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arXiv:cond-mat/9509039 · Physical review. B, Condensed matter

Laughlin wave function and one-dimensional free fermions

Prasanta Kumar Panigrahi, M. Sivakumar

Published Nov 15, 1995• 13 citations
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Abstract

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.

DOI
10.1103/physrevb.52.13742

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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.
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.
It has long been known that Laughlin's wave functions, describing the fractional quantum Hall effect at filling fractions ν=1/(2k+1), can be obtained as correlation functions in conformal field theory. We show how to generalize this approach to construct explicit trial wave functions for all states in the quantum Hall hierarchy corresponding to quasiparticle (as opposed to quasihole) condensates, including the recently observed state at ν= 4/11. At the filling fractions ν= n/(2np+1) this construction exactly reproduces Jain's composite fermion wave functions. An explicit connection is made to Wen's topological classification of FQH states.
A two-dimensional system of electrons with repulsive two-particle interaction is considered in a homogeneous magnetic field. Assuming the limit of high magnetic field the motion of the electrons is restricted to the lowest Landau level. We present quantum-mechanical wave functions for the many-particle system of N spin-polarized fermions on a parallelogram with periodic boundary conditions and with ${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$ quanta of magnetic flux flowing through its area. These wave functions are generalizations of the Laughlin wave function to more general rational values of the occupation fraction \ensuremath{\nu}=N/${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$. We divide the N electrons into n groups and assume Laughlin-Jastrow-type correlations for electrons within the same group, while electrons in different groups are correlated with a second Jastrow function. The corresponding wave functions exhibit a special structure that depends on \ensuremath{\nu} but is not affected by the interaction between the particles. Furthermore, we consider excited states with quasielectrons and quasiholes. Our theory leads to a unified view of the composite fermion theory of Jain and the hierarchy theory of Haldane, Laughlin, and Halperin. We present a qualitative argument why the fractions of Jain are observed preferably in the experiments.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
We consider the lowest Landau level on a torus as a function of its circumference L 1 . When L 1 → 0, the ground state at general rational filling fraction is a crystal with a gap—a Tao–Thouless state. For filling fractions ν = p /(2 pm + 1), these states are the limits of Laughlin's or Jain's wavefunctions describing the gapped quantum Hall states when L 1 → ∞ . For the half-filled Landau level, there is a transition to a Fermi sea of non-interacting neutral fermions (dipoles), or rather to a Luttinger liquid modification thereof, at L 1 ∼ 5 magnetic lengths. Using exact diagonalization we identify this state as a version of the Rezayi–Read state, and find that it develops continuously into the state that is believed to describe the observed metallic phase as L 1 → ∞ . Furthermore, the effective Landau level structure that emerges within the lowest Landau level is found to be a consequence of the magnetic symmetries.
We study the behavior of an interacting two-dimensional charged particle system in a transverse magnetic field and give a simple picture of the features that lead to the fractional quantum Hall effect. In particular, we discuss how the interparticle interactions give rise to an energy-level substructure in the lowest Landau level and develop a relatively simple model that gives insight into the analytical form of the Laughlin wave function. We analyze the model by transforming to a reference frame that rotates at half the cyclotron angular frequency. In the rotating frame, the vector potential for the magnetic field is replaced by a quadratic scalar potential. It is then evident that the system of interacting particles is a bound state, even in the absence of a charge-neutralizing background. We solve exactly a many-particle system in the rotating frame using a harmonic model interaction. In addition to being a good example for demonstrating basic methods in quantum mechanics, we show explicitly that the Laughlin function is an eigenstate of this model system.