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Modern Physics Letters B

THEORY OF NO-FIELD QUANTUM HALL EFFECT BASED ON PHASE-CHARGE BOSON WAVE FUNCTION

Masanori Sugahara, Nikolai N. Bogolubov

Published Dec 20, 2002• 1 citation
Publisher

Abstract

The derivation of the non-magnetic Laughlin state and other macroscopic quantum states in the semi-localized 2D electron system in the network of circular molecular orbits is made by the study of zero-point plasma oscillation. In the imaginary time representation, the electric field is transformed to the vector potential. After the cancellation of the mean-field component of the inter-electron repulsive field with the ion-lattice field, the boson Hamiltonian with respect to the phase-charge fluctuation is obtained using a Chern–Simons gauge field. Based on the resultant boson wave function, the macroscopic quantum state in hole doping is found to lead to a superfluidity that is described by a coherent function when λΘ > λQ, and to the particle-number-definite state described by a Laughlin function when λQ > λΘ, where λΘ is the phase-coherence length and λQ is the incompressibility length.

DOI
10.1142/s0217984902004615

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The derivation of the non-magnetic Laughlin state and other macroscopic quantum states in the semi-localized 2D electron system in the network of circular molecular orbits is made by the study of zero-point plasma oscillation. In the imaginary time representation, the electric field is transformed to the vector potential. After the cancellation of the mean-field component of the inter-electron repulsive field with the ion-lattice field, the boson Hamiltonian with respect to the phase-charge fluctuation is obtained using a Chern–Simons gauge field. Based on the resultant boson wave function, the macroscopic quantum state in hole doping is found to lead to a superfluidity that is described by a coherent function when λΘ > λQ, and to the particle-number-definite state described by a Laughlin function when λQ > λΘ, where λΘ is the phase-coherence length and λQ is the incompressibility length.
Recently, the authors discussed the possibility of the macroscopic quantum state similar to the Quantum Hall Effect in a semi-localized 2D electron system with a toroidal electron-wave amplitude in the absence of any magnetic field. In order to give the concrete statistical foundation of the study, the fermion-boson statistical transformation of the 2D electron system is made using a Chern–Simons gauge potential. Based on the solution of the resultant boson-type Hamiltonian, we construct the fermion-type solution via a unitary transformation. It is shown that the solution in the form of Laughlin function is stable when electrons form pairs. In the presence of hole doping, the pair Laughlin function leads to a representation of a superconducting state when the phase-coherence length λΘ exceeds the incompressibility length λQ, but when λΘ< λQ, it leads to a macroscopic quantum state characterized by particle-number definiteness.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2){U}_{q}{(su}_{2})Uq​(su2​) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
The Hall experiment can be explained by classical transport theory for lower magnetic fields (B ≲ 1 T). In Chapter 1 of this thesis, we review the Drude model to obtain cyclotron motion and to derive the conductivity tensor σ as a function of the magnetic field B. The quantum version of the classical Hall effect initiated a remarkable era in the search for new topological phases of two-dimensional materials and strongly correlated phases, treated via Landau quantization, which generates highly degenerate energy levels known as Landau levels (LLs). Strong, uniform magnetic fields lead to the quantization of the Hall resistivity ρ_H = h/(e^2*ν), where ν is the filling fraction of electrons, taking integer or fractional values depending on the system. This quantization reflects the presence of fully and partially filled LLs. Chapter 2 addresses the role of uniform magnetic and electric fields in a 2D system by choosing an appropriate vector potential A gauge. Also, we investigate the quantum mechanical current operator that allows us to quantize the conductivity tensor element σ_xy = e^2*ν/h. Subsequent sections present the derivations of the wave functions in both Landau and symmetric gauges and the corresponding eigenvalues. In the final sections of Chapter~2, we discuss the percolation concept, which explains Hall plateaux, and examine the role of edge states and random impurities in Hall samples by introducing confinement and random electrostatic potentials, which lead to backscattering effects at the boundaries and localized states in the bulk. Many-body physics plays a crucial role in the partially filled LL, highlighting the strong interactions between electrons in the context of the Fractional Quantum Hall Effect (FQHE). Chapter 3 begins with an exact solution for the two-body problem at filling fraction v = 1. Laughlin’s ansatz offers an elegant solution for the many-body problem at filling fraction v = 1/m, where m is even for bosons and odd for fermions. We review Laughlin’s incompressible wave function, which is an essentially exact solution for short-range interactions, and discuss the plasma analogy, which describes the Laughlin state as a quantum droplet with uniform density. The last section of Chapter 3 addresses the Haldane pseudopotential, which characterizes strong interactions in the FQHE regime. Rotating and trapped bosonic condensates exhibit unique properties for strongly correlated states in the FQHE regime, where the ground state lies in the lowest LL. The effective magnetic field generated by the Coriolis force mimics that of Quantum Hall (QH) systems, enabling the emergence of incompressible states for neutral bosons. In Chapter~4, we also study a three-dimensional rotating BEC in a harmonic trap, highlighting the connection between the FQHE and rapidly rotating BECs. In the final chapter, we investigate an equal-number binary bosonic gas with a fixed total angular momentum L in the z-direction. The tunable coupling ratio g_12/g allows us to identify a critical threshold beyond which the gas becomes unstable in the strongly attractive inter-species interaction regime in disc geometry. The collapse mode refers to the instability in which the binary gas undergoes spatial contraction. Overlap calculations for the ground state show that the Laughlin square wave function, which describes two species via separate Laughlin states, vanishes near the critical coupling ratio g_12/g ∼ -1, as expected. We introduce a molecular Laughlin ansatz that describes center-of-mass pairing between inter-species particles. Our molecular ansatz yields higher overlaps with both the ground state and low-lying excited states compared to the Laughlin square wave function. We also confirm that the ground state of the gas in this regime corresponds to a collapse mode, where the system rotates around its common center of mass. The dependence of the energy spectrum on g_12/g and the corresponding real-space density profiles are also presented.
A mean field theory for the quantum Hall liquid based on a bi-local collective field is presented. A self-consistent mean field and fluctuations due to low spin fields around the mean field are investigated. We find that the mean field energy is negative and its correction due to the fluctuations of the density field is finite and negative and the correction due to the vector fields cancells completely in the lowest order. The finite Hall conductance is computed and the massive Chern-Simons gauge theory is obtained as an effective long wave length theory.
Dedicated to Herbert Spohn on the occasion of his retirement from the TU München A bosonic analogue of the fractional quantum Hall effect occurs in rapidly rotating trapped Bose gases: There is a transition from uncorrelated Hartree states to strongly correlated states such as the Laughlin wave function. This physics may be described by effective Hamiltonians with delta interactions acting on a bosonic N-body Bargmann space of analytic functions. In a previous paper [RSY] we studied the case of a quadratic plus quartic trapping potential and derived conditions on the parameters of the model for its ground state to be asymptotically strongly correlated. This relied essentially on energy upper bounds using quantum Hall trial states, incorporating the correlations of the Bose-Laughlin state in addition to a multiply quantized vortex pinned at the origin. In this paper we investigate in more details the density of these trial states, thereby substantiating further the physical picture described in [RSY], improvingourenergyestimates andallowing toconsider moregeneral trappingpotentials. Our analysis is based on the interpretation of the densities of quantum Hall trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy). New estimates on the mean-field