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arXiv:2412.14879 · Physical review. B./Physical review. B

Spin fractionalization at the edge of quantum Hall fluids induced by bulk quasiparticles

Alexander Fagerlund, Alberto Nardin, Leonardo Mazza, Eddy Ardonne

Published Jul 14, 2025• 1 citation
arXivHTMLPDF

Abstract

We define a measurable spin for the edge of a lowest Landau level and incompressible fractional quantum Hall state in the presence of an Abelian or non-Abelian bulk quasiparticle. We show that this quantity takes a fractional value inherited from the fractional spin of the bulk quasiparticle. We present a geometric picture that does not rely on global symmetries of the wave function, but is able to treat quasiparticles and edges with different shapes. We study finite-size many-body wave functions on the cylinder with circular quasiparticles and straight edges. Our results are supported by matrix-product-state calculations for the Laughlin and the k = 3 Read-Rezayi states.

DOI
10.1103/t5ww-65lz

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We define a measurable spin for the edge of a lowest Landau level and incompressible fractional quantum Hall state in the presence of an Abelian or non-Abelian bulk quasiparticle. We show that this quantity takes a fractional value inherited from the fractional spin of the bulk quasiparticle. We present a geometric picture that does not rely on global symmetries of the wave function, but is able to treat quasiparticles and edges with different shapes. We study finite-size many-body wave functions on the cylinder with circular quasiparticles and straight edges. Our results are supported by matrix-product-state calculations for the Laughlin and the k = 3 Read-Rezayi states.
We use exact numerical diagonalization in the search of fractional quantum Hall states with non-Abelian quasiparticle statistics. For the (most promising) states in a partially filled second Landau level, the search is narrowed to the range of filling factors 7/3 < v e < 8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at v e = 5/2 (paired Moore–Read "pfafian" state) and 12/5 (clustered Read–Rezayi "parafermion" state). Outside of this range, the noninteracting composite fermion model with four attached flux quanta is validated, yielding the family of quantum liquids with fractional, but Abelian statistics. The borderline v e = 7/3 state is shown to be adiabatically connected to the Laughlin liquid, but its short-range correlations are significantly different.
We propose a method of realizing non-Abelian braiding of fractionalized quasiholes in the Laughlin fractional quantum Hall phase at ν=1/3ν=1/3ν=1/3 with realistic two-body interactions within the lowest Landau level. It is numerically shown that low-lying gapped excitations near ν=1/3ν=1/3ν=1/3 are contained almost entirely within the null space of the three-body Moore-Read model Hamiltonian. They are thus quantum fluids of non-Abelian quasiholes that are in principle physically accessible. In particular, Laughlin ground state can be described as a fluid of ``ψψψ-type" quasiholes formed by binding a magnetic flux with a Majorana fermion (MF), and the Laughlin quasiholes are described by the ``111-type'' quasiholes, which are magnetic fluxes without a MF attached. Within the Laughlin phase, Laughlin quasiholes can be locally fractionalized into non-Abelian quasiholes, when the strong attraction between them is overcome by properly designed one-body electronstatic trapping potentials. Extensive numerics with proper finite-size scaling corroborate this physical picture, and our study points to the possibility of realizing non-Abelian braiding within an Abelian topological phase in experiment without the need for fine-tuning realistic electron-electron interaction.
We propose to describe bulk wave functions of fractional quantum Hall states in terms of correlators of nonunitary b/cb/cb/c-spin systems. These yield a promising conformal field-theory analogon of the composite fermion picture of Jain. Fractional statistics are described by twist fields which naturally appear in the b/cb/cb/c-spin systems. We provide a geometrical interpretation of our approach in which bulk wave functions are seen as holomorphic functions over a ramified covering of the complex plane, where the ramification precisely resembles the fractional statistics of the quasiparticle excitations in terms of branch points on the complex plane. To extend Jain's main series, we use the concept of composite fermions pairing to spin singlets, which enjoys a natural description in terms of the particular $c=\ensuremath{-}2$ b/cb/cb/c-spin system as known from the Haldane-Rezayi state. In this way we derive conformal field-theory proposals for lowest Landau-level bulk wave functions for more general filling fractions. We obtain a natural classification of the experimentally confirmed filling fractions, which does not contain prominent unobserved fillings. Furthermore, our scheme fits together with classifications in terms of K matrices of effective multilayer theories leading to striking restrictions of these coupling matrices.
Advancing a microscopic field-theoretic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. Here we expand on such a program and, in particular, address the problems of Laughlin's quasiparticle(hole) operator fractionalization and condensation of non-local operators characterizing topological order in the lowest-Landau-level (LLL). We express the sequence of (both bosonic and fermionic) Laughlin second-quantized states in a manner that highlights the lack of local condensation. In addition, we present a rigorous constructive subspace bosonization dictionary for the bulk fluid and establish the universal long-distance behavior of edge excitations by formulating a conjecture complementary to Wen's.
Abstract We present a detailed microscopic study of edge excitations for n filled Landau levels. We show that the higher-level wavefunctions possess a non-trivial radial dependence that should be integrated over for properly defining the edge conformal field theory. This analysis let’s us clarify the role of the electron orbital spin s in the edge theory and to discuss its universality, thus providing a further instance of the bulk-boundary correspondence. We find that the values s i for each level, parameterize a Casimir effect or chemical potential shift that could be experimentally observed. These results are generalized to fractional and hierarchical fillings by exploiting the W-infinity symmetry of incompressible Hall fluids.
We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular T transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.