Relation between FQHE Plateau Width and Valley Energy

We have investigated the Fractional Quantum Hall Effect (FQHE) on the fundamental Hamiltonian with the Coulomb interactions between normal electrons without any quasi particle. The electron pairs placed in the Landau orbitals can transfer to many empty orbitals. The number of the quantum transitions decreases discontinuously when the filling factor ν deviates from the specific fractional number of ν0. The discontinuous decreasing produces the energy valley at the specific filling factors ν0 = 2/3, 4/5, 3/5, 4/7, 3/7, 2/5, 1/3 and so on. The diagonal elements of the total Hamiltonian and the number of the quantum transitions give the total energy of the FQH states. The energy per electron has the discontinuous spectrum depending on the filling factor ν. We obtain the function form of the energy per electron in the quantum Hall system. Then the theoretical Hall resistance curve is calculated near several filling factors. Therein the quantum Hall plateaus are derived from the energy valleys. The depths of the energy valleys are compared with the widths of the quantum Hall plateaus appearing in the experimental data of the Hall resistance. Our theoretical results are in good agreement with the experimental results.


Introduction
The quantum Hall effect is derived by the total Hamiltonian T H of a many-electron system which is composed of the single electron Hamiltonian 0,i

H
of the i-th electron and the Coulomb interaction between electrons as follows: ( ) ( ) ( ) where * m , e, i p and N are the effective mass, the elementary charge, the momentum and the total number of electrons.Therein ( ) z indicates the potential of the z-direction which confines the electrons to an ultrathin conducting layer.Also

( )
U y is the electric potential along the Hall voltage (y-direction).The vector potential, A , has the components, ( ) where B is the strength of the magnetic field.The last term of Equation ( 2 s is the z-component of the i-th electron spin operator.The details have been explained in the previous papers [1]- [14].When the Coulomb interaction between electrons is ignored in the quasi-2D electron system, the Hamiltonian of the single electron is exactly diagonalized same as in the Landau solution.At a filling factor 1 ν < , all the electrons are placed in the Landau orbitals with the Landau level number 0 L = .The residual Landau orbitals ( 0 L = ) are empty.So there are various electron-configurations in the Landau orbitals.We divide the total Hamiltonian T H into the diagonal part D H and the non-diagonal part I H as follows; We define two symbols W and C which are the expectation values of T H and the Coulomb energy, respec- tively.We call C "classical Coulomb energy" (the expectation value of the Coulomb interaction).Then the classical Coulomb energy becomes a minimum for only one electron configuration in the Landau orbitals.This property has been proven in the previous paper [9].That is to say W becomes a minimum at the only one electron configuration.The electron configuration gives the single ground state for each value of ν .The residual Coulomb interaction H I (non diagonal part of H T ) produces many quantum transitions from the ground state.We have examined the perturbation energy via these quantum transitions in details in the previous papers [1]- [14].Then all the electron pairs placed in the nearest Landau orbitals can transfer to all the empty orbitals at the specific filling factors 0 ν .When the filling factor ν deviates from the fractional numbers 0 ν the number of the transitions abruptly decreases.The reason comes from the combined effects of the momentum conservation along the x direction (current direction), the most uniform electron configuration and the Pauli exclusion-principle.The abrupt decreasing of the transition number yields the valley structure in the perturbation energy.That is to say the energy ( ) χ ν of the nearest electron pair takes a minimum at 0 ν ν = and the energy ( ) gives the energy gap (valley depth) as proven in the previous papers [9] [12] [13].
The total energy of the quantum Hall system is the sum of W (expectation value of T H ) and all the pair energy of electrons (placed in nearest orbital pairs and more distant orbital pairs), because the Coulomb interaction works between two electrons.We will study the function form of the expectation value W in the next section.Then we get the energy spectrum of the quantum Hall system which is quite different from the Halperin result [15].The valley depth in the pair energy and the function form of W give the quantum Hall plateaus at the specific filling factors 0 ν .The theoretical results are in good agreement with the experimental data.

Expectation Value of the Total Hamiltonian and Its ν-Dependence
We describe the expectation value of the total Hamiltonian T H by the symbol W which is the sum of the single electron energies and the expectation value of the classical Coulomb energy as follows; ( ) ( ) ( ) , , where ( ) E p is the single electron energy of the i-th electron and ( ) is the expectation value of the Coulomb interaction between electrons.The total number of Landau states with 0 L = is equal to 2π eB d   where  and d are the length and the width of a quantum Hall device respectively.The total charge of electrons at the filling factor ν is the product of ( ) e ν × − and the total number of Landau states.The total charge is divided by the area d  , and then we obtain the charge density as Figure 1 shows one of the experimental data [16].The upper figure indicates the Hall resistance H R divided by the Klitzing constant . The value of H K R R is equal to 1 ν which is almost proportional to the magnetic field strength B except the Hall plateau regions.This property is easily seen by comparing the data with the red line.That is to say, Bν is nearly equal to the constant value.Equation ( 5) means the charge den- sity σ to be proportional to Bν .Accordingly the macroscopic Coulomb energy ( ) σ may be treated to be a constant value in the experiment of Figure 1.
We next examine the microscopic charge-distribution in more details.Figure 2 shows the electron configuration with a minimum classical-Coulomb-energy at 2 3 ν = . Therein the bold lines express the occupied orbitals with electron and the dashed lines indicate the empty orbitals.The electron pair located at the orbitals AB is one example of the nearest-electron-pairs.Two electrons placed at B and C show the second nearest pair.The electron pair at A and C is the third nearest pair.The pair at A and D is the fourth nearest pair and so on.The classical Coulomb energy between two electrons is expressed by the symbols , , ξ η ς and κ respectively as in Fig- ure 2. Therein ξ is the largest one of the classical Coulomb pair energy, η is the second largest, ς is the third largest, κ is the fourth largest and so on.
The classical Coulomb energy between the pair (A, C) is weakened by the screening (shielding) effect of electron B. Also the classical Coulomb energy between the pair (A, D) is weakened by the screening effect of electrons B and C. Accordingly the ν-dependence of the classical Coulomb energy mainly comes from the first nearest and the second nearest pairs.The number of the more distant pairs (third, fourth, fifth and so on) are enormous many.The total number of electron pairs is ( ) . On the other hand the total number of the first and second nearest pairs is N. Accordingly the residual energies (namely the sum of all the more distant pair  and that between the second nearest pairs is equal to 2 N η ⋅ . Accordingly, the total classical Coulomb energy ( ) (Case of 4 7 ν = ) Figure 4 shows that the number of the first nearest pairs is equal to ( ) N and the number of the second nearest pairs is equal to ( ) . The total classical Coulomb energy is given by (Case of 5 7

ν =
) The number of the first nearest pairs is equal to ( ) (Any case of 1 2 1 ν < < ) We calculate the classical Coulomb energy for a general case of r q ν = ( ) As it is proven in Ref. [9] the electron-configuration with the minimum classical Coulomb energy is constructed by repeating the representative unit-configuration where r electrons exist in sequential q Landau orbitals.The number of empty orbitals per unit-configuration is q r − .All the empty orbitals are separated by one or more filled-orbitals at 1 2 1 ν < < .That is to say, all the empty orbitals are isolated as seen in Figures 2-5.Therefore the q r − sec- ond-nearest pairs exist per unit-configuration due to the presence of the q r − empty orbitals.The total num- bers of the first and the second nearest pairs is equal to the total number of electrons as easily seen in Figures 2-5.Therefore the number of nearest pairs becomes ( ) 2 r q r r q − − = − per unit-configuration.The total num- ber of nearest electron pairs is equal to ( )( ) ( )


This equation is expressed by using ν as , , 2 for From Equations ( 4) and (10) the expectation value of the total Hamiltonian is obtained as The single electron eigenenergy ( ) E p has been investigated in the previous papers and the result is the following form.
where λ expresses the ground state energy along the z direction (direction of the thickness in the thin con- ducting electron channel).Also

( )
i U α is the potential along the y direction (Hall voltage direction).Substitu- tion of Equation (12) into Equation ( 11) yields the expectation value of the total Hamiltonian as follows: We put together the constant parts as follows; ( ) Therein f is the constant value as where U is the mean value of the potential along the y-direction.Equation (13b) gives the function-form of W, which is illustrated in Figure 6.
The function W depends linearly upon 1 ν and the proportional coefficient ( ) N ξ η − − is negative, because the classical Coulomb energy between the first nearest electron pair, ξ , is larger than that between the second nearest pair, η .Thus the expectation value of the total Hamiltonian W changes continuously with 1 ν as in Figure 6.Accordingly the classical Coulomb energy has no energy-gap and so cannot produce the plateaus of Hall resistance.The confinement of the Hall resistance comes from another reason as studied in the previous papers [6] [9] [12] [13].The allowed transitions of electron-pairs decrease abruptly when the filling factor ν deviates from the specific filling factor 0 ν .This structure is named "valley structure".Summation of the valley energy and W gives the energy spectrum of the quasi 2D-electron system as examined in the next section.

ν-Dependence of the Total Energy
We already calculated the energy of electron pairs placed in the nearest orbitals by employing the perturbation calculation in the previous papers [1]- [13].The exact pair energy per electron is expressed by the following symbols ( ) χ ν and ( ) g ν where ( ) χ ν means the exact pair energy of the electrons placed in the nearest Landau orbitals and ( ) g ν means that of the more distant pairs.Then the total energy per electron ( ) ε ν is the sum of ( ) χ ν , ( ) g ν and the expectation value ( ) W N γ ν = as follows: The exact pair energies are obtained by summing all orders of the perturbation energies as follows: ( ) ( ) ( ) ( ) Consequently the total energy ( ) T E ν of the quasi 2D electron system has been expressed as Equations ( 13b) and (14a, b) derive the following relation: Accordingly the total energy is a sum of the following five terms: Also the energy per electron is given by If we change the gate voltage, then the value of the potential U varies.Accordingly the value of b can be controlled by changing the gate voltage.
We examine the higher order perturbation energies.In the previous articles [9] [12] [13], we calculated the pair energy for a filling factor with an even number denominator.Therein all the quantum transitions from the nearest pairs are forbidden at the filling factors ( ) ( ) ( ) . Therefore all order perturbation energies of the nearest electron (or hole) pair are zero; which gives the exact pair energy of the electrons (or holes) placed in the nearest neighboring Landau orbitals as follows; ( ) 0 for 2 1 2 and 1 2 Next, we investigate the pair energy at the filling factor with an odd number denominator.The results are ( ) ( ) Equations 25(a)-(d) mean that the n-th order term has the multipliers ( ) ( ) n j j − + , respectively.These multipliers become small for large n.We show several examples as follows; ( 1 2 1 0.04, 2 1 0.16, 1 2 1 0.11 for 3, 2 1 2 1 0.008, 2 1 0.064, 1 2 1 0.037 for 4, 2 The smallness of the multipliers means that the second order term is a main part of ( ) χ ν .We write again the second order term ( ) χ ν for various filling factors: We calculate the pair energies in the neighbourhood of ( ) ( ) ( ) For arbitrary fractional number ν , we can calculate ( ) 2 χ ν by using the same procedure.When the denomi- nator of the fractional number is large, the total number of the quantum transitions is calculated by a computer.
Because the higher order perturbation energy includes the small multipliers, ( )

Spectrum of the Total Energy versus Filling Factor
The energy spectra are examined which is given by Equation (30).Therein the term g indicates the non-nearest pair energy.Any non-nearest pair interleaves one or more Landau orbitals inside the pair.We have examined this effect in details in the article [12].The energies of the non-nearest pairs are smaller than that of the nearest pairs for 2 ν < .Accordingly we may ignore the ν-dependence of ( ) g ν in Equation (30).We draw four graphs of ( ) ε ν in the neighbourhood of ν = 2/3, 1/2, 2/5, 4/7 in Figure 7. , for the filling factors with odd number denominators is the same as that with even number denominators.So the difference of ( ) 2 χ ν between even and odd number denominators is negli- gibly small in the neighbourhood of 2 3 ν = .The energy spectra in Figure 7 and Figure 8 are drawn separately in the neighbourhood of the four filling factors.Some readers may want to know the spectrum in a wider region of the filling factor.We show it in Fig- ure 9.The ν dependence of ( ) ε ν (see Equation (30)) is induced by only two parameters Z and a which de- pend upon the size, thickness and shape of a quantum Hall device.(Note 1: ( ) The other parameters , , b g  don't yield ν -dependence.)We draw the energy spectrum in Figure 9 for the parameter ratio 0.5 Z a = as an example.We find many ranges of absent vertical-bar in Figures 7-9, because the present author doesn't calculate the value of ( ) χ ν yet in these ranges.Of course we can calculate ( ) 2 χ ν inside the ranges by using a computer and get the more dense vertical bars.
Hitherto, a few theorists have calculated the energy spectrum of FQH states.As an example Halperin's result [15] is shown in Figure 10 which has many cusps in the energy spectrum.The function-shape of Halperin's result is quite different from that of the present theory.Our theoretical spectrum-form is important to yield the Hall resistance curve in many experimental data as will be clarified in the next sections.zero in several ranges of magnetic field.But the actual experiments employ devices with impurities and lattice defects and are carried out at a finite temperature.Therefore, the diagonal resistance is very small but not zero.So, we roughly estimate the width ( ) W ν from the experimental data as follows; we take the width where XX R is lower than the green line in Figure 16.

Behaviour of Hall Resistance
Then, the experimental widths are Our second order calculations give the energy gaps (depths of the valleys) as; Neglecting the magnetic field dependence of Z and using the approximate relation H Z Z ≈ , Equations (47) gives the theoretical ratio of the energy gaps (depths of the valleys) as follows:  Thus, the theoretical result, namely ratio (48), is in good agreement with the experimental result (49) in spite of our rough estimation.

Discussion and Summary
There are two types of the traditional theories in the investigations of FQHE: one employs the quasi particle with a fractional charge [15] [26]- [29] and the other employs the quasi particle which is an electron binding to an even number of flax quanta namely composite fermion [30]- [41].The original Hamiltonian is described by the normal electrons and therefore the quasi-particles should be expressed by the wave function of many electrons.The wave function is unknown.The other problems are discussed in the Appendix.The present article has investigated the FQHE on the basis of the fundamental Hamiltonian without any quasi particle.When the filling factor deviates from the specific fractional numbers, the allowed quantum transitions decrease abruptly by the combined effect of the three properties which are the Fermi-Dirac statistics, the most uniform configuration of electrons and the momentum conservation along the x-axis.So the pair energy takes a minimum value at the specific filling factors discontinuously.Thus, the theoretical energy spectrum has the valley structure which yields the Hall resistance confinements.The theoretical results are in good agreement with the experimental data.
) indicates the Zeeman energy where * g is the effective g-factor, B µ is the Bohr magneton

Figure 1 .
Figure 1.Hall resistance R H and diagonal resistance R in ultrahighmobility device in [16].

)1 3 NFigure 3 .
Next we estimate the classical Coulomb energy for ν = 3/5, 4/7, 5/7 where the electron-configurations with the minimum classical Coulomb energy are shown in Figures 3The number of the first nearest pairs is equal to ( ) and the number of the second nearest pairs is equal to ( ) Then the total classical Coulomb energy is nearly equal to

5 Nas easily seen in Figure 5 .
and the number of the second nearest pairs is equal to ( ) Then the total classical Coulomb energy is

W 1 /Figure 6 .
Figure 6.Expectation value W of the total Hamiltonian.

Figure 16 .
Figure 16.Widths of the vanishing ranges in XX R in [17].