Magnetic monopoles in spin ice.docx

Magnetic monopoles in spin ice.docx

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Magneticmonopolesinspinice

Nature 451,42-45（3January2008）| doi:

10.1038/nature06433;Received27June2007;Accepted29October2007

Magneticmonopolesinspinice

C.Castelnovo1,R.Moessner1,2 &S.L.Sondhi3

1.RudolfPeierlsCentreforTheoreticalPhysics,OxfordUniversity,OxfordOX13NP,UK

2.Max-Planck-InstitutfürPhysikkomplexerSysteme,01187Dresden,Germany

3.PCTPandDepartmentofPhysics,PrincetonUniversity,Princeton,NewJersey08544,USA

Correspondenceto:

C.Castelnovo1 CorrespondenceandrequestsformaterialsshouldbeaddressedtoC.C.（Email:

 castel@physics.ox.ac.uk）.

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Abstract

Electricallychargedparticles,suchastheelectron,areubiquitous.Incontrast,noelementaryparticleswithanetmagneticchargehaveeverbeenobserved,despiteintensiveandprolongedsearches（seeref. 1 forexample）.Wepursueanalternativestrategy,namelythatofrealizingthemnotaselementarybutratherasemergentparticles—thatis,asmanifestationsofthecorrelationspresentinastronglyinteractingmany-bodysystem.Themostprominentexamplesofemergentquasiparticlesaretheoneswithfractionalelectriccharge e/3inquantumHallphysics2.Hereweproposethatmagneticmonopolesemergeinaclassofexoticmagnetsknowncollectivelyasspinice3, 4, 5:

thedipolemomentoftheunderlyingelectronicdegreesoffreedomfractionalisesintomonopoles.Thiswouldaccountforamysteriousphasetransitionobservedexperimentallyinspiniceinamagneticfield6, 7,whichisaliquid–gastransitionofthemagneticmonopoles.Thesemonopolescanalsobedetectedbyothermeans,forexample,inanexperimentmodelledaftertheStanfordmagneticmonopolesearch8.

Spin-icematerialsarecharacterizedbythepresenceofmagneticmoments 

iresidingonthesites i ofapyrochlorelattice（depictedin Fig.1）.ThesemomentsareconstrainedtopointalongtheirrespectivelocalIsingaxes 

 （thediamondlatticebondsin Fig.1）,andtheycanbemodelledasIsingspins 

i = 

Si

where Si = 

1and 

.Forthespin-icecompoundsdiscussedhere,Dy2Ti2O7 andHo2Ti2O7,（whereDyisdysprosiumandHoisholmium）themagnitude 

 ofthemagneticmomentsequalsapproximatelytenBohrmagnetons（

 10

B）.Thethermodynamicpropertiesofthesecompoundsareknowntobedescribedwithgoodaccuracybyanenergytermthataccountsforthenearest-neighbourexchangeandthelong-rangedipolarinteractions9, 10（forareviewofspinice,seeref. 4）:

Figure1:

 Thepyrochloreanddiamondlattices.

Themagneticmomentsinspiniceresideonthesitesofthepyrochlorelattice,whichconsistsofcorner-sharingtetrahedra.Theseareatthesametimethemidpointsofthebondsofthediamondlattice（black）formedbythecentresofthetetrahedra.Theratioofthelatticeconstantofthediamondandpyrochlorelatticesis 

.TheIsingaxesarethelocal[111]directions,whichpointalongtherespectivediamondlatticebonds.

Highresolutionimageandlegend（99K）

Thedistancebetweenspinsis rij,and a 

 3.54 Åisthepyrochlorenearest-neighbourdistance. D = 

0

2/（4

a3）=1.41 Kisthecouplingconstantofthedipolarinteraction.

Spinicewasidentifiedasaveryunusualmagnetwhenitwasnotedthatitdoesnotordertothelowesttemperatures T eventhoughitappearedtohaveferromagneticinteractions3.Indeed,spinicewasfoundtohavearesidualentropyatlow T （ref. 5）,whichiswell-approximatedbythePaulingentropyforwaterice, S 

 SP =（1/2）log（3/2）perspin.Pauling'sentropymeasuresthehugeground-statedegeneracyarisingfromtheso-calledicerules.Inthecontextofspinice,itsobservationimpliesamacroscopicallydegenerategroundstatemanifoldobeyingthe'icerule'thattwospinspointintoeachvertexofthediamondlattice,andtwoout.

Wecontendthatexcitationsabovethisground-statemanifold—thatis,defectsthatlocallyviolatetheicerule—aremagneticmonopoleswiththenecessarylong-distanceproperties.Fromtheperspectiveoftheseeminglylocalphysicsoftheicerule,theemergenceofmonopolesatfirstseemsrathersurprising.Wewillprobedeeperintohowthelong-rangemagneticinteractionscontainedinequation

（1）giverisetotheiceruleinthefirstplace.Wethenincorporateinsightsfromrecentprogressinunderstandingtheentropicphysicsofspinice,andthephysicsoffractionalizationinhighdimensions11, 12, 13, 14, 15,ofwhichourmonopoleswillprovetobeaclassicalinstance.

Weconsideramodestdeformationofequation

（1）,looselyinspiredbyNagle'swork16 onthe'unitmodel'descriptionofwaterice:

wereplacetheinteractionenergyofthemagneticdipoleslivingonpyrochloresiteswiththeinteractionenergyofdumbbellsconsistingofequalandoppositemagneticchargesthatliveattheendsofthediamondbonds（see Fig.2）.Thetwowaysofassigningchargesoneachdiamondbond