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Physical review. B, Condensed matter

Connection between spin-singlet and hierarchical wave functions in the fractional quantum Hall effect

D. Yoshioka, Allan H. MacDonald, S. M. Girvin

Published Aug 15, 1988• 68 citations
Publisher

Abstract

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 $\frac{1}{m}$. For $\ensuremath{\nu}=\frac{q}{p}$, the wave functions are related to the wave functions for the total $S=0$ 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 $S=0$, $\ensuremath{\nu}=\frac{1}{2}$ hollow-core-model ground state recently discovered by Haldane and Rezayi.

DOI
10.1103/physrevb.38.3636

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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 \ensuremathν\ensuremath{\nu}\ensuremathν not equal to 1m\frac{1}{m}m1​. For \ensuremathν=qp\ensuremath{\nu}=\frac{q}{p}\ensuremathν=pq​, the wave functions are related to the wave functions for the total S=0S=0S=0 states of objects with spin (q\ensuremath−1)2\frac{(q\ensuremath{-}1)}{2}2(q\ensuremath−1)​. In particular, for \ensuremathν=25\ensuremath{\nu}=\frac{2}{5}\ensuremathν=52​ the wave function is related to the total S=0S=0S=0, \ensuremathν=12\ensuremath{\nu}=\frac{1}{2}\ensuremathν=21​ hollow-core-model ground state recently discovered by Haldane and Rezayi.
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 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.
In this paper we study the phase diagram of the ground-state quantum numbers as functions of Haldane pseudopotential parameters V0{\mathit{V}}_{0}V0​ and V1{\mathit{V}}_{1}V1​, keeping all other Vm{\mathit{V}}_{\mathit{m}}Vm​'s at their Coulomb values. We study filling factors 1/2 and 5/2, using a system of eight electrons on a sphere. It is known that the Coulomb ground state is not a spin singlet. In the phase diagram, however, we find that there is a comparatively large region of L=0, S=0 states not far from the Coulomb point. Modifications necessary for the ground state to be in this sector are discussed. Some of the L=0, S=0 states in the phase diagram are incompressible because they have significant overlaps with the hollow core state or with the composite fermion states, which are known to be imcompressible.
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.
Laughlin's wave functions, which describe the fractional quantum Hall effect at filling factors $\ensuremath{\nu}=1/(2k+1)$, can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain's composite fermion wave functions at filling factors $\ensuremath{\nu}=n/(2kn+1)$. Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen's general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.