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NRG Study of an Inversion-Symmetric Interacting
Model: Universal Aspects of its Quantum Conductance
Axel Freyn, Jean-Louis Pichard
To cite this version:
Axel Freyn, Jean-Louis Pichard. NRG Study of an Inversion-Symmetric Interacting Model: Universal
Aspects of its Quantum Conductance. 2009. �hal-00373518�
Axel Freyn
and Jean-Louis Pi hard Servi e de Physique de l'
Etat Condense (CNRS URA 2464), IRAMIS/SPEC, CEA Sa lay, 91191 Gif-sur-Yvette, Fran e
We onsider s attering of spinless fermions by aninversion-symmetri intera ting model har-a terizedbythreeparameters(intera tionU,internal hopping t
d
and ouplingt
). Mapping this spinless modelonto anAnderson modelwithZeeman eld, we use thenumeri alrenormalization groupfor studying theparti le-hole symmetri ase. Weshow thatthe zero temperature limit is hara terizedbyalineoffree-fermion xedpointsandas ale (U;t
)oft
d
forwhi hthereisperfe t transmission. Thequantum ondu tan eandthelowenergyex itationsofthemodelaregivenby universalfun tions oftd= if td < and of td=t
2
if td > , = t 2
being the level widthof the s atterer. Thisuniversalregimebe omesnon-perturbativewhenU ex eeds .
PACSnumbers: 71.10.-w,72.10.-d,73.23.-b
In quantum transport theory, the ondu tan e G of ananosysteminside whi htheele tronsdonotintera t is given by g = G=(e 2 =h) = jt ns j 2
when the tempera-tureT !0, jt
ns j
2
beingthe probability for anele tron at the Fermi energy E
F
to be transmittedthrough the nanosystem. ThisLandauer-Buttikerformula anbe ex-tended to an intera ting nanosystem, ifit behavesasa non-intera ting nanosystem with renormalized parame-ters. Westudy su harenormalizationusing the numer-i al renormalization group(NRG) algorithm [1, 2℄ and an inversion-symmetri intera ting model (ISIM) whi h des ribesthes atteringofspin-polarizedele trons (spin-lessfermions) by anintera ting region hara terized by an internal hopping termt
d
, a ouplingterm t
and an intera tion strength U. This model wasused [3, 4℄ for studyingthee e tofanexternals attereruponthe e e -tive transmissionofan intera ting region,assuming the Hartree-Fo k(HF)approximation.WerevisitISIMwith the NRG algorithm for investigating non-perturbative regimes where other methods (NRG or DMRG algo-rithms)thantheHFapproa hbe omene essary.
Quantum impuritymodels[1℄,astheAndersonmodel whi hdes ribesalevelwithHubbardintera tionU ou-pled to a 3d bath of free ele trons, were introdu ed to study the resistan e minimum observed in metals with magneti impurities. The Kondo problem refersto the failure of perturbative te hniques to des ribe this min-imum. The solution of these models by the NRG al-gorithm, anon-perturbativete hnique [1,2℄ introdu ed by Wilson, is at the origin of the dis overy of univer-salbehaviorswhi h anemergefrommany-bodye e ts. The observation [5℄ of the Kondo e e t in semi ondu -torquantum dotshasopenedase ond eraforquantum impuritymodels,nowusedformodelingmesos opi ob-je ts (single [6℄ ordouble [7℄ quantum dot systems) in-side whi h ele trons intera t, in onta t with baths of freeele trons(large ondu ting nonintera tingleads).
ThoughtheKondo e e t isindu edbymagneti mo-ments, it is also at the origin of spinless models, su h
astheintera tingresonantlevelmodel[8℄(IRLM)whi h des ribesaresonantlevel(V
d d
y
d) oupledtotwobathsof spinlessele tronsviatunnelingjun tionsandan intera -tionU betweenthelevelandthebaths. IRLM,whi his oftenusedforstudyingnonequilibriumtransport[8,9℄,is relatedtothe Kondomodel, the hargestatesn
d =0;1 playing the role of spin states. Both ISIM and IRLM are inversion symmetri and an exhibit orbital Kondo e e ts. However, theZeeman eld a ting onthe impu-rity is played bythe hopping termt
d
forISIM, and by thesite energy V
d
for IRLM. Therefore, ISIM does not transmit the ele trons without eld, while IRLM does. Thetwo-parti lestateshavebeengivenforISIM[10℄.
Fortheparti le-holesymmetri ase[2℄,theAnderson model maps onto theKondo Hamiltonian ifU > , being theimpurity-level width. Inthat ase,there is a non-perturbative regime where the temperature depen-den eof physi al observablessu h as theimpurity sus- eptibility is given by universal fun tions of T=T
K , T
K beingthe Kondotemperature. If U < , theimpurity sus eptibility an be obtained by perturbation theory. MappingISIMontoan Andersonmodel with aZeeman eld t
d
, and assuming that the role of t d
should quali-tativelyresemblethatofa nitetemperature,weexpe t thefollowings enariofortheISIM ondu tan eg ofthe parti le-holesymmetri ase: IfU > /t
2
,weexpe t a non-perturbative regime where g should be given by auniversalfun tion of t
d
= independently ofthe values ofU and t , with as ale (t ;U)of t d
playingthe role ofaKondotemperatureT
K
. If U < , theHFtheory should orre tlygiveg. Thiss enariowillbemoreorless on rmedbyextensiveNRG al ulations.
ISIMHamiltonian: H =H ns +H l +H . The Hamil-tonianoftheintera tingregion(thenanosystem)reads:
H ns = t d y 0 1 + y 1 0 +V G (n 0 +n 1 )+Un 0 n 1 : (1) y x and x
are spinless fermion operators at site x and n
x =
y
LM
~
U
~
t
d
PO
FO
SC
~
t
c
FIG.1: Line offreefermion xedpoints( ~
U =0,thi ksolid line) hara terizing ISIM whenT ! 0 as t
d
in reases from t
d
= 0(SC xed point)towards t d
! 1(PO xed point). TheFO,LMandSC xedpointsandtheRGtraje tories[1 ℄ followedbyISIM as T de reases for t
d =0are indi ated in theplane e t d
=0,for >U (dashed)and <U (solid).
H l = t h X 0 1 x= 1 ( y x x+1 +H: :), where X 0 means
that x= 1;0;1areomittedfrom thesummation. The ouplingHamiltonianH = t ( y 1 0 + y 1 2 +H: :).
Mapping onto an Anderson model with Zeeman eld: Be ause ofinversionsymmetry,one anmap ISIMonto a semi-in nite 1d latti e where the fermions have a pseudo-spin and the double sitenanosystem be omes a single site with Hubbard repulsion U at the end point of the semi-in nite latti e. a
y e=o;x = ( y x+1 y x )= p 2
reating a spinless fermion in an even/odd (e=o) om-bination of the orbitals at the sites x and x + 1 of the in nite latti e, (or a fermion with pseudo-spin = e=o in a semi-in nite latti e), one gets H
ns = (V G t d )n e +(V G +t d )n o +Un e n o ,wheren =a y ;1 a ;1 and where the pseudo-spin \e" (\o") is parallel (anti-parallel) to the \Zeeman eld" t
d . In terms of the operators d y k ; = p 2= P 1 x=2 sin(k(x 1))a y ;x reat-ing a spinless fermion of pseudo-spin and momen-tum kin thesemi-in nitelatti e,H
l = P k ; k n k ; and H = P k ; V(k)(a y ;1 d k ;
+H: :),wherethek-dependent
hybridizationV(k) = t
p
2=sink yieldsan impurity
levelwidth =t 2 , n k ; =d y k ; d k ; and k = 2t h osk. ISIMisalmosttheAndersonmodel,ex eptthatthe im-purity has a Zeeman eld t
d
and is oupled to a semi-in nite 1d bath of free ele trons. When t
d
! 0, ISIM exhibitsanorbitalKondoe e tiftheequivalent Ander-sonmodel anberedu edto aKondomodel.
NRG pro edure: ISIM an be studied using Wilson's pro edure[1, 2℄developedfortheAndersonmodelafter minor hanges. First,weassume V(k) V(k
F
==2) and, taking =2,wedivide the ondu tion band (log-arithmi dis retization) of the ele tron bath into sub-bands hara terizedbyanindexn andanenergywidth d n = n (1 1
). Withinea hsub-band,weintrodu e a ompleteset oforthonormalfun tions
np
(),and ex-pandtheleadoperatorsinthisbasis. Droppingtheterms
with p 6= 0 and using a Gram-S hmidt pro edure, the original1d leadsgiveriseto anothersemi-in nite hain withnearestneighborhopping terms,ea hsitebeing la-belledbythesameindexnastheenergysub-bandfrom whi h it omes, and representing a ondu tion ele tron ex itationatalengths ale
n=2 k
1 F
enteredonthe im-purity. Inthistransformed1dmodel,thesu essivesites are oupledbyhoppingtermst
n;n+1 /
n=2
whi h van-ishasn!1. TheimpurityandtheN 1 rstsitesform aNRG hainoflengthN andofHamiltonianH
N . This length an be interpreted [2℄ as a logarithmi temper-atures ale. The NRG hain oupledto the impurityis iterativelydiagonalizedandres aled,thespe trumbeing trun atedtotheN
s
rststatesatea hiteration. The be-haviorof ISIMasT de reases anbeobtainedfromthe spe trum of H
N
as N in reases, the bandwidth of H N being suitably res aled at ea h step. A xed point of theRG ow orrespondstoanintervalofsu essiveeven (orodd) valuesof N wherethe res aledmany-body ex- itationsE
I
(N)donotvary. Ifitisafree-fermion xed point, E I = P , the
being one-body ex itations, andtheintera tingsystembehavesasanon-intera ting system(
~
U =0) withrenormalized parameters e t d and e t nearthe xedpoint. Moreover,ifonehas freefermions whenT !0,g anbeextra tedfromtheNRGspe trum.
Symmetri ase: UsingthisNRGpro edure,ISIM an be studied asafun tion of T for arbitraryvaluesof its bare parameters. Hereafter, we take t
h = 1, E F = 0 and V G
= U=2. This hoi e makes ISIM invariant under parti le-hole symmetry, with a uniform density (hn
x
i=1=2) and3e e tiveparameters( ~ U; e t ; e t d ).
Suppression of the LM xed point as t d
in reases: When t
d
= 0, ISIM is an Anderson model whi h has theRG owsket hedinFig.1fortheparti le-hole sym-metri ase. AtlowvaluesofN (highvaluesofT),ISIM islo atedinthevi inityoftheunstablefreeorbital(FO) xed point. As N in reases (T de reases), ISIM ows towardsthe stable strong oupling (SC) xed point. If t
2
<U,the ow anvisitanintermediateunstable xed point: the lo al moment(LM) xed pointbefore rea h-ingtheSC xedpoint. Inthat ase,ISIMisidenti alto aKondomodel hara terized by atemperatureT
K and by universal fun tions of the ratio T=T
K . If t
2
> U, the ow goes dire tly from the FO xed point towards theSC xed point, andthere isnoorbital Kondo e e t fort
d
!0. InFig.2(a),the rstmany-bodyex itations E
I
of ISIMaregivenforin reasingevenvaluesof N for t
d
=0. Sin e t 2
<U, onegets3plateaus orrespond-ingto the 3expe ted xedpoints. Insidethe plateaus, thespe traarefree-fermionsspe trawhi haredes ribed inRef. [2℄. However,betweentheplateaus, thereareno free-fermion spe tra and E
I 6= P . As t d in reases (Fig.2(a)),theLMplateaude reasesandvanisheswhen t
d U.
Evolution of the SC xed point as t d
in reases: In the limit N ! 1 (T ! 0), let us study the E as a
FO
(a3)
SC
t
d
= 5 ∗ 10
−
3
0
0
.5
1
E
I
0
20
40
60
N
FO
(a2)
LM
SC
t
d
= 10
−
6
0
0
.5
1
E
I
FO
(a1)
LM
SC
t
d
= 0
0
0
.5
1
E
I
(d)
(e)
U = 0
U = 0.04
10
−
4
10
−
2
t
10
0
d
τ
SC
PO
(c)
τ
SC
PO
(b)
0
2
4
ǫ
α
10
−
6
10
−
4
10
−
2
t
10
0
d
0
0
.5
1
g
∆ǫ
0
.01
1
100 t
d
/t
2
c
0
2
4
ǫ
α
10
−
6
10
−
4
10
−
2
t
10
0
d
0
0
.5
1
g
FIG.2: (Color online) Fig. 2(a): Many body ex itationsEI as a fun tionofN (evenvalues)for U =0:005 and t =0:01. Fortd=0(Fig.2(a1)),one anseethe3su essiveplateaus(FO,LMandSC xedpoints)oftheAndersonmodel[2 ℄. Astd in reases(Fig. 2(a2)and Fig.2(a3)),the LMplateau shrinksanddisappearswhent
d
U. Fig. 2(b): Onebodyex itations
(t
d
) (extra tedfromtheE I
(N!1;t d
))for U=0:1 andt
=0:1 (lefts ale). Thesolid(dashed)line orrespondstoNRG hainsofeven(odd)lengthN. Condu tan eg(td)extra tedfrom(td)usingEq.(2)(thi kred urve,rights ale). Fortd= , the
are independentofthe parity ofN andg=1. Fig.2( ): ForU =0, (t d =t 2 ) andg(t d =t 2
) extra tedfrom theNRG spe tra(). g= osh
2
(X)(redline)withX=ln(t d
=t 2
) is orre tlyreprodu ed. Figs.2(d),(e): g(t d
)fort =0:1andmany valuesofU, al ulated byNRGalgorithm(d)andbyHFtheory(e). InFig. 2(d),thelarger isU,thesmalleris td= where g=1. The urves orrespondrespe tivelytoU =0:25;0:2;0:15(3 leftpeaks)and U =0:1;0:09;:::;0:01;0(11rightpeaks). InFig. 2(e),theHF valuesarea urateforU =0:02;0:01;0(3rightpeaks),butbe omeina uratewhenU 0:04 . For U > ,theHF urves(dashedlines)areverydi erentofthe orrespondingNRG urves(Fig.2(d)).
fun tion of t d
. For t d
= 0, one has the SC limit [2℄ wheretheimpurityisstrongly oupledtothese ondsite (the ondu tion-ele tron state at the impurity site) of theNRG hain. Theimpurityand this siteform a sys-tem whi h an be redu ed to its ground state (a sin-glet), the N 2other sites arrying freefermions ex i-tations
whi h areindependentof thatsystem. Inthe presen eof aZeeman eld t
d
6=0, thefree-fermion rule E I (t d ) = P (t d
) remains valid (see Fig. 2(b)) and the T ! 0 limit of ISIM is given by a ontinuum line offree-fermion xedpoints where
~
U =0,assket hedin Fig.1. Whenthepseudo-spindegenera yisbroken,the rst (se ond) one-body ex itation
1 (
2
) arry respe -tivelyaneven(odd)pseudo-spinifNiseven. Thisisthe inverseifN is odd,
1 (
2
) arryingrespe tivelyanodd (even) pseudo-spin. Fort
d
!1, the impurity o upa-tion numbersn
e
=1and n o
= 0,and the N 1 other sites oftheNRG hainareindependentoftheimpurity. We allthis xedpoint\PolarizedOrbital"(PO),sin eit oin ideswiththeFO xedpointoftheAndersonmodel, ex eptthat thespinofthefreeorbital isfullypolarized in our ase. Sin e for N ! 1 and t
d
! 0 (SC xed point), thefreepart ofthe NRG hain hasN 2sites, while it has N 1 sites for t
d
! 1 (PO xed point), there is apermutation of the
(t d ) ast d in reases: as
shown in Figs.2(b) and ( ), the (t d !0) for N even be omethe (t d
!1)forN oddandvi e-versa.
Chara teristi energy s ale : We de ne the hara -teristi energy s ale (t
;U) of ISIM as the value of t d forwhi h the
(t
d
) areindependentof theparityofN when N ! 1. Be ause of parti le-holesymmetry, the nanosystem (the impurity of the NRG hain) is always o upied by one ele tron. Binding one ele tron of the leadswiththisele tronredu estheenergywhent
d < , whileitin reasestheenergywhent
d
> . Fort d
= ,it isindi erenttobind ornotanele tronoftheleadwith theoneofthenanosystem,makingISIMperfe tly trans-parent. This givestheproof that, for everyvaluesofU andt
, there is always avalue of t d
forwhi h g = 1. Theargumentisreminis enttothatgivingthe ondition for having a perfe tly transparent quantum dot in the Coulombblo kaderegime: t
d
in our ase,thegate volt-agein the other ase,have to be adjusted to valuesfor whi h it osts thesameenergyto put anextra ele tron outsideorinside thedot.
Extra tionofthe ondu tan egfromtheNRGspe tra: IfÆ
e (Æ
o
)aretheeven(odd)s atteringphaseshiftsatE F , g(t d )=sin 2 (Æ e Æ o )=sin 2 (t d ) (t !1) ; (2)
10−
12
10
−
6
10
0
10
6
τ /U
0
.001
1
U/t
2
c
10
−
4
10
−
3
10
−
2
10
−
1
10
0
g
10
−
2
10
0
10
2
10
4
10
6
10
8
t
d
/τ
t
c
= 1
t
c
= 0
.1
t
c
= 0
.01
FIG.3:(Coloronline)Condu tan egasafun tionoft d
= for 3valuesoft andmanyvalues0U 35. Inset: (U;t )=U asafun tionofU=t 2 (+)and t =t 2 exp (U=(t 2 ))(solid redline). where= 2 1
istheenergygapbetweenthetwo rst ex itations ofaNRG hainof evenlengthN !1 (see Fig.2(b)). WhenU =0,thisrelationisa onsequen eof Friedel sumrule,whi h anbewrittenfor ea h pseudo-spin hannel separately. In that ase, g = osh
2 (X) whereX =ln(t d =t 2 )and =t 2 . The (t d )givenbythe NRGalgorithmforU =0areshowninFig.2( )withthe orrespondingvaluesofgobtainedfromEq.(2),showing thatthispro eduregives orre tlygwhenU =0. Ithas beenshown[7,11,12℄thatEq.(2) analsobeusedwhen U 6=0,iftherearefreefermionswhenT !0.
Non-perturbativeregime (U > =A): InHFtheory,t d takes [3℄ a value v = t d +Uh y 0 1 (v;t )i and g = 1 if v = t 2
. This gives for the s ale a HF value HF = t 2 AU where A = h y 0 1 (v = t 2 ;t )i depends weakly on t , A =1= (1=4) for t =1 (0). When U ! t 2 =A, HF
!0,showingthatHFtheory annotbeusedabove an intera tion threshold whi h is almost the threshold giving the onset of the non-perturbative regime for theAnderson model. Thisbreakdown ofHF theory for U =A an be seen if one ompares Fig. 2(d) (NRG results)andFig.2(e)(HFresults).
Universality: The ondu tan e g extra ted from the NRG spe tra for t
= 0:01;0:1 and 1 and 0 U 35 is given as afun tion of t
d
= in Fig. 3. One an see 3 su essiveregimes. Whent
d
< ,there isasingle urve whi hisindependentofU andt
andwhi h orresponds to g = osh 2 (X)with X =ln(t d = ), and notln(t d =t 2 ) as for U = 0. When t d
> , another universal urve independentoft
andU des ribesthedataasafun tion oft
d
= asfarast d
doesnotex eed . Indeed,thesame data plotted as a fun tion of t
d
show that g be omes independent of U when t
d
> . In this third regime (parallellineswhi h anbeseeninFig.3forlargevalues oft d = )g= osh 2 (X)withX =ln(t d =t 2 )asifU=0. RolesofT andt d
: Wehaveassumedanalogiesbetween
man eldatT =0intheAndersonmodel,andeventually thee e t oft
d
at T =0in ISIM.Thiswasbasedonthe ideathatthesingletstateoftheSClimit ouldbebroken eitherifthetemperatureT ortheZeemanenergyt
d ex- eedsT
K
. Letussummarizetheinterestandthelimitof theseanalogies.In reasingT intheAndersonmodel(or in ISIMwith t
d
=0), onegets3regimes, ea h of them being hara terized by a single xed point (Fig. 2(a)). There are no free fermions for temperatures T T
K (SC{LM rossover) and T (LM{FO rossover). In ontrast, in reasing t
d
in ISIM at T = 0, one has al-ways free fermions (Fig. 2(b)), and not only around 3 xed points. However, there are 3 regimes in ISIM as t
d
in reases, as in the Anderson model as T in reases, delimited by 2 energys ales and . The behaviorof t 2 exp (U=(t 2
)) (inset of Fig. 3) resembles that ofT
K t
p
U=2exp (U=(t 2
))(inISIMunits),while these onds ale isgiven by inthe 2models. Eventu-ally,wepointoutthesimilaritybetweentheuniversality dis ussed in this letter for g and that whi h hara ter-izes[13℄alsoat T =0thebehaviorofthesinglet-triplet gap fora magneti impurity on ned in a boxof mean levelspa ing,asafun tion T
K =.
WethankDenisUllmo forveryusefuldis ussionsand the\TriangledelaPhysique"for nan ialsupport.
Present address: Institut Neel, 25avenue des Martyrs, BP166,38042Grenoble,Fran e.
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