Shosuke Sasaki

Energy gap in fractional quantum Hall effect

Abstract Binding energies of fractional quantum Hall states are calculated by a new method, in which we use Buttikers states whose orbits are parallel to the electric current. When the filling factor ν=(q−1)/q, we initially adopt such a many-electron state that (q−1) orbits in every q Buttikers orbits are filled with (q−1) electrons and the remaining one orbit is empty. Taking the Coulomb interaction between electrons into account, a pair of electrons in the nearest two orbits can be transferred to empty orbits for an odd integer q. The transition yields the second-order binding energy. However, for an even integer q, there is no transition because of the momentum conservation. The calculation at the filling factor (q−1)/q shows that the binding energy per electron is equal to Z/(q(q−1)) in any odd integer q, but is absent in any even integer q. When the filling factors = 3 5 , 4 7 , 5 7 , the binding energy per electron is equal to (2/15)Z, (3/28)Z, (2/35)Z , respectively. These calculated values of the binding energies are in accord with the experimental data of the fractional quantum Hall effect.

Gap Structure and Gapless Structure in Fractional Quantum Hall Effect

Higher-order composite fermion states are correlated with many quasiparticles. The energy calculations are very complicated. We develop the theory of Tao and Thouless to explain them. The total Hamiltonian is (𝐻𝐷

Energy Gaps in Fractional Quantum Hall States

Energy values versus filling factors v are examined for fractional quantum Hall states (FQHS). First, the classical Coulomb energy of nearest electron pairs in FQHS is shown to be linearly dependent upon 1/v. The residual Coulomb interaction produces quantum transitions. Examination of second order transitions indicates a discrepancy between the second order perturbation energy at v = 2/3 and the limiting energy value at v = (2s + 1)/(3s + 1) for an infinitely large value of s. Accordingly, an energy gap appears at v = 2/3. On the other hand, the second order perturbation energy 2(v = 3/4) is equal to the limiting energy value at v = (3s + 1)/(4s + 1) for an infinitely large s. Therefore, the energy spectrum is continuous near v = 3/4, meaning that it is gapless at v = 3/4. The same mechanisms appear in higher order perturbation calculations because the number of forbidden transitions in the higher order calculation is equal to that in the second order calculation. That is to say, the gap and gapless mechanisms can be extended to higher order calculations. In fractional filling factors other than v = 2/3 and 3/4, either a gap mechanism or a gapless mechanism appears for each filling factor. Consequently, our results can theoretically explain the precise confinement of Hall resistance at fractional filling factors with a gap mechanism

SPIN-POLARIZATION IN FRACTIONAL QUANTUM HALL EFFECT

Abstract Experimental data show a number of plateaus of varying widths in the magnetic-field dependence of the electron-spin-polarization for fractional quantum Hall states. We have calculated the magnetic-field dependence of the spin polarization using a new theory. We start by adopting the Landau gauge and ignoring Coulomb interactions between electrons; then we construct single electron states in equally spaced orbitals. For a number of filling factors we have examined the many-electron states with electron configurations having minimum classical Coulomb energy. The residual Coulomb interactions in each many-electron state produce spin-exchange-forces. We have solved the eigenvalue problem of the interaction Hamiltonian composed of nearest neighbor spin-exchange-interactions. From the eigenvalues we have calculated the magnetic-field dependence of the spin polarization. Our results are in good accord with the magnetic-field dependence in experimental results, including the number and shape of the plateaus.

Movement of diagonal resistivity in fractional quantum hall effect via periodic modulation of magnetic field strength

We examine the effect induced by periodic modulation of magnetic field strength in the fractional quantum Hall effect (FQHE). The classical Coulomb energy of the fractional quantum Hall (FQH) state is linearly dependent on 1/v, where v represents the fractional filling factor. This energy varies continuously with v. The residual Coulomb interactions produce quantum transitions. Then, the binding energy via the transitions yields an energy gap for specific fractional filling factors and no gap for the other fractional filling factors. The strength of the static magnetic field is fixed to yield an FQH state with gap energy. Then a periodic magnetic modulation is added to the system. Its resultant diagonal resistivity depends upon the oscillation frequency of the magnetic modulation. Examination of that dependence shows that the resistivity varies drastically at some frequency value, which can be evaluated for several fractional filling factors. These phenomena are strongly dependent upon the energy spectra of FQH states

Theory of Temperature Dependence of Elementary Excitation Energy

Assuming that the total Hamiltonian of liquid helium is completely diagonalized by a unitary operator, we can introduce “dressed-boson” operators as the unitary transformation of creation and annihilation operators of a helium atom. Then we can make a new viewpoint for the elementary excitation of liquid helium on the basis of the concept of the “dressed-boson”. The temperature dependence of the elementary excitation energy is studied. It is verified that the elementary excitation with a small momentum softens as the temperature approaches λ-point. The detection of the softening is proposed.

Spin Polarization Curve of Fractional Quantum Hall States with Filling Factor Smaller than 2

Kukushkin et al. have measured the electron spin polarization versus magnetic field in the fractional quantum Hall states. The polarization curves show wide plateaus and small shoulders. The 2D electron system is described by the total Hamiltonian (). Therein, is the sum of the Landau energies and classical Coulomb energies. is the residual interaction yielding Coulomb transitions. It is proven for any filling factor that the most uniform electron configuration in the Landau states is only one. The configuration has the minimum energy of . When the magnetic field is weak, some electrons have up-spins and the others down-spins. Then, there are many spin arrangements. These spin arrangements give the degenerate ground states of . We consider the partial Hamiltonian only between the ground states. The partial Hamiltonian yields the Peierls instability and is diagonalized exactly. The sum of the classical Coulomb and spin exchange energies has minimum for an interval modulation between Landau orbitals. Using the solution with the minimum energy, the spin polarization is calculated which reproduces the wide plateaus and small shoulders. The theoretical result is in good agreement with the experimental data.

Stabilization of precession‐free rotors supported by magnets

Two types of precession‐free rotors are proposed which are characterized by a magnetic suspension and by a precession damper. The type‐I rotor is composed of a contactless rotor with a magnetic precession damper, and the type‐II rotor is composed of a rotor with a mechanical precession damper. The classical equations of motion for these rotors are studied and solved. Then the solutions for the rotating motion of both rotors show that the inclination of the figure axis decreases spontaneously. Namely, these rotors have very stable rotation. The type‐II rotor is produced in the laboratory, and a revolution of 120 000 rpm is obtained in a preliminary experiment.

Experimental study on a precession‐free rotor suspended by magnets

A high‐speed rotor which is characterized by a mechanical precession eliminator and by a magnetic suspension has been constructed. In this instrument, a cylindrical rotor 2 cm (0.02 m) in diameter and 20 cm (0.2 m) long was made of aluminum. The moment of inertia of this rotor was approximately 86 g cm2 (8.6×10−6 kg m2) about the long axis. A stable rotation of 120 000 rpm was obtained in air as well as in vacuum. The rotor was suspended vertically by a set of suspension magnets at the top of the rotor and a set at the bottom. In addition, the bottom of the rotor is flexibly supported by a ruby bearing which moves smoothly on a plate but is constrained by friction. When precession of the rotor begins, the frictional force eliminates the precession intrinsically. Measurements of slowing rate for the revolution were made at several different air pressures. There is good agreement with the theory.

Plateaus in the Hall Resistance Curve at Filling Factors 2 < ν < 3

The fractional quantum Hall (FQH) states with higher Landau levels have new characters different from those with . The FQH states at are examined by developing the Tao-Thouless theory. We can find a unique configuration of electrons with the minimum Coulomb energy in the Landau orbitals. Therein the electron (or hole) pairs placed in the first and second nearest Landau orbitals can transfer to all the empty (or filled) orbitals at , 14/5, 7/3, 11/5, and 5/2 via the Coulomb interaction. More distant electron (or hole) pairs with the same centre position have the same total momentum. Therefore, these pairs can also transfer to all the empty (or filled) orbitals. The sum of the pair energies from these quantum transitions yields a minimum at . The spectrum of the pair energy takes the lowest value at and a higher value with a gap in the neighbourhood of because many transitions are forbidden at a deviated filling factor from . From the theoretical result, the FQH states with are stable and the plateaus appear at the specific filling factors .