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arXiv:cond-mat/0310153 · Journal of Physics A Mathematical and General

On the quasiparticle description ofc= 1 CFTs

Davide Controzzi, Kareljan Schoutens

Published Mar 23, 2004• 0 citations
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Abstract

We show that the description of c = 1 conformal field theory in terms of quasiparticles satisfying fractional statistics can be obtained from the sine-Gordon model with a chemical potential A , in the limit where A ≫ M . These quasiparticles are related to the excitations of the Calogero–Sutherland (CS) model. We provide a direct calculation of their two-particle S -matrix using Korepin's method. We also reconsider the computation of the CS S -matrix in terms of particles with fractional charge.

DOI
10.1088/0305-4470/37/14/008

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We show that the description of c = 1 conformal field theory in terms of quasiparticles satisfying fractional statistics can be obtained from the sine-Gordon model with a chemical potential A , in the limit where A ≫ M . These quasiparticles are related to the excitations of the Calogero–Sutherland (CS) model. We provide a direct calculation of their two-particle S -matrix using Korepin's method. We also reconsider the computation of the CS S -matrix in terms of particles with fractional charge.
The deformed Calogero-Sutherland (CS) model is a quantum integrable system with arbitrary numbers of two types of particles and reducing to the standard CS model in special cases. We show that a known collective field description of the CS model, which is based on conformal field theory (CFT), is actually a collective field description of the deformed CS model. This provides a natural application of the deformed CS model in Wen’s effective field theory of the fractional quantum Hall effect (FQHE), with the two kinds of particles corresponding to electrons and quasi-hole excitations. In particular, we use known mathematical results about super-Jack polynomials to obtain simple explicit formulas for the orthonormal CFT basis proposed by van Elburg and Schoutens in the context of the FQHE.
INTRODUCTION The Calogero-Sutherland model has recently attracted some attention mainly because, in spite of its simplicity, it yields nontrivial results which contribute to shape our understanding of fractional statistics [1]. Its most remarkable property is that its ground state is given by a Jastrow wave-function which is the one dimensional analogue of the Laughlin wave-function. The excitations are described by quasiparticles which carry a fraction of the quantum numbers of the fundamental particles. The wave functions are simple enough to allow the computation of physical quantities such as the dynamical correlation functions. Recently, it has become clear that the spin version of this model is closely related to the known spin chains such as the XXX chain. The wave functions of the spin models can be obtained as simply as those of the Calogero-Sutherland model by diagonalising simple differential operators known as the Dunkl operators [2]. In these lectures, I restrict to the
At coupling strengths \ensuremath{\lambda}=1/2, 1, or 2, the Calogero-Sutherland model (CSM) is related to Brownian motion in a Wigner-Dyson random matrix ensemble with orthogonal, unitary, or symplectic symmetry. Using this relation in conjunction with superanalytic techniques developed in mesoscopic conductor physics, we derive an exact integral representation for the CSM two-particle Green's function in the thermodynamic limit. Simple closed expressions for the single-particle Green's functions are extracted by separation of points. For the advanced part, where a particle is added to the ground state and later removed, a sum of two contributions is found: the expected one, with just one particle excitation present, plus an extra term arising from fractionalization of the single particle into a number of elementary particle and hole excitations.
One-dimensional fractional statistics is studied using the Calogero-Sutherland model (CSM) which describes a system of non-relativistic quantum particles interacting with inverse-square two-body potential on a ring.The inversesquare exchange can be regarded as a pure statistical interaction and this system can be mapped to an ideal gas obeying the fractional exclusion and exchange statistics.The details of the exact calculations of the dynamical correlation functions for this ideal system is presented in this paper.An effective low-energy one-dimensional "anyon" model is constructed; and its correlation functions are found to be in agreement with those in the CSM; and this agreement provides an evidence for the equivalence of the first-and the second-quantized construction of the 1D anyon model at least in the long wave-length limit.Furthermore, the finite-size scaling applicable to the conformally invariant systems is used to obtain the complete set of correlation exponents for the CSM.
The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.