scholarly journals $\bm{\widehat{{\rm U}(1)}\times\widehat{{\rm SU}({\lc m})}_{1}}$ THEORY AND c=m W1+∞ MINIMAL MODELS IN THE HIERARCHICAL QUANTUM HALL EFFECT

2000 ◽  
Vol 15 (06) ◽  
pp. 915-926 ◽  
Author(s):  
MARINA HUERTA
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
Degrees Of Freedom ◽  
Central Charge ◽  
Quantum Hall ◽  
Detailed Comparison ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field ◽  
Group Flow

Two classes of Conformal Field Theories have been proposed to describe the Hierarchical Quantum Hall Effect: the multicomponent bosonic theory, characterized by the symmetry [Formula: see text] and the W1+∞ minimal models with central charge c=m. In spite of having the same spectrum of edge excitations, they manifest differences in the degeneracy of the states and in the quantum statistics, which call for a more detailed comparison between them. Here, we describe their detailed relation for the general case, c=m and extend the methods previously published for c≤3. Specifically, we obtain the reduction in the number of degrees of freedom from the multicomponent Abelian theory to the minimal models by decomposing the characters of the [Formula: see text] representations into those of the c=mW1+∞ minimal models. Furthermore, we find the Hamiltonian whose renormalization group flow interpolates between the two models, having the W1+∞ minimal models as an infrared fixed point.

scholarly journals W1+∞ Field Theories for the Edge Extensions in the Quantum Hall Effect

1997 ◽  
Vol 12 (06) ◽  
pp. 1101-1111 ◽  
Author(s):  
Andrea Cappelli ◽  
Carlo A. Trugenberger ◽  
Guillermo R. Zemba
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
A Priori ◽  
Quantum Hall ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Universality Classes ◽  
Effective Theories

We briefly review these low-energy effective theories for the quantum Hall effect, with emphasis and language familiar to field theorists. Two models have been proposed for describing the most stable Hall plateaus (the Jain series): the multi-component Abelian theories and the minimal W1+∞ models. They both lead to a-priori classifications of quantum Hall universality classes. Some experiments already confirmed the basic predictions common to both effective theories, while other experiments will soon pin down their detailed properties and differences. Based on the study of partition functions, we show that the Abelian theories are rational conformal field theories while the minimal W1+∞ models are not.

THE ANNIHILATING IDEALS OF MINIMAL MODELS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 217-238 ◽  
Author(s):  
BORIS L. FEIGIN ◽  
TOMOKI NAKANISHI ◽  
HIROSI OOGURI
Keyword(s):  
Central Charge ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Finite Models ◽  
Conformal Field ◽  
Intimate Relation ◽  
Chiral Algebras

We describe several aspects of the annihilating ideals and reduced chiral algebras of conformal field theories, especially, minimal models of Wn algebras. The structure of the annihilating ideal and a vanishing condition is given. Using the annihilating ideal, the structure of quasi-finite models of the Virasoro (2,q) minimal models are studied, and their intimate relation to the Gordon identities are discussed. We also show the examples in which the reduced algebras of Wn and Wℓ algebras at the same central charge are isomorphic to each other.

scholarly journals On 2d CFTs that interpolate between minimal models

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Exactly Solvable ◽  
Structure Constants

We investigate exactly solvable two-dimensional conformal field theories that exist at generic values of the central charge, and that interpolate between A-series or D-series minimal models. When the central charge becomes rational, correlation functions of these CFTs may tend to correlation functions of minimal models, or diverge, or have finite limits which can be logarithmic. These results are based on analytic relations between four-point structure constants and residues of conformal blocks.

scholarly journals TOWARD A CONFORMAL FIELD THEORY FOR THE QUANTUM HALL EFFECT

1992 ◽  
Vol 06 (19) ◽  
pp. 3235-3247
Author(s):  
GREG NAGAO
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Hall Effect ◽  
Quantum Hall Effect ◽  
Order Parameter ◽  
Quantum Hall ◽  
Cyclotron Motion ◽  
Conformal Field ◽  
Large N ◽  
Collective Coordinate

An effective Hamiltonian for the study of the quantum Hall effect is proposed. This Hamiltonian, which includes a "current-current" interaction has the form of a Hamiltonian for a conformal field theory in the large N limit. An order parameter is constructed from which the Hamiltonian may be derived. This order parameter may be viewed as either a collective coordinate for a system of N charged particles in a strong magnetic field; or as a field of spins associated with the cyclotron motion of these particles.

CONNECTIONS BETWEEN THE QUANTUM HALL EFFECT AND CONFORMAL FIELD THEORY

1992 ◽  
Vol 07 (30) ◽  
pp. 2837-2849
Author(s):  
GREG NAGAO ◽  
QIAN NIU ◽  
JOSÉ GAITE
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Hall Effect ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Effective Field ◽  
Vertex Operators ◽  
Effective Hamiltonian ◽  
Cyclotron Motion ◽  
Conformal Field

The quantum Hall effect (QHE) is studied in the context of a conformal field theory (CFT). Winding state vertex operators for an effective field of N "spins" associated with the cyclotron motion of particles are defined. The effective field of spins may be used to define an effective Hamiltonian. This effective Hamiltonian describes the collective motion of the N particles (with coupling κ0) together with a current-current interaction (of strength κ1). Such a system gives rise to a CFT in the large-N limit when κ0=κ1. The Laughlin wave function is derived from this CFT as an N'-point correlation function of winding state vertex operators.

THE QUANTUM HALL EFFECT AT ARBITRARY RATIONAL FILLING: A PROPOSAL

1992 ◽  
Vol 07 (28) ◽  
pp. 2583-2591 ◽  
Author(s):  
G. CRISTOFANO ◽  
G. MAIELLA ◽  
R. MUSTO ◽  
F. NICODEMI
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Hall Effect ◽  
Quantum Hall Effect ◽  
Hierarchical Model ◽  
Quantum Hall ◽  
Coulomb Gas ◽  
Vertex Operators ◽  
Two Dimensional ◽  
Conformal Field

A description of the quantum Hall effect, already proposed for the fractional filling ν=1/m, based on the introduction of Coulomb gas-like vertex operators typical of a two-dimensional conformal field theory, is extended to the case ν=p/m. The resulting physical picture is compared with the hierarchical model.

scholarly journals FUSION AND TENSORING OF CONFORMAL FIELD THEORY AND COMPOSITE FERMION PICTURE OF FRACTIONAL QUANTUM HALL EFFECT

1996 ◽  
Vol 11 (01) ◽  
pp. 55-68 ◽  
Author(s):  
MICHAEL A.I. FLOHR
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Hall Effect ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Tensor Products ◽  
Composite Fermions ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
Lowest Landau Level

We propose a new way for describing the transition between two quantum Hall effect states with different filling factors within the framework of rational conformal field theory. Using a particular class of nonunitary theories, we explicitly recover Jain’s picture of attaching flux quanta by the fusion rules of primary fields. Filling higher Landau levels of composite fermions can be described by taking tensor products of conformal theories. The usual projection to the lowest Landau level corresponds then to a simple coset of these tensor products with several U(1)-theories divided out. These two operations — the fusion map and the tensor map — can explain the Jain series and all other observed fractions as exceptional cases. Within our scheme of transitions we naturally find a field with the experimentally observed universal critical exponent 7/3.

scholarly journals N = 2 SUPERSYMMETRIC YANG–MILLS AND THE QUANTUM HALL EFFECT

2006 ◽  
Vol 21 (23n24) ◽  
pp. 4807-4821 ◽  
Author(s):  
BRIAN P. DOLAN
Keyword(s):  
Phase Diagram ◽  
Hall Effect ◽  
Strong Coupling ◽  
Quantum Hall Effect ◽  
Modular Group ◽  
Degrees Of Freedom ◽  
Quantum Hall ◽  
Four Dimensions ◽  
Yang Mills

It is argued that there are strong similarities between the infrared physics of N = 2 supersymmetric Yang–Mills and that of the quantum Hall effect, both systems exhibit a hierarchy of vacua with a subgroup of the modular group mapping between them. The coupling flow for pure SU(2) N = 2 supersymmetric Yang–Mills in four dimensions is reexamined and an earlier suggestion in the literature, that was singular at strong coupling, is modified to a form that is well behaved at both weak and strong coupling and describes the crossover in an analytic fashion. Similarities between the phase diagram and the flow of SUSY Yang–Mills and that of the quantum Hall effect are then described, with the Hall conductivity in the latter playing the role of the θ-parameter in the former. Hall plateaux, with odd denominator filling fractions, are analogous to fixed points at strong coupling in N = 2 SUSY Yang–Mills, where the massless degrees of freedom carry an odd monopole charge.

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