State counting for excited bands of the fractional quantum Hall effect: Exclusion rules for bound excitons

Exact diagonalization studies have revealed that the energy spectrum of interacting electrons in the lowest Landau level splits, nonperturbatively, into bands, which is responsible for the fascinating phenomenology of this system. The theory of nearly free composite fermions has been shown to be valid for the lowest band, and thus to capture the low-temperature physics, but it overpredicts the number of states for the excited bands. We explain the state counting of higher bands in terms of composite fermions with an infinitely strong short-range interaction between an excited composite-fermion particle and the hole it leaves behind. This interaction, the form of which we derive from the microscopic composite-fermion theory, eliminates configurations containing certain tightly bound composite-fermion excitons. With this modification, the composite-fermion theory reproduces, for all well defined excited bands seen in exact diagonalization studies, an exact counting for $\nu >1/3$, and an almost exact counting for $\nu \le 1/3$. The resulting insight clarifies that the corrections to the nearly free composite-fermion theory are not thermodynamically significant at sufficiently low temperatures, thus providing a microscopic explanation for why it has proved successful for the analysis of the various properties of the composite-fermion Fermi sea.

Figure 1

Emergent “$\Lambda$ bands” in the FQHE spectra. The exact Coulomb energies are shown by dashes, obtained in the spherical geometry for $N$ electrons at total flux $2Q(hc/e)$. The left, middle, and right panels correspond to filling factors $\nu =1/3$, 2/5, and 3/7, respectively. The top panel shows the spectra for the incompressible state, while the middle and bottom panels show spectra of systems that contain a single CF particle or CF hole in the ground state. The energy is quoted in units of $e^2/\epsilon \ell$, where $\ell =\sqrt{\hslash c/eB}$ is the magnetic length and $\epsilon$ is the dielectric constant of the host medium. Alternate bands are colored in red and blue. The integers in brackets give the number of $L$ multiplets in the Landau bands of free fermions at $(N,2Q^{*})$ with $Q^{*}=Q-N+1$. The integers outside the bracket show the number of multiplets that remain after eliminating the $L=1$ CF exciton; these agree with the actual number of multiplets except in two cases in panels (a) and (g), marked in green. The dots are the CF spectra obtained by CF diagonalization.

Figure 2

Exact Coulomb spectra are shown for several systems whose ground band corresponds to several QPs or QHs: (a) two QPs of 2/5; (b) two QPs of 3/7; (c) two QHs of 3/7; (d) two QHs of 2/5; (e) three QPs of 3/7; (f) three QHs of 3/7. The energies levels are shown by dashes and alternate $\Lambda$ bands are colored in red and blue for ease of identification of different bands. The black dots show the CF energies obtained from CF diagonalization in the basis derived from all states up to the first excited Landau band. The integers in the brackets give the number of $L$ multiplets in the Landau bands of free fermions at $(N,2Q^{*})$, where $2Q^{*}=2Q-2(N-1)$. The integers outside the brackets show the number of multiplets that remain after excluding the single excitons listed in Eq. (2), and agree with the actual numbers of multiplets in all cases.

Figure 3

Exact Coulomb spectra for $(N,2Q)=(17,35)$ corresponding to a single QH at 3/7 (left panel) and $(16,32)$ corresponding to a QP at 3/7 (right panel); energies levels are shown by dashes and alternate $\Lambda$ bands are colored in red and blue. The black dots show the CF energies obtained from CF diagonalization in the basis derived from all states up to the second excited Landau band. The integers in the brackets give the number of $L$ multiplets in the Landau bands of free fermions at $(N,2Q^{*})$, where $2Q^{*}=2Q-2(N-1)$. The integers outside the brackets show the number of multiplets that remain after excluding the single excitons listed in Eq. (2), and agree with the actual numbers of multiplets in all cases.

Figure 4

This figure shows all configurations up to energy less than or equal to 3 units of CF cyclotron energy ($\hslash {\omega }_c^{*}$) at ${\nu }^{*}=3$ (which corresponds to the electron filling $\nu =3/7$). The angular momentum of each $\Lambda$ level is shown; it is $|Q^{*}|$ for the lowest $\Lambda$ level and increases by one unit for each successive $\Lambda$ level.

Figure 5

This figure depicts the missing CF excitons (their angular momenta $L$ are shown) up to 3 units of “CF cyclotron energy.” The top, middle, and bottom panels are for excitations out of one, two, and three filled $\Lambda$ levels, respectively.

Figure 6

CF spectra for $(N,2Q)=(18,37)$ obtained by performing CF diagonalization in the basis derived from all states up to the third excited Landau band. This corresponds to ${\nu }^{*}=3$ ($\nu =3/7$) and should be compared with Fig. 1c which includes all states up to the second excited band.