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arXiv:1603.06157 · Journal of Mathematical Physics

Deformed Calogero-Sutherland model and fractional quantum Hall effect

Farrokh Atai, Edwin Langmann A

Published Jan 1, 2017• 14 citations
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Abstract

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.

DOI
10.1063/1.4973509

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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.
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.
This thesis deals with a conformal field theoretical treatment of abelian anyons in (1 + 1)-dimensions and their relation to the integrable Calogero-Sutherland models. We generalize previous work relating anyons to the Calogero-Sutherland model by showing that the correlation function of the anyon field operators corresponds to the eigenfunctions of the deformed Calogero-Sutherland model. Our results suggest a physical application of the deformed Calogero-Sutherland model in the context of the fractional quantum Hall effect (FQHE). A key aspect for this work is the introduction of the dual anyon field operators, which obey a natural generalization of the canonical anti-commutation relation.
The 40-year-old Calogero-Sutherland (CS) model remains a source of inspirations for understanding 1d interacting fermions. At β=1,or0β=1, \text{or}0β=1,or0, the CS model describes a free non-relativistic fermion, or boson theory, while for generic βββ, the system can be interpreted either as interacting fermions or bosons, or free anyons depending on the context. However, we shall show in this letter that the fermionic picture is advantageous in diagonalizing the CS Hamiltonian. Comparing to the previously known multi-integral representation or the Dunkl operator formalism for the CS wave functions, our method depends on the (upper or lower) triangular nature of the fermion interaction, which is resolved in perturbation theory of the second quantized form. The eigenstate is constructed from a multiplet of unperturbed states and the perturbation is of finite order. The full construction is a similarity transformation from the free fermion theory, in the same spirit as the Landau Fermi liquid theory and the 1d Luttinger liquid theory. That means quasi-particles or anyons can be represented in terms of free fermion modes (or bosonic modes via bosonization). The method is applicable to other (higher than one space dimension) systems for which the adiabatic theorem applies.
We examine the application of c=1c=1c=1 conformal field theory to the description of the fractional quantum Hall effect (FQHE). It is found that the Gaussian model together with an appropriate boundary condition for the order parameter furnishes an effective theory for the Laughlin type FQHE. The plateau formation condition corresponds to taking the {\em chiral} portion of the theory.
We consider the known effective field theory of the Calogero-Sutherland model in the thermodynamic limit of large number of particles, obtained from the standard procedure in conformal field theory: the Hilbert space is constructed a priori in terms of irreducible representations of the symmetry algebra, and not by diagonalization of the hamiltonian, which is given in terms of fields that carry representations of the W 1 + ∞ algebra (representing the incompressibility of the Fermi sea). Nevertheless, the role of the effective hamiltonian of the theory is to establish a specific dynamics, which deserves further consideration. We show that the time evolution of the (chiral or antichiral) density field is given by the quantum Benjamin-Ono equation, in agreement with previous results obtained from the alternative description of the continuous limit of the model, based on quantum hydrodynamics. In this study, all calculations are performed at the quantum operator level, without making any assumption on the semiclassical limit of the fields and their equations of motion. This result may be considered as a reliable indication of the equivalence between the quantum field theoretic and quantum hydrodynamical formulations of the effective theories of the model. A one-dimensional quantum compressible fluid that includes both chiralities is the physical picture that emerges for the continuous limit of the Calogero-Sutherland model. • The chiral density field of the Calogero-Sutherland EFT satisfies a quantum Benjamin-Ono equation. • Agrees with quantum hydrodynamics results by Abanov and Wiegmann, extending their validity to quantum fields themselves. • Findings could be of interest to the QFT community, with new tools to deal with time non-locality within the realm of CFT. • Equivalence between the EFT and the hydrodynamic formulations of the thermodynamic limit of the Calogero-Sutherland model is supported.
We establish a one-to-one correspondence between the “composite particles” with N particles and the Young tableaux with at most N rows. We apply this correspondence to the models of Calogero–Sutherland and Ruijsenaars–Schneider and we obtain a momentum space representation of the “composite particles” in terms of creation operators attached to the Young tableaux. Using the technique of bosonization, we obtain a position space representation of the “composite particles” in terms of products of vertex operators. In the special case where the “composite particles” are bosons and if we add one extra quasiparticle or quasihole, we construct the ground state wave functions corresponding to the Jain series ν=p/(2np±1) of the fractional quantum Hall effect.