Rigorous microscopic derivation of Laughlin quasiparticle fractional charge and anyonic braiding statistics from realistic quantum Hall Hamiltonians

Arovas, Schrieffer and Wilczek (Phys. Rev. Lett. 53, 722 (1984)) used Berry phases of Laughlin-type variational wave functions to show that quasiparticles of the fractional quantum Hall effect carry fractional charge $e^*=\nu e$ and obey anyonic exchange statistics with phase $\pi\nu$ at filling factor $\nu=1/m$. The Letter attached three caveats to that derivation: the calculation used variational states rather than the exact adiabatic eigenstates of a microscopic Hamiltonian; with a third quasiparticle in the vicinity the phase change under interchange depends on the adiabatic path taken, so the pair-permutation definition used for Fermi and Bose statistics no longer suffices; and the peculiar statistics can be replaced by a more complicated effective Lagrangian describing particles with conventional statistics, which the Letter referred to a cited follow-up work rather than developed. Later literature settled the last two points at the effective-theory and experimental level: dynamical Chern-Simons effective field theories now encode flux attachment and statistical transmutation, the conformal-block/topological-field-theory framework gives the full braid-group representation, and experiments have confirmed the fractional charge and the anyonic statistical phase. The surviving open core is the first one: derive, rigorously or with controlled error bounds valid in the thermodynamic limit, the same fractional charge and the same statistical phases from the exact adiabatic eigenstates of a realistic microscopic Hamiltonian (e.g., two-dimensional electrons with Coulomb-type interactions), thereby putting the 1984 variational argument on fully microscopic footing.

Background

The fractional quantum Hall effect arises in a two-dimensional electron gas in a strong perpendicular magnetic field, where the kinetic energy is quantized into highly degenerate Landau levels and, at sufficiently high field, only the lowest of these is partially occupied. The filling factor $\nu$ (electron density divided by magnetic flux density, in units of one flux quantum $\phi_0=hc/e$ per electron) then takes fractional values such as $\nu=1/3$. Laughlin (1983) proposed variational ground-state wave functions

$$\psi_m=\prod_{j<k}(z_j-z_k)^m\,\exp\Bigl(-\tfrac{1}{4}\sum_l|z_l|^2\Bigr),$$

with $z_j=x_j+iy_j$ the electron coordinates, $m$ odd and $\nu=1/m$, describing an incompressible liquid (a fluid whose density cannot be changed without paying a finite energy cost, i.e. a state separated from its excitations by an energy gap), and argued that its localized excitations (quasiholes, deficits of electron density, and quasiparticles, excesses of electron density) carry fractional charge of magnitude $e/m$: in the convention in which the electron charge is $-e$, a quasihole carries $+e/m$ and a quasiparticle (quasielectron) carries $-e/m$.

Arovas, Schrieffer and Wilczek (1984) made the charge and statistics quantitative using the Berry phase — the geometric phase acquired by a quantum state when the parameters of its Hamiltonian are transported adiabatically around a closed loop. Moving the quasihole position $z_0$ in the quasihole state $\prod_i(z_i-z_0)\,\psi_m$ around a loop enclosing flux $\phi$ yields the Aharonov-Bohm phase of a particle of charge $e^*=\nu e$ (the Aharonov-Bohm phase is the phase $2\pi q\phi/\phi_0$ acquired by a charge $q$ encircling flux $\phi$). Transporting one quasihole around a second gives an extra phase $2\pi\nu$, so interchanging two quasiholes multiplies the state by $e^{i\pi\nu}$: for noninteger $\nu$ this is neither the bosonic $+1$ nor the fermionic $-1$. Particles in two dimensions whose exchange phase can take intermediate values are called anyons, and this was their first physical realization.

The authors themselves flagged three limitations. First, in a footnote, the calculation employed a variational wave function rather than the actual adiabatic eigenstate of a microscopic Hamiltonian; they argued the statistical properties "are not expected to be sensitive to this inconsistency" and offered the stand-in interpretation that the Laughlin state is an exact excited-state wave function of a model Hamiltonian. Second, when a third quasiparticle is nearby, the phase accumulated under interchange depends on the adiabatic path, so the pair-permutation notion used for Fermi and Bose statistics no longer suffices, and no complete algebra for the braid group — the group of topological classes of trajectories of particles that wind around one another in the plane, which replaces the permutation group in two dimensions — was derived. Third, the fractional statistics can be mimicked by a static, nondynamical "statistical" vector potential (a fictitious flux $\phi_0$ attached to each quasiparticle, replaced by $\phi_0(1-1/\nu)$ if quasiparticles are treated as fermions), but the "more complicated effective Lagrangian" describing conventional-statistics particles coupled to such a field was not developed in the Letter, which referred it to a cited follow-up work.

Two of these three points were subsequently settled at the effective-theory level. Zhang, Hansson and Kivelson (1989) and Lopez and Fradkin (1991) constructed the effective field theory of the fractional quantum Hall fluid: a Landau-Ginzburg theory (a field theory formulated in terms of an order-parameter field) coupled to a dynamical Chern-Simons gauge field, a topological gauge theory in two spatial dimensions plus time whose gauge field carries no propagating modes but implements flux attachment, i.e. the transmutation of statistics by binding flux to particles, and yields fractional charge. Moore and Read (1991) obtained multi-quasihole wave functions as conformal blocks (the holomorphic building blocks into which correlation functions of a two-dimensional conformal field theory, a theory invariant under angle-preserving coordinate changes, decompose), computed the braid-group representation acting on the degenerate multi-quasihole state space, and identified non-Abelian generalizations, in which successive exchanges act by non-commuting transformations on a degenerate state space. The topological quantum field theory framework (a field theory whose observables depend only on the topology of space-time paths, not on their geometry; reviewed by Nayak, Simon, Stern, Freedman and Das Sarma in 2008) supplies the complete, path-independent braid algebra, the Arovas-Schrieffer-Wilczek path dependence being decomposed into a universal topological statistical part plus the ordinary Aharonov-Bohm contribution. Dupont and Klevtsov (2026) verified mathematically, via the Chern classes (topological invariants) of the Laughlin quasihole vector bundle (the family of quasihole state spaces parametrized by the quasihole positions), that its curvature reproduces exactly this decomposition. Experiments confirmed the fractional charge $e/3$ by quantum shot noise (the fluctuation of the electrical current, whose size directly measures the charge of the current carriers; de Picciotto et al., 1997) and, much later, the anyonic statistical phase at $\nu=1/3$: coherent Fabry-Perot interferometry of fractional edge modes was established by Nakamura et al. (2019), the anyonic phase was then observed directly by interferometry as phase slips of $2\pi\nu=2\pi/3$ (Nakamura et al., 2020), and anyon-collision experiments measured the statistical phase independently (Bartolomei et al., 2020; Ruelle et al., 2023).

What has never been established is the first, foundational task. All rigorous or exact constructions live either at the level of the variational Laughlin function and fine-tuned model (parent) Hamiltonians for which the quasihole states are exact eigenstates by construction, or for tracer particles (external, distinguishable test particles) moving through a Laughlin liquid (Lundholm and Rougerie, 2016). For realistic interactions such as Coulomb, even the existence of a spectral gap (a nonzero energy separation between the ground state and all excited states) for the Laughlin state — the incompressibility that licenses the adiabatic argument — is unproven, although there is substantial mathematical progress on rigidity, incompressibility, frustration-free approximations (frustration-free Hamiltonians are sums of local terms that are all minimized simultaneously by the ground state) and, most recently, a rigorous derivation of the fractional Hall conductance from the Laughlin state on thin cylinders (Rougerie, 2022; Bo Yang, 2020; Bachmann, Schraven, Shapiro and Warzel, 2026). Numerical Berry-phase calculations with microscopic composite-fermion trial wave functions (Jain's composite fermions are electrons bound to an even number of magnetic-flux vortices, extending the variational ansatz beyond $\nu=1/m$; Jeon, Graham and Jain, 2003) confirm the predicted statistics in large finite systems but remain variational and finite-size. The open problem is therefore the microscopic foundation of fractional statistics: replacing the 1984 variational argument by a derivation from the exact eigenstates of a realistic Hamiltonian.

Problem Statement

For a two-dimensional electron gas in the plane at filling factor $\nu=1/m$ ($m$ odd; the filling factor is the electron density divided by the magnetic flux density, in units of the flux quantum $\phi_0=hc/e$), governed by a realistic microscopic Hamiltonian — electrons interacting through repulsive pair potentials of Coulomb type in a strong perpendicular magnetic field — establish, by rigorous proof or by an approximation scheme with explicit error bounds that vanish in a stated thermodynamic limit (the limit of infinitely many particles at fixed filling factor), that the exact adiabatic quasiparticle eigenstates of this Hamiltonian reproduce the fractional charge and the statistical phases that Arovas, Schrieffer and Wilczek derived from variational Laughlin wave functions. Concretely:

1. localized quasihole excitations (deficits of electron density) of the exact eigenstates carry fractional charge $e^*=+\nu e$, and the corresponding quasiparticle (quasielectron, excess of electron density) excitations carry $-\nu e$, in the convention in which the electron charge is $-e$ and the charge of an excitation is its integrated electron-density excess or deficit relative to the ground state; and

2. adiabatically transporting one quasihole around another accumulates the statistical phase $2\pi\nu$, so that exchange of two well-separated quasiholes multiplies the state by $e^{i\pi\nu}$, up to the ordinary Aharonov-Bohm phase from the physical magnetic field.

A fully realistic Hamiltonian also contains Landau-level mixing and finite-width effects, so a derivation that claims to cover them must control them, whereas a derivation carried out in the lowest-Landau-level projection must state the projection as an explicit approximation. The derivation must proceed from properties of the exact eigenstates, replacing the variational ansatz of the 1984 calculation, and must make explicit the role of the spectral gap (incompressibility) on which any adiabatic argument relies — establishing it, or stating precisely what is assumed. A derivation valid only for fine-tuned model (parent) Hamiltonians whose exact eigenstates are the Laughlin functions themselves does not answer the question: the source offered that substitution as a way of regarding the variational state, without establishing that the statistical properties are insensitive to the difference between that state and the actual adiabatic eigenstate of a realistic Hamiltonian. A lowest-Landau-level Coulomb Hamiltonian, for which the Laughlin state is only approximate, is not such a parent Hamiltonian.

References

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