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arXiv:cond-mat/9310020 · Physical review. B, Condensed matter

Half-integral spin-singlet quantum Hall effect

Lotfi Belkhir, Xiaofeng Wu, Jainendra K. Jain

Published Nov 15, 1993• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.48.15245

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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.
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.
We present numerical evidence in support of the view that the physical origin of the fractional quantum Hall effect at \ensuremath{\nu}=n/(2n+1) is associated with the fact that this is the largest filling factor at which an n-component system of electrons can avoid having any pair of electrons occur in a state of relative angular momentum 1. Our calculations, for n=2, are based on a model in which the Hilbert space is truncated to the two lowest Landau levels and the Landau-level separation, \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is a parameter of the model. For \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0 and short-range repulsive interactions, a zero-energy incompressible ground state occurs at \ensuremath{\nu}=2/5. Our numerical results suggest that, in agreement with arguments advanced by J. K. Jain, this incompressibility survives to the strong-field limit \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=\ensuremath{\infty}.
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 analyze properties of trial wave functions in fractional quantum Hall effect, generated by the "short-range", three-body repulsion and its mean field approximation.Ground states of electron repulsion Hamiltonians at filing factor ν = 1/2 are evaluated as a description of physically observed state ν = 5/2.We analyze overlaps between the Moore-Read state and its mean field approximation for different number of particles and compare both states with ground state of the Coulomb interaction in the first excited Landau level.Our study also includes examination of electron-electron correlation functions and electron densities of excited states for both interactions.
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.