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Explicit Relation of Quantum Hall Effect and Calogero-Sutherland Model

Azuma, H, Iso, S

Published Jan 1, 1993• 92 citations
OpenAlex

Abstract

Explicit relation between Laughlin state of the quantum Hall effect and one-dimensional(1D) model with long-ranged interaction ($1/r^2$) is discussed. By rewriting lowest Landau level wave functions in terms of 1D representation, Laughlin state can be written as a deformation of the ground state of Calogero-Sutherland model. Corresponding to Laughlin state on different geometries, different types of 1D $1/r^2$ interaction models are derived.

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Explicit relation between Laughlin state of the quantum Hall effect and one-dimensional(1D) model with long-ranged interaction (1/r21/r^21/r2) is discussed. By rewriting lowest Landau level wave functions in terms of 1D representation, Laughlin state can be written as a deformation of the ground state of Calogero-Sutherland model. Corresponding to Laughlin state on different geometries, different types of 1D 1/r21/r^21/r2 interaction models are derived.
We study the universal long-distance behaviour of the Laughlin state for the fractional quantum Hall effect and the ground state of the Calogero-Sutherland model (one dimensional 1/r^2 interaction model). In particular, it is shown that these two wave functions coincide exactly when Laughlin state is confined in a narrow cylinder geometry. The seeming difference of dimensionality is merely a difference of representation of wave functions. We also give a recipe to interpret operators acting on states in the lowest Landau level in terms of the usual one dimensional Fermion operators, which is important for extracting the Tomonaga-Luttinger liquid behaviour of the edge states.
We study the universal long distance behavior of the Laughlin state for the fractional quantum Hall effect and the ground state of the Calogero-Sutherland model (one-dimensional 1/r2 interaction model). In particular, it is shown that these two wave functions coincide exactly when Laughlin state is confined in a narrow cylinder geometry. The seeming difference of dimensionality is merely a difference of representation of wave functions. We also give a recipe to interpret operators acting on states in the lowest Landau level in terms of the usual one-dimensional fermion operators, which is important for extracting the Tomonaga-Luttinger liquid behavior of the edge states.
A simple one-dimensional model is proposed, in which N spinless interacting fermions occupy M>NM>NM>N degenerate states on a circle. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors νN/M=1/(2m+1)\nu N/M=1/(2m+1)νN/M=1/(2m+1). Within this model the complimentary wavefunction for ν=l−I/(2m+1)\nu =l-I/(2m+1)ν=l−I/(2m+1) is found explicitly.
Making use of the well-known phase-space reduction in the lowest Landau level, we show that the Laughlin wave function for the \ensuremath{\nu}=1/m case can be obtained exactly as a coherent-state representation of a one-dimensional (1D) wave function. The 1D system consists of m copies of free fermions associated with each of the N electrons, confined in a common harmonic well potential. Interestingly, the condition for this exact correspondence is found to incorporate Jain's parton picture. We argue that this correspondence between the free fermions and quantum Hall effect is due to the mapping of the 1D system under consideration to the Gaussian unitary ensemble in the random matrix theory.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.