1. 2. Hall resistance Magnetic Field Current I Quantum Hall Device Spin Polarization of Fractional Quantum Hall States 1. is nearly equal to zero at FQHE. No electron scattering. 1. V potential is nearly equal to zero at FQHE. No electron scattering. Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Japan Shosuke Sasaki QHE creates no heat. 3. FQHE are purely eigen-value problem of electrons. 4. The value of V H is extremely larger than. Therefore the gradient of V H cannot be ignored. 4. The value of V H is extremely larger than V potential. Therefore the gradient of V H cannot be ignored. Hall Voltage V H Poential Voltage V potential The FQHE is the famous phenomena. However there are some questions. One of them is the spin-polarization. We examine it in this talk. ICTF16@Dubrovnik Oct. 14, 2014 OR 77

2. Hamiltonian and its eigen-states in single electron system The eigen state is Landau wave function as follows: Hamiltonian of single electron This narrow potential realizes the appearance of only the ground state for z-direction as follows: the momentum We obtain the eigen equation for the y-direction where the center position c is proportional to for level L=0 where the variables separated. -eV H

3. Wave function in Many electron system Landau wave function of electron for L=0 Attention please. Hall voltage is extremely larger than potential voltage for IQHE and FQHE. Accordingly there is no symmetry between x and y directions. the center position moves to the right. This is the many electron state When the momentum increases, The position c is proportional to the momentum. the center position moves to the left. When the momentum decreases,

4. Total Hamiltonian of many-electron system We separate H T into where H D is the diagonal term and H I is non-diagonal part. At ν=2/3, the most uniform configuration is created by repeating (filled,empty,filled). This configuration gives the minimum value for W. y x Unit cell At ν=3/5, most uniform configuration is the repeat of (filled,empty,filled,empty, filled). Unit cell where where C(k 1k N ) expresses the diagonal part of the Coulomb interaction Hc which is called “classical Coulomb energy”. filled,empty,filled The dashed lines indicate the empty states Also this configuration yields minimum value for W. References: S. Sasaki, ISRN Condensed Matter Physics Volume 2014 (2014), Article ID 468130, 16 pages. S. Sasaki, Advances in Condensed Matter PhysicsVolume 2012 (2012), Article ID 281371, 13 pages.

5. Spin Polarization I. V. Kukushkin, K. von Klitzing, and K. Eberl, Phys. Rev. Lett. 82, (1999) 3665. I.V. Kukushkin, K. von Klitzing and K. Eberl have measured the spin polarizations for twelve filling factors. II.Their results give the very important knowledges for the polarization in FQHE shown below. Magnetic field

6. Special transitions via the Coulomb interactions before after Equivalent interaction electron orbital 1 12 2 Electron distribution after transition is exactly the same as that before transition. Therefore this partial Hamiltonian should be solved exactly because of the energy-degeneracy. electron A electron B electron orbitals transition 1 DC 23123 This interaction is equivalent to the spin exchange with next form: electron A transition electron B Total momentum conserves. C D Momentum A increases by. Momentum B decreases. Transition to the left

7. Most effective Hamiltonian and its equivalent form For  =2/3 Let us find the equivalent Hamiltonian by using the following mapping: Zeeman energy The strongest interaction is  because of the nearest pair. Also the second strongest interaction is  because of the second nearest pair. Number of orbitals per unit cell Number of electrons per unit cell

8. Renumbering of operators give the diagonalization of H We introduce new operators We calculate the Fourier transformation of H, then j is the cell number This Hamiltonian can be exactly diagnalized by using the eigen-values of the matrix M The exact eigen-energies yield polarization. Cell 1 Cell 2Cell 3

9. Energy spectra for ν=2/3, 3/5 and 4/7 ν=2/3 ν=3/5ν=4/7 where Energy spectrum of ν=2/3 Energy spectrum of ν=3/5 Energy spectrum of ν=4/7 Wave number energy Wave number energy Wave number energy

10. Polarization of Composite Fermion theory ν =3/5 ? Polarization of the present theory Present theory Effective magnetic field Applied magnetic field Spin direction is opposite against usual electron system. Recently J.K. Jain has written the article. A note contrasting two microscopic theories of the fractional quantum Hall effect Indian Journal of Physics 2014, 88, pp 915-929 He summarized the composite fermion theory. Indian Journal of Physics ν =4/7ν =3/5 Finite temperature 9 = 1 Ratio of critical field strength He wrote as follows: For spinful composite fermions, we write = + , where = and  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, where we expect, from a model that neglects interaction between composite fermions, a spin singlet state at very low Zeeman energies (with = 2 + 2), a partially spin polarized state at intermediate Zeeman energies ( = 3 + 1), and a fully spin polarized state at large Zeeman energies ( = 4 + 0). ν =4/7 ? Polarization Applied magnetic field Polarization

11. Theoretical curve of Spin Polarization Small shoulder We should explain the small shoulders. We try it. Red points indicate the experimental data by Kukushkin et al. Blue curves show our results. Thus the theoretical results are in good agreement with the experimental data. These small shoulders exist certainly in the data. Shosuke Sasaki, Surface Science 566 (2004) 1040-1046, ibid 532 (2003) 567-575.

12. Spin Peierls effect The interval becomes narrower in the second and fourth unit-cell and so on. =2/3 Let us find the value t which gives the minimum energy. The classical Coulomb energy is expressed by t as; where C is the parameter depending upon devices. where are the coupling constants for non-deformation, and are also dependent upon devices. narrower wider We express this deformation by the parameter t. We take account of the famous mechanism “spin Peierls effect” into consideration. The interval between Landau orbitals becomes wider in the first and third unit-cell like this.

13. Eigen-energy versus deformation t Lowest point t 2. Classical Coulomb energy is proportional to t 2. Eigen-energy of the above Hamiltonian H is shown by red curve. The total energy becomes minimum at the lowest point as follows: energy

14. Spin Polarization = 2/3 Calculated total energy for =2/3 Matrix of the Hamiltonian Polarization of =2/3

15. Spin Polarization : =3/5 Electron configuration of =3/5 Calculated total energy for =3/5 Polarization of =3/5 Matrix of the Hamiltonian

16. Theoretical curve of Spin Polarization These theoretical results are in good agreement with the experimental data. Thus the spin Peierls instabilities appear in the experimental data of Kukushkin et al. S. Sasaki, ISRN Condensed Matter Physics Volume 2013 (2013), Article ID 489519, 19 pages

17. Summary of theoretical calculation Our treatment is simple and fundamental without any quasi-particle. We have found a unique electron-configuration with the minimum classical Coulomb energy. For this unique configuration there are many spin arrangements which are degenerate. We succeed to diagonalize exactly the partial Hamiltonian which includes the strongest and second strongest interactions. Then the results are in good agreement with the experimental data. The composite fermion theory has some difficulties for the spin polarization. It is necessary to measure the polarization and its direction, especially, at  = 4/5 and 6/5. The shapes of polarization curves and the direction are very important to clarify the FQHE.

18. Acknowledgement Professor Masayuki Hagiwara Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Japan Professor Koichi Katsumata, Professor Hidenobu Hori, Professor Yasuyuki Kitano, Professor Takeji Kebukawa and Professor Yoshitaka Fijita Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

19. Thank you for your attention

20. For the detailed discussion of composite fermions, please come after this talk.

21. ν =8/5ν =4/3 ν =2ν = - 2/3 + ν =2ν = - 2/5 + Jain’s explanation for  >1 J.K. Jain, “A note contrasting two microscopic theories of the fractional quantum Hall effect” Indian Journal of Physics 2014, Vol. 88, pp 915-929 Indian Journal of Physics Thus polarizations of these states are not clarified in the composite fermion theory. Two flux quanta are attached to hole opposite The IQH state of electrons is combined with the composite fermion state of holes (not electrons).

22. Polarization of the present theory ν =4/3ν =8/5 Our results are in good agreement with the experimental data.

23. Composite fermions for ν =3/5, 3/7, 4/7, 4/9 Recently J.K. Jain has written the article :, A note contrasting two microscopic theories of the fractional quantum Hall effect Indian Journal of Physics 2014, 88, pp 915-929Indian Journal of Physics He summarized the composite fermion theory. ν =4/9ν =3/7 Two flux quanta are attached to each electron Blue dashed curves indicate the energies of composite-fermion with up-spin. Red for down-spin Note: Red (down-spin) energy is higher than that of up-spin. Magnetic field B energy Magnetic field B energy 1 4 9 16 ν =3/5 ν =4/7 The effective magnetic field is opposite to that with to that with ν =3/7, 4/9. The polarization with is opposite to that with The polarization with ν =3/5, 4/7 is opposite to that with ν =3/7, 4/9.

24. down-spin up-spin Detail Comparison for Polarization Composite Fermion theory Present theory Contenuous spectrum Contenuous spectrum Composite fermion result deviates from the experimental data. 2 23 2 Ratio = 1 2 Ratio of B = 1 22 2 opposite direction Dashed curves indicate the empty levels. Solid curves indicate the levels occupied with composite fermions. Effective magnetic field for composite fermion Applied magnetic field Composite fermion result deviates from the experimental data. Spin direction is opposite against usual electron case. Absolute zero temperature Absolute zero temperature Finite temperature ν =3/5 ν =4/7

25. ν =4/5 Polarization for FHQ states with ν =4/5 and 6/5 The polarization versus magnetic field should be measured. Electron bound with four flux quanta Hole bound with four flux quanta ν =1ν = - (1/5) ν =6/5 ν =1ν =1/5