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arXiv:2012.11611 · Physical review. B./Physical review. B

Composite particle construction of the Fibonacci fractional quantum Hall state

Hart Goldman, Ramanjit Sohal, Eduardo Fradkin

Published Jun 10, 2021• 5 citations
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Abstract

The Fibonacci topological order is the simplest platform for a universal topological quantum computer. While the $\ensuremath{\nu}=12/5$ fractional quantum Hall (QH) state has been proposed to support a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a QH system has been lacking. We use non-Abelian dualities to construct a Fibonacci state of bosons at filling $\ensuremath{\nu}=2$ starting from a trilayer of integer QH states. Our parent theory consists of bosonic composite vortices coupled to fluctuating $U(2)$ gauge fields, which is dual to the theory of Laughlin quasiparticles. The Fibonacci state is obtained by interlayer clustering of the composite vortices, along with flux attachment. We use this framework to motivate a wave function for the Fibonacci state.

DOI
10.1103/physrevb.103.235118

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The Fibonacci topological order is the simplest platform for a universal topological quantum computer. While the \ensuremathν=12/5\ensuremath{\nu}=12/5\ensuremathν=12/5 fractional quantum Hall (QH) state has been proposed to support a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a QH system has been lacking. We use non-Abelian dualities to construct a Fibonacci state of bosons at filling \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2 starting from a trilayer of integer QH states. Our parent theory consists of bosonic composite vortices coupled to fluctuating U(2)U(2)U(2) gauge fields, which is dual to the theory of Laughlin quasiparticles. The Fibonacci state is obtained by interlayer clustering of the composite vortices, along with flux attachment. We use this framework to motivate a wave function for the Fibonacci state.
For a given parent fractional quantum Hall (FQH) state at filling fraction $\ensuremath{\nu}$, the hierarchy construction produces FQH states at nearby filling fractions ${{\ensuremath{\nu}}_{n}}$ by condensing minimally charged quasiholes or quasiparticles of the parent state into their own FQH states. The hierarchy construction has been useful for relating families of FQH states and for the experimental identification of the topological order of parent states via the properties of daughter states. We reinterpret the hierarchy construction as a two-step procedure: stacking with a second FQH state and condensing a condensable algebra of bosons. This two-step procedure can be applied to both Abelian and non-Abelian FQH states, and it does not require calculations with a wave function. We show this construction reproduces the hierarchies for the Laughlin and Pfaffian states, and can be applied further to propose hierarchies for various non-Abelian FQH states.
It is an important open problem to understand the landscape of non-Abelian fractional quantum Hall phases which can be obtained starting from physically motivated theories of Abelian composite particles. We show that progress on this problem can be made using recently proposed non-Abelian bosonization dualities in 2+12+12+1 dimensions, which morally relate U(N)kU{(N)}_{k}U(N)k​ and $\mathrm{SU}{(k)}_{\ensuremath{-}N}$ Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings $\ensuremath{\nu}=k/(kM+2)$ by starting from kkk layers of bosons at $\ensuremath{\nu}=1/2$ with MMM Abelian fluxes attached. The Read-Rezayi states arise when kkk clusters of the dual non-Abelian bosons condense. We extend this construction by showing that Nf{N}_{f}Nf​-component generalizations of the Halperin (2,2,1) bosonic states have dual descriptions in terms of SU(Nf+1)1\mathrm{SU}{({N}_{f}+1)}_{1}SU(Nf​+1)1​ Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering kkk layers of these theories yields a non-Abelian SU(Nf)\mathrm{SU}({N}_{f})SU(Nf​)-singlet state at filling $\ensuremath{\nu}=k{N}_{f}/({N}_{f}+1+kM{N}_{f})$.
The Pfaffian fractional quantum Hall (FQH) states are incompressible non-Abelian topological fluids present in a half-filled electron Landau level, where there is a balanced population of electrons and holes. They give rise to half-integral quantum Hall plateaus that divide critical transitions between integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH states, such as the Laughlin state, that can be understood using partons, which are fermionic divisions of the electron. In this paper, we propose a family of incompressible paired parton FQH states at filling $\ensuremath{\nu}=\frac{1}{6}$ (modulo 1) that emerge from critical transitions between IQH states and Abelian FQH states at filling $\ensuremath{\nu}=\frac{1}{3}$ (modulo 1). These paired parton states are originated from a half-filled parton Landau level, where there is an equal amount of partons and holes. They generically support Ising-type anyonic quasiparticle excitations and carry non-Abelian Pfaffian topological orders (TO) for partons. We prove the principal existence of these paired parton states using exactly solvable interacting arrays of electronic wires under a magnetic field. Moreover, we establish a notion of particle-hole (PH) symmetry for partons and relate the PH symmetric parton Pfaffian TO with the gapped symmetric surface TO of a fractional topological insulator in three dimensions.
The recent proposal of non-Abelian boson-fermion dualities in 2+12+12+1 dimensions, which morally relate U(k)NU{(k)}_{N}U(k)N​ to $SU{(N)}_{\ensuremath{-}k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian ``composite fermions'' partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian, U(k)2U{(k)}_{2}U(k)2​ Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across NNN layers can also generate the family of Blok-Wen states with U(k)2U{(k)}_{2}U(k)2​ topological order.
Identifying and understanding interacting systems that can host non-Abelian topological phases with fractionalized quasiparticles have attracted intense attentions in the past twenty years. Theoretically, it is possible to realize a rich variety of such states by coupling two Abelian fractional quantum Hall (FQH) states together through gapping out part of the low energy degrees of freedom. So far, there are some indications, but no robust example has been established in bilayer systems for realizing the non-Abelian state in the past. Here, we present a phase diagram of a double-layer bosonic FQH system based on the exact diagonalization and density-matrix renormalization group (DMRG) calculations, which demonstrate a potential regime with the emergence of the non-Abelian bosonic Moore-Read state. We start from the Abelian phase with fourfold topological degeneracies on torus geometry when the two layers are weakly coupled. With the increase of interlayer tunneling, we find an intermediate regime with a threefold groundstate degeneracy and a finite fractional drag Hall conductance. We find the different topological sectors in consistent with Moore-Read state by inserting different fluxes in adiabatic DMRG study. We also extract the modular S−\mathcal{S}-S− matrix, which supports the emergence of the non-Abelian Ising anyon quasiparticle in this system.
The possibility of realizing non-Abelian statistics and utilizing it for topological quantum computation (TQC) has generated widespread interest. However, the non-Abelian statistics that can be realized in most accessible proposals is not powerful enough for universal TQC. In this Letter, we consider a simple bilayer fractional quantum Hall system with the 1/3 Laughlin state in each layer. We show that interlayer tunneling can drive a transition to an exotic non-Abelian state that contains the famous "Fibonacci" anyon, whose non-Abelian statistics is powerful enough for universal TQC. Our analysis rests on startling agreements from a variety of distinct methods, including thin torus limits, effective field theories, and coupled wire constructions. We provide evidence that the transition can be continuous, at which point the charge gap remains open while the neutral gap closes. This raises the question of whether these exotic phases may have already been realized at ν=2/3 in bilayers, as past experiments may not have definitively ruled them out.