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Anyons in the FQHE Aut.: Jernej Mravlje Adv.: Anton Ramsak Fermions, bosons, and … anyons (1,2) (1) (2) (2) (1) (2,1) (1,2) (1,2) (1) (2) (2) (1) (2,1) (1,2) fermions bosons 2D: anyons Crucial for understanding the properties of the FQHE! (2,1) exp (i ) (1,2) Spin… l , max. proj. m l S n m m m n m l n 2 Spin is quantized in half integers. But … S S x iS y [Si , S j ] i ijk Sk 2D – any spin Spin-statistics theorem s Pauli (1940)] 2 2D: any spin -> any statistics? … and statistics (2,1,3,4,..., N ) exp(i ) (1,2,..., N ) (1,2, ,..., N ) exp(i2 ) (1,2,..., N ) leads to statistics n if the wave-function is single-valued f ( ) 2D vs. 3D 3D : 2 0 2D : 2 0 Configurational space Redundancy in notation x ( x1 , x2 ,..., xn ) Leinaas & Myrheim (1977) xi X ex. : xi ( x, y, z, s z )i Configurational space of identical particles Credundant X N Ccorrect X N S N ex.: 1D ( x1 , x2 ) ( x2 , x1 ) Anyons in 1D 2 2 2 2 2 2 2 H ( ) 2m x12 22 4m x 2 m z 2 x z Boundary conditions: ( x,0) 0 ( x,0) 0 z * j z ( x,0) ( x,0) ( x,0) ( x,0) 0 z z * ( x,0) ( x,0) z k ( x, z ) exp( ix)(cos kz sin kz) k Model for anyon B E t B dS ( E ) dS E ds E 2r t q c lz M z qEr 2 2 C 2 q l z 4 4 0 h 2 e e Addition of 0 is equal to a boost for 2 in angular momentum of our model particle. Boson to fermion and vice-versa! Arbitrary flux -> anyon. Two particle anyonic wavefunction ( p1 qA1 ) ( p2 qA2 ) H 2m 2m P 2 ( p qArel ) H 4m m r A1, 2 zˆ 2 ; r r1 r2 2 r (r , ) (r , ) A A ; ( ) 2 P2 p2 H 4m m (r , ) exp( iq) (r , ) exp( (r, ) exp( iq 2) (r,) iq ) (r , ) 2 anyonic! HE and QHE eE ev B neE jB B U I ne RH B ne • HE: Hall (1879) charge and density of carriers • QHE: Von Klitzing et al. (1980) standard for resistance h (Von Klitzing const. RK 2 25812.807449(86) ) e measurement of fine structure constant IQHE: explanation Landau levels 2 1 H ( p eA) 2m B A ( y , x ) 2 H nj c (n 1 2) nj J nj j nj (r l B ) 2 nj (r , ) (r lB ) exp( ij ) L (r lB ) exp( ) 2 j j n jmax R 2 lB2 B S 0 0 Filling factor: N Z N jmax z x iy 0 j ( z ) z exp( j z 2 2 ) FQHE discovered in 1982 (Tsui et al. ) Stormer (1992) Laughlin ground state noninteracting z 0 j ( z ) z exp( j 2 2 ) ( z1 , z2 ) ( z1 ) ( z2 ) ( z2 ) ( z1 ) exp( 0 1 0 1 z1 z2 2 4 2 z1 z2 2 ) ( z1 z2 ) exp( filled Landau level Z ( z1 , z2 ,..., z N ) ( zi z j ) exp( i j 2 1 zj ) 4 j Laughlin ground state at filling: 1 m N ( z1 , z2 ,..., z N ) ( zi z j ) m exp( i j Laughlin(1983) 1 2 z1 ) 4 j - a guess - shows excellent overlap with numerical results 4 2 ) Fractional charge and fractional statistics of Laughlin excitations 0 : j j 1 j j 1 excitation of ground state m z0 ( z1 , z2 ,..., z N ) ( zi z0 )m i fractionally charged C 2 q l z 4 4 anyonic! Measurements of frac. charge and frac. statistics Shot noise measurements Resonant tunneling experiments Saminadayar & Glattli (1997) Quantum computation Averin & Goldman (2001) Conclusion Anyons in 2D Fractional charge of excitations of FQHE ground-state Fractional statistics -> anyons Possible use for quantum computation which is robust against decoherence