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Physical Review Letters

Theory of spin-singlet fractional quantum Hall effect at ν=1/2

Lotfi Belkhir, Jainendra K. Jain

Published Feb 1, 1993• 22 citations
Publisher

Abstract

Motivated by the observation of fractional quantum Hall effect (FQHE) at filling factor \ensuremath{\nu}=2+1/2 by Willett et al., we investigate the nature of the spin-singlet FQHE at \ensuremath{\nu}=1/2. We show that a simple trial wave function for the \ensuremath{\nu}=1/2 state, proposed earlier by one of us, is the exact nondegenerate ground state of a hard-core interaction model, and provide numerical evidence that it is adiabatically connected to the true Coulomb ground state.

DOI
10.1103/physrevlett.70.643

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Motivated by the observation of fractional quantum Hall effect (FQHE) at filling factor \ensuremath{\nu}=2+1/2 by Willett et al., we investigate the nature of the spin-singlet FQHE at \ensuremath{\nu}=1/2. We show that a simple trial wave function for the \ensuremath{\nu}=1/2 state, proposed earlier by one of us, is the exact nondegenerate ground state of a hard-core interaction model, and provide numerical evidence that it is adiabatically connected to the true Coulomb ground state.
We provide numerical evidence that the ground state of a short-range-interaction model at \ensuremath{\nu}=1/2 is incompressible and spin singlet for a wide range of repulsive interactions. Furthermore it is accurately described by a trial wave function studied earlier. For the Coulomb interaction we find that this wave function provides a good description of the lowest-lying spin-singlet state, and propose that fractional quantum Hall effect would occur at \ensuremath{\nu}=1/2 if this state became the global ground state. Some conclusions of an earlier paper are invalidated by the present numerical study of a larger system; these will be indicated.
Three different families of many-body wave functions which are singlet states of the total spin operator are presented. In particular, we propose a new strongly correlated singlet for the trial ground state at the filling factor \ensuremath{\nu}=2/3. Our wave function weakly mixes the states from higher Landau levels in order to minimize the potential energy, and accounts for recent experimental observations of the transition between different spin structures of the ground states at this filling factor.
We propose trial expressions for the many-body wave functions of the incompressible fluid states responsible for the fractional quantum Hall effect at filling factor $\ensuremath{\nu}$ not equal to 1m\frac{1}{m}m1​. For $\ensuremath{\nu}=\frac{q}{p}$, the wave functions are related to the wave functions for the total S=0S=0S=0 states of objects with spin $\frac{(q\ensuremath{-}1)}{2}$. In particular, for $\ensuremath{\nu}=\frac{2}{5}$ the wave function is related to the total S=0S=0S=0, $\ensuremath{\nu}=\frac{1}{2}$ hollow-core-model ground state recently discovered by Haldane and Rezayi.
Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$, 4/54/54/5, and 4/34/34/3, and partially spin-polarized states at filling factors 3/43/43/4 and 3/23/23/2, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$, 4/54/54/5, and 3/43/43/4, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.