Fractional Quantum Hall States for Filling Factors 2 / 3 < ν < 2

Fractional quantum Hall effect (FQHE) is investigated by employing normal electrons and the fundamental Hamiltonian without any quasi particle. There are various kinds of electron configurations in the Landau orbitals. Therein only one configuration has the minimum energy for the sum of the Landau energy, classical Coulomb energy and Zeeman energy at any fractional filling factor. When the strong magnetic field is applied to be upward, the Zeeman energy of down-spin is lower than that of up-spin for electrons. So, all the Landau orbitals in the lowest level are occupied by the electrons with down-spin in a strong magnetic field at 1 2 < < ν . On the other hand, the Landau orbitals are partially occupied by up-spins. Two electrons with up-spin placed in the nearest orbitals can transfer to all the empty orbitals of up-spin at the specific filling factors ( ) ( ) j j 0 3 1 2 1 = − − ν , ( ) ( ) j j 4 1 2 1 + + and so on. When the filling factor ν deviates from 0 ν , the number of allowed transitions decreases abruptly in comparison with that at 0 ν . This mechanism creates the energy gaps at 0 ν . These energy gaps yield the fractional quantum Hall effect. We compare the present theory with the composite fermion theory in the region of 2 3 2 < < ν .


Introduction
The composite fermion theory introduces a quasi-particle named composite fermion which is an electron bound by even number 2 p of flux quanta.The theory explains the fractional quantum Hall effect (FQHE) to be the integer quantum Hall effect (IQHE) of the composite fermions with an integer filling factor n. Then the filling factor of electrons becomes The other cases give the electron filling-factors 2 3 ν ≤ .Thus the original composite fermion theory cannot explain the fractional quantum Hall states with 2 3 ν > .In order to remove this difficulty, an extended theory has been considered as follows: 1) At the filling factor 1 ν > , the IQHE of composite fermions are combined with the IQHE of electrons.
Even number of the flux quanta attach to some electrons and the other electrons are not bound by flux quanta.The former electrons are affected by the effective magnetic field and the latter by the applied magnetic field.
2) In the region of 2 3 1 ν < < even number of flux quanta attach to a hole.Therein the electrons are not bound by flux quanta.
3) The effective magnetic field is anti-parallel to the applied magnetic field at ( ) + .Thus the direction of the effective field, the kind of particle (hole or electron) and the number of attached flux quanta are assumed to change with variation of the filling factor.This changing is very artificial.There is another investigation considered by Tao and Thouless [13] [14].They investigated the FQH states where the Landau orbitals in the lowest level are partially filled with electrons.
We have improved the Tao-Thouless theory on the basis of the fundamental method.There are many configurations of electrons in the Landau orbitals.The sum of the Landau energy, classical Coulomb energy and Zeeman energy takes the minimum value at only one configuration of electrons for any fractional filling factor.In the configuration the nearest electron pairs can transfer to all the empty orbitals for the specific filling factors.
We consider the 2D electron system under a low temperature and a strong magnetic field throughout the present article.When the direction of the magnetic field is upward, the Zeeman energy of down-spin is lower than that of up-spin for electrons.So, in the region 1 2 ν < < all the Landau orbitals in the lowest level are occupied by the electrons with down-spin.On the other hand the Landau orbitals are partially occupied by up-spins.(Note: In the previous papers [15]- [19] we have already examined the case of a weak magnetic field.In the case both down-and up-spin-electrons partially occupy the lowest Landau orbitals.This special case appears in a weak magnetic field by adjusting the gate voltage.In this paper we investigate only the case of a strong magnetic field.)The up-spin electron pair placed in the nearest orbitals can transfer to all the empty orbitals of up-spin at + , ( ) ( ) ) ( ) and so on.These energy gaps can explain the fractional quantum Hall effect in the region 1 2 ν < < as clarified in the following sections.(We have already succeeded to obtain the energy gaps for the specific filling factors in the regions 1 ν < and 2 ν > in the previous articles [20]- [30].)

The Fundamental Properties of a Quasi-2D Electron System
Figure 1 shows a quantum Hall device where the electric current flows along the x-axis and the Hall voltage appears along the y-axis.Therein the magnetic field is applied in the z-direction.
The narrow potential

( )
W z along the z-direction is expressed in Figure 2. Also Figure 3 shows the poten- tial ( ) U y of the y-direction.Therein the voltage 2 1 V V − is not zero because the confinement of Hall resistance is realized under a nonzero value of 2 1 V V − .The value of the Hall voltage is extremely larger than the potential voltage in the FQHE.So we cannot employ the x -y symmetry.Also we should take the potential The vector potential, A , has the components, ( ) where B is the strength of the magnetic field.(Here we cannot use the symmetric vector potential because of the potential ( ) U y .)We express the single electron Hamiltonian, 0 H , in the absence of the Coulomb interac- tion between electrons as, where ( ) s is the z-component of electron spin operator as, The potential along the x-axis doesn't exist in the Hamiltonian 0 H . (The impurity effect is ignored.)There- fore the eigen-states along the direction x is the plain wave.Then the Landau wave function of the single- electron is given by where ( ) is the wave function of the ground state along the z-direction, L H is the Hermite polynomial of L -th degree, L u is the normalization constant and  is the length of a quasi-2D electron system as shown in Figure 1.The integer L is called Landau level number hereafter.The momentum p of the x-direction satisfies the periodic boundary condition, and is related to the value J α as in Equation (4d).The eigen-energy is given by where λ is the ground state energy along the z-direction and ( ) is the potential energy in the y-direction.The energy difference between two Zeeman levels is equal to Here the effective g-factor for GaAs is about 0.22 times the g-factor of electron in vacuum, namely, 0.44 g * ≈ .The effective mass m * for GaAs is 0.067 times the electron mass m in vacuum.Therefore the energy difference between L and 1 L + Landau levels is about 67 times the Zeeman split energy for GaAs.
Many investigations of the FQHE have used the symmetric property between the x and y directions.However all the actual experiments have carried out in a nonzero voltage 2 1 V V − .Accordingly the potential

( )
U y cannot be ignored.
If we take the other types of the vector potential as ( ) the eigen-function in the y-direction is not a plain wave because of the y-dependence of ( ) U y .Consequently the actual quantum Hall system has no x y − symmetry.
Our treatment takes the potential ( ) U y into consideration which is the appropriate treatment in an actual system.In a many-electron system the total Hamiltonian is given by the following equation as where N is the total number of electrons, ε is the permittivity and 0, i H is the single particle Hamiltonian of the i-th electron without the Coulomb interaction as, ( ) ( ) ( )  , ; , , ; , ,  !  , ,  , , These states are the eigen-state of 0, 1 The expectation value of the total Hamiltonian is expressed by ( ) , , ; , , ; , , which is given as: , , ; , , ; , , , , , ; , , where C is the expectation value of the Coulomb interaction as follows: , ; , ,  , , ; , ,  4π   , , ; , ,  d d d  d d d Hereafter we call ( )  ) ( ) ( ) where H is composed of the off-diagonal elements only.Accordingly the total Hamiltonian T H of the quasi-2D electron system is a sum of D H and I H as Because the Coulomb interaction depends only upon the relative coordinate of electrons, the total momentum along the x-direction conserves in the quasi-2D electron system.That is to say the sum of the initial momenta i p and j p is equal to that of the final momenta i p′ and j p′ : We study the configuration of electrons in the Landau orbitals.The most uniform configuration of electrons is uniquely determined for any filling factor except at the both boundaries.The boundary effects can be neglected in a macroscopic system.
As seen in Equation ( 8), the 1 2 z s = − state has an energy lower than the 1 2 z s = state for 0 B > .There- fore at 1 2 ν < < all the Landau orbitals with 0 L = are occupied by electrons with down-spin and the orbitals are partially occupied by electrons with up-spin in a strong magnetic field and a low temperature.We introduce the total number M of the 0 L = orbitals and also express the number of electrons with down-spin and up-spin by N ↓ and N ↑ respectively.Then we get the following relations in 1 2 ν < < as The most uniform configurations will be examined for the case of 3 2 2 ν < < in Section 3 and for 1 3 2 ν < < in Section 4.

Electron Configurations and Energy Gaps for 2 3 2 < < ν
As an example we examine the FQH state with 5 3 ν = . Equation (15d) becomes Then the most uniform configuration of up-spin electrons is the repeat of (filled, empty, filled) for 5 3 ν = .
where p ∆ is the momentum transfer.All the allowed transitions are illustrated by the blue allow-pairs in Figure 4.
Accordingly the transfer momentum takes the following value: We introduce the following summation Z .
As shown in Figure 4, the transfer-momenta from AB (up-spin electron-pair) satisfies Equation (18).Then the number of the transfer-momenta is 1/3 of the total orbitals.Accordingly the perturbation energy AB ς of the pair AB is expressed by using Z as follows; ( ) because the momentum-interval, 2π  , is extremely small in a macroscopic size of a quantum Hall device.
The total number of the nearest electron pairs with up-spin is 1/2 of N ↑ .Therefore we obtain the nearest pair energy per up-spin-electron ε ↑ as ( ) When the filling factor deviates from 5 3 ν = , the electron configuration at 5 3 ν ≠ changes from the regular repeating of (filled, empty, filled).Accordingly the number of the Coulomb transitions decreases abruptly because the changing disturbs the Coulomb transitions.As an example, the 42 25 ν = state is illustrated in Figure 5 where the nearest orbitals with up-spin are indicated by red and brown colours.The sum of the nearest electron pairs with up-spin is ( ) The number of electrons with up-spin is seventeen in a unit sequence.Therefore the nearest pair energy per up-spin electron is ( ) ( ) When the filling factor ν is ( ) ( ) (s is a positive integer), the sum of the nearest-pair-energies inside the unit sequence is The filling factor ( ) ( ) is larger than 5/3.The number of up-spin-electrons inside a unit length is equal to ( ) s + and therefore the pair energy per up-spin-electron is given by ( When s becomes infinitely large, ε ↑ approaches ( ) ( ) Next we consider the filing factor 38/23 which is smaller than 5/3.The most uniform configuration is illustrated in Figure 6.
In this case, the sum of the nearest-pair-energies inside the unit sequence is ( ) At ( ) ( ) − , the sum of the nearest-pair-energies inside the unit sequence is ) Thus the energy gap appears between the energy value at 5 3 ν = and the limiting value from the left and right sides: At the filling factor ( ) ( ) + , the energy gaps are listed in the fourth column of Table 1.We consider the other cases.Figure 7 shows the most uniform configuration of electrons at the filling factor 8 5 ν = . The x-, y-, z-directions are indicated at the upper-left of Figure 7.The unit configuration is composed of five Landau orbitals and three electrons with up-spin.The number of the allowed transitions is two per unit length.Accordingly the perturbation energy AB ς of the pair AB is expressed by Z as ( ) The total number of the nearest electron pairs with up-spin is 1/3 times N ↑ .Therefore the nearest pair energy per up-spin-electron is When the filling factor deviates from   There are five up-spin-electron pairs placed in the nearest orbitals inside a unit length as in Figure 8.The number of allowed transitions is eleven for the pair AB, nine for EF and seven for CD in a unit length.Therefore the perturbation energies are obtained as follows: The sum of these pair energies is ( ) The number of electrons with up-spin is sixteen in a unit length and then the nearest pair energy per up-spinelectron is ( ) ( ) We examine more general cases of ( ) ( ) − .In the filling factor, the sum of the nearest-pair- energies inside a unit length is   The number of the nearest electron pairs with up-spin is 1/4 of N ↑ .Therefore the nearest pair energy per up-spin-electron is ( ) The total number of the nearest electron pairs with up-spin is 1 j times N ↑ .Therefore the nearest pair en- ergy per up-spin-electron is where the momentum transfer p ∆ takes the following values as ( ) . Then the second order perturbation energy of the hole pair AB is given by ( ) ,   3. The limiting values from both sides are calculated and written in the third column of Table 3. Subtractions of the limiting value from ( ) 0 ε ν ↑ give the energy gaps which are listed in the fourth column of Table 3. Tables 1-3 show the energy gaps at many filling factors.Thus the present theory can explain the confinement of the Hall resistance in the region of 1 2 ν < < .

Filling Factors with Even Integer for the Denominator
We examine the 7 4 ν = state as an example with even integer for the denominator of the filling factor.Figure 14 shows the most uniform configuration at 7 4 ν = .
There are many electron pairs in Figure 14.The pair AB is an example of the nearest electron pair.The . When the filling factor ν deviates from 0 ν , the number of allowed transitions decreases abruptly in comparison with that at 0 ν .This mechanism creates the energy gaps at 0 ν .These energy gaps yield the fractional quantum Hall effect.We compare the present theory with the composite fermion theory in the region of 2 -[12].The case of n = 1 . When the filling factor ν deviates from 0 ν , the number of allowed transitions decrease abruptly in comparison with that at 0 ν .This mechanism creates the en-
y and ( ) W z indicate the potentials confining electrons to an ultra-thin conducting layer.Therein m * is an effective mass of electron and momentum.The last term of Equation (2) indicates the Zeeman energy where g * is the effective g-factor, B µ is the Bohr magneton

Figure 4
Figure4shows the electron configuration in a 3D view where the x , y , z axes are drawn in the upper-left of the figure.Therein the tilted lines with the x-direction express the Landau orbitals of the lowest level schematically.All the orbitals are filled with down-spin electrons for a strong magnetic field because of the Zeeman energy.The up-spin electrons occupy the red-coloured orbitals.The empty orbitals for up-spin are drawn by dashed blue lines in Figure4.This electron configuration of up-spin has the minimum value for the classical Coulomb energy.We examine the quantum transitions via the Coulomb interaction I H .All the Coulomb transitions satisfy the momentum conservation along the x-axis.Figure4shows the quantum transitions from the electron pair AB
of the Coulomb transitions decreases abruptly because the electron configuration at 8 5 ν ≠ disturbs the Coulomb transitions.As an example, the 43 27 ν = state is illustrated in Figure 8.

. 9 .
The most uniform configuration is illustrated in Figure The perturbation energy of the pair AB is pair AB is obtained as
  is eliminated in the summation (50) because the diagonal element of I H is absent.The denominator in Equation (50) is negative and so H Z is positive.(The value of H Z is nearly equal to Z for the same magnetic field strength.)The interval of momentum transfer is very small for a macroscopic size of a device and therefore the perturbation energy, AB H ς , can be expressed by H 10/7 are illustrated in Figures 11-13, respectively.The perturbation energies, AB H ς , are also obtained by making use of H Z as follows: in the second column of Table

Table 1 .
Energy gaps for