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Acta Physica Polonica A

Quantum Hall State ν = 1/3 and Antilexicographic Order of Partitions

Bartosz Kuśmierz, Arkadiusz Wójs

Published Nov 1, 2016• 3 citations
Publisher

Abstract

We focus on a certain aspect of trial wave function approach in the fractional quantum Hall effect.We analyze the role of partition orderings and discuss the possible numerical search for the partition determining the subspace of the Hilbert space containing a particular quantum Hall wave function.This research is inspired by analogical properties of certain polynomials which are the object of interest of the symmetric function theory, especially the Jack polynomials (related to the so-called "Jack states").Presented method may be used in the search of candidate trial wave functions.We also justify (in certain cases) diagonalization of the Coulomb repulsion Hamiltonian restricted to certain subspaces.We focus on the states at filling factor ν = 1/3 in the lowest and second Landau level.

DOI
10.12693/aphyspola.130.1183

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We focus on a certain aspect of trial wave function approach in the fractional quantum Hall effect.We analyze the role of partition orderings and discuss the possible numerical search for the partition determining the subspace of the Hilbert space containing a particular quantum Hall wave function.This research is inspired by analogical properties of certain polynomials which are the object of interest of the symmetric function theory, especially the Jack polynomials (related to the so-called "Jack states").Presented method may be used in the search of candidate trial wave functions.We also justify (in certain cases) diagonalization of the Coulomb repulsion Hamiltonian restricted to certain subspaces.We focus on the states at filling factor ν = 1/3 in the lowest and second Landau level.
Fractional quantum Hall eect is a remarkable behaviour of correlated electrons, observed exclusively in two dimensions, at low temperatures, and in strong magnetic elds.The most prominent fractional quantum Hall state occurs at Landau level lling factor ν = 1/3 and it is described by the famous Laughlin wave function, which (apart from the trivial Gaussian factor) is an example of Jack polynomial.Fermionic Jack polynomials also describe another pair of candidate fractional quantum Hall states: MooreRead and ReadRezayi states, believed to form at the ν = 1/2 and 3/5 llings of the second Landau level, respectively.Bosonic Jacks on the other hand are candidates for certain correlated states of cold atoms.We examine here a continuous family of fermionic Jack polynomials whose special case is the Laughlin state as approximate wave functions for the 1/3 fractional quantum Hall eect.
We describe an occupation-number-like picture of fractional quantum Hall states in terms of polynomial wave functions characterized by a dominant occupation-number configuration. The bosonic variants of single-component Abelian and non-Abelian fractional quantum Hall states are modeled by Jack symmetric polynomials (Jacks), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a "squeezing rule" that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.
We consider spaces of trial wave functions for ground states and edge excitations in the fractional quantum Hall effect that can be obtained in various ways. In one way, functions are obtained by analyzing the entanglement of the ground-state wave function, partitioned into two parts. In another, functions are defined by the way in which they vanish as several coordinates approach the same value, or by a projection-operator Hamiltonian that enforces those conditions. In a third way, functions are given by conformal blocks from a conformal field theory (CFT). These different spaces of functions are closely related. The use of CFT methods permits an algebraic formulation to be given for all of them. In some cases, we can prove that there is a ground state, a Hamiltonian, and a CFT such that, for any number of particles, all of these spaces are the same. For such cases, this resolves several questions and conjectures: it gives a finite-size bulk-edge correspondence, and we can use the analysis of functions to construct a projection-operator Hamiltonian that produces those functions as zero-energy states. For a model related to the N=1\mathcal{N}=1N=1 superconformal algebra, the corresponding Hamiltonian imposes vanishing properties involving only three particles; for this we determine all the wave functions explicitly. We do the same for a sequence of models involving the M(3,p)M(3,p)M(3,p) Virasoro minimal models that has been considered previously, using results from the literature. We exhibit the Hamiltonians for the first few cases of these. The techniques we introduce can be applied in numerous examples other than those considered here.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
We obtain an exact matrix-product-state (MPS) representation of a large series of fractional quantum Hall (FQH) states in various geometries of genus 0. The states in question include all paired k=2k=2k=2 Jack polynomials, such as the Moore-Read and Gaffnian states, as well as the Read-Rezayi k=3k=3k=3 state. We also outline the procedures through which the MPSs of other model FQH states can be obtained, provided their wave function can be written as a correlator in a 1+11+11+1 conformal field theory (CFT). The auxiliary Hilbert space of the MPS, which gives the counting of the entanglement spectrum, is then simply the Hilbert space of the underlying CFT. This formalism enlightens the link between entanglement spectrum and edge modes. Properties of model wave functions such as the thin-torus root partitions and squeezing are recast in the MPS form, and numerical benchmarks for the accuracy of the new MPS prescription in various geometries are provided.