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The alternative to the incompressible fractional charge in the quantum Hall effect Comments

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arXiv:cond-mat/0210238v1 [cond-mat.mes-hall] 10 Oct 2002

The alternative to the incompressible fractional charge in quantum Hall e?ect: Comments on Laughlin and Schrie?er’s papers.
Keshav N. Shrivastava School of Physics, University of Hyderabad, Hyderabad 500046, India Laughlin has found “exactly” the wave function which is ascribed to an excitation of fractional charge, such as e/3. We ?nd that the exactness of the wave function is not destroyed by changing the charge to some other quantity, such as the magnetic ?eld. Thus e/3 and H can be replaced by e and H/3. Therefore, the wave function need not belong to a quasiparticle of charge e/3.

Corresponding author: keshav@mailaps.org Fax: +91-40-3010145.Phone: 3010811. 1

1. Introduction An e?ort is made to see if there is an alternative to the interpretation of the fractional charge in the quantum Hall e?ect. Laughlin has written the wave function for the quasiparticle of fractional charge. We examine to see if the wave function can be interpreted to describe integer charge and the blame of the fraction can be thrown on some other quantity. Laughlin1 has proposed a variational ground state which is a new state of matter, a quantum ?uid, the elementary excitations of which are fractionally charged. The correctness of the wave functions is veri?ed by direct numerical diagonalization of the many-body Hamiltonian. We wish to examine the alternative interpretation of the wave function without destroying the exactness of the calculation. In particular, we examine whether some other variable can take the blame of the fraction instead of the charge. If we bring a ?ux quantum near a solenoid, according to Laughlin, charge is a?ected whereas according to the correct answer, the ?eld is a?ected. The charge density should automatically give the correct electric and magnetic vectors so that the magnetic ?eld should be automatically correct. There are two variables, the charge and the ?eld in the ?ux quantization. Therefore, Laughlin’s throwing the blame only on the charge is not correct and the automatic correction to the electric and magnetic vectors does not occur. We ?nd that several quantities arise as a factor of the charge so that the e?ect of the fraction need not be thrown on the charge. Since the charge multiplies the magnetic ?eld, the factor arising in the ?eld can also be read as a factor of the charge, then there is no trouble in de?ning a fractional charge. However, if charge is corrected without correcting the magnetic ?eld, apparently there is no trouble, except that the choice of the variale is not unique. In this comment, we explain the various choices available without disturbing the exactness and point out that ?eld is better than the charge. 2. Theory Let us consider a two-dimensional electron gas in the x ? y plane subjected to a magnetic ?eld along z direction. The eigen states of the single-body Hamiltonian can be

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written as, |m, n >. The cyclotron frequency is, hωc =? (eHo /mc). The magnetic length ? h is ao =(? /mωc )1/2 which upon substituting the cyclotron frequency becomes, h ao = ( hc 1/2 ? ) . eHo (1)

The energy levels of the type of a harmonic oscillator are produced when the applied ?eld is along the z direction, 1 H |m, n >= (n + )|m, n > . 2 (2)

The wave function of the lowest Landau level is written as a function of z = x + iy with |m > as an eigen state of angular momentum with eigen value m. The manybody Hamiltonian consists of, the kinetic energy with the vector potential included in the momentum, a potential due to the positively charged nuclei, V(zj ), and the Coulomb repulsion between electrons. The wave functions composed of states in the lowest Landau level which describe the angular momentum m about the center of mass are of the form, 1 ψ = (z1 ? z2 )m (z1 + z2 )n exp[(? )(|z1 |2 + |z2 |2 )]. 4 (3)

Laughlin generalized this observation to N particles by writing product of Jastrow functions, 1 ψ = {Πj<k f (zj ? zk )}exp(? Σl |zl |2 ) 4 (4)

which minimizes the energy with respect to f . If ψ is antisymmetric f (z) must be an odd function, f (z) = zm with m = odd. To determine which m minimizes the energy, we write, |ψm |2 = exp(?βφ) (5)

where β=1/m and φ is the classical potential energy which describes a system of N identical particles of charge Q=m with the neutralizing back ground charge density, σ=(2π a2 )?1 per unit area. This is the classical one-component plasma (OCP). The o solution of which is well known. For Γ=2βQ2 =2m > 140, the OCP is a hexagonal crystal and ?uid otherwise. Laughlin’s wave function describes a ?uid of density, σm = 1 m(2πa2 ) o 3 (6)

which minimizes the energy. The charge density generated by (6) should be equal to that of the background charge so that there is overall charge neutrality. If we de?ne a new value of ao , then ma2 is replaced by a2 . Then the e?ect of m is o new not on the charge but it is in the distance. That means that the e?ect of m can occur on, (i)h, (ii)c,(iii) e or (iv) Ho . Thus there are at least four candidates to absorb the e?ect of m. Let us eliminate the Planck’s constant and the velocity of light. Then it is possible to a?ect either e or Ho . It will surely be interesting if the e?ect of m is thrown on the velocity of light. Then, a2 = new h(mc) ? eHo (7)

where for m=3, the quasiparticles will travel with the velocity of 3c which is faster than light. In the expression (1), the velocity of light c is the value in vacuum so that the particles with velocity 3c will be travelling faster than light which will be noncausal or causality violating. Another interpretation is obtained by throwing the value of m on the magnetic ?eld, a2 = new hc ? . e(Ho /m) (8)

Therefore, the number m of Laughlin can be absorbed as a factor of magnetic ?eld such as, gHo = Ho /m. (9)

It is perfectly allowed to devide the magnetic ?eld by m to de?ne a new ?eld. Out of the four options available, i.e., h, c, e and Ho , Laughlin selected e so that, the e?ective charge becomes, eef f = e/m (10)

where h, c and Ho are kept constant and m is an odd integer. Therefore, Laughlin’s choice of variables is not unique. Let us ask, if some one had selected the velocity of light, out of the four variables, then what would have happened? The answer to this question is that we would have obtained the particles faster than light. 3. Exactness

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The projection of ψm for three and four particles onto the lowest energy eigen state of angular momentum 3m for m=3 and m=5 states is about 0.98 which shows the exactness of the wave functions. Therefore, the exactness of the calculation is not a?ected by changing the variable from e to H, h or c. The calculation is exact for any of the four variables. The total energy per particle can be written in terms of the distribution function of the one-component plasma (OCP),


Utot = π

o

e2 [g(r) ? 1]rdr ? (4/3π ? 1)2e2 /R r

(11)

where integration domain is a disk of radius R=(πσm )?1/2 . At Γ=2, g(r) = 1- exp[(r/R)2 ] which shows that for m <9, the Utot is deeper than for charge density waves. The excitations of ψm are created by piercing the ?uid at zo with an in?nitely thin solenoid and passing through it a ?ux quantum ??=hc/e, adiabatically. This treatment does not treat the magnetic ?eld correctly. Here again, f ield × area = hc/e (12)

so that the variable is e×?eld×area and hence the entire blame can not be thrown on e. It may appear that changing the e will automatically take care of the electric vector E and the magnetic ?eld H of the electromagnetic ?eld but there is H in the ?ux quantization which is unchanged. A change in the charge density should change both E and H. According to the adiabatic approximation, the ?ux quantum produces the quasiparticles. If this approximation is not valid, ?eld will be produced not the quasiparticles. In fact the variables, h and c are also very good candidates. Increased value of the Planck’s constant will increase the cyclotron absorption energy and increased c will give tycheons: noncausal faster than light particles. The e?ect of passing a ?ux quantum is to change the single-body wave function from, 1 1 (z ? zo )m exp(? |z|2 ) to (z ? zo )m+1 exp(? |z|2 ). 4 4 (13)

An approximate representation of these states is chosen by Laughlin for the quasielectron and the quasihole. Laughlin writes |ψ +zo |2 as exp(-βφ′) with β=1/m and φ′=φ-2Σl ln|zl ? 5

zo | where φ′ describes an OCP interacting with a phantom point charge at zo . The plasma screens this phantom by accumulating an equal and opposite charge near zo . The particle charge in the plasma is 1, rather than m so the accumulated charge is 1/m, i.e., meef f √ = 1. The energy required to create a particle of Debye length ao / 2 is given by, e2 π . ?D = √ 4 2 m2 ao (14)

Note that the quantity which enters is again m2 ao and not just e/m so that the charge can be changed from e to e/m or the charge may be kept constant at e and ao changed to m2 ao . Therefore, two equivalent possibilities exist. One is to de?ne the e?ective charge, eef f =e/m and the other is to keep e unchanged and change ao to aef f = m2 ao . The state described by ψm is incompressible because compressing it is equal to injecting particles and particles carry charge so the fractional charge will be destroyed. The incompressibility is not built in the hamiltonian and is extranuously imposed. It is easy to impose an external boundary condition that there is incompressibility but in the expression (14) such an incompressibility can not be enforced. Therefore, the incompressibility enforcement is not contained in the theory. The compressibility creates sound waves so the incompressibility causes the sound to be absent. This is equivalent to eliminating phonons, which has not been done, so the resistivity can touch zero value. In the BCS theory of superconductivity, the zero resistance is obtained not by introducing incompressibility but by eliminating phonons which makes the electrons attactive. We do not discuss the impurities or rotations because these are not contained in our hamiltonian. The Hall conductance is (1/m)e2 /h so that the e?ective charge is eef f =(1/m)e and the factor e/h is caused by the units. The origin of this e?ective charge is not clear because of the other candidates and for m=3, the charge e/3 is not generated by the hamiltonian discussed by Laughlin but the antisymmetry surely requires that m is odd, such as 1, 3 , 5, 7, ... etc. Therefore, odd aspect is a result of antisymmetry. Laughlin does not use spin so that there is spin-charge decoupling automatically built in the calculation. The spinless electrons may give unphysical results. Below eq.(13) it is argued that introducing a ?ux quantum ?φ=hc/e results into accumulation of charge 1/m. In fact, the charge need not 6

accumulate and only the ?eld is modi?ed as in (8). Therefore, the factor 1/m does not determine the charge of the excitations. In a di?erent context Laughlin has pointed out that the bulk modulus of the Hartree-Fock ground state was zero. The bulk modulus is actually ?nite, as is usually the case for Jastrow-type trial wave functions for helium as admitted in an errata2 . 4. Missing charge. If we agree to the choice of the charge and change it from e to e/3, then what happened to the remaining charge of 2/3? If we say that e remains e and H changes to H/3, then we need not look for the missing charge. The wave function (4) does not conserve charge. Laughlin does not provide any prescription to ?nd the missing charge. 5. Conclusions The exact wave function, the excitations of which are fractionally charged does not determine the quasiparticle charge uniquely. The blame of the fraction need not be thrown on the quasiparticle charge. The introduction of the alternative choice of variable, instead of charge, does not destroy the exactness of the calculation. The magnetic ?eld of the ?ux quantization condition has been left out. It is clear that Laughlin’s wave function can not explain the experimental observations of the quantum Hall e?ect. It may be pointed out that the composite fermion model (CF) which requires that the ?ux quanta be attached to the electron is also not correct.3?5 In the expression (13) of Arovas et al6 the quantity e? Aφ can be replaced by (e/m)Aφ . Now, multiply Aφ by g so that the quantity of interest becomes (e/m)gAφ with g=1. It is clear that (e/m) and (g/m) are not resolvable because (e/m)gAφ is exactly equal to e(g/m)Aφ . Therefore, whether the charge should be fractionalized or the unit ?ux should be corrected are not resolved. If charge is corrected, then fractional statistics will come otherwise the unit ?ux will be changed. This means that the fermion need not obey the “fractional statistics” and φo will be modi?ed. The correct theory of the quantum Hall e?ect is given in ref.7.

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6. References 1. R. B. Laughlin, Phys. Rev. Lett. 50, 1395(1983). 2. R. B. Laughlin, Phys. Rev. Lett. 60, 2677 (1988);61, 379(E)(1988). 3. M. I. Dyakonov, cond-mat/0209206. 4. B. Farid, cond-mat/0003064. 5. K. N. Shrivastava, cond-mat/0209666. 6. D. Arovas, J. R. Schrie?er and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984). 7. K. N. Shrivastava, Introduction to quantum Hall e?ect, Nova Science Pub. Inc., N. Y. (2002). Note: Ref.7 is available from: Nova Science Publishers, Inc., 400 Oser Avenue, Suite 1600, Hauppauge, N. Y.. 11788-3619, Tel.(631)-231-7269, Fax: (631)-231-8175, ISBN 1-59033-419-1 US$69. E-mail: novascience@Earthlink.net

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