Controlled microscopic foundation of the dipolar composite-fermion Fermi liquid at the half-filled Landau level ($\nu=1/2$)

At filling factor $\nu=1/2$ of the lowest Landau level, interacting two-dimensional electrons form a compressible metallic state whose quasiparticles are composite fermions: neutral dipoles consisting of an electron bound to two vortices, with dipole vector perpendicular to the momentum. Pasquier and Haldane (1998) gave this dipole picture a microscopic starting point through a model of repulsive bosons at $\nu=1$ mapped onto fermion-vortex pairs, deriving a dipole Fermi-liquid Hamiltonian with an interaction-generated effective mass and $(\ell q)^2$-damped response — but at the cost of dropping the antisymmetrization / physical-subspace projection of their Eq. (3) (the square-root factor $\chi^+\sqrt{1+:\chi\chi^+:}$), using an uncontrolled mean-field pairing at full density, leaving the extension to the fermionic $\nu=1/2$ problem heuristic, and leaving the effective mass $m^*$ and Landau parameters undetermined. The open problem is to establish the dipolar composite-fermion liquid of the $\nu=1/2$ lowest Landau level on a controlled microscopic footing directly in the fermionic problem: retaining the physical-subspace projection exactly or with controlled error, justifying the Fermi sea beyond mean field, computing $m^*$ and the Landau parameters quantitatively from the microscopic projected interaction, and assessing gauge-field fluctuation corrections to the response. Despite three decades of development — Hamiltonian and constraint theories, the particle-hole-symmetric Dirac composite fermion, and large-scale numerics — no controlled derivation with first-principles parameters exists, and the 2025–2026 controversy over the long-wavelength structure factor shows the response question remains live.

Background

In the fractional quantum Hall effect, a two-dimensional electron gas in a strong perpendicular magnetic field has its kinetic energy collapsed into highly degenerate Landau levels, and the residual dynamics is governed purely by the electron-electron interaction. The filling factor $\nu$ is the ratio of electron density to magnetic flux density, and the magnetic length $\ell=\sqrt{\hbar c/eB}$ sets the microscopic scale. Unlike the incompressible liquids at odd-denominator fractions, the state at $\nu=1/2$, where the lowest Landau level (LLL) is half filled, is compressible and metallic. Jain's composite-fermion approach explains this: an electron bound to two quantized vortices ('two flux quanta') sees, on average, zero magnetic field, so these composite objects form a Fermi sea. Halperin, Lee and Read (HLR) developed this into a fermionic Chern-Simons field theory — a statistical gauge theory implementing flux attachment — which accounts for the linear-in-$q$ longitudinal conductivity probed by surface-acoustic-wave experiments (in which an acoustic wave travelling along the sample surface probes the electron conductivity at a finite wavelength). Read subsequently observed that the vortex bound to the electron is spatially separated from it, so the composite fermion should be viewed as a neutral dipole whose dipole vector $\ell^2\hat{z}\times\mathbf{k}$ is perpendicular to its momentum $\mathbf{k}$; in this dipole picture the quasiparticle couples to a slowly varying external potential only through gradients, producing the characteristic $(\ell q)^2$ damping of low-momentum response. Trial wave functions for the $\nu=1/2$ Fermi sea studied by Rezayi and Read give strong numerical support to this picture. Pasquier and Haldane gave the dipole picture a microscopic starting point. They considered repulsively interacting bosons at $\nu=1$ confined to the LLL and mapped the bosonic dynamics onto that of interacting fermions: to each LLL orbital one adjoins canonical fermion operators, and pair operators $\chi^+_{ij}$ generate the bosonic Hilbert space inside an enlarged fermionic space. The physical subspace is recovered by antisymmetrizing over the fermion indices (Eq. (3) of their paper), which in the operator dictionary appears as a square-root factor $\chi^+\sqrt{1+:\chi\chi^+:}$. Pairing each boson with a fermionic vortex (the vortex carrying the opposite charge) produces a dipole bound state, and within a mean-field, low-pair-density approximation they derived an effective dipole Hamiltonian (their Eq. (15)) describing a Fermi sea of neutral dipoles with an effective mass $m^*$ of order $V(\ell)$ — the interaction evaluated at the magnetic-length scale — generated purely by the interactions, with no band mass. The static and dynamic response functions they obtain (their Eq. (17)) are governed by the first Landau parameter $F_1$ — the $l=1$ harmonic of the Fermi-liquid interaction function, the standard phenomenological parameter of Landau's theory of Fermi liquids — and show the $(\ell q)^2$ damping expected for neutral dipoles. The authors themselves flag the gaps that define the open problem. First, they keep only the leading term of the square root, replacing $\chi^+\sqrt{1+:\chi\chi^+:}$ by $\chi^+$, which amounts to disregarding the antisymmetrization/projection of Eq. (3); the effect of retaining it exactly is not established. Second, the pairing of bosons with vortices is a low-density approximation that becomes uncontrolled at $\nu=1$, where every boson is paired ($N_2/N=1$). Third, the derivation is for bosons at $\nu=1$, and the claim that the same low-energy theory describes the physical fermionic $\nu=1/2$ state is argued heuristically ('it is tempting to assume'), not derived. Fourth, $m^*$ and $F_1$ are not determined quantitatively from the microscopic projected interaction $V(\mathbf{x}-\mathbf{y})$, including possible nonperturbative enhancements (although the model suggests no divergence of $m^*$). Fifth, the finite-$N$ discrete-mode formulation introduces a lattice cutoff $a$ and a three-scale regime $a\ll\ell\ll\xi$ (with $\xi$ the physically relevant scale) whose continuum-limit control is not analyzed. Sixth, the response is obtained from a mean-field transport equation, and gauge-field or fluctuation corrections are not assessed. In the modern era, particle-hole symmetry — the exact symmetry of a half-filled Landau level under exchanging electrons and holes — has sharpened the target further: Son's Dirac composite fermion describes the same Fermi liquid with manifest particle-hole symmetry, but it too is an infrared effective theory whose mass remains a free parameter.

Problem Statement

Establish the dipolar composite-fermion description of the compressible state at filling factor $\nu=1/2$ on a controlled microscopic footing, directly in the fermionic lowest-Landau-level problem. Starting from the projected microscopic Hamiltonian $H=\frac{1}{2}\int\rho(\mathbf{x})\,V(\mathbf{x}-\mathbf{y})\,\rho(\mathbf{y})\,d^2x\,d^2y$ acting in the lowest Landau level (with $\rho$ the projected density and $V$ the bare interaction), a solution must jointly deliver: (a) a treatment in which the projection onto the physical subspace is retained exactly, or in which the error committed by dropping it is explicitly controlled (in the Pasquier-Haldane bosonic representation that projection is the full antisymmetrization, i.e. the square-root operator factor $\chi^+\sqrt{1+:\chi\chi^+:}$ of their Eq. (3); a derivation working directly in the fermionic problem must instead state the projection on which it relies and show that it is retained exactly or that the error of dropping it is explicitly bounded); (b) a justification of the formation of the dipolar Fermi sea beyond mean field at full pair density, where the low-density pairing argument used by Pasquier and Haldane fails; (c) a quantitative determination of the effective mass $m^*$ and the Landau parameters (notably $F_1$ entering the response) from the microscopic projected interaction $V(\mathbf{x}-\mathbf{y})$, including whether nonperturbative effects enhance or renormalize them (the Pasquier-Haldane model suggests no divergence, but does not establish it); and (d) a quantitative assessment of the gauge-field and fluctuation corrections to the low-$q$ response functions that the mean-field transport treatment of Pasquier and Haldane omits. These four components are jointly needed for the single overarching objective — a first-principles microscopic theory of the $\nu=1/2$ composite-fermion liquid. Here 'controlled' is to be read component by component: for each approximate step the answer must identify the small parameter or extrapolation procedure that governs it and state an explicit bound on the error of that step. A resolution may also be negative: an answer that establishes, with the same standard of control, that one or more of (a)–(d) cannot be achieved as required also addresses the objective.

References

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