Particle-hole symmetry and composite fermions in fractional quantum Hall states

We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi-liquid description, we demonstrate the appearance of the Girvin-Macdonald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the modified random-phase approximation (MRPA) of the Halperin-Lee-Read (HLR) theory is particle-hole symmetric, correlation functions obtained from this scheme are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the MPRA of the HLR theory violate it.

Figure 1

A deformed Fermi surface.

Figure 2

Dispersion relation of neutral excitations of the Jain series $\nu =\frac{1}{2}\pm \frac{1}{2(2N+1)}$. The solid and dashed lines represent the dispersions with and without the Coulomb interactions. The $N$ dependence of these curves is only through the rescaling of the $x$ axis. (a) $F_1=0.5,F_{n>1}=0,\lambda =1$. (b) $F_n=0,\lambda =0.05$.