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1、Eur.J.Phys.21(2000)499–510.PrintedintheUKPII:S0143-0807(00)14578-7QuantumspinliquidsFred´ericMila´InstitutdePhysiqueTheorique,Universit´edeLausanne,BSP,CH-1015Lausanne,Switzerland´Received7June2000Abstract.Thispaperpresentsanoverviewofthepropertiesofmagnetici
2、nsulators,withemphasisonquantumeffects.Inparticular,theconsequencesofstrongquantumfluctuationsonthelow-energypropertiesofanumberofsystemsarereviewed:theoccurrenceofaspingapinspin-1chainsandotherlow-dimensionalmagnets;thepresenceoflow-lyingsingletexcitationsins
3、everalfrustratedmagnets;theinterplayoforbitalandspinfluctuationsinorbitallydegenerateMottinsulators.Thisreviewisintendedasapedagogicalintroductiontothefield,andtheformalismhasbeenkeptataminimallevel.1.IntroductionThelow-temperaturepropertiesofmanysystems,inpart
4、iculartransitionmetaloxides,canbeverypreciselydescribedbyaneffectivespinHamiltonianknownastheHeisenbergmodel[1]:H=JijSi·Sj(1)(i,j)Inthisexpression,thethreecomponentsofSsatisfytheusualcommutationrulesofangularmomentum([Sx,Sy]=iSz,etc)andS2=S(S+1),wherethespin
5、Scanbeintegerorhalf-integer.Inthesesystems,themagneticmomentsarecarriedbytransitionmetalatoms,andchargefluctuationsoccuratenergiesmuchhigherthanthelargestexchangeintegralsJijbecauseofthestrongCoulombrepulsionbetweenelectronsinlocalizedorbitals.ThesesystemsareM
6、ottinsulators[1,2].ThepropertiesoftheHeisenbergmodeldramaticallydependonvariousparameters,inparticularuponthefollowing.(a)Thesizeofthespin.Ingeneral,magneticmomentshavebothanorbitalandaspincontributionand,dependingonthesystem,theresultingspincanbeminimal(S=1/
7、2)orquitelarge(inprincipleuptoS=15/2inrareearths).(b)Thesignandtherangeofthecouplings.Thecouplingismostoftenpositive(orantiferromagnetic)becauseofthekineticexchangethatallowsanelectrontohoptoaneighbouringorbitalalreadyoccupiedbyanotherelectronaslongastheirspi
8、nsareantiparallel.However,iftheorbitalsonneighbouringsitesareorthogonal,thisprocessisforbiddenbysymmetry,andthecouplingisthennegative(orferromagnetic):beinggloballyantisymmetric,thewavefu
