains less than αc if h̄ωc ≫ e/ǫlB holds. Then in this situation it is possible that the pairing state which results in the quantum Hall effect occurs. The effect of the real spin degrees of freedom and the Zeeman energy is also discussed. 73.40.Hm, 71.10.Pm Typeset using REVTEX 1 The two-dimensional electron systems with partially filled Landau level have rich structures. One of them is the fractional quantum Hall effect observed at odd integer denominator Landau level filling factor [1]. The ground state of this system is the incompressible liquid [2]. Contrary to these odd denominator filling fraction, the quantization of the Hall conductance is not observed at even integer denominator filling fraction [3,4] except for the case of ν = 5/2 [5,6]. The possibility of the quantum Hall effect at these filling fraction is still controversial problem [7]. One of the theoretical framework to understand the system with ν = 1/2 is the composite fermion (CF) picture [8]. In this theory, electrons are mapped into CFs which have charge e and the fluxes 2Φ0 (Φ0 = ch/e). Halperin, Lee, and Read (HLR) [9] studied this CF problem within the random phase approximation. At the mean field level, the fictitious fluxes attached to CFs cancel the external magnetic field and the system is described as the fermion system in the absence of the magnetic field. The system is, however, not an ordinary Fermi liquid due to the fluctuation of the Chern-Simons gauge field. Including this gauge field, the effective mass of CFs seems to diverge [9]. On the other hand, Greiter, Wen, and Wilczek (GWW) discussed the possibility of the pairing state of CFs at ν = 1/2 [10]. However, GWW used an approximation that the number of CF’s fluxes is small and they retained first order term for Chern-Simons gauge filed and neglected the quadratic term and the Coulomb interaction term. Hence how neglected terms affect the pairing state is unclear and the condition of quantum Hall effect is not discussed [11]. In this paper, we discuss the possibility of the pairing state of CFs taking into account all interaction terms in the absence of the disorder. First, we derive the Hamiltonian which takes into account all interaction terms for CFs. From this Hamiltonian, we derive the gap equation for the spinless CFs and analyzing it, we show that the ground state of the system is the p-wave BCS [12] pairing state of CFs. With increasing the ratio α ≡ (e/ǫlB)/ǫF , where lB is the magnetic length and ǫF is the Fermi energy of the composite fermions, the gap of the pairing state goes to zero. The pairing state occurs in the region of α < αc, where αc ∼ 8.2. The value of α is calculated by estimating the Fermi energy ǫF contained as the 2 parameter of CF theory. Using the result of HLR, α is ∼ 6.7 if the condition h̄ωc ≫ e/ǫlB holds. This value of α is less than αc. Hence if the condition h̄ωc ≫ e/ǫlB holds, the pairing state of CFs which results in the quantum Hall effect occurs. In GaAs samples, it will be realized in more strong magnetic field than one in present experiments. We also discuss the effect of the real spin degrees of freedom and the Zeeman energy and there we naturally understand the polarization of real spin. To begin with, we derive the effective Hamiltonian for the BCS pairing of CFs. The second quantized Hamiltonian for the two-dimensional spinless electron system in the presence of an external uniform magnetic field perpendicular to the layer is given by H = ∫ drψ(r) [ 1 2m (−i∇ +A) − μ ] ψ(r) + VC , (1) where, VC = 1/2 ∫ dr1 ∫ dr2 (1/ǫ|r1 − r2|) δρ(r1)δρ(r2). Here we take the units h̄ = c = e = 1 and δρ(r) = ψ(r)ψ(r) − ρ̄ with ρ̄ the average particle density. We introduce the generalized CF field operators [13] by φ(r) = eψ(r), π(r) = ψ(r)e, (2) where J(r) = 2 · ∫ drρ(r) log(z − z)− 1 4lB 2 |z|, (3) with z = x + iy. The factor 2 in the first term of r.h.s. of Eq. (3) denotes the number of fluxes attached to CFs. The operators φ(r) and π(r) satisfy the fermion anticommutation relations. In terms of these operators, the Hamiltonian is described by H = ∫