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Fractional charge revealed in computer simulations of resonant tunneling in the fractional

Fractional charge revealed in computer simulations of resonant tunneling in the fractional
Fractional charge revealed in computer simulations of resonant tunneling in the fractional

a r X i v :c o n d -m a t /0608252v 1 [c o n d -m a t .m e s -h a l l ] 10 A u g 2006

Fractional charge revealed in computer simulations of resonant tunneling in the

fractional quantum Hall regime

E.V.Tsiper

School of Computational Sciences,George Mason University,Fairfax,VA 22030

Center for Computational Materials Science,Naval Research Laboratory,Washington,DC 20375

etsiper@https://www.wendangku.net/doc/9f2219026.html, (May 2,2006)The concept of fractional charge is central to the theory of the fractional quantum Hall e?ect (FQHE).Here I use exact diagonalization as well as con?guration space renormalization (CSR)to study ?nite clusters which are large enough to contain two independent edges.I analyze the conditions of resonant tunneling between the two edges.The “computer experiment”reveals a periodic sequence of resonant tunneling events consistent with the experimentally observed fractional quantization of electric charge in units of e/3and e/5.

accepted to Phys.Rev.Lett.(2006)

Perhaps,the most intriguing feature of the FQHE 1is the existence of quasiparticles whose electric charge is a simple fraction of the elementary charge e .2Quasiparti-cles of charge e ?=e/3and e/5have been ?rst observed experimentally in the ν=15fractional states,respectively,using resonant tunneling via a quantum an-tidot (a potential hill).3–5

Since the bulk fractional state is an insulator,the most interesting transport properties of the system are asso-ciated with the edges,particularly with tunneling into or between the edges.3–8Due to cluster size limitations,computational studies of edge physics have focused on the properties of a single edge,such as non-universality of the tunneling exponent 9,10or reconstruction of the charge density.11–14The properties of the Laughlin wave-function describing a dual-edge system have been studied in cylindrical 15and disk geometries.16Study of edge to edge tunneling through a bulk fractional state requires clusters large enough to contain two independent edges.Exact diagonalization (ED)of ?nite clusters has been very fruitful in helping to understand the physics of FQHE.2,9–16Ordinary electronic structure methods fail for this system because the kinetic energy is quantized by the magnetic ?eld.ED,or “Full CI”in quantum chemi-cal terminology,imposes severe restrictions on the cluster size,since the dimensionality of the Hilbert space grows exponentially with the number of particles N :

L N ≈

12πLf (1?f )

1

2.

In the past we have perfected the Lanczos

technique,17,18both Hermitian 19,20and not,21–23to work with matrices up to ~109×109.Equation (1)translates this into about N =12particles at f =12.Whereas exact solutions for up to N =22are sometimes possible,20N >~12normally require approximate meth-ods.Here I use ED and also an approximate method to model the resonant tunneling experiments.3–5

In 3–5a periodic sequence of resonant tunneling events was observed as either the magnetic ?eld H or the back-gate voltage V BG were varied.The tunneling events are thought of in terms of a quasiparticle tunneling through the bulk of the fractional state between the outer edge of the sample and the inner edge formed around the an-tidot.The periodicities ?H and ?V BG were related 3to the quasiparticle charge e ?.

190 200 210 220 230 240 250 260 270 280 290

M

E M

??E E M M

Fig.1Exact ground-state energy of N =12electrons in 2D con?ned by Coulomb attraction to a uniformly-charged

annulus (inset)of charge density σ=1

5

of the ?lled Landau level.The units are e 2

/?H .The upper panel also shows CSR results for K =104and 105.Full Hilbert space dimensionality is 108?109and varies with M .26The 2

E M [the dotted purple line].

In order to mimic the experimental setup I consider a planar FQHE sample with two unconnected edges (in-

1

set in Fig.1).N electrons in the lowest Landau level are con?ned by the potential of a uniformly-charged disk with a hole in the center,positioned in the plane of the two-dimensional(2D)electron gas.The positive charge densityσand the inner radius R1of the disk are free pa-rameters.The outer radius R2is always chosen such that the whole system is neutral.The electronic densityρ(r) con?nes itself between R1and R2,falling o?sharply be-

yond this range.Settingσto a fractionν=1

5,etc.of

the densityσ1of the completely?lled Landau level con-

trols the fractional state,withρ(r)approachingνσ1(for N→∞)far from both edges.Near the edgesρ(r)is

known to exhibit oscillatory behavior thought to decay slowly into the bulk.13Such behavior prevents formation

of a well-de?ned density plateau between the edges in the ?nite clusters studied here numerically.

An increase in R1strengthens the antidot and expels charge from inside of the antidot towards the outer edge.

The charge expelled does not accumulate in the bulk be-cause of neutrality considerations and because of incom-pressibility of the bulk fractional state.I prefer to use the

“missing charge”Q=σπR21/e as a variable,instead of R1.When Q is continuously increased,the ground state

of the system reconstructs via a step-like process.The reconstruction events correspond to ground state degen-

eracies,when it costs no energy to transfer charge from the inner to the outer edge.This is precisely the condi-

tion for resonant tunneling through the antidot.

In the disk geometry the single-particle statesψm in

the lowest Landau level are characterized by the angu-lar momentum m=0,1,...,and the total angular mo-

mentum M= m is conserved.The Coulomb matrix elements are known.24,25The matrix elements of the con-

?ning potential are V m=V m(R2)?V m(R1),where V m(R)= ρ

ˉh c/eH is the magnetic length,Z= R2/4?2H,q±00=1,q±0m=(2m±1)q0,m?1,q±im=(2m?2i±1)q±i,m?1+2q±i?1,m?1?2q?i?1,m?1.

For a given set of N,Q,andσI?nd the lowest en-ergy E M(Q)at each M.The ground state energy is then E(Q)=min E M(Q).The ground state reconstruction events occur via level crossings of branches with di?erent M and lead to a step-wise function M(Q).The number p of the steps that occur per?Q=1may be related to the charge e/p that is moved from the inner to the outer edge per one reconstruction event.

Figure1shows E M(Q)for N=12,Q=2.The se-quence of sharp cusps on the right curve are the1

3σ1,whereas the states with

M=270±12,M=270±24,etc.are candidates for the ground state at di?erent Q.The quasi-periodicity with ?M=12is due to the approximate invariance of the an-tidot Hamiltonian with respect to the Laughlin’s quasi-

hole creation operator2A0,also known to be the gener-ator of in?nitesimal magnetic translations.27Applied to

an arbitrary many-electron wave functionΨ,it translates it in the angular momentum space,m→m+1.The total angular momentum then transforms as M→M+N:

ΨM+N≈A0ΨM(3) At Q=0the ground state occurs,approximately,at the Laughlin’s angular momentum28

M?(Q=0)=

N(N?1)

2ν?

Q(Q?1)

2m,where?H is the magnetic length.There-

fore,in a macroscopic system whose density approaches a constantνσ1in the bulk,the operator A0pushes the density out of the center,creating an e?ective positive charge e?=νe.This is an exact formal property of A0 but relates to the physical system via the approximate invariance(3).Indeed,Eq.(5)can be obtained from(4)

by applying A(Q/ν)

:

M?=M?(Q=0)+NQ/ν.(6) Figure1,therefore,suggests that the ground state of the1

3

.

Remarkably,the range of M that corresponds to the2

5

σ1shows up also for N=11,10, 9,and8,though it is less pronounced for smaller N. Figure1(b)exposes the tiny structure in E M(Q)by subtracting its greatest convex minorant

3

σ1 and2

?M=N/3at3

5

and3

5

σ1, M=216,K=105.The inset shows convergence with K when K is increased by a factor1.2at every iteration.“+”(red)and“×”(green)data sets indicate insigni?cance of the starting value of K except for the initial iterations.The stars (blue)give overlap with the exact eigenvector.The squares (purple)show the magnitude of projection of the exact eigen-vector onto the CSR subspace(that is,maximum overlap with any vector in the subspace),and as such characterize the qual-ity of the subspace.34

The resulting basis truncation is essentially many-body,and cannot be achieved by truncating or rotating the single-particle basis.ED performed in the subspace yields a variationally-stable ground state energy that con-verges to the exact value as K is increased.In practice, I do not keep K constant,but increase it from iteration to iteration(Fig.2,inset),monitor the convergence,and extrapolate as1/K→0.35

Figure1compares CSR results for K=104and105 against the exact solution.We see that the essential FQHE structure survives the basis truncations of sev-eral orders of magnitude,and that a reasonable accuracy is achieved for K=105in the whole range of M.Qual-itative results are obtained already with K as small as 104.

I used CSR to compute E M(Q)for Q in steps of0.25, interpolated between these points with a cubic spline and found min E M(Q)over M at every Q(Fig.3).I used K=200,000for N≤13,K=500,000for N=14, and K=1,000,000for N=15.K was doubled in some calculations where the extrapolation to1/K→0seemed ambiguous.

0 1 2 3

M

Q

σ = ? σ

1

1

3

N=11

N=12

N=13

N=14

N=15

150

200

250

300

350

400

450

0 1 2 3

Q

σ = ? σ

1

2

5

N=11

N=12

N=13

N=14

N=15

0 1 2 3

Q

σ = ? σ

1

3

7

N=13

N=14

N=15

Fig.3Angular momentum M(Q)of the ground state changes in steps as the“missing charge”Q is tuned continu-ously.The steps occur at the ground state degeneracies,when it costs no energy to move a quasiparticle from the inner to the outer edge,and can be associated with the resonant tun-neling between the edges through the bulk of the FQH state.

Figure3shows steps M(Q)for N from11through 15,as the“missing charge”Q is continuously increased. The left panel shows that three steps typically occur per ?Q=1for all N.The general slope is consistent with Eq.(6),and most of the steps in the left panel have ?M=N precisely.The data for the2

5

data shows that the change M→M+N occurs usually in two steps,although these steps are not always equal to N/2.The data for3

shows behavior similar to the2

7fraction.It could also indicate a genuine property of

the3

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1)/N?1≤m≤(M?M??1)/N+3N?2,beyond which

the occupation numbers are small.This restriction reduces the actual matrix size from~109to about1.6×107.In CSR calculations the above restriction on m is lifted,how-ever I used m≤50for technical reasons.

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34The CSR calculation in Fig.2took11minutes.A single point ED run for the same cluster(N=12,M=216)took 24hours.The CPU time comparison is meaningful only to

a degree,because ED calculations are disk intensive.

35Technically,the basis expansion procedure is possible since the Hamiltonian matrix is sparse and has a limited number of matrix elements for each row(con?guration).I“expand”

the con?gurations one by one in the order of decreasing their weight.For each con?guration a number of new con-?gurations related to it by a matrix element is formed and added to the basis.The process continues until the target K is achieved,and all the remaining unexpanded con?gu-rations are discarded.In order to enhance the K′/K ratio I limit the number of new con?gurations by considering only the matrix elements with?m=±1.Di?erent schemes for choosing“large”matrix elements are possible and yield similar results.

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4

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