arXiv:2105.05625 · Physical review. B./Physical review. B

Adiabatic construction of hierarchical quantum Hall states

Martin Greiter, Frank A. Wilczek

Published Sep 20, 2021• 8 citations

Abstract

Intuitive arguments and numerical simulations have long been used to argue for the incompressibility of fractionally quantized Hall states at the most prominent odd-denominator Landau-level filling fractions $\ensuremath{\nu}$ = $p$/($p$ ($m$ - 1) + 1), where $p$ is an integer and $m$ an odd integer. Here, the authors strengthen the case by deriving these states more rigorously via adiabatic localization of magnetic flux onto the particles. They arrive at wave functions suggested by Jain's composite-fermion picture.

DOI: 10.1103/physrevb.104.l121111

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arXiv:2105.05625 · Physical review. B./Physical review. B

Adiabatic construction of hierarchical quantum Hall states

Martin Greiter, Frank A. Wilczek

Published Sep 20, 2021• 8 citations

arXiv HTML PDF

Abstract

\ensuremath{\nu}

p

p

(

m

- 1) + 1), where

p

is an integer and

m

an odd integer. Here, the authors strengthen the case by deriving these states more rigorously via adiabatic localization of magnetic flux onto the particles. They arrive at wave functions suggested by Jain's composite-fermion picture.

DOI: 10.1103/physrevb.104.l121111

Related papers

Quantum Hall system in Tao-Thouless limit

Emil J. Bergholtz, A. Karlhede

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where ${L}_{1}$ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd $q$ , the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd $q$ , are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as ${L}_{1}$ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite ${L}_{1}$ ; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where ${L}_{1}$ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd $q$, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd $q$, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as ${L}_{1}$ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite ${L}_{1}$; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.

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Study of composite fermions: Beyond few particle systems

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N. Read, Edward H. Rezayi

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arXiv:cond-mat/9809384Physical review. B, Condensed matterMar 15, 1999943 citations

\ensuremath{\nu}

p

p

(

m

- 1) + 1), where

p

is an integer and

m

{L}_{1}

is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd

q

, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd

q

, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as

{L}_{1}

increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite

{L}_{1}

; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.

p

. There is a collective mode in these states whose phenomenology is in agreement with standard arguments coming, e.g., from the composite-fermion picture. At filling factor 1, the potential Fermi sea of composite fermions is replaced by a paired state, the Moore-Read state. This is most clearly seen from the half-flux nature of elementary excitations. We find that the hierarchy picture does not extend up to the point of transition towards a vortex lattice. While we cannot come to any definite conclusions, we investigate the clustered Read-Rezayi states and show evidence for incompressible states at the expected ratio of flux versus number of Bose particles.

k+1

-body interactions, for all integers

k>~1.

The remarkably simple wave functions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The

k=2

case is the Pfaffian. For

k>~2,

the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For

k=3,

these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors $\ensuremath{\nu}=2+3/5,2+2/5.$