s. v. iordanskiy- fractional quantum hall effect and vortex lattices

8/3/2019 S. V. Iordanskiy- Fractional Quantum Hall Effect and vortex lattices

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Fractional Quantum Hall Effect and vortexlattices.

Published JETP Lett.87,p669 (2008)

S. V. Iordanskiy

Landau Institute for Theoretical Physics, Russian Academy ofSciences,

S. V. Iordanskiy Fractional Quantum Hall Effect and vortex lattices
http://goforward/http://find/http://goback/


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introduction

The experimental discovery of Integer Quantum Hall Effect (IQHE)by v.Klitzing (1980)and Fractional Quantum Hall Effect (FQHE)by Tsui,Stormer and Gossard (1982) was one of the outstanding

achievements in condensed matter physics of the last century. Thequalitative physical explanation of IQHE was given soon afterexperimental discovery but the theory of FQHE is far from beingcomplete. In pioneer works of Laughlin (1981,1983) the variationalfunction for some odd Ll fillings was suggested. He put forward the

idea of the excitations with the fractional electron charge.

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other fractions

The other fractions existence was explained by the construction ofthe daughter states (Haldane 1983, Laughlin 1984, B.Halperin1984). The most successful phenomenological description was

given by Jains (1989-1990) model of composite fermions givingthe majority of the observed fractions. According to Jains model2d electrons in the perpendicular magnetic field are dressed by 2flux quanta of magnetic field opposite to the external one. Theinclusion of this additional field in a formalized theory gives

Chern-Simons Hamiltonian (B.Halpern, P.Lee, N.Read, 1993)



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plateau

However this model introduces artificial 6-fermionic model andthere are experimental fractions which can not be found by thismodel. QHE consists in the existence of the plateau of Hallconductivity near some special values of the electron density. At

integer fillings of Ll we have large energy gaps in high magneticfield with the macroscopic degeneracy of the levels. That results inthe electron motion along lines where the impurity potential isconstant. Those lines are closed near the minima or maxima of

impurity potential and therefore the corresponding electron stateshave not macroscopic current.



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plateau 2

If the electron chemical potential is somewhere between Ll thedensity is changed due to localized states without changing of theHall current defined by the delocalized states near the percolation

threshold. We see that for IQHE it is important the existence oflarge energy gaps and localized states developed in the weakimpurity potential due to the degeneracy of the one particle states.The experimental data near the fractional fillings are quite close tothose at integer fillings. Therefore the general physical picture

must be close for IQHE and FQHE.



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picture



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extension

I shall show how to remove the restrictions of the Jains model andto obtain a more general and more simple model with the standardCoulomb interaction. The main conception is associated with thepossibility to have topological textures in 2d electron system. Thevortices are wide spread in condensed matter physics. The simpleand general definition can be done using the canonicaltransformation for the field operators of second quantization(r) = ei(r)(r) + = +ei

with having vortex like singularities.



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zeta

curl = 2Kn,n

(r rnn) where rnn = n1 + n2 form alattice, K is some integer. I assume full spin polarization and omit

spin indices. It is evident that both ,+ and ,+ satisfy Fermicommutation relations. The problem to find and ei can besolved by Weierstrass function.



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zeta2

It is a convergent series

=

1

z +

Tnn=0 1

zTnn +

1

Tnn +

z

T2nn

where z= x+ iy, Tnn = n1 + n2 and 1, 2 are the minimal

complex periods of the vortex lattice.



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real

In the real representation = K(Re, Im) and = K

r

r0(Redx+ Imdy)

The phase factor ei

will be simple function on 2d plain for anyinteger K. The substitution of the above expressions into thestandard hamiltonian of the interacting electrons in magnetic fieldgives the new hamiltonian



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hamiltonian

H= 2

2m

+

i+ ec

A2

d2r+12

U(r r)+(r)|chi+(r)(r)(r)d2rd2r

where U(r) is Coulomb interaction, A(r) is the vector potential ofthe external uniform magnetic field. I take the gauge linear incoordinates. therefore A(r + ) = A(r) + A()-function has the same property (z+ ) = (z) + ().



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effective potential

If we introduce effective vector potential Aeff = A ce anytranslation along the periods of the vortex lattice makes a changein the effective vector-potential which can be removed by thechange of the gauge. Therefore the obtained hamiltonian isinvariant under magnetic translationsTm()(r) = (r + )exp

ie

cAeff()r

for any real period of the

vortex lattice.



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flux

It is easy to connect the effective vector-potential with theeffective

magnetic flux through the unit cell of the vortex lattice =

AeffdR = Aeff(1)2 Aeff(2)1 = B01 2 + K0where the flux quantum 0 =

2|e|c

, B0 is the external magneticfield normal to 2d plain.



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representation

Magnetic translations are non commutative and give rayrepresentations of ordinary translation group. The simple finiterepresentation and simple spectral properties are possible only forthe rational number of the flux quanta per the unit sell (E.Brown,

1964),(J.Zak, 1964) = l

N0 = B0s+ K0

l,N are integers, s is the area of the unit cell.Thus the situation isisomorphic to the case of the magnetic field with the rational fluxnumber per the unit cell plus the periodic magnetic field with thezero flux.


fi ld


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strong field

Consider the case of the weak interaction when kinetic one particlehamiltonian is the main term. That corresponds to the largemagnetic field when Coulomb interaction grows as

B0 and the

kinetic part grows as B0 and is possible to use a perturbationtheory. Any magnetic translation gives the state with the sameenergy and these translations does not commute. As aconsequence one has highly degenerate spectrum. In order to findthe one particle states one must put some boundary conditions.The simplest are generated by magnetic translations

Tm(L)(r) = (r)here L = L1, L2which define the sample size,like Born-v.Karmanconditions.


d


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groundstate

The simplest realization of the finite representation is the sampleofN N unit cells and the general case M1NM2N unit cells. Itmust be M1M2N

2 states of the same energy as the consequence ofthe magnetic translations non commutativity. The ground state

energy according to the perturbation theory isE0 = N

2M1M20 +12

U(r r) < +(r)+(r)(r)(r)d2rd2r

where 0 is the ground state energy for the kinetic part and theangle brackets denote the average over the filled states. If oneadds more electrons they go to the state with the largerenergy.The gaps must be found numerically.


l t d iti


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electron densities

The set of the observed electron densities depend on the value ofthe gaps, temperatures, and sample purity. To take into accountthese numerous factors is a difficult task. But it is easy to solve

the opposite problem- to find the vortex lattices corresponding tothe observed densities using the expression for the electron densityne =

B00

N

lNK .The observed fractions are given by the following tables.


l tti


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lattices

K = 2, l = 1q 1 2 3 -5 -2 -3 -4 4

13

25

37

59

23

35

47

49

12

That fractions correspond to the celebrated Jains rule. Especiallymust be noted the half filling of the Ll corresponding to limNand the effective flux equal to zero. In this case we have anordinary group of the translations with a normal band structure.

This state can be achieved by the increasing of electron densityN> 0 as well as by decreasing N< 0. This circumstance mustcreate the symmetry vanishing of the gap for electron and holes.


d


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end

Other observed fractions correspond to the table

K = 1, l = 1q -4 4 2

43

45

23

where one have the double of the fraction 2/3, and also

K = 1, l = 2q -7 -5 5 2

7553

57

12

here one have not observed double of the fraction 1/2 with thegap. The exclusion of the doubles requires extensive numerical

calculations.S. V. Iordanskiy Fractional Quantum Hall Effect and vortex lattices

end2


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end2

All observed fractions correspond to vortex lattices with 1 or 2 fluxquanta per unit cell. The one particle ground state degeneracy forthe periodic magnetic field was found initially in the work ofB.Dubrovin and P.Novikov(1980).


s. v. iordanskiy- fractional quantum hall effect and vortex lattices

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