Hamiltonian Description of Composite Fermions Magnetoexciton

A microscopic Hamiltonian theory of the FQHE, developed by Shankar and myself based on the fermionic Chern-Simons approach, has recently been quite successful in calculating gaps in Fractional Quantum Hall states, and in predicting approximate scaling rela

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a r 1999Hamiltonian Description of Composite Fermions:Magnetoexciton Dispersions

Ganpathy Murthy

Physics Department,Boston University,Boston MA 02215

and Department of Physics and Astronomy,Johns Hopkins University,Baltimore MD 21218

(February 1,2008)

A microscopic Hamiltonian theory of the FQHE,developed by Shankar and myself based on the

fermionic Chern-Simons approach,has recently been quite successful in calculating gaps in Fractional

Quantum Hall states,and in predicting approximate scaling relations between the gaps of di?erent

fractions.I now apply this formalism towards computing magnetoexciton dispersions (including

spin-?ip dispersions)in the ν=1

5,and 35and 35state,in contrast to the spin-polarized 1√2p 2p +1?il 2( j (q ×Πj )e ?iqx j )(1)where A,A ?refer to the annihilation and creation oper-ators of the magnetoplasmon oscillators,l =1/√

A microscopic Hamiltonian theory of the FQHE, developed by Shankar and myself based on the fermionic Chern-Simons approach, has recently been quite successful in calculating gaps in Fractional Quantum Hall states, and in predicting approximate scaling rela

the CF,and a dipole piece which alone survives at

ν=1/2and has the value proposed

by Read20

.

(A number

of recent constructions

have

emphasized

this dipolar aspect23–25).

?We also?nd that that as q→0all transition matrix

elements of¯ρfrom the HF ground state vanish at

least as q2.

The?nal property is an essential property of physi-

cal charge density matrix elements from incompressible

liquid ground states in the LLL22.We will present argu-

ments in the next section to show that if one intends to

use the Hartree-Fock approximation ignoring constraints,

these properties of¯ρare essential.They make it plausible

that¯ρdoes not su?er vertex corrections.

The Hamiltonian of the low-energy sector(dropping

the magnetic moment term)is

H=

1

(2π)2

v(q)¯ρ(?q)¯ρ(q).(2)

where v(q)is the electron-electron interaction.We cir-

cumvent the fact that¯ρis to be trusted only for small

q as follows.Consider real samples which have a?nite

thicknessΛof the same order as l,so that the Coulomb

interaction is cuto?at large wavevectors26.It was re-

alized by Haldane and Rezayi27that this has a large

e?ect on the gap,while leaving the wavefunctions es-

sentially unchanged.We will focus on such interactions,

parametrized byλ=Λ/l for which numerical results for

the transport gaps are available28–30.The advantage is

that asλbecomes large only small-q matrix elements

of the density are invoked in computing gaps,and we

expect our theory to become more accurate.It is possi-

ble that beyond some largeλthe incompressible liquid

might cease to be the ground state27.Our theory,which

is based on a liquid state with uniform density,can be

expected to work up to thisλ.In fact,the magentoex-

citon dispersions can be used to infer these instabilities,

as will be shown later.

This hamiltonian is to be supplemented by n con-

straints which identify the physical subspace.We will

expand on this issue in the next section.In two

earlier papers we presented calculations of gaps for a

few fractions31in the Hartree-Fock approximation,and

tested certain scaling relations that arise naturally in the

our theory against numerical results obtained using CF

wave functions32.Emboldened by the good agreement we

found between the predictions of our theory and numeri-

cal results,in this paper I present results for magnetoex-

citon(ME)dispersions for the polarized FQHE states at

ν=1

5

,and3

3

,

and the use of CF wavefunctions for excited states38.

Field-theoretic approaches include the seminal work of

Lopez and Fradkin12for the collective modes of gapped

fractions based on a fermionic CS approach,and an RPA

treatment39based on the formalism of Halperin,Lee,and

Read(HLR)14.These?eld-theoretic approaches su?er

from the problem that the bare mass(actually the band

mass m b)12or a phemenological mass m?enters the dis-

persions,whereas in the LLL the scale should be set solely

by e2/εl.

Our approach essentially solves the problem of strong

correlation by rewriting the theory in terms of CFs,which

are the quasiparticles of the theory in the same sense as

those in Landau’s Fermi Liquid theory:The quasiparti-

cles can interact strongly with a Fermi-Liquid-like inter-

action,but matrix elements which scatter them out of

their states are small at low energies,allowing them to

be long-lived.Furthermore,the only scale in our low-

energy hamiltonian is indeed e2/εl,as desired.For the

reasons to be explained below,we can use a formalism

such as TDHF,which ignores vertex corrections,with our

formula for¯ρ.

It must be emphasized that the TDHF approach

of MacDonald35,36which includes many Landau levels

(LLs)is absolutely essential for …… 此处隐藏：41377字，全部文档内容请下载后查看。喜欢就下载吧 ……