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arXiv:2503.06914 · SciPost Physics

Spin of fractional quantum Hall neutral modes and “missing states” on a sphere

Dung Xuan Nguyen, Dam Thanh Son

Published Nov 18, 2025• 1 citation
arXivHTMLPDF

Abstract

A low-energy neutral quasiparticle in a fractional quantum Hall system appears in the latter’s energy spectrum on a sphere as a series of many-body excited states labeled by the angular momentum L L and whose energy is a smooth function of L L in the limit of large sphere radius. We argue that the signature of a nonvanishing spin (intrinsic angular momentum) s s of the quasiparticle is the absence, in this series, of states with total angular momentum less than s s . We reinterpret the missing of certain states, observed in an exact-diagonalization calculation of the spectrum of the \nu=7/3 ν = 7 / 3 FQH state in a wide quantum well as well as in many proposed wave functions for the excited states as a consequence of the spin-2 nature of the zero-momentum magnetoroton.

DOI
10.21468/scipostphys.19.5.131

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A low-energy neutral quasiparticle in a fractional quantum Hall system appears in the latter’s energy spectrum on a sphere as a series of many-body excited states labeled by the angular momentum L L and whose energy is a smooth function of L L in the limit of large sphere radius. We argue that the signature of a nonvanishing spin (intrinsic angular momentum) s s of the quasiparticle is the absence, in this series, of states with total angular momentum less than s s . We reinterpret the missing of certain states, observed in an exact-diagonalization calculation of the spectrum of the \nu=7/3 ν = 7 / 3 FQH state in a wide quantum well as well as in many proposed wave functions for the excited states as a consequence of the spin-2 nature of the zero-momentum magnetoroton.
The effect of Landau level (LL) mixing on the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state is studied directly in a minimal, yet still numerically tractable, approximation. Specifically, the inter-LL excitations with one unit of cyclotron energy are included in the exact diagonalization of a small yet representative number of electrons on a sphere. Significant reduction of the excitation gap found already in this simple model indicates that LL mixing must play an important role in more realistic systems, with a full spectrum of allowed inter-LL excitations. This suggests a possible route to the explanation of the troubling discrepancy between earlier calculations and experiments. On the other hand, even within our limited approach the LL mixing is found to considerably reduce overlaps with the Moore-Read wave function, raising the question of the actual realization of non-Abelian quasiparticles (QPs) at $\ensuremath{\nu}=5∕2$. In view of the discussed limitations of exact numerics, conclusive resolution of this question will probably require an experiment directly probing the QP statistics.
We numerically study the spin polarization of the fractional quantum Hall state at filling factor $\ensuremath{\nu}=5/2$. By using both exact diagonalization and the density matrix renormalization group methods on the sphere, we are able to analyze more values of partial spin polarization (in addition to fully polarized and unpolarized) than any previous studies. We find that for the Coulomb interaction the exact finite-system ground state is fully polarized, for shifts corresponding to both the Moore-Read Pfaffian state and its particle-hole conjugate (anti-Pfaffian). This result is found to be robust against small variations in the interaction and change of shift. The low-energy excitation spectrum is consistent with spin-wave excitations of a fully magnetized ferromagnet.
We demonstrate the experimental feasibility of incompressible fractional quantum Hall-like states in ultracold two-dimensional rapidly rotating dipolar Fermi gases. In particular, we argue that the state of the system at filling fraction nu = 1/3 is well described by the Laughlin wave function and find a substantial energy gap in the quasiparticle excitation spectrum. Dipolar gases, therefore, appear as natural candidates of systems that allow us to realize these very interesting highly correlated states in future experiments.
Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings ν=n/(2n±1)ν{=}n/(2n{\pm}1)ν=n/(2n±1) and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at ν=n/(4n±1)ν{=}n/(4n{\pm}1)ν=n/(4n±1). We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at ν=2/5ν{=}2/5ν=2/5. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.
The low-energy physics of fractional quantum Hall (FQH) states-a paradigm of strongly correlated topological phases of matter-to a large extent is captured by weakly interacting quasiparticles known as composite fermions.In this paper, based on numerical simulations and effective field theory, we argue that some high-energy states in the FQH spectra necessitate a different description based on parton quasiparticles.We show that Jain states at filling factor ν ¼ n=ð2pn Æ 1Þ with integers n, p ≥ 2 support two kinds of collective modes: In addition to the well-known Girvin-MacDonald-Platzman (GMP) mode, they host a high-energy collective mode, which we interpret as the GMP mode of partons.We elucidate observable signatures of the parton mode in the dynamics following a geometric quench.We construct a microscopic wave function for the parton mode and demonstrate agreement between its variational energy and exact diagonalization.Using the parton construction, we derive a field theory of the Jain states and show that the previously proposed effective theories follow from our approach.Our results point to partons being "real" quasiparticles which, in a way reminiscent of quarks, become observable only at sufficiently high energies.
We study a model fractional quantum Hall (FQH) wave function called the Gaffnian state, which is believed to represent a gapless, strongly correlated state that is very different from conventional metals. To understand this exotic gapless state better, we provide a representation based on work of Halperin in which the pairing structure of the Gaffnian state becomes more explicit. We employ the single-mode approximation introduced by Girvin, MacDonald, and Platzman, here extended to three-body interactions, in order to treat a neutral collective excitation mode in order to clarify the physical origin of the gaplessness of the Gaffnian state. We discuss approaches to extract systematically the relevant physics in the long-distance, large-electron-number limit of FQH states using numerical calculations with relatively few electrons. In Appendices, we provide second-quantized expressions for many-body Haldane pseudopotentials in various geometries including the plane, sphere, cylinder, and torus based on the proper definition of the relative angular momentum.