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arXiv:0712.1461 · Physical Review B

Trapping Abelian anyons in fractional quantum Hall droplets

Zi-Xiang Hu, Xin Wan, Peter Schmitteckert

Published Feb 28, 2008• 16 citations
arXivHTMLPDF

Abstract

We study the trapping of Abelian anyons (quasiholes and quasiparticles) by a local potential (e.g., induced by an atomic force microscopy tip) in a microscopic model of fractional quantum Hall liquids with long-range Coulomb interaction and edge confining potential. We find, in particular, that at Laughlin filling fraction $\ensuremath{\nu}=1∕3$, both quasihole and quasiparticle states can emerge as the ground state of the system in the presence of the trapping potential. As expected, we find that the presence of an Abelian quasihole has no effect on the edge spectrum of the quantum liquid, unlike in the non-Abelian case [X. Wan et al., Phys. Rev. Lett. 97, 256804 (2006)]. Although quasiholes and quasiparticles can emerge generically in the system, their stability depends on the strength of the confining potential and the strength and the range of the trapping potential. We discuss the relevance of the calculation to the high-accuracy generation and control of individual anyons in potential experiments, in particular, in the context of topological quantum computing.

DOI
10.1103/physrevb.77.075331

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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.

•Jan 1, 2024•0 citations
We study the trapping of Abelian anyons (quasiholes and quasiparticles) by a local potential (e.g., induced by an atomic force microscopy tip) in a microscopic model of fractional quantum Hall liquids with long-range Coulomb interaction and edge confining potential. We find, in particular, that at Laughlin filling fraction \ensuremathν=1∕3\ensuremath{\nu}=1∕3\ensuremathν=1∕3, both quasihole and quasiparticle states can emerge as the ground state of the system in the presence of the trapping potential. As expected, we find that the presence of an Abelian quasihole has no effect on the edge spectrum of the quantum liquid, unlike in the non-Abelian case [X. Wan et al., Phys. Rev. Lett. 97, 256804 (2006)]. Although quasiholes and quasiparticles can emerge generically in the system, their stability depends on the strength of the confining potential and the strength and the range of the trapping potential. We discuss the relevance of the calculation to the high-accuracy generation and control of individual anyons in potential experiments, in particular, in the context of topological quantum computing.
In a Laughlin fractional quantum Hall state, one- and two-quasihole states can be obtained by diagonalizing the many-body Hamiltonian with a trapping potential or, for larger systems, from the linear combination of the edge Jack polynomials. The quasihole states live entirely in the subspace of the lowest-energy branch in the energy spectrum with a fixed number of orbits, or a hard-wall confinement. The reduction in the Hilbert space dimension facilitates the study of time evolution of the quasihole states after, say, the removal of the trapping potential. We explore the quench dynamics under a harmonic external potential, which rotates the quasiholes in the droplet, and discuss the effect of long-range interaction and more realistic confinement. Accurate evaluation of the mutual statistics phase of anyons for a wide range of anyon separation can be achieved from the Berry-phase calculation.
We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $\ensuremath{\nu}=5∕2$, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle (with charge $\ifmmode\pm\else\textpm\fi{}e∕4$) when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around $4\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement (smooth edge), while the Moore-Read Pfaffian state is favored in strong confinement (sharp edge).
We study quench dynamics of bosonic fractional quantum Hall systems in small lattices with cylindrical boundary conditions and low particle density. The states studied have quasiholes or quasiparticles relative to the bosonic Laughlin state at half-filling. Pinning potentials are placed at edge sites (or sites close to the edges) to trap quasiholes and qausiparticles. The potentials are then turned off, and because the edges of fractional quantum Hall systems host chiral edge modes, we expect chiral dynamics of the quasiholes and quasiparticles. We numerically show that chiral motion of the density distribution is observed and robust for the case with positive potentials (quasiholes), but that there is no noticeable chiral motion for negative potentials (quasiparticles). The comparison of the numerical ground states with model lattice Laughlin wave functions suggests that both positive and negative potentials do create and pin anyons that are not necessarily well separated on small lattices. Initializing the dynamics with the model state also shows the lack of chiral dynamics of quasiparticles. Our results suggest that, in small lattices with low particle density, quasiparticles are strongly adversely affected in dynamical processes, whereas quasiholes are dynamically robust. Published by the American Physical Society 2024
We study the ground state and low-energy excitations of fractional quantum Hall systems on a disk at a filling fraction nu = 5/2, with Coulomb interaction and a background confining potential. We find the Moore-Read ground state is stable within a finite but narrow window in parameter space. The corresponding low-energy excitations contain a fermionic branch and a bosonic branch, with widely different velocities. A short-range repulsive potential can stabilize a charge +e/4 quasihole at the center, leading to a different edge excitation spectrum due to the change of boundary conditions for Majorana fermions, clearly indicating the non-Abelian nature of the quasihole.
We propose a method of realizing non-Abelian braiding of fractionalized quasiholes in the Laughlin fractional quantum Hall phase at ν=1/3ν=1/3ν=1/3 with realistic two-body interactions within the lowest Landau level. It is numerically shown that low-lying gapped excitations near ν=1/3ν=1/3ν=1/3 are contained almost entirely within the null space of the three-body Moore-Read model Hamiltonian. They are thus quantum fluids of non-Abelian quasiholes that are in principle physically accessible. In particular, Laughlin ground state can be described as a fluid of ``ψψψ-type" quasiholes formed by binding a magnetic flux with a Majorana fermion (MF), and the Laughlin quasiholes are described by the ``111-type'' quasiholes, which are magnetic fluxes without a MF attached. Within the Laughlin phase, Laughlin quasiholes can be locally fractionalized into non-Abelian quasiholes, when the strong attraction between them is overcome by properly designed one-body electronstatic trapping potentials. Extensive numerics with proper finite-size scaling corroborate this physical picture, and our study points to the possibility of realizing non-Abelian braiding within an Abelian topological phase in experiment without the need for fine-tuning realistic electron-electron interaction.
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.