Extended Data Fig. 3: RILS measurements at v = 1/3 with θ = 10°.

a, RILS spectra at v = 1/3 in the unpolarized geometry as a function of ω_L. Similar to those in Fig. 1e, the red and blue dashed lines indicate magnetoroton and spin-wave excitations, respectively. Compared with the result at θ = 25°, \({\varDelta }_{{\rm{s}}}^{0}\) at θ = 10° has a lower energy but remains at E_z, confirming its assignment. b, Calculated dispersions of collective excitations at v = 1/3 that support the assignment of the modes. The red dashed line is scaled down from the ideal zero-width result²⁹ by a factor of 0.305, accounting for the finite-thickness effect. The blue dashed line represents a generic dispersion for the spin-wave excitations. c, RILS spectra of the \({\varDelta }_{{\rm{m}}}^{{\rm{R}}}\) excitation at v = 1/3 in the unpolarized geometry at different ω_L. The well-resolved peaks are marked by the vertical red dashed line. We mention that the \({\varDelta }_{{\rm{m}}}^{{\rm{R}}}\) mode energy at 25° is larger than that at 10°, since a larger tilted angle induces a higher in-plane magnetic field, causing the electrons to behave in a more two-dimensional manner. On the other hand, the \({\varDelta }_{{\rm{m}}}^{0}\) energies at two tilted angles are closed. It is because a smaller tilted angle also gives a reduced kl_B in the magnetoroton dispersion, which corresponds to an increased \({\varDelta }_{{\rm{m}}}^{0}\) energy, as shown in the red dashed line in b. The two factors interplay in the case of \({\varDelta }_{{\rm{m}}}^{0}\). d, At v = 1/3, magnetoroton modes could be understood as excitations of composite fermions from the topmost (the lowest) occupied composite-fermion Landau level to the next unoccupied one.