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Spin Polarization of Fractional Quantum Hall States ICTF16@Dubrovnik Oct. 14, 2014 OR 77 Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Japan Shosuke Sasaki The FQHE is the famous phenomena. However there are some questions. One of them is the spin-polarization. We examine it in this talk. z Quantum Hall Device Source Magnetic Field y Drain x Current I width d Poential Voltage Vpotential length of 2DES Hall Voltage VH 1. Vpotential is nearly equal to zero at FQHE. No electron scattering. VH I where VH I 2. Hall resistance QHE creates no heat. 3. FQHE are purely eigen-value problem of electrons. 4. The value of VH is extremely larger than Vpotential. Therefore the gradient of VH cannot be ignored. Hamiltonian and its eigen-states in single electron system Potential W(z) -eVH Fig.6 Gate Potential -eV1 W(z) z Potential for z-direction (Potential width is very narrow) Potential Probe U(y) Magnetic field -eV2 y=0 Hall Probe y Fig.7 Potential U(y) y=d Hamiltonian of single electron 2 H 0 p eA 2m U y W z A yB, 0, 0, rot A (0,0, B) H 0 E This narrow potential realizes the appearance of only the ground state for z-direction as follows: x, y, z 1 exp ikx y z where the variables separated. We obtain the eigen equation for the y-direction k eBy 2 2 2 1 1 U y exp ikx y E exp ikx y 2 2m 2m y k eBy 2 (eB) 2 y k /(eB)2 (eB) 2 y c 2 The eigen state is Landau wave function as follows: where the center position c is proportional to ikx y c 2 e e z for level L=0 the momentum k 2 c integer eB eB Wave function in Many electron system Landau wave function of electron for L=0 Gate ikx y c 2 e e Potential Probe Magnetic field y Hall Probe b z k 2 c integer eB eB The position c is proportional to the momentum. Attention please. x Hall Probe Potential Probe Hall voltage is extremely larger than potential voltage for IQHE and FQHE. Accordingly there is no symmetry between x and y directions. electron orbital When the momentum decreases, increases, This is the manymoves electron state the center position to the right. left. Total Hamiltonian of many-electron system 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. N H T H 0 xi , y i , z i H C We separate HT into H T H D H I where HD is the diagonal term and HI is non-diagonal part. i1 HD k ,, k W k ,, k k ,, k where k1, , kN ,k E k C k , ,k 1 N 1 N 1 N k1 ,, k N W k1 , N N 0 i 1 N i1 1 N! where C(k1•••kN) expresses the diagonal part of the Coulomb interaction Hc which is called “classical Coulomb energy”. k1 xN , yN , zN x1, y1, z1 kN xN , yN , zN k1 x1, y1, z1 kN filled,empty,filled y At ν=2/3 , the most uniform configuration is created by repeating (filled,empty,filled). This configuration gives the minimum x value for W. Unit cell At ν=3/5 , most uniform configuration is the repeat of (filled,empty,filled,empty, filled). Also this configuration yields minimum value for W. Unit cell The dashed lines indicate the empty states Spin Polarization I. V. Kukushkin, K. von Klitzing and K. Eberl have measured the spin polarizations for twelve filling factors. Magnetic field z II. Their results give the very important knowledges for the polarization in FQHE shown below. y Source x Drain width d length of 2DES I. V. Kukushkin, K. von Klitzing, and K. Eberl, Phys. Rev. Lett. 82, (1999) 3665. Special transitions via the Coulomb interactions before transition Total momentum conserves. electron A 1 electron electron B B Momentum B decreases. 1 Transition to the left 2 electron orbital This interaction is equivalent to the spin exchange with next form: after Momentum A increases by 2 electron A 2 D 1 2 3 D C 1 2 3 electron orbitals hermite conjugate 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. transition C . 1 2 Equivalent interaction 1 3 hermite conjugate Most effective Hamiltonian and its equivalent form Number of electrons per unit cell For n=2/3 Number of orbitals per unit cell c3 c4 c1 c2 The strongest interaction is because of the nearest pair. Also the second strongest interaction is because of the second nearest pair. Let us find the equivalent Hamiltonian by using the following mapping: 0, H c*n 0 c 2 2 1 c1*c2 c2*c1 1 c c *2 j c 2 j 1 c *2 j c 2 j 1 c *2 j 1c2 j * 2 j 1 2 j j 1,2,3 i1,2,3 B g 1 * B 2c i c i 1 0 2 Zeeman energy Renumbering of operators give the diagonalization of H We introduce new operators a1, j a2 , j c1 c2 c3 c4 c5 c6 a1,1 a2,1 a1, 2 a2, 2 a1,3 a2,3 Cell 2 Cell 3 j is the cell number We calculate the Fourier transformation of H, then Cell 1 H a1* pa2 p a*2 pa1 p e ip a*2 pa1 p e ip a1* pa2 p p p B g 1 B 2a1* pa1 p a*2 pa2 p 2 0 2 This Hamiltonian can be exactly diagnalized by using the eigen-values of the matrix M B g * B e ip M ip * e g B B The exact eigen-energies yield polarization. Energy spectra for ν=2/3, 3/5 and 4/7 ν=3/5 ν=2/3 B g * B e ip M ip * B g B e energy ν=4/7 B g * B 0 e ip ip B g * B e * g B 0 B M B g *B M * 0 g B B eip * B g B * eip 0 g B B energy Wave number Energy spectrum of ν=2/3 1 1 e d 2 energy Wave number Energy spectrum of ν=3/5 Wave number Energy spectrum of ν=4/7 d tanh s p 2kBT dp s 1 where polarizati on e dimension of matrix d Polarization of the present theory Present theory Polarization Polarization ν =4/7 ν =3/5 Applied magnetic field Applied magnetic field Applied magnetic field He wrote as follows: For spinful composite fermions, we write n* = n*+ n* , where = n* and n* are the filling factors of up and down spin composite fermions. possible spin polarizations of theAvarious FQHE states then predicted by analogy to the IQHE of spinful electrons. Recently J.K.The Jain has written the article. note contrasting two are microscopic theories of the fractional quantum Hall effect For example, the 4/7 state maps into n* = 4, where we expect, from a model that neglects interaction between composite fermions, a spin singlet Indian of Physics 2014, 88,npp summarized theory. state at very Journal low Zeeman energies (with * =915-929 2 + 2), a He partially spin polarizedthe statecomposite at intermediatefermion Zeeman energies (n* = 3 + 1), and a fully spin polarized state at large Zeeman energies (n* = 4 + 0). Polarization of Composite Fermion theory Effective magnetic field ν =4/7 ? Spin direction is opposite against usual electron system. Ratio of critical field strength = Finite temperature 1 9 ν =3/5 ? Theoretical curve of Spin Polarization 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. Shosuke Sasaki, Surface Science 566 (2004) 1040-1046, ibid 532 (2003) 567-575. n23 Small shoulder n 3 5 n 4 7 Small shoulder 0.25, kBT 0.2 0.25, kBT 0.2 n 7 5 n 1 2 kBT 5.5 0.25, kBT 0.1 Small shoulder 0.35, kBT 0.1 n 8 5 0.1, kBT 0.1 These small shoulders exist certainly in the data. explain the small shoulders. We try it. We should Spin Peierls effect We take account of the famous mechanism “spin Peierls effect” into consideration. The between Landau orbitalsinbecomes widerand in the first and third and unit-cell like this. Theinterval interval becomes narrower the second fourth unit-cell so on. wider narrower wider narrower n =2/3 ' ' ' ' We express this deformation by the parameter t. The classical Coulomb energy is expressed by t as; W N 0 C t 2 where C is the parameter depending upon devices. 0 1 t , 0 1 t 0 1 t , 0 1 t where 0 , 0 are the coupling constants for non-deformation, and are also dependent upon devices. Let us find the value t which gives the minimum energy. Eigen-energy versus deformation t a1* p a 2 p a 2* p a1 p a 2* p a3 p a3* p a 2 p H * * ip * ip * p a3 p a 4 p a 4 p a3 p e a 4 p a1 p e a1 p a 4 p g B1 22a p a p a p a p a p a p a p a p 4 * B * 1 1 * 2 * 3 2 3 * 4 p energy 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: Lowest point 4 Spin Polarization n = 2/3 ' ' ' ' B g *B 0 e ip Matrix of * B g B 0 the Hamiltonian M 0 B g *B * * * * a1 p a 2 p a 2 p a1 p a 2 p a3 p a3 p a 2 p e ip H 0 B g * B * * ip * ip * a3 p a 4 p a 4 p a3 p e a 4 p a1 p e a1 p a 4 p B g * B1 2 2 a1* p a1 p a 2* p a 2 p a3* p a3 p a 4* p a 4 p 4 p p Calculated total energy for n=2/3 Polarization of n=2/3 1 1 4 e dp tanh s p 2k BT 4 2 s 1 Spin Polarization : n =3/5 Electron configuration of n=3/5 ' ' ' Calculated total energy for n=3/5 BgB 0 0 0 0 e ip Matrix of the Hamiltonian 0 0 0 BgB 0 BgB 0 0 BgB 0 0 0 0 0 0 0 0 0 BgB 0 e ip 0 0 0 BgB 0 Polarization of n=3/5 1 6 e tanh p 2 k T dp s B 6 2 s 1 Theoretical curve of Spin Polarization S. Sasaki, ISRN Condensed Matter Physics Volume 2013 (2013), Article ID 489519, 19 pages 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. 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 n= 4/5 and 6/5. The shapes of polarization curves and the direction are very important to clarify the FQHE. 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 Thank you for your attention For the detailed discussion of composite fermions, please come after this talk. Jain’s explanation for n>1 ν =4/3 ν =8/5 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 The IQH state of electrons is combined with the composite fermion state of holes (not electrons). ν =2 + ν = - 2/3 Two flux quanta are attached to hole Effective magnetic field Applied magnetic field + ν =2 + ν = - 2/5 Two flux quanta are attached to hole opposite Effective magnetic field Applied magnetic field + Thus polarizations of these states are not clarified in the composite fermion theory. Polarization of the present theory ν =4/3 ν =8/5 Our results are in good agreement with the experimental data. Composite fermions forarticle ν =3/5, 3/7, 4/7, 4/9 Recently J.K. Jain has written the :, 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. Two flux quanta are attached to each electron Effective magnetic field Applied magnetic field Applied magnetic field Blue dashed curves indicate the energies of composite-fermion with up-spin. Red for down-spin Effective magnetic field energy Magnetic field ν =3/7 ν =3/5 Applied magnetic field ν =4/9 The effective magnetic field is opposite to that with ν =3/7, 4/9. ν =4/7 Effective magnetic field (opposite direction) Applied magnetic field Effective magnetic field (opposite direction) 1 4 9 16B Note: Red (down-spin) energy is higher than that of up-spin. energy Magnetic field B The polarization with ν =3/5, 4/7 is opposite to that with ν =3/7, 4/9. Detail Comparison for Polarization Absolute zero temperature ν =3/5 ν =4/7 Spin direction is opposite against usual Composite fermion result deviates electron case. from the experimental data. Finite temperature Composite Fermion theory Composite fermion result deviates from the experimental data. Dashed curves indicate the empty levels. Solid curves indicate the levels occupied with composite fermions. Applied magnetic field Effective magnetic field for composite fermion down-spin 2 Ratio of B = 1 Absolute zero temperature opposite direction Finite temperature Present theory Contenuous spectrum 2 2 up-spin Ratio = 1 2 2 2 Contenuous spectrum 3 2 Polarization for FHQ states with ν =4/5 and 6/5 ν =4/5 ν =1 ν = - (1/5) Applied magnetic field Effective magnetic field Hole bound with four flux quanta + Applied magnetic field Effective magnetic field + ν =1 ν =6/5 Electron bound with four flux quanta ν =1/5 The polarization versus magnetic field should be measured.