Fatih Balli Department of Physics, University of South Carolina 11/6/2015. Fatih Balli, Department of Physics UofSC

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1 Fatih Balli Department of Physics, University of South Carolina 11/6/2015 1

2 Timeline and Motivation Hall Effect Landau Problem on Planar System Quantum Hall Effect Incompressibility and Multiparticle Wavefunction Effect on Metrology 2

3 In 1879 Edwin Hall discovered Hall Effect. In 1930 Landau problem(quantization) is constructed for single electron in 2D geometry under magnetic field. In 1980 integer quantum hall effect is discovered by Klaus von Klitzing. In 1982 fractional quantum hall effect is discovered by Stormer, Tsui and Gossard In 1983 R.B. Laughlin constructed multi particle wavefunctions in 2D systems. The problem was that the Laughlin wavefunctions are not translationally invariant. 3

4 In 1983 F.D.M. Haldane proposed an electron gas system on under the influence on Dirac monopole located at the center of. In 2000 Hu and Zhang generalized the QHE on under a SU(2) gauge field. However, infinite number of SU(2) internal degrees of freedom is needed to obtain finite spatial density. In 2004 Nair and Karabali formulated QHE on complex projective spaces. 4

5 In 2013 F. Balli, S. Kurkcuoglu, A.R. Behtash and G. Unal formulate QHE on Grassmann Manifolds. Quantum Hall Effect (QHE) on complex Grassmann manifolds Gr2(CN) are formulated. We setup the Landau problem in Gr2(CN) and solve it using group theoretical techniques and provide the energy spectrum and the eigenfunctions in terms of SU(N) Wigner D functions for electrons on Gr2(CN) under the influence of abelian U(1) and non abelian SU(2) background magnetic monopoles or a combination of these thereof. We identify the Lowest Landau Level (LLL) for the single particle states and give their degeneracy in terms of the dimension of the relevant irreducible representations (IRR's) of SU(N). 5

6 First observed by Edwin Hall in 1879 At steady state condition Lorentz force law gives "Resistivity" on Hall Systems is = 6

7 Electrons are trapped in a thin layer such as interface between GaAs and GaAlAs Strong Magnetic Field an Low Temperature Spin frozen state due to Zeeman Effect Single electron (or weakly interacting electron system) Ψ(t,x,y,z) = ψ (x,t) (z) x = (x,y) (z) is the wave function of ground state 7

8 Hamiltonian of the system is,, 8

9 We can define annihilation and creation operators as, b X+iY) The corresponding Hamiltonian and energy eigenvalue equations are 9

10 Since the Hamiltonian is gauge dependent, corresponding wave functions differ with respect to the gauge chosen. For the disk geometries, symmetric gauge is a proper choice. The wave functions reads is the extremum of the probability density function. 10

11 Quantization of Resistivity A New Phase of the Matter Requirements: I. Strong Magnetic Field II. Low Temperature 11

12 February/5/1980 Resistivity Filling Factor 12

13 Band Diagrams of Structure containing AlGaAs andgaas 13

14 The Effect of Disorder 14

15 Regions of localized states in the Landau levels which are broadened by the impurity potential is as 15

16 a) at each electron is pierced by a flux quanta b) When one electron is added it is placed in a higher energy level. 16

17 Incompressibility factor is: Incompressibility condition: =0. If the system is incompressible, chemical potential changes discontinuously as a response to change in number density. This is possible if Multiparticle wave function is Wave functions are not translationally invariant since momentum operator does not commute with Hamiltonian. 17

18 Von Klitzing constant is being used as the standard resistance unit since 1990 Ω QHE allows a high precision measurement of the ne structure constant A result is 18

19 "...It is, however, my hope that the individuum will remain the center because the creative act remains very personal despite the increasing need for collaborative efforts... " "... I am happy, that I have so many friends all over the world who contributed to my research work, and I believe that also in the future basic research offers the best opportunity of reaching across borders and overcoming ideological barriers... " Klaus von Klitzing s speech at the Nobel Banquet; 12/10/

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