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arXiv:1902.04173 · Physical review. B./Physical review. B

Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states and beyond

Ajit Coimbatore Balram, Maissam Barkeshli, Mark Spencer Rudner

Published Jun 19, 2019• 46 citations
arXivHTMLPDF

Abstract

The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\ensuremath{\nu}=k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted ${\overline{2}}^{k}{1}^{k+1}$, and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the ${\overline{2}}^{k}{1}^{k+1}$ state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence $\overline{n}{\overline{2}}^{2}{1}^{4}$, with $n=1,2,3$, to describe the FQH states observed at $\ensuremath{\nu}=2+1/2$, $2+2/5$, and $2+3/8$.

DOI
10.1103/physrevb.99.241108

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cond-mat.str-elCross-listedarXiv:2608.05140•Aug 6, 2026•0 citations
The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling \ensuremathν=k/(k+2)\ensuremath{\nu}=k/(k+2)\ensuremathν=k/(k+2). Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1, and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a [U(1)k+1\ifmmode×\else\texttimes\fiU(2k)\ensuremath−1]/[SU(k)\ensuremath−2\ifmmode×\else\texttimes\fiU(1)\ensuremath−1][U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}][U(1)k+1​\ifmmode×\else\texttimes\fiU(2k)\ensuremath−1​]/[SU(k)\ensuremath−2​\ifmmode×\else\texttimes\fiU(1)\ensuremath−1​] coset conformal field theory describes the edge excitations of the 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1 state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence n‾2‾214\overline{n}{\overline{2}}^{2}{1}^{4}n2214, with n=1,2,3n=1,2,3n=1,2,3, to describe the FQH states observed at \ensuremathν=2+1/2\ensuremath{\nu}=2+1/2\ensuremathν=2+1/2, 2+2/52+2/52+2/5, and 2+3/82+3/82+3/8.
The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N\ensuremath{-}1)$-singlet FQH states at filling fraction $\ensuremath{\nu}=k(N\ensuremath{-}1)/[N+k(N\ensuremath{-}1)m]$ labeled by integers kkk and mmm, which are potentially realized in multicomponent quantum Hall systems or SU(N)SU(N)SU(N) spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an $(N\ensuremath{-}1)$-component free-boson CFT and the $SU{(N)}_{k}/{[U(1)]}^{N\ensuremath{-}1}$ CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an SU(N)SU(N)SU(N) Heisenberg model.
Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for the exotic fundamental physics they would exhibit, and for universal quantum computing applications. The fractional quantum Hall (FQH) state at a filling factor of ν=12/5 is a promising candidate; however, its precise nature is still under debate and no consensus has been achieved so far. Here, we investigate the nature of the FQH ν=13/5 state and its particle-hole conjugate state at 12/5 with the Coulomb interaction, and we address the issue of possible competing states. Based on a large-scale density-matrix renormalization group calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at ν=13/5 and 12/5 is captured by the k=3 parafermion Read-Rezayi state (RR_{3}), including a robust excitation gap and the topological fingerprint from the entanglement spectrum and topological entanglement entropy. Furthermore, by considering the infinite-cylinder geometry (topologically equivalent to torus geometry), we expose the non-Abelian GS sector corresponding to a Fibonacci anyonic quasiparticle, which serves as a signature of the RR_{3} state at 13/5 and 12/5 filling numbers.
We calculate the dominant excitations for the kkk-level (k∈Nk\in\mathbb{N}k∈N) Read-Rezayi (RR) states and their particle-hole conjugates, the anti Read-Rezayi (RRˉ\bar{\textrm{RR}}RRˉ), proposed for quantum Hall states. These states are supposed to be build over the second Landau level with total filling factor ν=2+ν∗ν=2+ν^*ν=2+ν∗ with ν∗=k/(k+2)ν^*=k/(k+2)ν∗=k/(k+2) for RR and ν∗=2/(k+2)ν^*=2/(k+2)ν∗=2/(k+2) for RRˉ\bar{\textrm{RR}}RRˉ. In the kkk-level RR states, based on Zk\mathbb{Z}_kZk​ parafermions, the dominant excitations are the fundamental quasiparticles with fractional charge ek∗=e/(k+2)e^*_k= e/(k + 2)ek∗​=e/(k+2), with eee the electron charge, if k=2,3k=2,3k=2,3. For k=4 the single-qp and the 2-agglomerate, with charge 2ek∗2e^*_k2ek∗​, have the same scaling and both dominate, while for k>4k>4k>4 the 2-agglomerates are dominant. Anyway the dominance of the 2-agglomerates can be affected by the presence of environmental renormalizations. For all the kkk-level RRˉ\bar{\textrm{RR}}RRˉ states the single-qp and the 2-agglomerate have the same scaling and both dominate. In this case only the presence of environmental renormalizations can make one dominant over the other. We determine the conditions where the environmental renormalizations of the charged and neutral modes make dominant the Abelian 2-agglomerates over the non-Abelian single-quasiparticles in the two models and for any value of kkk. We conclude observing that, according these predictions, the dominance of 2-agglomerates, at very low energies for the ν=5/2ν=5/2ν=5/2, can be an interesting indication supporting the validity of the anti-Pfaffian model in comparison to the Pfaffian.
We compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling nu = k/r. For r = 2, these states are the Zk Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k+1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5,3/7,4/9,.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.
Non-Abelian anyons arise as exotic excitations in fractional quantum Hall (FQH) matter and have proved very elusive to realize in conventional platforms. In this work, we show that on a programmable quantum hardware platform, the more exotic FQH excitations are the less costly ones to prepare: clustered non-Abelian FQH states admit parallel quantum preparation circuits whose two-qubit depth is independent of system size, while constructing the more common Abelian Laughlin state requires a sequential circuit chain with linear depth. The centerpiece of this work is our new systematic framework for cataloging possible FQH states and preparing them on quantum circuits at unprecedented scale and variety. Our prepared parafermionic Read--Rezayi Z3\mathbb{Z}_3Z3​ state holds depth 3 from 8 to 118 qubits, and full root sampling extends to a 154-qubit, 104-electron Read--Rezayi Z4\mathbb{Z}_4Z4​ state. In all, our demonstrated 18-family catalog of prepared FQH states extends to all 156 qubits of an IBM Heron processor, limited only by existing hardware scale. Measurements on the prepared states recover the expected fractional quasihole charges, with the charge estimator exact in every symmetry-selected shot for the clustered states, and braiding data of the non-Abelian e/4e/4e/4 quasihole measured via interferometric extensions. Our work establishes a scalable route to studying FQH physics on quantum processors and opens new avenues for preparing and probing non-Abelian topological matter far beyond the reach of conventional platforms.