Nature of composite fermions and the role of particle-hole symmetry: A microscopic account

Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half-filled Landau level, Son has made an intriguing proposal [Phys. Rev. X 5, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well-established microscopic theory of the fractional quantum Hall effect and the $\frac{1}{2}$ state, which is based on nonrelativistic composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor $\nu =\frac{1}{2}$ corresponds to an effective time-reversal transformation (i.e., $\left\{{\mathit{k}}_j\right\}\to \{-{\mathit{k}}_j\}$) for composite fermions, and discuss how this connects to the absence of $2k_{\mathrm{F}}$ backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion $\mathrm{\Lambda }$ levels (analogs of electronic Landau levels), we determine the $\mathrm{\Lambda }$ level spacing and find it to be very nearly independent of the $\mathrm{\Lambda }$ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a “normal form” for the unprojected wave functions of the $n/(2pn-1)$ states that correctly capture the topological properties even without lowest Landau level projection.

Figure 1

The exact Coulomb spectrum (dashes) of interacting electrons at $\nu =\frac{3}{7}$ for 18 electrons, reproduced from Ref. [33]. Several bands are evident, marked alternately by blue and red colors for ease of depiction. The second excited band (colored blue) is well explained in terms of a combination of the excitations $2\to 4$, $1\to 3$, and ${(2\to 3)}^2$ of nonrelativistic composite fermions. The CF theory based on nonrelativistic composite fermions predicts the correct number of states in the band at each quantum number and, furthermore, predicts energies shown by black dots (obtained without any adjustable parameter).

Figure 2

The upper panel shows bare CF hole in various $\mathrm{\Lambda }$ levels at $\nu =\frac{3}{7}$ (right panels), analogous to the hole in various Landau levels at $\nu =3$. The lower panel shows the thermodynamic Coulomb energy of the bare CF hole excitation as a function of its $\mathrm{\Lambda }\mathrm{L}$ index for various filling factors in the sequence $n/(2n+1)$. All energies are quoted in Coulomb units of $e^2/\epsilon \ell$ and measured relative to the energy of the bare CF hole in the topmost filled $\mathrm{\Lambda }\mathrm{L}$. The filled symbols are obtained using Jain's wave functions for the CF hole. The solid straight lines indicate the prediction if composite fermions are taken as nonrelativistic fermions.

Figure 3

Comparison of the non-normalized density of the generalized CF quasihole state at $\nu =\frac{1}{3}$ for different values of $\alpha$ for $N=15$ electrons. The coordinate $r$ is the chord distance on the sphere. The horizontal dotted magenta line shows the density of the corresponding uniform incompressible state.