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arXiv:cond-mat/0310273 · Physical Review B

Geometric phases and quantum entanglement as building blocks for non-Abelian quasiparticle statistics

Ady Stern, Felix von Oppen, Eros Mariani

Published Nov 23, 2004• 186 citations
arXivHTMLPDF

Abstract

Some models describing unconventional fractional quantum Hall states predict quasiparticles that obey non-Abelian quantum statistics. The most prominent example is the Moore-Read model for the $\ensuremath{\nu}=5∕2$ state, in which the ground state is a superconductor of composite fermions, the charged excitations are vortices in that superconductor, and the non-Abelian statistics is closely linked to the degeneracy of the ground state in the presence of vortices. In this paper we develop a physical picture of the non-Abelian statistics of these vortices. Considering first the positions of the vortices as fixed, we define a set of single-particle states at and near the core of each vortex, and employ general properties of the corresponding Bogolubov--de Gennes equations to write the ground states in the Fock space defined by these single-particle states. We find all ground states to be entangled superpositions of all possible occupations of the single-particle states near the vortex cores, in which the probability for all occupations is equal, and the relative phases vary from one ground state to another. Then, we examine the evolution of the ground states as the positions of the vortices are braided. We find that as vortices move, they accumulate a geometric phase that depends on the occupations of the single-particle states near the cores of other vortices. Thus, braiding of vortices changes the relative phase between different components of a superposition, in which the occupations of these states differ, and hence transform the system from one ground state to another. These transformations, that emanate from the quantum entanglement of the occupations of single-particle states and from the dependence of the geometric phase on these occupations, are the source of the non-Abelian statistics. Finally, by exploring a ``self-similar'' form of the many-body wave functions of the various ground states, we show the equivalence of our picture, in which vortex braiding leads to a change in the relative phase of components in a superposition, and pictures derived previously, in which vortex braiding seemingly affects the occupations of states in the cores of the vortices.

DOI
10.1103/physrevb.70.205338

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Some models describing unconventional fractional quantum Hall states predict quasiparticles that obey non-Abelian quantum statistics. The most prominent example is the Moore-Read model for the \ensuremathν=5∕2\ensuremath{\nu}=5∕2\ensuremathν=5∕2 state, in which the ground state is a superconductor of composite fermions, the charged excitations are vortices in that superconductor, and the non-Abelian statistics is closely linked to the degeneracy of the ground state in the presence of vortices. In this paper we develop a physical picture of the non-Abelian statistics of these vortices. Considering first the positions of the vortices as fixed, we define a set of single-particle states at and near the core of each vortex, and employ general properties of the corresponding Bogolubov--de Gennes equations to write the ground states in the Fock space defined by these single-particle states. We find all ground states to be entangled superpositions of all possible occupations of the single-particle states near the vortex cores, in which the probability for all occupations is equal, and the relative phases vary from one ground state to another. Then, we examine the evolution of the ground states as the positions of the vortices are braided. We find that as vortices move, they accumulate a geometric phase that depends on the occupations of the single-particle states near the cores of other vortices. Thus, braiding of vortices changes the relative phase between different components of a superposition, in which the occupations of these states differ, and hence transform the system from one ground state to another. These transformations, that emanate from the quantum entanglement of the occupations of single-particle states and from the dependence of the geometric phase on these occupations, are the source of the non-Abelian statistics. Finally, by exploring a ``self-similar'' form of the many-body wave functions of the various ground states, we show the equivalence of our picture, in which vortex braiding leads to a change in the relative phase of components in a superposition, and pictures derived previously, in which vortex braiding seemingly affects the occupations of states in the cores of the vortices.
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Quasihole excitations in fractional quantum Hall (FQH) systems exhibit fractional statistics and fractional spin, but how the spin-statistics relation emerges from many-body physics remains poorly understood. Here we prove a spin-statistics relation using only FQH wave functions, on both the sphere and disk geometry. In particular, the proof on the disk generalizes to all quasiholes in realistic systems, which have a finite size and could be deformed into arbitrary shapes. Different components of the quasihole spins are linked to different conformal Hilbert spaces (CHSs), which are null spaces of model Hamiltonians that host the respective FQH ground states and quasihole states. Understanding how the intrinsic spin of the quasiholes is linked to different CHSs is crucial for the generalized spin-statistics relation that takes into account the effect of metric deformation. In terms of the experimental relevance, this enables us to study the effect of deformation and disorder that introduces an additional source of Berry curvature, an aspect of anyon braiding that has been largely neglected in previous literature.
Fractional quantum Hall (FQH) fluids host quasiparticle excitations that carry a fraction of the electronic charge. Moreover, in contrast to bosons and fermions that carry exchange statistics of 0 and $\ensuremath{\pi}$ respectively, these quasiparticles of FQH fluids, when braided around one another, can accumulate a Berry phase, which is a fractional multiple of $\ensuremath{\pi}$. Deploying the spherical geometry, we numerically demonstrate that composite fermion particle (CFP) excitations in the Jain FQH states carry Abelian fractional statistics. Previously, the exchange statistics of CFPs were studied in the disk geometry, where the statistics get obscured due to a shift in the phase arising from the addition of another CFP, making its determination cumbersome without prior knowledge of the shift. We show that on the sphere this technical issue can be circumvented and the statistics of CFPs can be obtained more transparently. The ideas we present can be extended to determine the statistics of quasiparticles arising in certain non-Abelian partonic FQH states.
The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing. This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory. The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n-1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n-k quasiholes and k quasielectrons. In the third approach, we study the topological nature of the degeneracy 2n-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n-1 for 2n vortices.
We numerically investigate non-Abelian braiding dynamics of vortices in two-dimensional topological superconductors, such as sss-wave superconductors with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the vortices constitute a topological qubit, which offers a fundamental building block of topological quantum computation. As the MZMs are protected by Z2{\mathbb{Z}}_{2}Z2​ invariant, however, the Majorana qubit and quantum gate operations may be sensitive to intrinsic decoherence caused by MZM hybridization. Numerically simulating the time-dependent Bogoliubov--de Gennes equation without assuming a priori existence of MZMs, we examine quantum noises on the unitary operators of non-Abelian braiding dynamics due to interactions with neighboring MZMs and other quasiparticle states. We demonstrate that after the interchange of two vortices, the lowest vortex-bound states accumulate the geometric phase $\ensuremath{\pi}/2$, and errors stemming from dynamical phases are negligibly small, irrespective of interactions of MZMs. Furthermore, we numerically simulate the braiding dynamics of four vortices in two-dimensional topological superconductors, and discuss an optimal braiding condition for realizing the high performance of non-Abelian statistics and quantum gate operations of Majorana-based qubits.
The non-Abelian geometric phases of the robust degenerate ground states were proposed as physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We find a way to determine even the pure U(1) phase of the non-Abelian geometric phases. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the chiral central charge Delta c mod 24 of the edge states, as well as the Hall viscosity. Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) provide a quite complete way to characterize 2D topological order.