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Anomalous Cross Section Induced by Topological Quantum Interference De-Hone Lin Department of Physics, NSYSU 23 December 2004 Partial wave theory of a two dimensiona l scattering problem for an arbitray short range potential and a nonlocal A - B magnetic flux is establishe d. The scattering process of a hard disk like potential and the magnetic flux is examined. Since the nonlocal influence of magnetic flux on the charged particles is universal, the nonlocal effect in hard disk case is expected to appear in quite general potential system and will be useful in understand ing some phenomena in mesoscopic phyiscs. Partial wave theory of a three dmensional scattering problem for an arbitrary short range potential and a nonlocal A - B magnetic flux is establishe d. An anomalous total cross section is revealed at the specific quantized magnetic flux at low energy whi ch helps explain composite fermion and boson model in the fractional quantum Hall effect. Since the nonlocal quantum interferen ce of magnetic flux on the charged is universal, the nonlocal effect is expected to appear in quite general potential system and will be useful in understand ing some other phenomena in mesoscopic phyiscs. Fractional Quantum Hall States •2-D electron system inside the GaAs/AlGaAs heterostructure •High magnetic fields (B~10T) •Low temperatures (T~0.1K) Coulomb forces flux quantum attachment Electron vortex attraction at fractional Landau - level filling 1/3. (a) Placing three vortices onto each electron reduces electron - electron (Coulomb) repulsion. (b) Vortex attachment can be viewed as the attachment of magnetic flux quanta to the electrons, transform ing them to composite bosons. An even - denominato r fractional Hall plateau h / e2 at xy is observed 5/ 2 at tempera tures below 100 mK R. willett, J.P. Eisenstein, H.L. Stormer, D.C. Tsui , A.C. Gossard, and J.H. English, PRL, vol. 59, 1776, 1989. FQHE at 5/2. A FQHE state at such an even denominato r fraction should not be allowed. The origin of the state remains unclear. An exciting possibilit y for the origin of this state is the formation of composite fermion pairs. H.L. Stormer, Nobel Lecture : The fractional quantum Hall effect RMP, 71, 875, 1998. Nature, Vol. 406, 863 (2000). Scarola et al.' s calculatio n is a beautiful demonstrat ion that a Fermi sea of composite fermions can form Cooper pairs, and gives some insight into why such pairing occurs at 5 / 2 but not 1/2. Like Cooper' s 1956 calculatio n, this is an important step forward in our understand ing of the 5 / 2 state, but it is not yet a full BCS theory of composite fermion pairing. It is possible that such a theory, when it arrives, may help explain another phenomenon in which pairing arising from purely repulsive interactio ns - - high - temperatur e supercondu ctivity. N. Bonesteel, Nature, Vol. 406, 841 (2000). A.K. Geim etc, Nature 407, 55, 2000. At first pointed out by Bardeen and Ginzburg in the early sixties, the amount of magnetic flux carried by vortice s in supercondu cting materials depends on their distance from the sample edge, and can be smaller th an one flux quantum, h / 2e. In bulk supercondu ctors, this reduction of flux becomes negligible at submicrome tre distances from the edge, but in thin films the effect may survive much farther into the material. Here we measure the amount of flux introduced by individual vortices in a supercondu cting film, finding that the flux always differs substantia lly from . We have observed vortices that carry as little as 0.001 , as well as negative vortices, whose penetratio n leads to the expulsion of magnetic field. Summary Quantum interference of magnetic flux Quantum interference in partial wave theory and anomalous cross section in two dimensions Quantum interference in partial wave theory and anomalous cross section in three dimensions Composite bonsons and fermions Introduction V () Bound states therein 1 and V () 0 2 for a for a Radius Phase shifts A charged particle D. Bohm and Y. Aharonov in 1959 found AB effect Interference pattern Charged particle Magnetic flux ( x) 1 ( 0) 1 ( x) ( 0) 2 ( x) ( 0) 2 ( 0) ( 0) ( x ) 1 ( x ) 2 ( x ) ( 0) x ie 1( 0) ( x ) exp A( x ) dx c 1 1 (x ) 2 ie x ( x ) exp A( x ) dx c 2 ( 0) 2 (x ) x x ie ie ( 0) (0) 1 ( x ) exp A( x ) dx 2 ( x ) exp A( x ) dx. c 1 c 2 ie ( 0) (0) ( x ) 1 ( x ) 2 ( x ) exp A( x ) dx c ie ( 0) (0) 1 ( x ) 2 ( x ) exp . c The periodic cycle of interferen ce is given by (c / e)2 . The four-vector formulation of the non-integrable phase factor is given by ie exp A dx . c C.N. Yang, and T.T. Wu, Phys. Rev. D 12, 2845 (1975). 1. It is non-local in the sense that it exists even when the interfering beams pass through a field free region and is associated with the entire closed curve C. 2. It is topological in the sense that the phase shift is unaffected when is deformed within the field free region. 3. It is geometrical in the sense that the above phase factor represents parallel transport (holonomy transformation) around with respect to the electromagnetic connection gauge. Aharonov-Bohm magnetic flux V () Bound states therein 1 and V () 0 2 for a for a Radius Phase shifts A charged particle The system is very important in understanding the quantum Hall effect, superconductivity, and the transport properties of nano structures. Quantum interference in partial wave theory and anomalous cross section in two dimensions Partial Wave Method for a Short Range Potential and an Aharonov-Bohm Flux ( 0 ) 2 E H 0 x , i G x , x ; E ( x x ), 2 2 where H 0 V ( x ). 2 In polar coordinates for the cylindrically symmetric system: (0) G x , x ; E 1 im( ') G , ' ; E e 2 m (0) m Magnetic field exists in the system, then ie (0) Gx , x ; E G x , x ; E exp A(r ) dr x c x For the Aharonov-Bohm Flux yeˆ x xeˆ y A( x ) 2 g , 2 2 x y the magnetic field B3 4g ( x ), and the magnetic flux g / 4 . 2 d 2 1 d 2 2 V ( ) G ( , ; E ) ( ) E 2 d 2 d Where m 0 is a real number wit h 0 2eg / c / 0 The corresponding radial wave equation reads 2 d 2 1 d 2 2 V ( ) Rk ( ) 0 E 2 d 2 d with k 2E / . The solution in exterior region a Rk ( ) k cos (k ) J (k ) sin (k ) N (k ) The general solution of a scattering particle reads k ( x ) m k cos (k ) J (k ) sin (k ) N (k )e im At x i ie k ( x ) Asymp exp ik x exp A( x ) dx f ( ) exp ik c The scattering amplitude 1 f ( ) 2k e m i ( / 4 ) 2i sin e The total cross section 4 2 t sin k m im For identical bosons (fermions) carrying the magnetic flux, the differenti al cross section is given by ( ) f ( ) f ( ) . 2 Thus the total cross section reads ( )d which yield - t (bosons) 16 t (fermions) 2 m ,even 16 sin 2 sin m ,odd 包含AB effect 的分波散射理論所繪的短範圍位能相互作用的散射截 面圖,圖一橫軸是能量的大小,縱軸是散射截面的大小,可看出低 能量時散射截面產生驚人的下降現象;圖二橫軸是磁通的大小,可看 到散射截面隨著磁通以週期性變化的神奇現象。這些效應對於納米 量子傳輸系統和納米量子光電系統有許多重要的應用。 Total cross sections for identical bosons carrying the magnetic flux. The cross section t 0 at the quantized magnetic flux ( 2n 1) 0 . Periodic structures of cross sections of bosons carrying the magnetic flux. t 0 when the magnetic flux is quantized at (2n 1) 0 for ka 0.5. Total cross sections of identical fermions carrying the magnetic flux. t 0 when the magnetic flux is quantized at 2n 0 for ka 0.5. Periodic structures of total cross sections for identical fermions carrying the magnetic flux. t 0 when the magnetic flux is quantized at 2n 0 for ka 0.5. Quantum Interference and Anomalous Cross Section in Three Dimensions Plane wave exp{ik r } l 0 l ~ l m 4i l jl (kr)Ylm*~ ( k , k )Ylm~ ( , ) Quantum interference of magnetic flux leads to r ie exp{ik r } exp{ A( r ) dr } c P q 0 * ~ C j ( kr ) Z q ,m l qm ( k , k ) Z qm ( , ) m ~ ~ ~ l where l q m 0 , and Cq ,m i (2l 1). The angular part is defined as Z q,m ( , ) ~ m ( q 1)( l m 0 1) 0 ( m 0 , m 0 ) im cos sin P (cos ) e ~ q 2 2 2 ( l 1) ( m 0 , m 0 ) q where P (cos ) is the Jacobi function. The general solution for a charged particle moving in a short range potential, and an Aharonov-Bohm magnetic flux is found to be k (r ) q 0 m u ~l (r ) r Z ( k , k ) Z qm ( , ) * qm At large distance, we expect it to become like r ie exp{ikr} k ( r ) F exp{ik r } exp{ A( r ) dr } f ( , ) c P r The asymptotic behaviorufor ~ (u r~l)(r ) is given by l ~ C ( k ) l u~l ( r ) sin kr ~l 2 k r 1 ~ l ~ ~ l sin ~l sin( l ) sin kr 2 The Scattering amplitude is found to be 1 f ( , ) k q 0 i ~l 2~ e sin ~l cos l * ~ (2l 1) ~ ~ Z q ,m ( k , k ) Z q ,m ( , ) i ( ~l l ) 1 e m sin ~l sin l At the quantized values of flux, the result reduces to the well-known amplitude 1 i l f ( ) (2l 1)e sin l Pl (cos ) k q 0 Magnetic flux y e x e In the case of the incident direction perpendicu lar to the magnetic 2, ktotal 0)cross section d ( , ) reads flux, i.e. k / 2, (kk 0,/ the t t d ( , ) 4 t 2 k q 0 m ~2 2 4 (2 1) sin cos ( ) Z q ,m 2 2 1 2 sin sin cos( ) sin sin where ~2 (2q m 0 ), and Z q ,m ~ (q 1 / 2)( l 1 / 2) . ~ (q 1)( l 1) The total cross sections for the identical bosons (fermions) t ( , ) f ( , ) ( , ) carrying the magnetic flux are given by t ( , ) f ( , ) ( , ) t (bosons ) 16 k2 q 0 16 t ( fermions) 2 k q 0 ~ (2 1) sin 2 cos 4 ( ) Z q2,m 2 2 m ,even 1 2 sin sin cos( ) sin sin ~ (2 1) sin 2 cos 4 ( ) Z q2,m 2 2 1 2 sin sin cos( ) sin sin m ,odd Hard Sphere Potential The phase shift is given by tan j (ka) / n (ka) Accordingly, the total cross sections is given by 16 t 2 k ~ (2 1) cos 2 ( ) J 2 1/ 2 (ka) Z q2,m q 0 m J 2 1/ 2 (ka) J 2 1/ 2 (ka) 2 sin( ) J 1/ 2 (ka) J 1/ 2 (ka) The total cross - section for a charged particle scattering by a hard sphere, and a magnetic flux with 0 2a 2 The periodic structure of the scattering total cross - sections for hard sphere, and a magnetic flux along the z - axis. Anomalous appear at the quantized values of flux (2n 1) 0 / 2, n 0,1,2, . The total cross - sections for the identical bosons carrying the magnetic flux. Anomalous appear (disappear ) at the quantized magnetic flux (2n 1) 0 ( 2n 0 ). The periodic stucture of AB oscillatio n for the cross - sections of identical bosons carrying the magnetic flux. The cross section approaches zero when the flux is quantized at (2n 1) 0 for ka 0.5. The total cross - sections for the identical bosons Carrying the magnetic flux. Anomalous appear (disappear ) at the quantized magnetic flux 2n 0 ( (2n 1) 0 ). The periodic stucture of AB oscillatio n for the cross - sections of identical fermions carrying the magnetic flux. The cross section approaches zero when the flux is quantized at 2n 0 for ka 0.5. Conclusions: (1) The total cross section is drastically decreased in the long wave length limit and(or) sufficient short range potential. This phenomenon may ascribed to the magnetic flux induced transparency (FIT). (2) The cross section is symmetric around magnetic flux n 0 / 2 with the oscillating period 0 ,where n is the positive integer, and is the fundamental magnetic flux quantum 2c / e. (3) For identical “Bosons” (“Fermions”), there exists the phenomenon of FIT only for odd (even) number multiple of 0 , and the cross section is symmetric around the odd (even) number multiple of with the oscillating period 2 0 . Such effect is similar to the picture of the composite Boson (Fermion) in two dimensional fractional quantum Hall effect, and is useful in the question on pinning force in superconductor. Thank you! Composite Particles in Fractional Quantum Hall Effect Nature, Vol. 406, 863 (2000). Scarola et al.' s calculatio n is a beautiful demonstrat ion that a Fermi sea of composite fermions can form Cooper pairs, and gives some insight into why such pairing occurs at 5 / 2 but not 1/2. Like Cooper' s 1956 calculatio n, this is an important step forward in our understand ing of the 5 / 2 state, but it is not yet a full BCS theory of composite fermion pairing. It is possible that such a theory, when it arrives, may help explain another phenomenon in which pairing arising from purely repulsive interactio ns - - high - temperatur e supercondu ctivity. N. Bonesteel, Nature, Vol. 406, 841 (2000).