Singular or non-Fermi liquids(18)

Singular or non-Fermi liquids

284C.M.Varmaetal./PhysicsReports361(2002)

267–417

Fig.10.Single-particleenergy kinonedimension,intheapproximationthatthedispersionrelationislinearizedaboutkF.NotethattheFermisurfaceconsistsofjusttwopoints.Thespectrumofparticle–holeexcitationsisgivenby!(q)= (k+q) (k)=kFq=m.Low-energyparticle–holeexcitationsareonlypossibleforqsmallorforqnear2kF.

Fig.11.Phasespaceforparticle–holeexcitationspectruminonedimensioncomparedwiththesameinhigherdimensions,Fig.8.Forlinearizedsingle-particlekineticenergy k=±vF(k kF),particle–holeexcitationsareonlypossibleonlinesgoingthroughk=0andk=2kF.

perturbatively7

ZkF≈1 2g2N(0)=EF:

(24)

Thusinaperturbativecalculationofthee ectofinteractionsthebasicanalyticstructureoftheGreen’sfunctionisleftthesameasfornon-interactingfermions.ThegeneralproofofthevalidityofLandautheoryconsistsinshowingthatwhatwehaveobtainedtosecondorderingremainsvalidtoallordersing.Theoriginalproofs[4]areself-consistencyarguments—wewillconsiderthembrie yinSection2.4.TheyassumeaÿniteZintheexactsingle-particleGreen’sfunctionsande ectivelyshowthattoanyorderinperturbationtheory,thepolarizabil-ityfunctionsretaintheanalyticstructureofthenon-interactingtheory,whichinturnensuresaÿniteZ.

Inonedimension,phase-spacerestrictionsonthepossibleexcitationsarecruciallydi erent.8HeretheFermisurfaceconsistsofjusttwopointsintheone-dimensionalspaceofmomenta—seeFig.10.Asaresult,whereasind=2and3acontinuumoflow-energyexcitationswithÿniteqispossible,inonedimensionatlow-energyonlyexcitationswithsmallkork≈2kFarepossible.ThesubsequentequivalentofFig.8fortheone-dimensionalcaseistheoneshowninFig.11.Uponintegratingoverthemomentumkwithacut-o ofO(kF)thecontributionfrom

ThisquantityhasbeenpreciselyevaluatedbyGalitski[104]forthemodelofadiluteFermigascharacterizedbyascatteringlength.8

Itmightappearsurprisingthattheyarenotdi erentinanyessentialwaybetweenhigherdimensions.

7