Anderson localization versus charge-density-wave formation in disordered electron systems

0.27 MB·English

by  S. Nishimoto

Anderson localizationversus charge-density-wave formationindisordered electron systems
S. Nishimoto,1 S. Ejima,2 and H. Fehske2
1Institutefor Theoretical Solid State Physics, IFW Dresden, 01171 Dresden, Germany
2Instituteof Physics, ErnstMoritz Arndt UniversityGreifswald, 17489 Greifswald, Germany
(Dated:October31,2012)
Westudytheinterplayofdisorderandinteractioneffectsincludingbosonicdegreesoffreedomintheframe-
workof agenericone-dimensional transport model, theAnderson-Edwards model. Usingthedensity-matrix
renormalizationgrouptechnique, weextractthelocalizationlengthandtherenormalizationoftheTomonaga
Luttinger liquid parameter from the charge-structure factor by a elaborate sample-average finite-size scaling
2 procedure. ThepropertiesoftheAndersonlocalizedstatecanbedescribedintermsofscalingrelationsofthe
1 metallicphasewithoutdisorder. Weanalyzehowdisordercompeteswiththecharge-density-wavecorrelations
0
triggeredbythebosonsandgiveevidencethatstrongdisorderwilldestroythecharge-orderedstate.
2

t PACSnumbers:71.23.An,71.27.+a,71.30.+h,71.45.Lr
c
O
Disorder is an inherent part of any solid state system [1]. ial way [15]. This has beendemonstratedfor the Anderson-
0
3 Low-dimensionalmaterialsareexceedinglysusceptibletodis- Holsteinmodelwithinthestatisticaldynamicalmeanfieldand
order. In one dimension (1D), theory predicts that all carri- momentumaverageapproximations[16,17]. Atfinitecarrier

] ers are strongly localized for arbitrary energies and arbitrar- density,theMott-AndersontransitionforCoulombcorrelated
l
e ilyweakdisorder. ThisholdsforAnderson’snon-interacting electrons was investigated by self-consistent mean-field the-
-
r tight-bindingHamiltonianwith a diagonal(i.e., on-site) ran- oryinD = ∞andD = 3[18–20], aswellasbyvariational
st dom potential [2, 3]. The coherent backscattering from the Gutzwiller ansatz based approaches [21]. Electron-electron
. randomlydistributedimpuritiestherebytransformsthemetal interactionsmayscreenthedisorderpotentialinstronglycor-
t
a intoaninsulator. relatedsystems,stabilizingtherebymetallicity[22].Exactre-
m
In1D themutualinteractionof theparticlesislikewise of sults are rare however. In 2D, Lanczos and quantum Monte
- significance; here even weak interactions can cause strong Carlo data suggest a disorder-induced stabilization of the
d
n correlations. The instantaneousCoulomb repulsion between pseudogap,alsoawayfromhalf-filling[23]. Thedensityma-
o theelectrons,forinstance,tendstoimmobilizethechargecar- trixrenormalizationgroup(DMRG)[24]allowsthenumerical
c riers as well. As a consequence, at half-filling, a Mott insu- exactcalculationofground-statepropertiesofdisordered,in-
[
lating(spin-density-wave)phaseisenergeticallyfavoredover teractingfermionsystemsin1D,onfairlylargesystems. Ex-

1 themetallicstate[4]. Theretardedelectron-phononcoupling, ploitingthistechnique,thepropertiesofdisorderedLuttinger
v ontheotherhand,mayleadtostructuraldistortionsaccompa- liquids have been analyzed in the framework of the spinless
2
niedbypolaronformation[5],andisthedrivingforcebehind fermion Anderson-t-V model (AtVM) [25] and the spinful
4
1 the metal-to-Peierlstransition, establishinga charge-density- Anderson-Hubbardmodel[26].
8 wave(CDW)order[6].
Inthispaper,weaddresshowmany-bodyAndersonlocal-
0. Tounderstandhowdisorderandinteractionacttogetheris ization competes with CDW formation triggered by bosonic
1 ofvitalimportancenotonlydiscussingthemetal-insulatorit- degrees of freedom in the framework of the Anderson-
2 selfbutalsoanalyzingtheelectronicpropertiesofmanyquasi- Edwardsmodel(AEM). The Edwardsmodel[27] represents
1
1D materials of current interest, such as conjugated poly- a very general two-channel fermion-bosonHamiltonian, de-
:
v mers, organic charge transfer salts, ferroelectric perovskites, scribing quantum transport in a background medium. Its
Xi halogen-bridged transition metal complexes, TMT[SF,TF] fermion-bosoninteractionpart
chains, Qn(TCNQ) compounds, or, e.g., the quite recently
r 2
a studiedvanadiumdioxidePeierls-Mottinsulator[7–10]. Car-
bonnanotubes[11]andorganicsemiconductors[12]areother H =−t f†f (b†+b ) (1)
fb b j i i j
exampleswheredisorderandbosonicdegreesoffreedomare hXi,ji
ofimportance.Regardinginteractingbosons,ultracoldatoms
trappedin opticallattices offerthe uniquepossibility totune
boththedisorderandinteractionstrength[13]. mimics the correlations/fluctuations inherent to a spinful
fermionmany-particlesystembyaboson-affectedtransferof
Unfortunatelythe subtle interplay of disorder and interac-
tion effectsis one of the most challengingproblemsin solid spinless charge carriers. In Eq. (1), a fermion fi(†) creates
statetheoryand—despitefiftyyearsofintenseresearch—still (orabsorbs)alocalbosonb(†) everytimeithopstoanearest
i
an area of uncertainty, see [14] and references therein. In neighbor(NN)sitej. Therebyitcreatesalocalexcitationin
the limit of vanishing charge carrier density only the inter- thebackgroundwith energyω : H = ω b†b . Because
0 b 0 i i i
actionwiththelatticevibrationsmatters. ThenAndersondis- ofquantumfluctuationsthebackgrounddisPtortionsshouldbe
ordermayaffectthepolaronself-trappinginahighlynontriv- abletorelaxwithacertainrateλ. TheentireEdwardsHamil-
2
easily generalized to treat systems including bosons [32], in
1
ordertoobtainunbiasedresultsforthefullAEM,
∆=0 ∆>0
0.8
repulsiveTLL Andersoninsulator H =∆ ε nf +H , (3)
AE i i E
-100.6 ? Xi
ω
0.4 and the related AtVM, HAtV = ∆ iεinfi +HtV, where
disorderofstrength∆isinducedbyinPdependentlydistributed
0.2 CDW randomon-sitepotentialsε ,drawnfromtheboxdistribution
i
P(ε ) = θ(1/2 − |ε |). Within the pseudosite approach, a
0 i i
0 10 20 30 boson is mapped to n pseudosites [32, 33]. In the numer-
b
λ-1 ical study of the AEM we take into account up to n = 4
b
pseudosites and determine n by the requirement that local
FIG. 1. (Color online) DMRG metal-insulator phase boundary for b
bosondensityofthelastpseudositeislessthan10−7foralli.
the1Dhalf-filledEdwardsmodelwithoutdisorder(solidline).CDW
orderissuppressedifthebackgroundfluctuationsdominate[ω <1] Furthermorewe keepuptom = 1200density-matrixeigen-
0
orifthesystem’sabilityforrelaxationishigh[λ > λc(ω0)]. The statesintherenormalizationstepstoensurethatthediscarded
blue crosses denote theparameter setsconsidered in thispaper for weightissmallerthan10−8. Thecalculationsareperformed
thedisorderedEdwardsmodel. forfinitesystemswithlengthsL=16to128andopenbound-
ary conditions (OBC). For the simpler effective AtVM we
reachL=192withOBC.Heretheuseofm=1000density-
tonianthenreads
matrixeigenstatesmakesthediscardedweightnegligible. To
H =H −λ (b†+b )+H . (2) gainrepresentativeresultsforourdisorderedsystemswepro-
E fb i i b
Xi ceed as follows. We first compute the physical quantity of
interestatfixedLfornumeroussamples{ε },thensetupan
i
Aunitarytransformationb 7→b +λ/ω eliminatestheboson
i i 0 appropriate statistical average, and finally perform a careful
relaxationterminfavorofasecondfermionhoppingchannel:
finite-sizescaling.
H = H − t f†f + H . We like to emphasize
E fb f hi,ji j i b Animportantquestionis, ofcourse, whichphysicalquan-
that (i) this free-fePrmion transfer takes place on a strongly tity to use in the finite-size scaling of the Anderson transi-
reduced energy scale t = 2λt /ω however, and (ii) co-
f b 0 tion. For this purpose the localization length ξ seems to be
herentpropagationof a fermionis possible even in the limit
promising, because it is sensitive to the nature, localized or
λ=t =0bymeansofasix-stepvacuum-restoringhopping
f extended, of the electron’s eigenstate [34, 35]. So far ξ has
process[28],actingasadirectnextNNtransfer“fi†+2fi”.The beendeterminedfromthephasesensitivityoftheground-state
Edwardsmodelrevealsa surprisinglyrich physics. Depend- energy[17, 25]. Quite recently, Berkovits demonstratesthat
ingontherelativestrengthstf/tb ofthetwotransportmech- alsotheentanglemententropycanbeusedtoextractthelocal-
anisms and the rate of bosonic fluctuations tb/ω0 it repro- izationlength[36]. However,inbothmethodsthesystemsize
duces Holstein and t-J model-like lattice- and spin-polaron Lshouldbealwayslargerthanξ.
transport, respectively, in the single-particle sector [28, 29]. Advantageously,thelocalizationlengthcanbeextractedby
Forthehalf-filledbandcase,ametal-insulatorquantum-phase a finite-size scaling analysis of the charge density structure
transition from a repulsive Tomonaga luttinger liquid (TLL) factor even for L ≪ ξ, which works equally well for inter-
to a CDW has been reported [30, 31], see Fig. 1. Note that actingsystems[26],andthereforeallowsustodiscussthein-
the CDW is a few-boson state that typifies rather a corre- terplay between Anderson localization and CDW formation
lated (Mott-Hubbard-type)insulatorthan a Peierls state with in a consistent manner. The charge structure is defined as
manybosons(phonons)involved[30, 31]. Since inthelimit C˜(q) = 1 L [hnfnf i−hnfihnf i]eiqr,. Assuming
L i,r=1 i i+r i i+r
ω0 ≫ 1 ≫ λ (here, and in what follows tb is taken as an exponenPtial decay of the equal-time density-density cor-
the unitof energy)backgroundfluctuationsare energetically relationsintheAndersoninsulatingphase[37], the structure
costly, charge transport is hindered and an effective Hamil- factorscaleswith
tonian with NN fermion repulsion results. To leading or-
2
−detrf, inhai,rjeidfuj†cfeid+(zVero-bionsfionn)fi+H1ilwbeitrht sVpa=ce,t2bw/eωg0.etTHhitsVso=- C˜(q)=−2Kπρ∗2e−eπ6π6ξ2ξL−−11q2, (4)
callePd t-V model canPbe mapped onto the exactly solvable
XXZ model, which exhibits a TLL-CDW quantum phase where q = 2π/L ≪ 1 [26]. Equation (4) contains two un-
transition at V/t = 2, i.e., at λ−1 = 4. This value is knownparameters:thelocalizationlengthξandthedisorder-
f c
smallerthanthoseobtainedfortheEdwardsmodelinthelimit modifiedTLLinteractioncoefficientK∗. Hence,ifthecharge
ρ
ω−1 ≪ 1, where λ−1 ≃ 6.3 (see Fig. 1 and Ref. [31]), be- structure factor is determined numerically, ξ and K∗ can be
0 c ρ
causealreadythree-siteandeffectivenext-NNhoppingterms easilyderivedbyfittingthenumericaldatawithEq.(4). For
wereneglectedinthederivationofthetV-model. vanishingdisorder∆→0,ξdivergesandK∗becomestheor-
ρ
We now employthe DMRG technique[24], which can be dinaryTLLparameterK . Weareawarethatadisordered1D
ρ
3
0.02 105 1
AEM AtVM
AEM ω0=1
∆∆==00..05 0.01 ∆∆//ttff==00..05 104 0.8 ω0=2
∆=1.0 ∆/tf=1.0 0.6
q) ∆=2.0 ∆/tf=2.0 ξ103 K*ρ 1
(0.01
v 0.4
˜Ca 2 0.8 ω0=3
10
0.2
0.6
101 (a) λ=0.1 0 1∆/tf2 3 (b)
0
105 1
0 0 λ=1
0 0.01 0.02 0 0.005 0.01 0.015
AEM
1/(L-δ) 1/(L-δ) 0.9
4
10 λ=0.2
FIG.2.(Coloronline)ChargestructurefactoroftheAEMwithλ= 0.8
0.1andω0 = 2(leftpanel)andtheAtVMwithV/tf = 1.5(right ξ103 K*ρ 1 λ=0.1
panel)sampledover300and500disorderrealizations,respectively. 0.7 0.9
Dashedlinesgivethefinite-sizescalingofC˜av(q)accordingto(4). 2 0.8
10
0.6
0.7
(c) ω0=2.5 0 1∆/t2 3 (d)
systemisnolongeraTLLandconsequentlytheTLLparam- 101 0.5 f
0.1 1 2 0 1 2
eter is ill-defined in the strict sense. Nevertheless, if the lo- ∆ ∆
calizationlengthsignificantlyexceedsthelatticeconstant,the 1
AtVM
short-range correlation functions should still show a power-
4
10
lawdecay. Thereforewe mightgainsomevaluableinforma-
tionaboutthelocalmotionoffermionsfromK∗. 0.8
ρ
Figure2demonstratesthatthefinite-sizescalingoftheav- ξ K*ρ
3
eragedchargestructurefactorC˜ (q)bymeansof(4)works 10
av
0.6
tothebestandequallywellforthe1DAEMandAtVM(this
(f)
appliestoallparametervaluesdiscussedbelow). Toaccomo- 2 (e)
10
datethemissingcorrelationsowingtotheOBC,wehaveplot-
0.4
tedC˜ (q)asafunctionof1/(L−δ)insteadof1/L(thisway 0.1 1 2 0 1 2
av ∆/t ∆/t
ofplottingthedataisnonessentialbutgivesaquantitativere- f f
finementofthefit). Theparameterδ isadjustedtoreproduce
FIG.3. (Coloronline)Leftpanels: Log-logplotofthelocalization
K∗ = K and ξ = ∞ at ∆ = 0. We note the generalten-
ρ ρ length versus disorder strength for the AEM at fixed λ = 0.1 (a),
dency that the charge correlations arising at finite L will be fixedω = 2.5(c),andfortheAtVM(e). Dashedlinesarefitsto
0
suppressedthemorethelarger∆. Eq. (5) with: (a) ξ = 620, 98, 50 and γ = 1.75, 1.65, 1.22 for
0
Ina nextstep, we extractthelocalizationlengthξ andthe λ = 1, 0.2,0.1; (c)ξ0 = 500, 65, 40andγ = 1.81, 1.35,1.1for
modifiedTLLparameterforthedisorderedEdwardsandt-V ω0 = 1,2,3;(e)ξ0 = 440,350,230,190,150andγ = 1.95,1.5,
models. Figure3 showsthe dependenceof ξ andK∗ on the 1.3, 1.1, 0.95 for V/tf = 0, 0.5, 1. 1.5, 2 (from top to bottom),
ρ respectively. Right panels: Corresponding results for the modified
disorderstrength∆. Firstofall,wefindapowerlawdecayof
TLLparameterK∗intheAEM[(b),(d)],andAtVM(f).
ξwith1/∆inthewhole(λ,ω ;V)parameterregime: ρ
0
ξ/ξ =∆−γ (5)
0
Theright-handpanelsdisplaystrikingdifferencesinthe∆-
[notethat∆isgiveninunitsoft (t )inFig.3fortheAEM dependenceofK∗forthemodelsunderconsideration.These
b f ρ
(AtVM)].Theestimatedvaluesofthe(bare)decaylengthξ can be attributed to the fact, that the Edwards model con-
0
andtheexponentγ aregiveninthecaptionofFig.3forchar- tainstwoenergyscalesλandω whilethephysicsofthet-V
0
acteristic model parameters. As expected, the localization model is merely governed by the ratio V/t = t /2λ, i.e,.
f b
length decreases with increasing disorder strength. Stronger ω dropsout. Far away from the CDW instability, however,
0
electroniccorrelations,i.e,smallerλorlargerω0(largerV)in bothmodelsdescribeaweaklycorrelatedTLLwithKρ <∼ 1
theAEM(AtVM),alsotendtoreduceξ. Inanycase,ξ turns andK∗slowlydecaysasthedisorder∆increases(seethered
ρ
outtobefiniteassoonas∆>0,indicatingthattherepulsive opentrianglesinFig.3;tomakethecomparisonwiththet-V
TLL,ifrealizedfor∆=0,makeswayforanAndersoninsu- model data easy, we have shown K∗ versus ∆/t in the in-
ρ f
lator.Therebythelocalizationlengthbecomescomparablethe sets). IfwemovetowardstheCDWinstabilitybydecreasing
latticespacingat∆=2intheAEMwithλ=0.1,ω =2.5, λat fixedω > ω or increaseω atλ < λ (cf. Fig. 1) a
0 0 0,c 0 c
whileitisstillabout102fortheAtVMwithV/t =2. non-monotonousbehavior develops. At small ∆, K∗ is sig-
f ρ
4
103 2 renormalizationequation: d(∆2)/dl = (3−2Kρ)∆2 (with
AtVM
ξ0 1.8 scalequantityl). Thiscausesthescalingrelation[37,39,40]
1020 1 2 1.6 γ =2/(3−2K ). (6)
104 V/tf γ1.4 ρ
γ/tf AEM 1.2 TherightpanelofFig.4displaysthatγ basicallydependson
ξ0103 1 Kρ as predicted by Eq. (6). This means that the long-range
localization propertiesof the AEM can be understoodin the
0 1 2 0.5 0.6 0.7 0.8 0.9 1
Veff/tf Kρ frameworkofAtVMwithaneffectiveintersiteinteractionin-
ducedbythebosonicdegreesoffreedom.Sincethe(effective)
FIG.4. (Coloronline)Leftpanels: Decaylengthξ0asafunctionof CoulombrepulsiontendsinresulttoalesserKρ,γ decreases
the(effective)CoulombrepulsionV(eff) fortheAtVMandAEM. with increasing V(eff) (cf. Fig. 4). In this way Anderson
Rightpanel:Correspondingγ-exponentvs.Kρ,incomparisontothe localizationissuppressedbythe2k -CDWfluctuationstrig-
scalingrelation(6)(dashedline).Forfurtherexplanationseetext. F
gered by V, which on their part also lead to charge carrier
immobilization however. While γ = 2 in the free-fermion
limit(V, 1/λ → 0), it scalesto unityapproachingthe CDW
nificantlyenhancedasthedisorderincreases.Obviouslyweak
transitionpoint, located, e.g., atλ ≃ 0.07forω = 2.5 re-
disorderdestabilizesthe2k -CDWcorrelationslocally,since c 0
F spectivelyatω ≃3.1forλ=0.1.
disorder-induced second- (and higher-) order boson-assisted 0,c
The question how disorder affects the insulating CDW
(inelastic) hopping processes are possible in the AEM, even
state could not be addressed by the above TLL-based scal-
forω ≫ 1. ThisinsharpcontrasttotheAtVM,whereonly
0
ing approach. Here, increasingthe strength∆ of disorder, a
elasticscatteringtakesplaceandtheintersiteCoulombrepul-
crossoverisexpectedfromtheCDW-regimedominatedbythe
sionishardlyaffectedby∆. Asaresult,inthedisorderedt-V
scaleofthesingle-particle(charge)gap∆ (∆ = 0)[31] to
modeltheCDWcorrelationswillbestrongerandmorerobust. c1
Hence, for the AtVM, K∗ appearsto be nearly independent theAnderson-regime,where∆≫∆c1.Intheformercasethe
ρ
from∆ for0.5 <∼ V/tf <∼ 2. Thisalso notablydiffersfrom CDWsurvivesmoderatedisorder(asexperimentallyobserved
fordisorderedPeierls-Mottinsulators[10]),inthelattercase
the behavior found for the disordered Hubbard model [26],
a pseudogapscale maydevelopasinthe stronglydisordered
where the umklapp scattering is effectively enhanced by the
Hubbardmodel[23]. Figure5, showingthespatialvariation
formationofMottplateausappearingduetodisorder[38]. If
∆exceedsacertainvalueintheAEM,K∗ startstodecrease of the local fermion/boson densities for a specific but typi-
ρ
caldisorderrealization(notethatanyrealexperimentisper-
and finally the whole scaling procedure breaks down when
ξ >∼1(seethepointat∆=2intheupperrightpanelwithKρ∗ finosrimdeedthoenCaDpWartpichualsaer(sλam−1pl=e),1i0ll0u,stωra−te1s=the0.s4i;tucaft.ioFnigd.e1e)p.
wellbelow0.5). Inthisregimethewavefunctionsofthepar- 0
One realizes thatlong-rangechargeorderceases to exist but
ticlesarestronglylocalizedandtheTLL-behaviorisasmuch
short-range CDW correlations may locally persist whenever
suppressedastheCDWcorrelations. Letuspointoutthatthe
enhancementofK∗ triggeredbythebosonicdegreesoffree- neighboringonsite potentials will differ not much (see, e.g.,
ρ
theregioni=45...55inthelowerpanelofFig.5).
dommightserveasaexplanationfortheobservedincreasing
chargevelocitynearanegativelychargeddefectinthesingle- To summarize, using an unbiased numerical DMRG ap-
wall carbon nanotubes [11], since the TLL parameter K is proach,forthefirsttimeweinvestigatedtheinterplayofdisor-
ρ
proportionaltothechargevelocity. derandinteractioneffectsincludingbosonicdegreesoffree-
We now focus on the localization behavior at large dis- domintheframeworkofthe1DspinlessfermionAnderson-
tances [O(ξ ≫ 1)], and therefore made an attempt to an- Edwards model. Although the TLL phase disappears owing
alyze the decay length ξ and the exponent γ, for both the tothedisorder,thelocalizationpropertiesoftheAndersonin-
0
AEM and AtVM, in terms of the interaction exponent K sulator state can be understood in terms of scaling relations
ρ
andthechargesusceptibilityχ ofstandardTLLtheory[37]. containing the charge susceptibility and the Luttinger liquid
c
Weexpectthatξ isstronglyinfluencedbythestrengthofthe parameterofthemetallicphasewithoutdisorderonly,asinthe
0
chargefluctuationsquantifiedbyχ , whichisgiven—forthe caseofthespinlessfermionAnderson-t-V model. However,
c
the Anderson-Edwardsmodel reveals a more complex inter-
t-V model—asχ = 2Kρ = 4 π −1
withchargeveloccityvπ.vcFigurπeq41s−h(o2Vwtfs)2th(cid:20)aatrcξcosn(−ic2eVtlfy)scale(cid:21)s rdeiltaiotinoanlsbceatwtteereinngdicshoarndneerlasn,dinvCoDlvWingcobroresloantiiocnesxbcietcaatiuosneaandd-
c 0
withV/t , i.e., ξ ∝ χ infact(seeupperpanel). Thesame annihilationprocesses, appear. Thisoffersapromisingroute
f 0 c
holds—perhaps surprisingly—for the AEM when ξ /tγ is for adapting the description of low-dimensional transport in
0 f
plotted versusan effectiveintersite repulsionestimated from many disordered materials. Disorder also affects the CDW
V /t = −2cos(π/2K )(seelowerpanel). Moreover,for state inthattruelong-rangeordervanishesforstrongon-site
eff f ρ
thedisorderedt-V model,theexponentγ isconnectedtoK potentialfluctuationsthoughlocalCDWcorrelationssurvive.
ρ
ofthe(spinlessfermion)TLLsystemwithoutdisorderviathe ThisworkwassupportedbyDFGthroughSFB652.
5
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