Fig. 7: Number of Haldane modes for the Jain states at N/(2nN + η) with different structures of CHSs.

a n = 1, N > 1, η = ± 1, e.g., the FQH state at 2/3 and 2/5. The white point denotes the Laughlin states (N = 1, η = 1) in the corresponding CHS (blue circle). Because \(\delta {\bar{\rho }}_{{{{{{{{\bf{q}}}}}}}}}=\delta {\bar{\rho }}_{{{{{{{{\bf{q}}}}}}}}}^{*}\), where \(\delta {\bar{\rho }}_{{{{{{{{\bf{q}}}}}}}}}^{*}\) is the density operator projected to the CHS denoted by the gray circle, these states will show the same behavior as the Laughlin state at 1/3, i.e., one peak with no Haldane mode required. b n > 1. In this case, a single Haldane mode is needed in the effective field theory despite two peaks observed in the spectral function, among which, given n, the states with N = 2, η = − 1 can be regarded as the particle-hole conjugate partner of the corresponding Laughlin state within some specific CHS (gray circle), and the states with N > 1, η = 1 are the states in higher CF levels as shown in Fig. 2b. c, d More possible CHS structures. Thus, the number of Haldane modes added to the effective theory cannot be easily reckoned from the number of peaks in the spectral function I(E), which is instead related to the microscopic Hamiltonian used.