SolidStateCommunications,Vol.24,pp. 317—318,1977. PergamonPress. Printedin GreatBritain

ON ThE THEORYOFTHE CONDUCTIVITY OFA QUASI-ONE-DIMENSIONAL METAL

A.A. Abrikosovandl.A. Ryzhkin

L.D. LandauInstitutefor TheoreticalPhysics,Vorobjevskoye,Shosse2, Moscow 117334,USSR

(Receivedl2June1977 byE.A.Kaner)

An exactcalculationis presentedof the longitudinalconductivityof aquasi-one-dimensionalmetalat T= 0. The resultdiffers from the oneobtainedby meansof a scalingargument[1].

IN THE PREVIOUSarticle [l]the longitudinaland considera function GRor GA andseparateout apiecetransverseconductivitieshavebeencalculatedof a of it betweenthe closestoperators~ and~k (or ~ andquasi-one-dimensionalmetalwithan energyspectrum flk etc.)with i ~ k.Thenthefactor dependingon a is

e(p) — ii = V(Ip~I Po) + a(p~,p~) (1) exp [± ia(p)(z, Zk)/V].This factor is averagedindependentlyfrom all the

wheref adp~dp~= 0. It wasassumedT = 0, hencethe others.But if the closestare ~ and ~ or two p1’sso thatonly scatteringmechanismwastheinteractionwith they arelinked in the impurity averagingthey do notimpurities,which wassupposedshort-ranged:U(r) -÷ 0 if produceanya dependentfactor.Taking into accountallr ~ r0, r0/a1~ 1. suchlinks occuringbetweensomecertain~ and~k (or

The longitudinalconductivityin casecr12 /v ~ 1 ~j andflk etc.)with i � k we obtaina factor(c.f. [1],(12 — back-scatteringpathlength,v — Fermivelocity) [2])wasobtainednotfrom an exact theorybutby meansof

exp [±ia(p)(zj—zk)] exp [—Iz~—zkI(/~’ +l~’)/2]a scalinginvarianceassumption.In this articlewe presenttheexact theory.The result differs from theone (3)obtainedin [1] andthis demonstratestheabsenceof wherethedashdenotesthep-averaging.Sinceal/v ~ Iscalinginvarianceof thekind usedin [1]. thefirst exponentvariesslowly with z. — zk relatively

The startingexpressionfor the longitudinalconduc- to the secondandwe thereforesubstituteit by itstivity obtainedin [l]is averagewith thesecondexponentasweight function.

After thatweget a factore

2v2 d2pazz = ISpG[GR(zzl)U

3GA(zlz)U3] (2) (1 —y/2)exp [—Iz1—zkI(l~’ +l~1)/2] (4)iT

HeretheGR andGA dependon a(p)and on therandom withfields ~andi~describingtheinteractionwith impurities, ~, 8~(l~+ /-

1)~2/v2 (5)the latterbeingfunctionsactingon thep-variable(see [1]) Here we did not takeinto accountthe possibility

of appearanceof a2 from a’s enteringdifferentGreen~ j z)P~d2k= Pka(p) = a(p— k)Pk. functions(onefrom GR,the otherfrom GA). Onecan

(2~.)2 howevershowthat suchaverageslead to verysmall

Consideraas someexternalfield togetherwith ~- contributionsto the conductivity(relativeorder(al/v)4.andi~.The a-field is alsoaveragedandthisis achieved We shall omit the proofof thatsinceit is ratherby thep-integration,howeverthis field is non-Gaussian. cumbersome.Theessentialpropertyof this averagingis that dueto Soit appearsthateveryGreenfunctionbetweenthetheshort rangeof the impurity potentialthe averaging closest~ and~k (I * k) is renormalizedaquiringa factorof thea’sappearingon different sidesof a certain~or 1 — 7/2. Insteadof t~hatwe cankeepthebareGreenO operatormustbe doneindependently.Indeed functionsrenormalizingthe fields ~ andi~.Thereis

howeveran exception.Thoseoperatorsenteringthed2k links of thetype ~ between~, and~k with i * k

a~~am-~ a’~(p)f ~~am(J, — k) ~ remainunrenormalized.Theselinks producedthefactor

Since(It,, 2) varieswith k only when ki ~ r~1,one exp [— Z~— Zk (1~’+ 1j’ )/2] in (3). Ifwe passseesthat a” andam areaveragedindependently.Now everywhereto renormalizedoperators~ andi~so every-thinghasto be expressedin termsof renormalizedpath

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318 CONDUCTIVITY OFA QUASI-ONE-DIMENSIONAL METAL Vol. 24,No.4

lengths1~and‘2. Hencethe exponentialfactortakes l/T = 7v(121 + i~1) = 8~(l~’+ lj~’)~/v. (8)

theformBut this correspondsto the conductivityof a purely one-

exp [ Iz, Zk (i2

1 + l~’)(l + 7)121 dimensionalmetalat a finite frequencyw = i/r. Substi-

= exp [—(z1 — Zk I(1j’ + 1;’ )/2] tuting into thecorrespondingformula(seee.g. [2]) we

obtainx exp [—Iz,—zkI(1~+i~’)~/2]. (7) F ~

a = S’I—Q(w)Nowwe considerthefirst of thesetwo factorsas arising ZZ ~ litfrom thelinks ~ (and0?)andthe secondasresulting 64~(3)e

2~ l~l~from thedampingof bareelectrons. = s ~ ~ (9)

Therenormalizationof the operatorsandhencethe iT 1 2

pathlengthsleadsto a small correctionin theconduc- Herewe tookinto accountrelativeto [2] two pro-tivity. Thedampinghoweverplaysan essentialrole. If we jectionsof the spinanddivided by S — thexy areaperpassto the Fouriercomponentsin zthenthe damping onechain.The formula(7) correspondsto thediffusionalappearsasan imaginaryfrequencyi/2r in GR and estimate(lOb)in [1] anddiffers from (42)in [1]— i/2r in GA where obtainedfrom the scalinghypothesis.

REFERENCES

1. ABRIKOSOV A.A. & RYZHKIN I.A.,J. Exp. Theor. Phys.72, 225 (1977).

2. ABRIKOSOVA.A. & RYZHKIN I.A.,J.Exp. Theor. Phys.71, 1916 (1976).