Particle-hole mirror symmetries around the half-filled shell: Quantum numbers and algebraic structure of composite fermions

W. C. Haxton, Daniel J. Haxton, and Byungmin Kang
Phys. Rev. B 98, 115140 – Published 21 September 2018

Abstract

Composite fermions (CFs) of the fractional quantum Hall effect (FQHE) are described as spherical products of electron and vortex spinors, built from underlying L=1/2 ladder operators aligned so that the spinor angular momenta, Le and Lv, are maximal. We identify the CF's quantum numbers as the angular momentum L in (LeLv)L, its magnetic projection mL, the electron number N [Lv=(N1)/2], and magnetic ν spin, mν=LeLv. Translationally invariant FQHE states are formed by fully filling p subshells with their respective CFs, in order of ascending L for fixed Le and Lv, beginning with the lowest allowed value, L=|mν|. CF subshells are contained entirely within the first Landau level (FLL). Alternatively, we provide an equivalent hierarchical wave function in which the underlying objects are vortices with Lv=p2, correlated pairwise via Lvi·Lvj. We show that CFs can be written as a valence operator carrying the angular momentum quantum numbers L,mν,mL acting on a scalar half-filled intrinsic state. This scalar state serves as the vacuum for the valence electron (b,b̃) and vortex (v,ṽ) ladder operators that create FQHE states. With respect to this vacuum, FQHE states can be grouped into ν-spin multiplets mirror symmetric around mν=0, in which N is held constant. mν0 states have a net electron particle or hole number. Particle-hole conjugation with respect to this vacuum is identified as the mirror symmetry relating FQHE states of the same N but distinct fillings ν=p/(2p+1) and ν¯=p/(2p1), e.g., 2/52/3. Alternatively, mirror symmetric ν-spin multiplets can be constructed in which the magnetic field strength is held fixed: the valence states are electron particle-vortex hole excitations relative to the half-filled vacuum (mν>0) and their mirror conjugates (mν<0). Multiplet members are linked by the ν-spin raising/lowering operators, Ŝ±ν. Particle-hole (PH) symmetry—relating the N-particle FQHE state Ψ of filling ν=p/(2p+1) to the N¯-particle state Ψ¯ of filling ν¯=(p+1)/(2p+1), e.g., 2/53/5—is shown to be equivalent to electron-vortex exchange, bv and b̃ṽ. The N-particle states Ψ and ŜΨ¯ are connected by this mirror symmetry. In this construction, N¯N CFs of the state Ψ¯ occupy an extra zero-mode subshell that is annihilated by Ŝν. We link this structure, familiar from supersymmetric quantum mechanics, to the CF Pauli Hamiltonian, which we show is isospectral, quadratic in the ν-spin raising and lowering operators Ŝ±ν, and fourfold degenerate in Ψ,ŜνΨ,Ψ¯, and ŜνΨ¯. On linearization, it takes a Dirac form similar to that found in the integer quantum Hall effect (IQHE).

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
2 More
  • Received 12 February 2018
  • Revised 21 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.115140

©2018 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

W. C. Haxton1,2,*, Daniel J. Haxton1,3,†, and Byungmin Kang1,4,‡

  • 1Department of Physics, University of California, Berkeley, California, USA
  • 2Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA
  • 3KLA-Tencor, Fremont, California, USA
  • 4Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA

  • *haxton@berkeley.edu
  • danhax@gmail.com
  • bkang119@berkeley.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 98, Iss. 11 — 15 September 2018

Reuse & Permissions
Access Options
CHORUS

Article Available via CHORUS

Download Accepted Manuscript
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review B

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×