Critical Spectral Statistics at the Metal-Insulator Transiti(2)

1.0

η

0.5

0.0U/U

*

FIG.3.SchematicevolutionofηasafunctioninteractionU/Uintheregimeneff≈2

oftherenor-malized

The spectral properties of a disordered system with few interacting three-dimensional spinless fermions are investigated. We show the existence of a critical spacings distribution which is invariant upon increase of the system size, but strongly depends on

trixbecomessparserandsparserandthecriticalspacingsdistributionexhibitslessandlesslevelrepulsion.

Inthethermodynamic(TD)limit,forsuchFermisys-tems,itisnaturaltode neatemperatureasneffT=T2/ 1～δ ,andasTmustbeconstantasL→∞,thisimpliesthatT/B1≈neff 1/B1=const.Twodi erentregimesarethenofinterest:

Theone-particlecriticalregimeclosetotheAndersontransition,wherethesystemsizeissigni cantlysmallerthantheone-particlelocalizationlength.There,boththeone-particlecorrelationvolumeandneffincreaseas～Ld ωuntilL≈l1andsincel1canbeverylarge,soisneffandthereisalmostnolevelrepulsionatthetran-sitionintheTDlimit.Inthiscasethescalingatthetransitionisnotobservablesinceηcincreaseswithnandtheη-curvesintersectatdi erentpoints(seeFig.3). Theone-particlelocalizedregime1 l1<L,wherewe

d

have 1≈B1/l1=const.ThisregimecorrespondstotheresultspresentedinFig.1forwhichneffisconstantandonehasa xed-pointatawell-de nedvalueofUc.Sinceitisexpectedthatalargeinteractionwillulti-matelytransformthesystembackintoaninsulator,itisimportanttoestimatetheparametricalwidthofthemetallicphase.Inthiscase,itseemsnaturaltocarryoutthesubstitution 1→Qin(2)sothatinthelocal-izedregimewithneff=const,theconditionQc～ cleadstoametallicphaseintherange

1

1

≤n2effT

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