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International Journal of Modern Physics B

Incompressible Quantum Fluids of Ideal Anyons in a Strong Magnetic Field

Fu‐Chun Zhang, Michael Ma

Published Jun 1, 1991• 0 citations
Publisher

Abstract

Ideal anyons with statistics ν in a strong magnetic field are studied by means of a similarity transformation. The ground state exhibits "integer" quantum Hall effect at filling factor 1/ν with quasiparticle excitations of charge q/ν and statistics -1/ν. Certain electron FQH states can be considered as realization of this, for example, the sequence 2/5, 3/7, … hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum Hall hierarchy.

DOI
10.1142/s0217979291001620

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Ideal anyons with statistics ν in a strong magnetic field are studied by means of a similarity transformation. The ground state exhibits "integer" quantum Hall effect at filling factor 1/ν with quasiparticle excitations of charge q/ν and statistics -1/ν. Certain electron FQH states can be considered as realization of this, for example, the sequence 2/5, 3/7, … hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum Hall hierarchy.
We study the problem of anyons with statistics \ensuremath{\nu} in a strong magnetic field by means of a similarity transformation. The ground state of this problem exhibits an ``integer'' quantum Hall effect at filling factor 1/\ensuremath{\nu} with quasiparticle excitations of charge q/\ensuremath{\nu} and statistics -1/\ensuremath{\nu}. Certain electron fractional quantum Hall states can be considered as a realization of this, for example, the 2/5, 3/7,. . . hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum hall hierarchy.
The quantum Hall effect has been the prototypical example for many emerging fields of Physics, including the study of topological phases of matter, topological order, and topological insulators. The objects of interest in this thesis are anyons, which are elementary excitations in fractional quantum Hall (FQH) fluids. Anyons are quasiparticles with fractional charge and fractional statistics. The latter prop- erty makes it a subject of immense interest due to application in fault-tolerant quantum computing. Here, we employ a perspective that anyons are the appro- priate degree of freedom to describe the relevant physics of FQH systems. This is because when a topological phase is assumed and the energy gap goes to infinity, anyons (“quasiholes”) are the only possible elementary excitations, since they are not energetically punished by the correspondng model Hamiltonian. With respect to a realistic Hamiltonian, however, this choice of degree of freedom depends on the exact effective interaction in the system. For example, while the Laughlin state is the model state at filling factor n + 1/3 (where n is some integer), when the effective electron-electron interaction is more long-ranged, the quasiholes of a Gaffnian state may be excited and become the better degree of freedom to describe physics in the same filling factor. This perspective results in a surprising observa- tion that a Laughlin quasihole may fractionalize under certain conditions, leading to a quasiparticle of charge e/6 that can potentially be detected in experiments. Another important aspect of anyons is their intrinsic spin and the corresponding spin-statistics relation, which must account for the fact that FQH anyons are not point particles, but objects with finite sizes and intricate internal structures. Our investigation leads to a rigorous derivation of the general spin-statistics relation for anyons in Abelian FQH phases. Furthermore, we show it is possible to fully capture the important features of the statistics by a conformal Hilbert space (CHS) description. Our analysis provides the necessary tools to study such geometric effects, which we use to investigate how disorder in a realistic system may affect different experimental schemes for measuring anyon statistics.
We show the low-lying excitations at filling factor ν=n+1/3 with realistic interactions can be understood as quantum fluids with "Gaffnian quasiholes" as the proper elementary degrees of freedom. Each Laughlin quasihole can thus be understood as a bound state of two Gaffnian quasiholes, which in the lowest Landau level (LLL) behaves like "partons" with "asymptotic freedom" mediated by neutral excitations acting as "gluons." Near the experimentally observed nematic FQH phase in higher LLs, quasiholes become weakly bound and can fractionalize with rich dynamical properties. By studying the effective interactions between quasiholes, we predict a finite temperature phase transition of the Laughlin quasiholes even when the Laughlin ground state remains incompressible, and derive relevant experimental conditions for its possible observations.
An anyon wavefunction (characterized by the statistical factor n ) projected onto the lowest Landau level is derived for the fractional quantum Hall effect states at filling factor ν = n /(2 pn +1) ( p and n are integers). We study the properties of the anyon wavefunction by using detailed Monte Carlo simulations in disc geometry and show that the anyon ground-state energy is a lower bound to the composite fermion one. Our results suggest that the composite fermions can be viewed as a combination of anyons and a fluid of charge–neutral dipoles.
We obtain integer and fractional quantum Hall crystals as ground states of a two-dimensional electron system subject to a strong perpendicular magnetic field and a periodic potential. For certain fractional states, we show that the Hall crystal can constitute an anyon crystal, with a periodic ordering of well-defined anyons. We find that the latter states can be stabilized at odd denominator Landau level filling fractions when Landau level mixing is sufficiently weak, and near half-filling of the underlying lattice. These phases are obtained from a mean-field analysis of an effective lattice model of bosons attached to an odd number of flux quanta, which transmutes their statistics to that of electrons. In boson coordinates, the Hall crystal is a supersolid: a superfluid with charge order. Under strong interactions, vortex-anti-vortex pairs spontaneously nucleate in the supersolid, realizing a crystalline state of anyons.
A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.