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arXiv:1810.12288 · Physical review. B./Physical review. B

Quasielectrons in lattice Moore-Read models

Sourav Manna, Julia Wildeboer, Anne E. B. Nielsen

Published Jan 28, 2019• 7 citations
arXivHTMLPDF

Abstract

Moore-Read states can be expressed as conformal blocks of the underlying rational conformal field theory, which provides a well explored description for the insertion of quasiholes. It is known, however, that quasielectrons are more difficult to describe in continuous systems, since the natural guess for how to construct them leads to a singularity. In this work, we show that the singularity problem does not arise for lattice Moore-Read states. This allows us to construct Moore-Read Pfaffian states on lattices for filling fraction $5/2$ with both quasiholes and quasielectrons in a simple way. We investigate the density profile, charge, size, and braiding properties of the anyons by means of Monte Carlo simulations. Further we derive an exact few-body parent Hamiltonian for the states. Finally, we compare our results to the density profile, charge, and shape of anyons in the Kapit-Mueller model by means of exact diagonalization.

DOI
10.1103/physrevb.99.045147

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Moore-Read states can be expressed as conformal blocks of the underlying rational conformal field theory, which provides a well explored description for the insertion of quasiholes. It is known, however, that quasielectrons are more difficult to describe in continuous systems, since the natural guess for how to construct them leads to a singularity. In this work, we show that the singularity problem does not arise for lattice Moore-Read states. This allows us to construct Moore-Read Pfaffian states on lattices for filling fraction 5/25/25/2 with both quasiholes and quasielectrons in a simple way. We investigate the density profile, charge, size, and braiding properties of the anyons by means of Monte Carlo simulations. Further we derive an exact few-body parent Hamiltonian for the states. Finally, we compare our results to the density profile, charge, and shape of anyons in the Kapit-Mueller model by means of exact diagonalization.
This work concerns Ising quasiholes in Moore-Read type lattice wave functions derived from conformal field theory. We commence with constructing Moore-Read type lattice states and then add quasiholes to them. By use of Metropolis Monte Carlo simulations, we analyze the features of the quasiholes, such as their size, shape, charge, and braiding properties. The braiding properties, which turn out to be the same as in the continuum Moore-Read state, demonstrate the topological attributes of the Moore-Read lattice states in a direct way. We also derive parent Hamiltonians for which the states with quasiholes included are ground states. One advantage of these Hamiltonians lies therein that we can now braid the quasiholes just by changing the coupling strengths in the Hamiltonian since the Hamiltonian is a function of the positions of the quasiholes. The methodology exploited in this article can also be used to construct other kinds of lattice fractional quantum Hall models containing quasiholes, for example, investigation of Fibonacci quasiholes in lattice Read-Rezayi states.
From an experimental point of view, quasielectrons and quasiholes play very similar roles in the fractional quantum Hall effect. Nevertheless, the theoretical description of quasielectrons is known to be much harder than that of quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here, we demonstrate that the same problem does not arise in lattice fractional quantum Hall models. This result allows us to make detailed investigations of the properties of quasielectrons, including their braiding statistics and density distribution on lattices on the plane and on the torus. We show that some of the states considered have high overlap with certain fractional Chern insulator states. We also derive few-body Hamiltonians, for which various states containing quasielectrons are exact ground states.
Quasielectrons in fractional quantum Hall systems are known to be much harder to describe theoretically than quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here we demonstrate that the same problem does not arise in lattice fractional quantum Hall models, so that quasielectrons can indeed be obtained as the inverse of quasiholes. Using this significantly simplified description, which can be applied quite generally, we compute braiding properties of quasielectrons and quasiholes and show that the charge distribution of quasielectrons is minus the charge distribution of quasiholes for different models in the disc and the torus geometry. We also derive few-body Hamiltonians, for which various states containing quasielectrons are ground states.
From an experimental point of view, quasielectrons and quasiholes play very similar roles in the fractional quantum Hall effect. Nevertheless, the theoretical description of quasielectrons is known to be much harder than that of quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here, we demonstrate that the same problem does not arise in lattice fractional quantum Hall models. This result allows us to make detailed investigations of the properties of quasielectrons, including their braiding statistics and density distribution on lattices on the plane and on the torus. We show that some of the states considered have high overlap with certain fractional Chern insulator states. We also derive few-body Hamiltonians, for which various states containing quasielectrons are exact ground states.
Matrix product state techniques provide a very efficient way to numerically evaluate certain classes of quantum Hall wave functions that can be written as correlators in two-dimensional conformal field theories. Important examples are the Laughlin and Moore-Read ground states and their quasihole excitations. In this paper, we extend the matrix product state techniques to evaluate quasielectron wave functions, a more complex task because the corresponding conformal field theory operator is not local. We use our method to obtain density profiles for states with multiple quasielectrons and quasiholes, and to calculate the (mutual) statistical phases of the excitations with high precision. The wave functions we study are subject to a known difficulty: the position of a quasielectron depends on the presence of other quasiparticles, even when their separation is large compared to the magnetic length. Quasielectron wave functions constructed using the composite fermion picture, which are topologically equivalent to the quasielectrons we study, have the same problem. This flaw is serious in that it gives wrong results for the statistical phases obtained by braiding distant quasiparticles. We analyze this problem in detail and show that it originates from an incomplete screening of the topological charges, which invalidates the plasma analogy. We demonstrate that this can be remedied in the case when the separation between the quasiparticles is large, which allows us to obtain the correct statistical phases. Finally, we propose that a modification of the Laughlin state, that allows for local quasielectron operators, should have good topological properties for arbitrary configurations of excitations.
The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.