Dynamical Symmetries for Nanostructures: Implicit Symmetries in Single-Electron Transport Through Real and Artificial Molecules

366 Pages·2012·4.73 MB·English

by  Konstantin Kikoin

Konstantin Kikoin
Mikhail Kiselev
Yshai Avishai
Dynamical Symmetries
for Nanostructures
Implicit Symmetries in Single-Electron Transport
Through Real and Artificial Molecules
Ph.D. Konstantin Kikoin Ph.D. Mikhail Kiselev
School of Physics and Astronomy The Abdus Salam Intl. Center
Tel Aviv University for Theoretical Physics
69978 Tel Aviv Strada Costiera 11
Israel 34151 Trieste
[email protected] Italy
[email protected]
Ph.D. Yshai Avishai
Ben Gurion University
84105 Beer Sheva
Israel
[email protected]
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Dedicatedto thememoryof YuvalNe’eman
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whoseideasarethecornerstonesof the
theoriespresentedin thisbook.

Preface
Themaingoalofthismonographistodemonstratetherelevanceofdynamicalsym-
metryandits breakingto therapidlygrowingfield ofnanophysicsin general,and
nanoelectronicsinparticular.Itisintendedtoamalgamateseeminglyhighlyabstract
conceptsofGrouptheorywiththephysicsofrecentlyfabricatednanoobjectssuch
assingleelectrontransistors.Inallthesesystems,dynamicalsymmetriesareshown
to be intimately related with many-bodyphysics,and in particular,the ubiquitous
Kondoeffectandotherhallmarksofquantumimpurityproblems.Thereby,weex-
poseyetanotherfacetoftheexistingdeepandprofoundrelationsbetweenquantum
fieldtheoryandcondensedmatterphysics.
The conceptof symmetry in quantum mechanicshas had its golden age in the
middleofthelastcentury.Inthatperiod,thebeauty,eleganceandefficiencyofgroup
theoreticalphysicshasbeenexposedinnumerousremarkablerevelations,fromclas-
sification of hadron multiplets, isospin in nuclear reactions, the orbital symmetry
inRydbergatoms,point-groupsincrystallography,translationalsymmetryinsolid
statephysics,andsoon.Atthefocusofallthesestudiesstandsthesymmetrygroup
oftheunderlyingHamiltonian.Usingthepowerfulformalismofgrouptheory,the
energy spectrum of the physical system possessing the pertinent symmetry could
beextractedwithinanelegantandtimesavingformalism.Exploitingtheproperties
of discrete and infinitesimal rotation and translation operators, general statements
aboutthebasicpropertiesofquantummechanicalsystemscouldbeformulatedina
formoftheorems(Wignertheorem,Blochtheorem,Goldstonetheorem,Adlerprin-
ciple, etc). The intimaterelation betweengrouptheoryand quantummechanicsis
thereforewellestablishedandhasbeenexposedinnumerousexcellenthandbooks.
Asomewhatmoresubtleaspectfeaturinggrouptheoryandquantummechanics
emergedandwasformulatedlateron,thatis,theconceptofdynamicalsymmetry.
Thenotionofdynamicalsymmetrygroupisdistinctfromthatofthefamiliarsym-
metrygroup.Tounderstandthisdistinctioninanheuristicwayletusrecallthatall
generators of the symmetry group of the Hamiltonian Hˆ encode certain integrals
ofthemotion,whichcommutewithHˆ.Theseoperatorsinducealltransformations
whichconservethesymmetryoftheHamiltonian,andmayhavenon-diagonalma-
trixelementsonlywithinagivenirreduciblerepresentationspaceofHˆ.Ontheother
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viii Preface
hand,dynamicalsymmetryofHˆ isrealizedbytransformationsimplementingtran-
sitionsbetweenstatesbelongingtodifferentirreduciblerepresentationsofthesym-
metrygroup.Onemaythensaythatthegeneratorsofdynamicalsymmetrygroupof
a quantummechanicalsystemare infactthe generatorsoftheenergyspectrumor
somepartofit.Specialexamplesofdynamicalsymmetriesinquantummechanics
emergeashiddensymmetries,whereadditionaldegeneracyexistsduetoanimplicit
symmetryof the interaction.Anotherexampleis supersymmetry, where the group
algebraincludesbothcommutationandanticommutationrelations.
ThestartingpointinmostofouranalysisisageneralizedAndersonHamiltonian
which,undercertainconditionscanbeapproximatedbyageneralizedspinHamil-
tonian encodinga myriad of exchange interactionsbetween localized electrons in
nano-objects(suchasquantumstatesincomplexquantumdots)anditinerantelec-
tronsinthereservoirsmadeincontactwiththelocalizedelectrons.Theseexchange
interactionsmaybeduetospinaswellastoorbitaldegreesoffreedom.Theylead
to effective exchange Hamiltonians that display a rich pattern of dynamical sym-
metries. Mathematically, these symmetries are exposed as the pertinent exchange
Hamiltonianincludes,inadditiontothestandardspinoperators,newsetsofvector
operatorswhichformthebasisfortherepresentationofirreducibletensoroperators
enteringtheeffectiveHamiltonian.Theseoperatorsinducetransitionsbetweendif-
ferentspinmultipletsandgeneratedynamicalsymmetrygroups(suchasSU(n)and
SO(n))thatare notexposedwithinthe bareAndersonHamiltonian.Like in quan-
tum field theory,the most dramaticaspects of dynamicalsymmetryin the present
contextisnotitsrelationwiththespectrumbut,rather,themannerinwhichitisbro-
ken.Anindispensabletoolformanipulatingthepertinentmathematicsrequiredfor
identifyingtherelevantdynamicalsymmetrygroupsisthesuperalgebraofHubbard
operators,uponwhichwewillheavilyrely.
Theroleofdynamicalsymmetriesandtheirmanifestationswillbereviewedand
analyzed in several systems such as complex quantum dots (planar, vertical and
self-assembled), molecular complexesadsorbed on metallic surfaces and attached
to quantum wires, cold gases confined in magnetic traps. It will be shown how
these dynamical symmetries are activated by Coulomb and exchange interactions
withitinerantelectronsinthemacroscopicFermiorBosereservoirs(metallicleads
and substrates in various nanodevices). We will then develop the concept within
numerousphysicalsituations,includingtheKondocotunnellinginvariousenviron-
ments.Thenotionofdynamicalsymmetryismeaningfulalsoforthesystemsoutof
equilibrium,inpresenceofelectromagneticfieldandstochasticnoiseandin time-
dependentproblemslikeLandau–Zenereffect.
Thus, the main goal of this book is to generalize the principles of dynamical
symmetries formulated for the integrable systems to the many-body systems, for
whichonlythelow-energypartoftheexcitationspectrumisknown.
TelAviv-Trieste-BeerSheva, KonstantinKikoin
October31,2011 MikhailKiselev
YshaiAvishai
Acknowledgements
We acknowledgefruitfuldiscussionswith our colleaguesBorisAltshuler, Jan von
Delft,PeterFulde,YuriGalperin,YuvalGefen,LeonidGlazman,VladimirGritsev,
David Khmelnitskii, Il’ya Krive, Tetiana Kuzmenko, Stefan Ludwig, Laurens W.
Molenkamp,Florina Onufrieva,MichaelPustilnik, Jean Richert, RobertShekhter,
MaartenWegewijs.
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