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arXiv:2305.00291 · Physical Review A

Refermionized theory of the edge modes of a fractional quantum Hall cloud

Alberto Nardin, Iacopo Carusotto

Published Nov 6, 2025• 1 citation
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Abstract

Making use of refermionization techniques, we map the nonlinear chiral Luttinger liquid model of the edge modes of a spatially confined fractional quantum Hall cloud developed in our recent work [Nardin et al., Phys. Rev. A 107, 033320 (2023)] onto a one-dimensional system of massive and interacting chiral fermions, whose mass and interactions are set by the filling factor of the quantum Hall fluid and the shape of the external confining potential at the position of the edge. As an example of the predictive power of the refermionized theory, we report a detailed study of the dynamic structure factor and of the spectral function of a fractional quantum Hall cloud. Among other features, our refermionized theory provides a physical understanding of the effective decay of the edge excitations and of the universal power-law exponents at the thresholds of the dynamic structure factor. The quantitative accuracy of the refermionized theory is validated against a full two-dimensional calculation based on a combination of exact diagonalization and Monte Carlo sampling.

DOI
10.1103/2jzb-k48j

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Making use of refermionization techniques, we map the nonlinear chiral Luttinger liquid model of the edge modes of a spatially confined fractional quantum Hall cloud developed in our recent work [Nardin et al., Phys. Rev. A 107, 033320 (2023)] onto a one-dimensional system of massive and interacting chiral fermions, whose mass and interactions are set by the filling factor of the quantum Hall fluid and the shape of the external confining potential at the position of the edge. As an example of the predictive power of the refermionized theory, we report a detailed study of the dynamic structure factor and of the spectral function of a fractional quantum Hall cloud. Among other features, our refermionized theory provides a physical understanding of the effective decay of the edge excitations and of the universal power-law exponents at the thresholds of the dynamic structure factor. The quantitative accuracy of the refermionized theory is validated against a full two-dimensional calculation based on a combination of exact diagonalization and Monte Carlo sampling.
We report numerical studies of the linear and nonlinear edge dynamics of a non-harmonically-confined macroscopic fractional quantum Hall fluid. In the long-wavelength and weak excitation limit, observable consequences of the fractional transverse conductivity are recovered. The first nonuniversal corrections to the chiral Luttinger liquid theory are then characterized: for a weak excitation in the linear response regime, cubic corrections to the linear wave dispersion and a broadening of the dynamical structure factor of the edge excitations are identified; for stronger excitations, sizable nonlinear effects are found in the dynamics. The numerically observed features are quantitatively captured by a nonlinear chiral Luttinger liquid quantum Hamiltonian that reduces to a driven Korteweg--de Vries equation in the semiclassical limit. Experimental observability of our predictions is finally discussed.
We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.
The rapid development of artificial gauge fields in ultracold gases suggests that atomic realization of fractional quantum Hall physics will become experimentally practical in the near future. While it is known that bosons on lattices can support quantum Hall states, the universal edge excitations that provide the most likely experimental probe of the topological order have not been obtained. We find that the edge excitations of an interacting boson lattice model are surprisingly sensitive to interedge hybridization and edge-bulk mixing for some confining potentials. With properly chosen potentials and fluxes, the edge spectrum is surprisingly clear even for small systems with strong lattice effects such as bandwidth. Various fractional quantum Hall phases for bosons can be obtained, and the phases $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=2/3$ have the edge spectra predicted by the chiral Luttinger liquid theory.
We study the quantum dynamics in response to time-dependent external potentials of the edge modes of a small fractional quantum Hall fluid composed of few particles on a lattice in a bosonic Laughlin-like state at filling ν=1/2. We show that the nonlinear chiral Luttinger liquid theory provides a quantitatively accurate description even for the small lattices that are available in state-of-the-art experiments, away from the continuum limit. Experimentally accessible data related to the quantized value of the bulk transverse Hall conductivity are identified both in the linear and the non-linear response to an external excitation. The strong nonlinearity induced by the open boundaries is responsible for sizable quantum blockade effects, leading to the generation of nonclassical states of the edge modes.
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.