Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2105.12814 · WORLD SCIENTIFIC eBooks

Mean Field Theories of Quantum Hall Liquids Justified: Variations on the Greiter–Wilczek Theme

T. H. Hansson, Steven Allan Kivelson

Published Mar 31, 2022• 1 citation
arXivHTMLPDF

Abstract

We present a field theoretic variant of the Wilczek - Greiter adiabatic approach to Quantum Hall liquids. Specifically, we define a Chern-Simons-Maxwell theory such that the flux-attachment mean field theory is exact in a certain limit. This permits a systematic way to justify a variety of useful approximate approaches to these problems as constituting the first term in a (still to be developed) systematic expansion about a solvable limit.

DOI
10.1142/9789811251948_0009

Related papers

A Mean Field Theory for the Quantum Hall Liquid. I

K. Ishikawa

A mean field theory for the quantum Hall liquid based on a bi-local collective field is presented. A self-consistent mean field and fluctuations due to low spin fields around the mean field are investigated. We find that the mean field energy is negative and its correction due to the fluctuations of the density field is finite and negative and the correction due to the vector fields cancells completely in the lowest order. The finite Hall conductance is computed and the massive Chern-Simons gauge theory is obtained as an effective long wave length theory.

Progress of Theoretical Physics•Nov 1, 1992•1 citation

Fermion liquids as quantum Hall liquids in phase space: A unified approach for anomalies and responses

Jaychandran Padayasi, Ken K. W., Kun Yang

The discovery of many strongly correlated metallic phases has inspired different routes to generalize or go beyond the celebrated Landau Fermi liquid theory. To this end, from universal consideration of symmetries and anomalies, Else, Thorngren, and Senthil (ETS) have introduced a class of theories called ersatz Fermi liquids which possess a Fermi surface and satisfy a generalized Luttinger's theorem. In this work, we view all such fermion liquids obeying the Luttinger theorem as incompressible quantum Hall liquids in higher-dimensional phase space and use it as the starting point to derive their effective low-energy field theory. The noncommutativity of phase space motivates us to use the Seiberg-Witten map to derive the field theory in an ordinary (commutative) space and naturally leads to terms that correspond to the correct topological Chern-Simons action postulated by ETS in one, two, and three dimensions. Additionally, our approach also reproduces all the nontopological terms that characterize important contributions to the response, including the semiclassical equations of motion. Finally, our derivations of Chern-Simons terms from the Seiberg-Witten map also verify a longstanding conjecture in noncommutative field theory.

arXiv:2501.08379•Physical review. B./Physical review. B•Mar 17, 2025•1 citation

Anyon Mean Field as an Exact Limit of a Gauge Theory

Roberto Iengo A

We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions ’ charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the-translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. This lecture is based on ref.[1] and on previous work done on the problem of a translational invariant formulation of the Anyon mean field theory and its properties with respect to superfluidity and possibly superconductivity (for a general review of the previous work see ref.[2]). Actually, to discuss those issues in a proper way it is necessary to introduce a suitable mathematical formalism:

•Jan 1, 2008•0 citations

Generalized flux states and the quantum Hall effect (abstract)

Gergely T. Zimányi, Franco Nori, Elihu Abrahams

We investigate certain generalized flux phases arising in a mean-field approach to the t-J model. First we establish that the energy of noninteracting electrons moving in a uniform magnetic field has an absolute minimum as a function of the flux at exactly one flux quantum per particle. Using this result we show that if the hard-core nature of the hole bosons is taken into account, then the generalized flux states in the mean-field approximation constitute a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. We elaborate the relation to the integer and fractional quantum Hall effect. Based on this analogy one expects that only states with certain fractional fillings (ν=1/odd integer) are stable against the fluctuations of the (gauge) field. We speculate that at arbitrary fillings the system minimizes its energy by phase separating into these stable states. Finally we work out the ground state energy for nonuniform (staggered) flux phases and establish that the often claimed equivalence of frustration and doping does not hold for these states.

Journal of Applied Physics•Apr 15, 1991•0 citations

Microscopic description of the chiral Tomonaga-Luttinger liquid at the fractional quantum Hall edge

Huang, Wei, Zhao-Long Wang, Mu-Lin Yan

The effective field theory of the fractional quantum Hall edge is reformulated from microscopic dynamics. Noncommutative Chern-Simons theory is a microscopic description for the quantum Hall fluid. We use it for reference. Considering relabeling symmetry of the electrons and incompressibility of the fluid, we obtain a constraint and derive a chiral Tomonaga-Luttinger liquid theory containing interaction terms. We calculate one-loop corrections to the phonon and electron propagators and get a new tunneling exponent. It agrees with experiments.

arXiv:0906.4319•Jun 23, 2009•0 citations

Quantum wires, Chern-Simons theory, and dualities in the quantum Hall system

Julio Toledo, Renann Lipinski Jusinskas, Carlos A. Hernaski +1 more

Over the years, many theoretical frameworks have been developed to understand the remarkable physics of the quantum Hall system. In this work we discuss the interplay among quantum wires, Chern-Simons theory, bosonization, and particle-vortex duality, which enable a unified approach for the Laughlin states of the fractional quantum Hall system. Starting from the so-called quantum wires system, which is a semimicroscopic description in terms of $(1+1)$-dimensional theories, we discuss the emergence of $(2+1)$-dimensional low-energy effective field theories by using different maps connecting the microscopic degrees of freedom with the macroscopic ones. We show the origin of these maps by embedding the bosonized version of the original quantum wires model in a gauge-invariant theory and using particle vortex duality.

arXiv:2205.08488•Physical review. B./Physical review. B•Aug 9, 2022•2 citations
We present a field theoretic variant of the Wilczek - Greiter adiabatic approach to Quantum Hall liquids. Specifically, we define a Chern-Simons-Maxwell theory such that the flux-attachment mean field theory is exact in a certain limit. This permits a systematic way to justify a variety of useful approximate approaches to these problems as constituting the first term in a (still to be developed) systematic expansion about a solvable limit.
A mean field theory for the quantum Hall liquid based on a bi-local collective field is presented. A self-consistent mean field and fluctuations due to low spin fields around the mean field are investigated. We find that the mean field energy is negative and its correction due to the fluctuations of the density field is finite and negative and the correction due to the vector fields cancells completely in the lowest order. The finite Hall conductance is computed and the massive Chern-Simons gauge theory is obtained as an effective long wave length theory.
The discovery of many strongly correlated metallic phases has inspired different routes to generalize or go beyond the celebrated Landau Fermi liquid theory. To this end, from universal consideration of symmetries and anomalies, Else, Thorngren, and Senthil (ETS) have introduced a class of theories called ersatz Fermi liquids which possess a Fermi surface and satisfy a generalized Luttinger's theorem. In this work, we view all such fermion liquids obeying the Luttinger theorem as incompressible quantum Hall liquids in higher-dimensional phase space and use it as the starting point to derive their effective low-energy field theory. The noncommutativity of phase space motivates us to use the Seiberg-Witten map to derive the field theory in an ordinary (commutative) space and naturally leads to terms that correspond to the correct topological Chern-Simons action postulated by ETS in one, two, and three dimensions. Additionally, our approach also reproduces all the nontopological terms that characterize important contributions to the response, including the semiclassical equations of motion. Finally, our derivations of Chern-Simons terms from the Seiberg-Witten map also verify a longstanding conjecture in noncommutative field theory.
We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions ’ charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the-translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. This lecture is based on ref.[1] and on previous work done on the problem of a translational invariant formulation of the Anyon mean field theory and its properties with respect to superfluidity and possibly superconductivity (for a general review of the previous work see ref.[2]). Actually, to discuss those issues in a proper way it is necessary to introduce a suitable mathematical formalism:
We investigate certain generalized flux phases arising in a mean-field approach to the t-J model. First we establish that the energy of noninteracting electrons moving in a uniform magnetic field has an absolute minimum as a function of the flux at exactly one flux quantum per particle. Using this result we show that if the hard-core nature of the hole bosons is taken into account, then the generalized flux states in the mean-field approximation constitute a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. We elaborate the relation to the integer and fractional quantum Hall effect. Based on this analogy one expects that only states with certain fractional fillings (ν=1/odd integer) are stable against the fluctuations of the (gauge) field. We speculate that at arbitrary fillings the system minimizes its energy by phase separating into these stable states. Finally we work out the ground state energy for nonuniform (staggered) flux phases and establish that the often claimed equivalence of frustration and doping does not hold for these states.
The effective field theory of the fractional quantum Hall edge is reformulated from microscopic dynamics. Noncommutative Chern-Simons theory is a microscopic description for the quantum Hall fluid. We use it for reference. Considering relabeling symmetry of the electrons and incompressibility of the fluid, we obtain a constraint and derive a chiral Tomonaga-Luttinger liquid theory containing interaction terms. We calculate one-loop corrections to the phonon and electron propagators and get a new tunneling exponent. It agrees with experiments.
Over the years, many theoretical frameworks have been developed to understand the remarkable physics of the quantum Hall system. In this work we discuss the interplay among quantum wires, Chern-Simons theory, bosonization, and particle-vortex duality, which enable a unified approach for the Laughlin states of the fractional quantum Hall system. Starting from the so-called quantum wires system, which is a semimicroscopic description in terms of (1+1)(1+1)(1+1)-dimensional theories, we discuss the emergence of (2+1)(2+1)(2+1)-dimensional low-energy effective field theories by using different maps connecting the microscopic degrees of freedom with the macroscopic ones. We show the origin of these maps by embedding the bosonized version of the original quantum wires model in a gauge-invariant theory and using particle vortex duality.