Spin Polarization of Fractional Quantum Hall States with ν < 2

The spin polarization of a fractional quantum Hall state shows very interesting properties. The curve of polarization versus magnetic field has wide plateaus. The fractional quantum Hall effect is caused by the Coulomb interaction because the 2D electron system without the Coulomb interaction yields no energy gap at the fractional filling factor. Therefore, the wide plateau in the polarization curve is also caused by the Coulomb interaction. When the magnetic field is weak, some electrons have up-spins and the others down-spins. Therein the spin-exchange transition occurs between two electrons with up and down spins via the Coulomb interaction. Then the charge distribution before the transition is the same as one after the transition. So these two states have the same classical Coulomb energy. Accordingly, the partial Hamiltonian composed of the spin exchange interaction should be treated exactly. We have succeeded in diagonalizing the spin exchange interaction for the first and second nearest electron pairs. The theoretical results reproduce the wide plateaus very well. If the interval modulations between Landau orbitals are taken into the Hamiltonian, the total energy has the Peierls instability. We can diagonalize the Hamiltonian with the interval modulation. The results reproduce wide plateaus and small shoulders which are in good agreement with the experimental data.


Introduction
As well known, the integer and fractional quantum Hall effects (IQHE and FQHE) appear in a quasi-two dimensional electron system under a strong magnetic field perpendicular to the 2D electron channel the thickness of which is extremely thin [1]- [17].Only the ground state along the z-direction of Figure 1 is realized actually for a low temperature and so we can treat the system as a 2D electron system.The confinement of the Hall resistance at IQHE is extremely accurate and then the confined resistance is employed as the resistance standard [18].The IQHE and FQHE are observed in various materials namely Si, GaAs, graphene and so on.The QHE appears independently of materials and is caused only by many electrons in a thin film.The Hall resistance is the ratio between Hall voltage and electric current.In the IQHE the Hall voltage is larger than about 10 −4 Volts and the diagonal (potential) voltage is less than 10 −11 Volt.Also in the FQHE the Hall voltage is extremely large compared with the potential voltage.Therefore this asymmetry of x and y directions in Figure 1 should be taken into consideration in the investigations of QHE.Consequently the total Hamiltonian of many electrons should be composed of three kinds of interactions namely the strong magnetic field interaction, the Coulomb interactions between electrons, and the electric field produced by the Hall voltage.However, many theories ignore the electric field along the Hall voltage.We take account of the electric potential along the y-direction in Figure 1.
We have developed the theory of Tao and Thouless [19] [20] and have found out the most uniform configuration of electrons in the Landau orbitals which yields the minimum expectation value of the total Hamiltonian T H [21]- [25].Since the Coulomb interaction depends only upon the relative coordinate between two electrons, the x component of the total momentum is conserved in the Coulomb transition.Because of the momentum conservation and the most uniform configuration, the number of the allowed transitions takes the local maximum at the specific filling factors ( ) j j+ , ( ) ) ( ) 2 2 2 1 j j + + and so on.The transitions produce the binding energy of the electron pair.The larger the number of transitions, the pair energy is lowered.If the filling factor ν deviates from the specific filling factors 0 ν , many Coulomb transitions are forbidden.The forbidden mechanism yields the energy gap at the specific filling factors 0 ν .Thereby, the Hall confinement appears at the specific filling factors.Thus the present theory can explain the FQHE well without any quasi-particle [21]- [26].
The spin-polarizations of FQHE are obtained by V. Kukushkin, K. von Klitzing, and K. Eberl [27].They clarified the magnetic field dependence of the polarization under fixing the filling factors.Although their experiments are rather old, the polarization curves versus magnetic field give us the very important knowledge.The experimental polarization-curves have the wide plateaus and small shoulders.
There are many spin-arrangements in the most uniform configuration of electrons in the Landau orbitals.These spin-arrangements have the same minimum-expectation-value of T H .That is to say many electron states with various spin-arrangements are perfectly degenerate.Therefore, we should exactly diagonalize the residual Coulomb interaction.We examine the partial Hamiltonian composed of the strongest and second strongest interactions between electrons.Then we have succeeded to diagnalize the Hamiltonian exactly.The theoretical results are in good agreement with the experimental data.Recently J. K. Jain has written the article [28] and compared the composite fermion (CF) theory [29]- [44] with the Haldane-Halaperin (HH) hierarchy theory [45] [46].Also he summarized the composite fermion theory and showed the theoretical results on the spin-polarization of FQH states.We compare the polarizations by the CF theory with that by our theory in Section 5.
Next we observe the experimental data [27] in more details.Then we find small shoulders on the curves of the polarization versus magnetic field.The small shoulders appear at the middle between two wide plateaus.This property is similar to the famous feature in the spin-Peierls effect [47].The spin-Peierls effect is caused by the lattice distortion with the period doubling the unit cell.The sum of spin-energy and distortion-energy (namely total energy) becomes lower than that without distortion.We applied the distortion to the intervals between Landau orbitals in the previous works [48] [49] where the magnitude of distortion was treated to be a fixed value.However the magnitude should be determined by minimizing the total energy in each field strength.Accordingly the magnitude of distortion depends on the field strength.The minimizing of the total energy needs a very complicated program and a long cpu-time.In this article we succeed to make the Mathematica program with parallel computation.The calculated results show the Peierls instability and the polarization curves are in better agreement with the experimental data than the previous results.

Fundamental Properties
A typical quantum Hall device is illustrated in Figure 1 where the current flows along the x-direction and the magnetic field B has the z-direction, namely ( ) . In the absence of the Coulomb interaction between electrons, the Hamiltonian of a single electron is given by , where , 0, 0 2 is the electron momentum and m * is the effective mass of elec- tron, the value of which differs from material to material.The value of m * in GaAs is about 0.067 times the free electron mass.
The eigen-value problem of 0 H is solved and the single-electron wave function , L J ψ is expressed as where L the Landau level number, ( ) z φ the wave function of the ground state along z-direction, L H the Hermite polynomial of L -th degree and L u is the normalization constant.Also J α is given by The eigenenergy is given by where λ is the ground state energy along the z-direction and ( ) is the potential energy in Figure 2. Next let us consider the many electron system.The total Hamiltonian is given by ( ) ( ) ( ) where N is the total number of electrons and 0, i H is the single particle Hamiltonian: The eigenstate of 0, 1 , , ; , , , , ; , , where ( ) , , ; , ,   ) ( ) ( ) Because the Coulomb interaction depends upon only the relative coordinate, the total momentum of the x-direction is conserved in this system as follows: where i p and j p are the initial momenta and i p′ and j p′ are the final momenta via I H .All the electrons in the ground state of 0, 1 exist in the lowest Landau levels ( ) for a filling factor 2 ν ≤ because electrons can take up-and down-spin states.If the electrons with the momenta 1  , , , as in Equation ( 3).For any filling factor, we can find only one electronconfiguration with the minimum energy of D H .The proof has been done in Ref. [21].The residual Hamilto- nian I H acts between two electrons.All the initial states with the momenta 1 2 , p p are , , where 1 p′ and 2 p′ are the final momenta via the interaction I H . Now we consider the Coulomb transitions of only between the degenerate ground states of D H .The ground states have the most uniform configuration.Therefore the electron- configuration of the final state is equivalent to that of the initial state.So the final momentum set is the same as the initial one.Then we obtain ( ) . The case of ( ) is removed because the diagonal matrix elements of I H are zero.Accordingly the final momenta become Even if Equation ( 9) is satisfied, the initial state is identical to the final state for the same spin direction of the two electrons.Therein the transition matrix element is zero.Accordingly non-zero matrix elements are , , where We examine following three cases: Figure 3 shows the case A.
This transition is equivalent to the following process: the spin at site 1 flips from up to down, and the spin at site 2 flips from down to up without changing the momenta.Thus the Coulomb transition of the case A is equivalent to a spin exchange process which is described by the interaction 1 2 ξσ σ − + .Therein σ + is the spin trans- formation operator from down to up-spin state and σ − is the adjoint operator of σ + .Another Coulomb tran- sition given by Equation (10b) is equivalent to 1 2 ξσ σ + − .Accordingly Coulomb transition between sites 1 and 2 is ( ) where ξ was already defined by Equation (11a).In this Coulomb transition, the classical Coulomb energy of the initial state is exactly equal to that of the final state.
Next Figure 4 shows the case B.
Coulomb interactions in Cases B and C are equivalent to the following spin exchange interactions where η and ς are the coupling constants defined by Equation (11b) and (11c), respectively.

(
) ( ) We examine the third strongest interaction at 2 3 ν = . Therein another electron is inserted as seen in Figure 5(a).Therefore the interaction between the third nearest electrons becomes weak by the screening effect of the inserted electron in between the pair.At  becomes weak by the screening effect of the interposing electron.Consequently the most effective interaction is obtained for 2 3 This Hamiltonian yields the quantum transition between the degenerate ground states.We take account of the Zeeman interaction and get the most effective Hamiltonian as where g * is the effective g-factor, B is the magnetic field, ( ) , for 3 5. , for 4 7.
These three Hamiltonians can be exactly diagonalized by using the method of Ref. [50].

Diagonalization of the Most Effective Hamiltonian for FQH States
We introduce a new mapping from a spin state to a fermion state.The down-spin state ↓ is mapped to the vacuum state 0 and the up-spin state ↑ is mapped to the fermion state 0 c * where c * is the creation operator.Thereby the spin operators σ + , σ − and z σ are mapped as; ( ) Many electron states are mapped to the many fermion states where the multiplying order of the creation operators is the same as the order of the up-spins: where the operators i c and i c * satisfy the anti-commutation relations.The spin operators 2 1 2 are mapped to the products of the fermion operators as follows: It has been verified in Ref. [50] that the mapping ((20a), (20b), (20c)) is isomorphic.Accordingly Hamiltonian Equation ( 15) is equivalent to the following form: We can exactly solve the eigen-value problem of H by introducing new operators j a and j b : , , and where j is the cell number.Thereby the Hamiltonian (21) becomes ( ) ( ) ( ) ( ) where J is the total number of cells given by 2 J N = ( N is the total number of electrons).We apply a Fourier transformation to Equation ( 23) and obtain as where ( ) In Equation ( 24) the term with one value of p is expressed by the following matrix: This matrix has two eigen-values ( ) We introduce new annihilation operators ( ) A p and ( )

2
A p as follows: Hamiltonian ( 24) is expressed by making use of ( ) A p and ( )

Spin-Polarization
The electron-spin polarization e γ is defined by the thermodynamic mean value as follows: ( ) where  is the thermal average and the minus sign comes from the negative charge of electron.All the ei- gen-states are specified by the set of the numbers s n p = where B k is the Boltzmann constant and T is the temperature.Accordingly, the probability of ( ) . Then the thermal average is obtained as follows; Substitution of Equation ( 31) into Equation (30) gives . Because the momentum interval is extremely small, the electron-spin polarization is expressed by the integration as ( ) ( ) Similarly we obtain the spin-polarization at the filling factors of 3/5 and 4/7 as follows: The calculated results are shown in Figure 6 which are in good agreement with the experimental data.ν * = , where we expect, from a model that neglects in- teraction between composite fermions, a spin singlet state at very low Zeeman energies (with 2 2

Comparison between Present Theory and Composite Fermion Theory
ν * = + ), a partially spin polarized state at intermediate Zeeman energies ( ) ν * = + , and a fully spin polarized state at large Zeeman energies ( ) The CF cyclotron energy is proportional to B and the Zeeman energy is proportional to the applied mag- netic field B as explained in the article [28].Then the CF energy CF ε is equal to where α and β are the coefficients.Therein + where L * is the CF Landau level number ( ) When the filling factor ν of electrons is samller than 2/3, ν is expressed by using the CF level number ν * the number of attached flux quanta 2 p as follows: ( ) When the sign in the denominator is plus, the effective magnetic field is parallel to the applied magnetic field.
So the coefficient β becomes positive.On the other hand the effective magnetic field has opposite direction against the applied field for the minus sign in Equation ( 35).Then β is negative.We obtain the sum of the CF cyclotron energy and the Zeeman energy for various CF levels and CF spins.results are shown in Figure 7 where we find the crossing points between the energies for spin-up and down states.The ratio of the field strengths at the crossing points is 1:4:9:16 and so on.For an example 4 7 the composite fermions occupy the lowest four levels which are indicated by the bold curves as in Figure 8(a).The ratio of the strength 1 2 , B B and 3 B is 1:4:9.Accordingly the spin-polarization depends on the magnetic field strength at zero temperature as in Figure 8(b).coupling constants ξ , ξ ′ , η and η′ .The value of ξ ′ is larger than ξ because the ξ ′ interaction path is shorter than that of ξ .Also, η η ′ > holds.We express the distance between nearest Landau orbitals by the symbol 0 r in non-distortion case.The dis- tance becomes 0 r d + for an odd cell number and the other one becomes 0 r d − for an even cell number.Then, the classical Coulomb energy W increases.The increasing value per electron is proportional to 2 d as ( ) where C f is the constant parameter.We next examine d -dependence of ξ and ξ ′ .
When 0 d > , the coupling constant ξ is weaker than ξ ′ because the ξ interaction path is longer than that of ξ ′ .When 0 d < , ξ is stronger than ξ ′ because the ξ interaction path is shorter than that of ξ ′ .So the d -dependence in ξ and ξ ′ is given by ( ) ( ) where 0 ξ is the coupling constant for the non-distortion case and f ξ is the proportionality constant.A new dimensionless quantity t is introduced as follows ( )( ) Thereby the coupling constants ξ and ξ ′ is expressed as The increasing classical Coulomb-energy W ∆ is also described by the distortion parameter t as: where C is the dimensionless coefficient as We introduce new operators 1, , , , and make the Fourier transformation of the operators: , , , where J is the total cell number namely 4 J N = , and ( ) 41) and (42) into Equation ( 40) yields For each value of p , the right hand side of Equation ( 43 Here we assume that the ratio ξ ξ ′ is almost equal to the ratio η η ′ because the interval modulation gives the same effects to ξ and η .Then Equation ( 38) yields the following ratios: ( ) ( ) Four eigen-values of M are expressed by the symbols λ 1 (p), λ 2 (p), λ 3 (p) and λ 4 (p) ( ) λ λ λ λ ≤ ≤ ≤ .The eigen-energies is obtained by using Equations (45a) and (45b): Then the diagonal form of the Hamiltonian ( 43) is expressed by the eigen-energies λ 1 , λ 2 , λ 3 , λ 4 as: The thermal average ( ) ( ) is expressed by the eigen-energies as derived in Equation (31). ( Then the spin exchange energy is We can replace the summation in Equation ( 49) by the integration because the total number of electrons is a macroscopic value.
The total energy per electron is the sum of the classical Coulomb energy and the spin energy as ( ) ( ) Figure 12 shows the classical Coulomb energy for the parameter 0.5 C = in Equation ( 39) by the dashed black curve.Also the spin-exchange energy ( )

Spin Polarization with Interval Modulation
We have obtained the diagnal forms of the Hamiltonians which have the Peierls instability.The spin-polarization The total momentum along the x-direction conserves via the Coulomb interactions.When the electron 1 placed in the Landau orbital A transfers to the orbital B, the electron 2 placed in the orbital B should transfer to the orbital A because of the total momentum conservation and the relation between momentum and orbital position.We have found the most effective Hamiltonian which is composed of the strongest and second strongest Coulomb interactions.Then we have succeeded to diagonalize the most effective Hamiltonian exactly.The calculated eigen-energies yield the theoretical value of the polarization which has reproduced the wide plateaus on the spin polarization curves.Furthermore we have applied the interval modulation between Landau orbitals and have solved the eigen-equation of the new Hamiltonian.The total energy decreases by the interval modulation.So the modulation appears actually.Thereby the calculated polarization curves have the small shoulders in addition to the wide plateaus.The theoretical results are in good agreement with the experimental data [27].

= and y d = as in Figure 2 .
the potential along the z-direction confining electrons to the thin conducting layer.The po- tential ( ) U y has large potential difference between both ends, 0 y In the right hand side of Equation (1),

DH
is composed of all diagonal parts and I H is constructed only by the off-diagonal elements as follows:

↓
where the symbols ↑ and ↓ indicate up and down spins, re- spectively.The final states are 1 2 5. The most uniform configuration is inFigure 5(a) and Figure 5(b).The spin-states are numbered sequentially from left to right as shown by green color.It is noteworthy that the site-number of spin is different from the orbital number.At the 2 3 ν = , the strongest interaction acts between the nearest electron pairs which has the coupling constant ξ .The second strongest interaction has the coupling constant η as shown in Figure 5(a).At 2 5 ν = , the strongest and the second strongest interactions have the coupling constants η and ζ as shown in Figure 5(b).

Figure 3 .
Figure 3. Equivalence of specific Coulomb transition and spin exchange interaction for case A.

4 Figure 4 .Figure 5 .
Figure 4. Equivalence of specific Coulomb transition and spin exchange interaction for case B. ξ η ξ η ξ η ξ η ξ x y 2 z σ is the electron spin operator of the z-direction and B µ is the Bohr magneton.
, , , , , , , , 0 c c c c cc * * * * * * J. K. Jain examined the spin-polarization in the CF theory[28].He wrote as follows: "For spinful composite fermions, we write ν ν ν * * * ↑ ↓ = + , where ν * ↑ and ν * ↓ are the filling factors of up and down spin composite fermions.The possible spin polarizations of the various FQHE states are then predicted by analogy to the IQHE of spinful electrons.For example, the 4/7 state maps into 4

Figure 7 .Figure 8 .Figure 11 .
Figure 7. (a) Energy of CF for total energy produced by this distortion.Use of the isomorphic mapping namely Equations ((20a), (20b), (20c)) give the spin exchange Hamiltonian for 2 ) is expressed by the following matrix M : .2 T B =is expressed by the red dashed curve.The blue curve indicates the total energyTotal E N ∆ which has a minimum value at a non-zero t .Thus the interval modulation occurs actually.
Landau orbitals, the classical Coulomb energy takes the lowest value.Thereby the many-electron state becomes the ground state of )