@article{69f1446c2f184a158976c58d82294a0b,

title = "Hall resistance for fermions in a flux-tube and a possible violation of perfect Aharonov-Bohm periodicity",

abstract = "We study the Hall resistance of a two dimensional quantum system with reflecting walls threaded by a magnetic flux tube of small radius. The interaction of the electron spin with the magnetic field inside the flux tube is taken explicitly into account. Owing to the lack of rotational invariance, this interaction leads to a small (albeit qualitatively remarkable) deviation from perfect Aharonov-Bohm periodicity even when the flux tube shrinks into a flux line. We explain how this result is compatible with gauge invariance.",

author = "Y. Avishai and Band, {Y. B.}",

note = "Funding Information: We suggestt hat in contrastw ith the free space Aharonov-Bohmsc atterinrge sult,e lectricarle sistance (and in fact, electront ransmissioinn general)in an Aharonov-Bohmflu x is sensitivteo the spin degreeo f freedome venfor unpolarizede lectronsD. ue to the lack of rotationasly mmetryit, is not possibleto relate the scalara nds pinora mplitudebsy a merep haseT. he effecto f spin is quite sizable.M oreovert, his leadst o a small deviation from perfect Aharonov-Bohm of the tube is very small, the discussionf ollowing periodicity. equation(2 ) indicatesth atthe dominanct ontribution For the Hall resistanceo f an unpolarizedt wo to this lack of periodicityw ill come from the presence dimensionael lectrong as in the symmetrigc eometry, of the singulars olutionw hichin turn is relatedt o the as in Fig. (1), the following points hold: (1) Our effecto f spin. In this case,a precursofro r the limiting numericalr esultsare consistentw ith time reversal case is expectedto emergeT. o be more concretew, e symmetryw hich requires that R,~(q~b) e an anti-suggesta measuremenotf the Hall resistancea t symmetrifcu nctiono f flux, i.e. R,,(qS) = - R~,(-~b). 4) = _+5 to study the deviation from Aharonov-(2) lfq~ = N, the flux is unseen(s eer emarkf ollowing Bohm periodicity as the radius of the flux tube equation(3 )), R,~(4)= N) = R,~(4) = 0), as required becomess mallera nd smaller. by gauge invariance.( 3) If (1~# N, R,,(q~) breaks Aharonov-Bohmpe riodicityw henthe additiono f an integern umbero f flux units changest he sign of the total flux. Thus, if~b > 0then R,,(~b + N) = R,,(~), and similarlyf or negativvea luesR ,,(-• - N) = R,~ (-O), but if q5 > 0and q5 - Nthen R,,(~b - N) # R,,(qS). In particulaRr ,, (qS)f ailsto vanisha t q5 = 0.5. Havingp resenteedv idenceth atp erfectp eriodicity is lost eveni n the limit wheret heflux tubes hrinksi nto a singlef lux line, we muste xplainh ow this resultd oes not violateg augei nvarianceO. ne may argue that if the limit R --* 0 is takeno n the original vector potential, if impliesa (locally)p ureg augep otentiaAl = (O/27rr) O, and hence,i f the changeq ) ~ qb + nq~0is observable,t his violatesg augein varianceH. owevert, hecurl of such vectorp otentiails the singularm agneticfi eld H = (@/2~) 6(r)~. In more than one dimension,a delta function potential in the wave equation is problematict;h e wave-functionm ust vanish at the origin in order to maintain a finite energyd ensity integral.B ut then the wave functionc annotc ontaina Bessefl unctiono f negativefr actionaol rdera s we have argued above \[3, 4\]. The way out of this cycle of contradictinagr gumentiss to notice that the limiting procedurwe hichwe havea doptedis takeno n the wave .\['unction and not on the vector potential. These two limiting procedureds o not commutea, nd it is only the first one which makess ense.I n other words, the vector potentials(O /2~r)0 and qb + nOo/2~r 0 are not distinguishablea,s dictatedb y gaugei nvariance. On the other hand, therea re situationsfo r which Acknowledgements - We would like to thank Ph. de SousaG erbertB, . Horovitz,M . Kaveh and J. Polonyi for very helpfuld iscussionsT.h is researchis supported in part by a grant from the U.S. Israeli Binational ScienceF oundation.",

year = "1991",

month = jan,

day = "1",

doi = "10.1016/0038-1098(91)90431-T",

language = "English",

volume = "77",

pages = "77--81",

journal = "Solid State Communications",

issn = "0038-1098",

publisher = "Elsevier Ltd.",

number = "1",

}