Braiding Non-Abelian Quasiholes in Fractional Quantum Hall States

Abstract

Quasiholes in certain fractional quantum Hall states are promising candidates for the experimental realization of non-Abelian anyons. They are assumed to be localized excitations, and to display non-Abelian statistics when sufficiently separated, but these properties have not been explicitly demonstrated except for the Moore-Read state. In this work, we apply the newly developed matrix product state technique to examine these exotic excitations. For the Moore-Read and the Z3 ReadRezayi states, we estimate the quasihole radii, and determine the correlation lengths associated with the exponential convergence of the braiding statistics. We provide the first microscopic verification for the Fibonacci nature of the Z3 Read-Rezayi quasiholes. We also present evidence for the failure of plasma screening in the nonunitary Gaffnian wave function.PACS numbers: 05.30. Pr, 73.43.Cd, 03.67.Mn Non-Abelian anyons have been the focus of much theoretical and experimental interest due to the exciting prospect of topologically fault-tolerant quantum computing . As noted by Kitaev [1], the topological degeneracy of these exotic excitations allows nonlocal storage of quantum information, while adiabatic braiding implements unitary quantum gates. Candidates [22][23][24] for their physical realization are the quasiholes in certain fractional quantum Hall states [25], in particular, those around the plateaus at fillings ν = 5/2 and 12/5 [26][27][28]. Their model wave functions, namely, the Moore-Read [29] (MR) and the Z 3 Read-Rezayi [30] (RR) states, enjoy an elegant first-quantized rewriting [29,31] in terms of conformal field theory [32-34] (CFT) correlators, from which many physical properties can be predicted. The strengths of this approach rest on a crucial conjecture [29]: quasihole braiding statistics can be directly read off from the monodromy of the CFT correlators. Under this conjecture, the MR quasiholes are Ising anyons, while the Z 3 RR ones are Fibonacci anyons. But the proof of the conjecture itself is lacking.The relation between statistics and monodromy was originally established for the Laughlin state [35] through the plasma analogy [36]. Assuming sufficient quasihole separations, the statistics-monodromy equivalence holds true when the plasma is in the screening phase. With considerable effort, this line of argument was recently extended to the MR state [37][38][39], in agreement with finite-size numerics [40][41][42]. More complicated states like the Z 3 RR still remain uncharted territory for both analytics and numerics, despite their capacity for universal quantum computation [2][3][4]. Moreover, wave functions constructed from nonunitary field theories (such as the Gaffnian [43]) are conjectured not to give rise to sensible statistics [38], yet the microscopic symptom of such pathology is still under investigation [44].In this Letter, we aim to settle the aforementioned issues through numerical studies of certain model wave functions. Until very recently, this was a daunting task due to the exponentially larg...

Variational Ansatz for an Abelian to Non-Abelian Topological Phase Transition in ν=1/2+1/2 Bilayers

“…They identified a large class of model WFs and their quasihole excitations with Conformal Field Theory (CFT) correlators from which the topological content of the phase may be read off (under the generalized screening assumption [58]). It furthermore allows for an exact Matrix Product State (MPS) description of these strongly correlated phases of matter [59][60][61], allowing for large scale numerical study of their relevance and properties [62,63].…”

Braiding properties of paired spin-singlet and non-Abelian hierarchy states

“…We consider the master formula for the case m 1 = 4, Eq. (20), since it yields all braid matrices. The master formula is:…”

Quasielectrons as inverse quasiholes in lattice fractional quantum Hall models

“…In the last years, there has been significant progress for explicitly computing braiding properties of various types of quasiholes [14][15][16][17][18][19][20], but the quasielectrons are again more complicated to work with. For the latter it has been found that the braiding properties are as expected for the composite fermion version of the Laughlin quasielectrons [21,22], while for Laughlinʼs proposal for the quasielectron wavefunction the expected braiding statistics is obtained in the thermodynamic limit when considering a finite size braiding loop [23,24].…”