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arXiv:1207.4604 · Physica Scripta

Quasiparticle agglomerates in the Read–Rezayi and anti-Read–Rezayi states

Alessandro Braggio, Dario Ferraro, Nicodemo Magnoli

Published Nov 1, 2012• 3 citations
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Abstract

We calculate the dominant excitations for the $k$-level ($k\in\mathbb{N}$) Read-Rezayi (RR) states and their particle-hole conjugates, the anti Read-Rezayi ($\bar{\textrm{RR}}$), proposed for quantum Hall states. These states are supposed to be build over the second Landau level with total filling factor $ν=2+ν^*$ with $ν^*=k/(k+2)$ for RR and $ν^*=2/(k+2)$ for $\bar{\textrm{RR}}$. In the $k$-level RR states, based on $\mathbb{Z}_k$ parafermions, the dominant excitations are the fundamental quasiparticles with fractional charge $e^*_k= e/(k + 2)$, with $e$ the electron charge, if $k=2,3$. For k=4 the single-qp and the 2-agglomerate, with charge $2e^*_k$, have the same scaling and both dominate, while for $k>4$ the 2-agglomerates are dominant. Anyway the dominance of the 2-agglomerates can be affected by the presence of environmental renormalizations. For all the $k$-level $\bar{\textrm{RR}}$ 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 $k$. We conclude observing that, according these predictions, the dominance of 2-agglomerates, at very low energies for the $ν=5/2$, can be an interesting indication supporting the validity of the anti-Pfaffian model in comparison to the Pfaffian.

DOI
10.1088/0031-8949/2012/t151/014052

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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.
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 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 $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the 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 $\ensuremath{\nu}=2+1/2$, 2+2/52+2/52+2/5, and 2+3/82+3/82+3/8.
We study the $\ensuremath{\nu}=\frac{2}{k+2}$ quantum Hall states which are particle-hole conjugates of the $\ensuremath{\nu}=\frac{2}{k+2}$ Read-Rezayi states. We find that equilibration between the different modes at the edge of such a state leads to an emergent SU(2)k\mathrm{SU}{(2)}_{k}SU(2)k​ algebra in the counter-propagating neutral sector. Heat flow along the edges of these states will be in the opposite direction of charge flow. In the k=3k=3k=3 case, which may be relevant to $\ensuremath{\nu}=2+\frac{2}{5}$, the thermal Hall conductance and the exponents associated with quasiparticle and electron tunneling distinguish this state from competing states such as the hierarchy and/or Jain state.
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 present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $\ensuremath{\nu}=5∕2$, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle (with charge $\ifmmode\pm\else\textpm\fi{}e∕4$) when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around $4\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement (smooth edge), while the Moore-Read Pfaffian state is favored in strong confinement (sharp edge).
The Read-Rezayi wave function is one of the candidates for the fractional quantum Hall effect at filling fraction $\ensuremath{\nu}=2+⅗$, and thereby also its hole conjugate at 2+⅖2+⅖2+⅖. We study a general class of tripartite composite fermion wave functions, which reduce to the Rezayi-Read ground state and quasiholes for appropriate quantum numbers, but also allow a construction of wave functions for quasiparticles and neutral excitations by analogy to the standard composite fermion theory. We present numerical evidence in finite systems that these trial wave functions capture well the low energy physics of a four-body model interaction. We also compare the tripartite composite fermion wave functions with the exact Coulomb eigenstates at 2+⅗2+⅗2+⅗, and find reasonably good agreement. The ground state as well as several excited states of the four-body interaction are seen to evolve adiabatically into the corresponding Coulomb states for N=15N=15N=15 particles. These results support the plausibility of the Read-Rezayi proposal for the 2+⅖2+⅖2+⅖ and 2+⅗2+⅗2+⅗ fractional quantum Hall effect. However, certain other proposals also remain viable, and further study of excitations and edge states will be necessary for a decisive establishment of the physical mechanism of these fractional quantum Hall states.
Quaternate generalization of Pfaffian state at ν=5/2ν=5/2ν=5/2
We consider a quaternately generalized Pfaffian QGPf(1J(zi,zj,zk,zl))[J(z1,...,zN)]2(\frac{1}{J(z_i,z_j,z_k,z_l)})[J(z_1,...,z_N)]^2(J(zi​,zj​,zk​,zl​)1​)[J(z1​,...,zN​)]2 in which the square of Vandermonde determinant, [J(z1,...,zN)]2[J(z_1,...,z_N)]^2[J(z1​,...,zN​)]2, implies the upmost Landau level is half filled. This wave function is the unique highest density zero energy state of a special short range interacting Hamiltonian. One can think this quaternate composite fermion liquid as a competing ground state of Moore-Read (MR) Pfaffian state at ν=5/2ν=5/2ν=5/2. The degeneracy of the quasihole excitations above the QGPf is higher than that of Moore-Read even Read-Rezayi quasiholes. The QGPf is related to a unitary conformal field theory with Z2×Z2Z_2\times Z_2Z2​×Z2​ parafermions in coset space SU(3)2/U(1)2SU(3)_2/U(1)^2SU(3)2​/U(1)2 . Because of the level-rank duality between SU(3)2SU(3)_2SU(3)2​ and SU(2)3SU(2)_3SU(2)3​ in conformal field theory, these quasiholes above this QGPf state obeying non-abelian anyonic statistics are expected to support the universal quantum computation at ν=5/2ν=5/2ν=5/2 as Read-Rezayi quasiholes at ν=13/5ν=13/5ν=13/5. The edge states of QGPf are very different from those of the Pfaffian's.