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arXiv:2407.19013 · Physical Review Research

Quantum Hall edges beyond the plasma analogy

Per Moosavi, Blagoje Oblak, Bastien Lapierre, Benoit Estienne, Jean-Marie Stéphan

Published May 12, 2025• 1 citation
arXivHTMLPDF

Abstract

We demonstrate that the widely used plasma analogy is unreliable at predicting edge properties of quantum Hall states. This discrepancy arises from a fundamental difference between quantum Hall droplets and plasmas (Coulomb gases): The former are incompressible liquids subject to area-preserving deformations, while the latter are governed by electrostatics and thus involve conformal transformations. Consequently, the plasma analogy fails at the edge, except in fine-tuned geometries, as it does not account for the emergent local edge velocity. We quantitatively show how the analogy's failure affects physical quantities, such as fluctuations of local observables and absorption rates in microwave spectroscopy, measurable in both solid-state experiments and quantum simulators.

DOI
10.1103/physrevresearch.7.023134

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We demonstrate that the widely used plasma analogy is unreliable at predicting edge properties of quantum Hall states. This discrepancy arises from a fundamental difference between quantum Hall droplets and plasmas (Coulomb gases): The former are incompressible liquids subject to area-preserving deformations, while the latter are governed by electrostatics and thus involve conformal transformations. Consequently, the plasma analogy fails at the edge, except in fine-tuned geometries, as it does not account for the emergent local edge velocity. We quantitatively show how the analogy's failure affects physical quantities, such as fluctuations of local observables and absorption rates in microwave spectroscopy, measurable in both solid-state experiments and quantum simulators.
We study bulk and edge correlations in the compressible half-filled state [B. I. Halperin et al., Phys. Rev. B 47, 7312 (1993) and, E. Rezayi and N. Read, Phys. Rev. Lett. 72, 900 (1994); 73, 1052 (1994)] using a modified version of the plasma analogy. The corresponding plasma has anomalously weak screening properties, and as a consequence, we find that the correlations along the edge do not decay algebraically as in the Laughlin (incompressible) case, while the bulk correlations decay in the same way. The results suggest that due to the strong coupling between charged modes on the edge and the neutral fermions in the bulk, reflected by the weak screening in the plasma analog, the (attractive) correlation hole is not well defined on the edge. Hence, the system there can be modeled as a free Fermi gas of electrons (with an appropriate boundary condition). We finally comment on a possible scenario in which the Laughlin-like dynamical edge correlations may nevertheless be realized.
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.
Abstract The description of chiral quantum incompressible fluids by the W∞ symmetry can be extended from the edge, where it encompasses the conformal field theory approach, to the non-conformal bulk. The two regimes are characterized by excitations with different sizes, energies and momenta within the disk geometry. In particular, the bulk quantities have a finite limit for large droplets. We obtain analytic results for the radial shape of excitations, the edge reconstruction phenomenon and the energy spectrum of density fluctuations in Laughlin states.
A quantum Hall (QH) interface is different from an ordinary QH edge, as the latter has its location determined by the confining potential, while the former can be unpinned and behave like a free string. In this paper, we demonstrate this difference by studying three different interfaces formed by (i) the Laughlin state and the vacuum, (ii) the Pfaffian state and the vacuum, and (iii) the Pfaffian and the anti-Pfaffian states. We find that stringlike interfaces propagating freely in the QH system lead to very different dynamical properties from edges. This qualitative difference gives rise to fascinating physics and suggests a different direction for future research on QH physics. We also discuss briefly possible analogies between QH interfaces and concepts in string theory.
We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction $\ensuremath{\nu}=5/2$. We complete the program started in V. Gurarie and C. Nayak, [Nucl. Phys. B 506, 685 (1997)]. and show that the degenerate four-quasihole and six-quasihole wave functions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a c=1+12c=1+\frac{1}{2}c=1+21​ conformal field theory. As a consequence, this proves that the non-Abelian statistics of the excitations in this state are given by the explicit analytic continuation of these wave functions. Our proof is based on a plasma analogy derived from the Coulomb gas construction of Ising model correlation functions involving both order and (at most two) disorder operators. We show how this computation also determines the non-Abelian statistics of collections of more than six quasiholes and give an explicit expression for the corresponding conformal block-derived wave functions for an arbitrary number of quasiholes. Our method also applies to the anti-Pfaffian wave function and to Bonderson-Slingerland hierarchy states constructed over the Moore-Read and anti-Pfaffian states.