Singular or non-Fermi liquids

270C.M.Varmaetal./PhysicsReports361(2002)

267–417

Fig.1.Schematicphasediagramnearaquantumcriticalpoint.Theparameteralongthex-axiscanbequitegeneral,likethepressureoraratioofcouplingconstants.Wheneverthecriticaltemperaturevanishes,aQCP,indicatedwithadotintheÿgure,isencountered.Inthevicinityofsuchapointquantummechanical,zero-point uctuationsbecomeveryimportant.However,whenTcisÿnite,criticalslowingdownimpliesthattherelevantfrequencyscalegoesas!～|Tc T| z,dwarÿngquantume ects;thestandardclassicalcriticalmethodologythenapplies.AnexampleofaphasediagramofthistypeforMnSiisshowninFig.34below.

diagramaroundthequantumcriticalpoint(QCP)wherethetransitiontemperaturetendstozeroasafunctionofsomeparameter,seeFig.1.Thisphenomenonhasbeenextensivelyinvestigatedforawidevarietyofmagnetictransitionsinmetalswherethetransitiontemperaturecanbetunedthroughtheapplicationofpressureorbyvaryingtheelectronicdensitythroughalloying.Heavyfermions,withtheirclosecompetitionbetweenstatesofmagneticorderwithlocalizedmomentsanditinerantstatesduetoKondoe ects,appearparticularlypronetosuchQCPs.EquallyinterestingarequestionshavingtodowiththechangeinpropertiesduetoimpuritiesinsystemswhicharenearaQCPinthepurelimit.

Thedensity–densitycorrelationsofitinerantdisorderedelectronsatlongwavelengthsandlowenergiesmusthaveadi usiveform.Intwodimensionsthisleadstologarithmicsingularitiesinthee ectiveinteractionswhentheinteractionsaretreatedperturbatively.Theproblemofÿndingthegroundstateandlow-lyingexcitationsinthissituationisunsolved.Ontheexperimentalside,thediscoveryofthemetal–insulatortransitionintwodimensionsandtheunusualpropertiesobservedinthemetallicstatemakethisanimportantproblemtoresolve.

Theone-dimensionalelectrongasrevealslogarithmicsingularitiesinthee ectiveinteractionseveninasecond-orderperturbationcalculation.Avarietyofmathematicaltechniqueshavebeenusedtosolveawholeclassofinteractingone-dimensionalproblemsandonenowknowstheessentialsofthecorrelationfunctionseveninthemostgeneralcase.Animportantissueiswhetherandhowthisknowledgecanbeusedinhigherdimensions.