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arXiv:0804.4691 · Physical Review Letters

Pfaffian Statistics through Adiabatic Transport in the 1D Coherent State Representation

Alexander Seidel

Published Nov 4, 2008• 32 citations
arXivHTMLPDF

Abstract

Recent work has shown that the low energy sector of certain quantum Hall states is adiabatically connected to simple charge-density-wave patterns that appear, e.g., when the system is deformed into a thin torus. Here it is shown that the patterns emerging in this limit already determine the non-Abelian statistics of the nu=1 Moore-Read state. Aside from the knowledge of these patterns, the method only relies on the principle of adiabatic continuity, the effectively noncommutative geometry in a strong magnetic field, and topological as well as locality arguments.

DOI
10.1103/physrevlett.101.196802

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In the search for states with non-Abelian statistics, we explore the fractional quantum Hall effect in a system of two-dimensional (2D) charge carrier holes. We propose a method of mapping states of holes confined to a finite width quantum well in a perpendicular magnetic field to states in a spherical shell geometry. We take into account strong coupling between the spin and motion of charge parallel and perpendicular to the 2D layer. This method gives the single-particle hole states used in the exact diagonalization of systems with a small number of holes in the presence of Coulomb interactions, density matrix renormalization group, and topological entanglement entropy calculations. The hole quantum Hall state at half filling of the ground state in a magnetic field near the crossing of single-hole states is likely the Moore-Read Pfaffian state.

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Recent work has shown that the low energy sector of certain quantum Hall states is adiabatically connected to simple charge-density-wave patterns that appear, e.g., when the system is deformed into a thin torus. Here it is shown that the patterns emerging in this limit already determine the non-Abelian statistics of the nu=1 Moore-Read state. Aside from the knowledge of these patterns, the method only relies on the principle of adiabatic continuity, the effectively noncommutative geometry in a strong magnetic field, and topological as well as locality arguments.
Many trial wave functions for fractional quantum Hall states in a single Landau level are given by functions called conformal blocks, taken from some conformal field theory. Also, wave functions for certain paired states of fermions in two dimensions, such as px+ipy{p}_{x}+i{p}_{y}px​+ipy​ states, reduce to such a form at long distances. Here we investigate the adiabatic transport of such many-particle trial wave functions using methods from two-dimensional field theory. One context for this is to calculate the statistics of widely separated quasiholes, which has been predicted to be non-Abelian in a variety of cases. The Berry phase or matrix (holonomy) resulting from adiabatic transport around a closed loop in parameter space is the same as the effect of analytic continuation around the same loop with the particle coordinates held fixed (monodromy), provided the trial functions are orthonormal and holomorphic in the parameters so that the Berry vector potential (or connection) vanishes. We show that this is the case (up to a simple area term) for paired states (including the Moore-Read quantum Hall state) and present general conditions for it to hold for other trial states (such as the Read-Rezayi series). We argue that trial states based on a nonunitary conformal field theory do not describe a gapped topological phase, at least in many cases. By considering adiabatic variation in the aspect ratio of the torus, we calculate the Hall viscosity, a nondissipative viscosity coefficient analogous to Hall conductivity, for paired states, Laughlin states, and more general quantum Hall states. Hall viscosity is an invariant within a topological phase and is generally proportional to the ``conformal spin density'' in the ground state.
In the theory of the fractional quantum Hall effect, much attention is paid to the correspondence between fractional quantum Hall wave functions and conformal blocks in certain rational conformal field theories (CFT). This correspondence is powerful, enabling the calculation of the fractional statistics of a state that would be difficult if not impossible to calculate directly from the wave functions. But it is, in general, conjectural, remaining without microscopic justification in many cases of interest, and involves heavy mathematical machinery. We detail an alternative method to calculate Abelian and non-Abelian fractional statistics, the coherent state method. The method relies on assumptions which are independent of those underlying the CFT correspondence, so it serves as an independent check of results for the statistics where they exist, and an alternative source of results when a CFT is not known. We show how the coherent state method can be used to derive the statistics of several increasingly complicated trial wave functions: ν=1/2 Laughlin, Moore-Read, and k=3 Read-Rezayi. We discuss implications of our method for a possible notion of braiding statistics of the Gaffnian state, a "nonunitary" quantum Hall state for which the ramifications of the CFT correspondence are less well understood. We go on to derive formulas for the counting of zero modes of all these states on the torus.
Fractional quantum Hall states are a cornerstone of topological physics, hosting fractionally charged quasiparticles with exotic statistics that promise to enable topologically protected quantum information processing. Among these, the Pfaffian state introduced by Moore and Read implements a p-wave pairing structure that supports excitations with non-Abelian exchange statistics. Despite extensive study in electronic systems, direct access to its pairing structure has remained limited. Here we realize a three-particle bosonic Pfaffian state of ultracold 87Rb^{87}\mathrm{Rb}87Rb atoms in an optical lattice subject to a Floquet-engineered synthetic magnetic field. Using a Bayesian-optimized adiabatic protocol, we prepare a state exhibiting Pfaffian pairing correlations. Site-resolved measurements of multi-point density correlations reveal a pronounced suppression of short-range three-body coincidences, reflecting the underlying pairing structure. We further probe the state's transport response through Hall drift measurements. Our results establish a bottom-up approach to engineering non-Abelian topological order and lay the groundwork for future explorations of anyonic braiding in synthetic matter.
We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction ν = 5/2.We complete the program started in V. Gurarie and C. Nayak, [Nucl. Phys. B 506, 685 (1997)].and show that the degenerate four-quasihole and six-quasihole wave functions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a c = 1 + 1 2 conformal field theory.As a consequence, this proves that the non-Abelian statistics of the excitations in this state are given by the explicit analytic continuation of these wave functions.Our proof is based on a plasma analogy derived from the Coulomb gas construction of Ising model correlation functions involving both order and (at most two) disorder operators.We show how this computation also determines the non-Abelian statistics of collections of more than six quasiholes and give an explicit expression for the corresponding conformal block-derived wave functions for an arbitrary number of quasiholes.Our method also applies to the anti-Pfaffian wave function and to Bonderson-Slingerland hierarchy states constructed over the Moore-Read and anti-Pfaffian states.
Inspired by the recent theoretical discovery of robust fractional topological phases without a magnetic field, we search for the non-abelian quantum Hall effect in lattice models with topological flat bands. Through extensive numerical studies on the Haldane model with three-body hard-core bosons loaded into a topological flat band, we find convincing numerical evidence of a stable ν=1 bosonic non-abelian quantum Hall effect, with the characteristic threefold quasidegeneracy of ground states on a torus, a quantized Chern number, and a robust spectrum gap. Moreover, the spectrum for two-quasihole states also shows a finite energy gap, with the number of states in the lower-energy sector satisfying the same counting rule as the Moore-Read pfaffian state.
In the search for states with non-Abelian statistics, we explore the fractional quantum Hall effect in a system of two-dimensional (2D) charge carrier holes. We propose a method of mapping states of holes confined to a finite width quantum well in a perpendicular magnetic field to states in a spherical shell geometry. We take into account strong coupling between the spin and motion of charge parallel and perpendicular to the 2D layer. This method gives the single-particle hole states used in the exact diagonalization of systems with a small number of holes in the presence of Coulomb interactions, density matrix renormalization group, and topological entanglement entropy calculations. The hole quantum Hall state at half filling of the ground state in a magnetic field near the crossing of single-hole states is likely the Moore-Read Pfaffian state.