Magnetic induction dependence of Hall resistance in Fractional Quantum Hall Effect

Quantum Hall effects (QHE) are observed in two-dimensional electron systems realised in semiconductors and graphene. In QHE the Hall resistance exhibits plateaus as a function of magnetic induction. In the fractional quantum Hall effects (FQHE) the values of the Hall resistance on plateaus are h/e2 divided by rational fractions, where −e is the electron charge and h is the Planck constant. The magnetic induction dependence of the Hall resistance is the strongest experimental evidence for FQHE. Nevertheless a quantitative theory of the magnetic induction and temperature dependence of the Hall resistance is still missing. Here we constructed a model for the Hall resistance as a function of magnetic induction, chemical potential and temperature. We assume phenomenological perturbation terms in the single-electron energy spectrum. The perturbation terms successively split a Landau level into sublevels, whose reduced degeneracies cause the fractional quantization of Hall resistance. The model yields all 75 odd-denominator fractional plateaus that have been experimentally found. The calculated magnetic induction dependence of the Hall resistance is consistent with experiments. This theory shows that the Fermi liquid theory is valid for FQHE.

The basic mechanism of the integer QHE (IQHE) 1,2 and FQHE 3,4 is non-uniform distribution of electron density caused by the Lorentz force acting on the electrons 5 . Theoretically the non-uniform distribution can be taken into account by using the method of subsystem [6][7][8][9][10] , in which the system is divided into many strips of rectangular-shaped subsystems parallel to the direction of the bias current. The electron density in each subsystem may be different, but the chemical potential takes the same value. In each subsystem we derive the relation between the bias current and the transverse potential difference using the many-electron quantum field theory [11][12][13] . The dynamics of the electrons is described in terms of the second quantised field operators that satisfy the equal-time anti-commutation relations. The Lorentz force acting on the electrons can be calculated by the Heisenberg equations for the mechanical momentum of the electrons. We assume a model Hamiltonian  , where H 0 is the kinetic energy term with the external perpendicular magnetic field, H spin is the Zeeman spin term, H e is the coupling to the electric field, and H int is the electron-electron interaction term. Calculating the statistical ensemble average of the Heisenberg equations for the mechanical momentum and assuming the steady state condition, we obtain the bias current as a function of Hall voltage in each subsystem. Assuming that the statistical ensemble average of the electron number density is given by the Fermi distribution function 11 and calculating the sum of the bias currents of all subsystems, we obtain the inverse of Hall resistance

Model of FQHE
To construct a model of the FQHE let us first examine the theoretical mechanism of plateaus in IQHE, which can be quantitatively explained by adopting the Landau level  ε ε ω ζα as the energy spectrum in equation (1) In the zero temperature limit the inverse of Hall resistance becomes Considering the above analysis, let us inspect the Hall resistance data in the FQHE experiment reported in ref. 14 . The quantisation unit of − R H 1 on e 2 /3h, 2e 2 /3h and 4e 2 /3h plateaus observed in the FQHE experiment 14 is e 2 /3h. In view of equation (4) the most plausible explanation for this is that a Landau level is split into three sublevels. Each sublevel has the degeneracy D 1 = D 0 /3. We assume that the level-splitting is caused by a perturbation Hamiltonian 1  ′ , which yields new quantum numbers m 1 = −1, 0, 1 for sublevels. Let us call these sublevels the m 1 sublevels. The 2e 2 /5h, 2e 2 /5h, 4e 2 /5h, and 7e 2 /5h plateaus in FQHE can be explained by assuming an additional perturbation Hamiltonian ′ 2  that splits each m 1 sublevel into five sublevels. Let us call these sublevels the m 2 sublevels. Each sublevel has the degeneracy D 2 = D 1 /5.We assume that ′ 2  is small perturbation to ′ 1  . The 3e 2 /7h and 4e 2 /7h plateaus in FQHE can be explained by assuming an additional perturbation Hamiltonian ′ 3  that splits each m 2 sublevel into seven sublevels. Let us call these sublevels the m 3 sublevels. Each sublevel has the degeneracy D 3 = D 2 /7.We assume that  ′ 3 is small perturbation to  ′ 2 . Hence, the quantised values of FQHE resistance at fractional plateaus can be attributed to the degeneracies of sequentially split sublevels. This analysis indicates a model energy spectrum where m l is an integer ranging − ≤ ≤ l m l l . We have defined m = (m 1 , m 2 , m 3 , …). The parameters λ l are assumed to satisfy the condition |λ l+1 | < |λ l |. Using the Hall resistance formula given by equation (1), we can determine the parameters λ l from the experiment. If λ l are independent of B, in the zero-temperature limit, the locations of step edges on the B axis are given by equation (3) as

Results
Odd-denominator fractional plateaus. Because the number of possible m l 's for a given l is 2l + 1, the degeneracy of an energy level with quantum numbers (N, α, m) is . This formula yields the values of Hall resistance on plateaus as

Hence the inverse of Hall resistance for FQHE is given as
where j is a positive integer. This formula yields all 75 observed odd-denominator fractional plateaus 15,16 . It should be noted that the Hall resistance given by equation (8) is consistent with the result obtained on the basis of the fractal geometry 17 .

Magnetic induction dependence of Hall resistance.
For the practical purpose of calculating the magnetic induction dependence of Hall resistance we assume l max = 3 in the model energy spectrum given by equation Note that the parameters λ l may depend on B. For instance if we assume B 1/ j λ ∝ then the energy gaps corresponding to the fractional plateaus become proportional to B . Here we consider the simplest case that λ l are independent of B. Then the three parameters λ l in equation (9) are fitted to the experimental Hall resistance curve in ref. 14 . Their values are λ 1 = 0.25, λ 2 = 0.14, and λ 3 = 0.003. Considering the Hall resistance data for the IQHE experiment in ref. 18 , the effective g-factor is adjusted to g 12 = ⁎ . The effective mass is M = 0.067 M 0 . The chemical potential is determined by the slope of experimental Hall resistance curve for weak magnetic induction. The value is μ = . × − 13 14 10 15 erg. The theoretical resistance curve as a function of B is plotted in Fig. 1, using equations (7) and (9) for T = 85 mK which is the experimental temperature in ref. 14 . In order to see the plateaus clearly the theoretical resistance curve for T = 5 mK is plotted in Fig. 2. The theoretically calculated Hall resistance plateaus 1/3, 2/5, 3/7, 4/7, 3/5, 2/3, 4/5, 1, 4/3, 7/5, 5/3, and 2 are consistent with the experiment 14 . Although the theoretical curve agrees with experiment fairly well, it seems necessary to consider the B-dependence of λ l to improve agreement. In Fig. 3 the magnetic induction and temperature dependence of the Hall resistance is shown in a 3D plot. It shows the Hall resistance curve given by equation (7) becomes classical as temperature increases. Hence the formula (7) can yield FQHE, IQHE and classical Hall effects.

Discussion
The quantum number m l introduced in the model perturbation energy spectrum (5) ranges − ≤ ≤ l m l l . Therefore, it is plausible that these quantum numbers m l and l correspond to angular momentum. Because the orbital angular momentum operators cannot be defined in the 2-dimensional space, it is necessary to consider the problem in the 3-dimensional space. We adopt θ φ r ( , , ) for the 3-dimensional polar coordinates. Then the 3-dimensional lowest Landau level (LLL) wave function is 19 where N m is the normalisation factor, =  a c eB / is the magnetic length, Y lm is spherical harmonics, d is the thickness of the 2-dimensional system, and z r d /2 The expansion (10) shows that the lowest Landau level in the three-dimensional space is a superposition of angular momentum eigenstates of different l. The allowed values of m in equation (10) are only non-positive integers 19 . Because the quantum number m l ranges from l to −l, it cannot belong to the unperturbed state given by equation (10). Therefore, the quantum number m l possibly corresponds to new degree of freedom of the Landau orbitals in the three-dimensional space.
A model based on magnetoplasmon excitations in the three-dimensional space may explain the energy spectrum given by equation (5). The experimentally observed quantised plateaus in the magnetic induction dependence of magnetoplasmon dispersion 20 clearly indicate significance of magnetoplasmons in the quantum Hall ohm. The experimental Hall resistance at T = 85 mK from ref. 14 is also plotted in gray. effects 6,9 . In the fractional quantum Hall regime the electron system can be regarded as a liquid of electrons in LLL orbitals. In order to examine how the magnetoplasmon fields affect the dynamics of a single-electron energy spectrum, we assume an electron P in a LLL orbital whose center is located at the origin of the coordinates. The Maxwell equations for magentoplasmon fields have source terms due to the electrons and uniform positive background 13 . In the following discussions we shall use the Lorentz gauge and four vector notations for simplicity of mathematical expressions and to avoid the decomposition of vector fields into transverse and longitudinal components 21,22 . Then the current densities in source terms in the Maxwell equations can be written as P i nduced where μ j P is due to the electron P and μ j induced is due to the other electrons. The Greek subscripts denote the components in the four-dimensional space 23 . The background uniform charge is included in these terms. In the self-consistent linear response approximation (SCLRA) 6,21,24,25 the latter is given as    (7) is shown as a 3D plot for 0 < T < 10 K and 0 < B < 30 T.
We consider magnetoplasmons whose wavelengths are much larger than Landau radius. Then, the current density j P μ is well localised in the vicinity of the origin of the coordinates. This type of equations with a localised source term has been intensively studied in the theories of multipole fields such as radiation of electromagnetic fields 27,28 , and it is known that spherical harmonics Y lm (θ, φ) are most relevant orthogonal basis to expand the field variables, particularly stationary waves. Therefore, the stationary modes of magentoplasmons given by equation (13) are labeled with (l, m). The propagator for the quantised magnetoplasmon field can be calculated from equation (13). Then the self-energy of the temperature Green function for the electron field can be expressed in terms of the magnetoplasmon propagator by virtue of the finite temperature generalised Ward-Takahashi relations 12,29 . Consequently, the electron energy spectrum will acquire perturbation terms labeled with (l, m).
The essential features of the SCLRA equations are determined by the retarded current-current response functions of the many-electron system 21,26 . Therefore, in order to elaborate on this magnetoplasmon model of the spectrum given by equation (5) it is necessary to calculate these response functions, or to investigate their mathematical properties. The physical idea of this magnetoplasmon model is very similar to that of the Lamb shift in quantum electrodynamics 30,31 . While an electron bound to an atomic orbital interacts with electromagnetic field in the Lamb shift, here an electron in LLL orbital interacts with magnetoplasmon. In both cases the reaction of the electromagnetic fields is the essential cause of the phenomena. Although magnetoplasmon is electromagnetic field, its modes are much more complicated than the electrogamgnetic field in the Lamb shift.
Since the discovery of the fractional quantum Hall effect, there have been a number of interesting theoretical models 32 . Among them the fractal geometry model 17 seems to be deeply related to the present model. The Hall resistance formula given by equation (8) is consistent with the results given in ref. 17 . It is an interesting theoretical problem to investigate whether the current-current response functions can contain a geometrical structure such as the fractal geometry discussed in ref. 17 .
We explained the fractional quantised values of the Hall resistance on plateaus in terms of the degeneracies of sublevels created from Landau levels by the phenomenologically introduced perturbation in the single-electron energy spectrum. The present theory yields all 75 odd-denominator fractional values observed experimentally to date 15,16 . The simple model with l max = 3 yields twelve plateaus whose magnetic induction dependence is consistent with the experiment. No existing theories can yield this quantitative fit to the experiment. By calculating the temperature dependence of the Hall resistance, we plotted a 3D graph that explicitly shows how FQHE and IQHE disappear and become classical Hall effect as temperature increases. Because the Hall resistance formula (1) depends only on the single-electron energy spectrum via Fermi distributions and can explain both IQHE and FQHE, this theory clearly shows that the Fermi liquid theory 11,12,33 is valid for IQHE and FQHE.

Methods
Derivation of Hall resistance formula. The x 1 axis is taken along the direction of the bias current, and the where ψ α and † ψ α are the second quantized electron field operators in the Heisenberg picture. The integral notation where L and ΔL are the length and width of a subsystem Ω i . We use Einstein convention for the summation over the spin variables. The superscript i denotes a subsystem. We also define the electron density operator † ρ ψ ψ = α α and the electric current density operator  (15) give the equations of motion for the current density operators 10 . The next steps are to take the statistical mechanical ensemble average of each term in the equations and to introduce a phenomenological damping terms W k i , which are necessary to ensure the Ohm's law. We obtain Ω Ω Although the electron-electron interaction H int is rigorously included in the derivation, the electron-electron potential does not appear explicitly in this equation. The details of the damping term W 2 are not necessary. The only required condition for the damping terms is that they must vanish when there is no current.
To calculate the Hall resistance it is necessary to define macroscopic currents I k that correspond to experimentally measurable currents. We assume <ρ> i , <J 1 > i and <J 2 > i in each subsystem are uniform. We first define macroscopic currents I k i in a subsystem Ω i such that We also define the Hall voltage in each subsystem such that V E L The electron energy spectrum in the subsystem Ω i consists of an i-independent part ε q and an i-dependent part δε i , where q is the quantum number of a quasi-electron state. The total current I 1 is The single-electron energy spectrum is denoted by ε q with a quantum number q. The degeneracy of energy level q is denoted by D(q). where N m 0, 2D is the normalisation factor, and  a c eB / = is the magnetic length. Here we use (ρ, φ) for the 2-dimensional polar coordinates and (r, θ, φ) for the 3-dimensional polar coordinates. It is also necessary to consider explicitly the confining potential V conf. (x 3 ) and the 3-dimensional kinetic energy in the Hamiltonian 26,34 . We assume the electrons are in the ground state of V conf. (x 3 ) with a simple Gaussian wave function