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:cond-mat/9810311 · Journal of Physics A Mathematical and General

Renormalization of modular invariant Coulomb gas and sine-Gordon theories, and the quantum Hall flow diagram

David Carpentier

Published Jan 1, 1999• 8 citations
arXivHTMLPDF

Abstract

Using the renormalization group (RG) we study two-dimensional electromagnetic Coulomb gas and extended sine-Gordon theories invariant under the modular group SL (2, ). The flow diagram is established from the scaling equations, and we derive the critical behaviour at the various transition points of the diagram. Following the proposal for a SL (2, ) duality between different quantum Hall fluids, we discuss the analogy between this flow and the global quantum Hall phase diagram.

DOI
10.1088/0305-4470/32/21/301

Related papers

Global phase diagrams for charge transport in two dimensions

C.A. Lütken

It is suggested that it is not necessary to solve the full non-perturbative problem of two-dimensional charge transport in order to obtain global information about the phase and flow diagrams for the quantum Hall system and its relatives. It is argued that the effective quantum field theories encoding the macroscopic properties of these systems are invariant under a symmetry contained in the modular group. New phase and flow diagrams are obtained which exhibit a hierarchy of only odd or only even phases (Hall plateaus). The Kramers-Wannier-like symmetry determines the exact location of all renormalization group fixed points, which appear to be in agreement with available scaling experiments on the quantum Hall system. It also explains the observed 'super-universality', of the delocalization exponent.

Journal of Physics A Mathematical and General•Sep 7, 1993•24 citations

Renormalization of the Periodic Scalar Field Theory by Polchinski's Renormalization Group Method

I. Nándori, Kornél Sailer, Ulrich D. Jentschura +1 more

The renormalization group (RG) flow for the two-dimensional sine-Gordon model is determined by means of Polchinski's RG equation at next-to-leading order in the derivative expansion. In this work we have two different goals, (i) to consider the renormalization scheme-dependence of Polchinski's method by matching Polchinski's equation with the Wegner-Houghton equation and with the real space RG equations for the two-dimensional dilute Coulomb-gas, (ii) to go beyond the local potential approximation in the gradient expansion in order to clarify the supposed role of the field-dependent wave-function renormalization. The well-known Coleman fixed point of the sine-Gordon model is recovered after linearization, whereas the flow exhibits strong dependence on the choice of the renormalization scheme when non-linear terms are kept. The RG flow is compared to those obtained in the Wegner-Houghton approach and in the dilute gas approximation for the two-dimensional Coulomb-gas.

•Jan 1, 2002•20 citations

Wavefunction renormalization for the Coulomb gas by the Wegner-Houghton renormalization group method

I. Nándori, János Polonyi, Kornél Sailer

The renormalization group (RG) flow for the sine-Gordon model is determined by means of the method of Wegner and Houghton in the next to leading order of the derivative expansion. For smali values of the fugacity this agrees with the well-known RG flow of the two-dimensional Coulomb gas found in the dilute-gas approximation and a systematic way of obtaining higher-order corrections to this approximation is given.

arXiv:hep-th/0012208•Philosophical Magazine B•Oct 1, 2001•26 citations

Gamma(2) modular symmetry, renormalization group flow and the quantum hall effect

Yvon Georgelin, Thierry Masson, Jean-Christophe Wallet

We construct a family of holomorphic -functions whose renormalization group (RG) flow preserves the (2) modular symmetry and reproduces the observed stability of the Hall plateaus. The semicircle law relating the longitudinal and Hall conductivities that has been experimentally observed is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependence of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occurring in the crossover of the permitted transitions is discussed.

arXiv:cond-mat/9906193•Journal of Physics A Mathematical and General•Dec 15, 1999•16 citations

Phase Transition in 2-D Sine-Gordon System

T. Ohta

The renormalization group equations for the sine-Gordon system with a short wave-length cutoff in two dimensions are derived by means of a momentum-shell recursion method. In a special limit they agree with the Kosterlitz renormalization group equations which describe the critical properties of the phase transitions of the two-dimensional (2-D) neutral Coulomb gas with single charges ±1 and of the 2-D isotropic X-Y model in the absence of an external field. The results, therefore, give rise to a possibility that certain classes of phase transitions in low dimensions can be described in a unified way with the sine-Gordon Hamiltonian.

Progress of Theoretical Physics•Oct 1, 1978•38 citations

Comparison of renormalization group schemes for sine-Gordon-type models

I. Nándori, S. Nagy, Kornél Sailer +1 more

The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.

arXiv:0903.5524•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 17, 2009•30 citations
Using the renormalization group (RG) we study two-dimensional electromagnetic Coulomb gas and extended sine-Gordon theories invariant under the modular group SL (2, ). The flow diagram is established from the scaling equations, and we derive the critical behaviour at the various transition points of the diagram. Following the proposal for a SL (2, ) duality between different quantum Hall fluids, we discuss the analogy between this flow and the global quantum Hall phase diagram.
It is suggested that it is not necessary to solve the full non-perturbative problem of two-dimensional charge transport in order to obtain global information about the phase and flow diagrams for the quantum Hall system and its relatives. It is argued that the effective quantum field theories encoding the macroscopic properties of these systems are invariant under a symmetry contained in the modular group. New phase and flow diagrams are obtained which exhibit a hierarchy of only odd or only even phases (Hall plateaus). The Kramers-Wannier-like symmetry determines the exact location of all renormalization group fixed points, which appear to be in agreement with available scaling experiments on the quantum Hall system. It also explains the observed 'super-universality', of the delocalization exponent.
The renormalization group (RG) flow for the two-dimensional sine-Gordon model is determined by means of Polchinski's RG equation at next-to-leading order in the derivative expansion. In this work we have two different goals, (i) to consider the renormalization scheme-dependence of Polchinski's method by matching Polchinski's equation with the Wegner-Houghton equation and with the real space RG equations for the two-dimensional dilute Coulomb-gas, (ii) to go beyond the local potential approximation in the gradient expansion in order to clarify the supposed role of the field-dependent wave-function renormalization. The well-known Coleman fixed point of the sine-Gordon model is recovered after linearization, whereas the flow exhibits strong dependence on the choice of the renormalization scheme when non-linear terms are kept. The RG flow is compared to those obtained in the Wegner-Houghton approach and in the dilute gas approximation for the two-dimensional Coulomb-gas.
We construct a family of holomorphic -functions whose renormalization group (RG) flow preserves the (2) modular symmetry and reproduces the observed stability of the Hall plateaus. The semicircle law relating the longitudinal and Hall conductivities that has been experimentally observed is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependence of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occurring in the crossover of the permitted transitions is discussed.
The renormalization group equations for the sine-Gordon system with a short wave-length cutoff in two dimensions are derived by means of a momentum-shell recursion method. In a special limit they agree with the Kosterlitz renormalization group equations which describe the critical properties of the phase transitions of the two-dimensional (2-D) neutral Coulomb gas with single charges ±1 and of the 2-D isotropic X-Y model in the absence of an external field. The results, therefore, give rise to a possibility that certain classes of phase transitions in low dimensions can be described in a unified way with the sine-Gordon Hamiltonian.
The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.