Charge density waves in disordered media circumventing the Imry-Ma argument

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by  Hitesh J. Changlani

ChargedensitywavesindisorderedmediacircumventingtheImry-Maargument
Hitesh J. Changlani,1 Norm M. Tubman,2,1 and Taylor L. Hughes1
1DepartmentofPhysics, UniversityofIllinoisatUrbana-Champaign, Urbana, Illinois61801, USA
2Department of Chemistry, University of California, Berkeley, California 94720, USA
(Dated:January14,2016)
Twopowerfultheoreticalpredictions,AndersonlocalizationandtheImry-Maargument,imposesignificant
restrictions on the phases of matter that can exist in the presence of even the smallest amount of disorder in
one-dimensional systems. These predictions forbid conducting states and ordered states respectively. It was
thusremarkablethatamechanismtocircumventAndersonlocalizationrelyingonthepresenceofcorrelated
disorderwasfound, thatisalsorealizedincertainbiomolecularsystems. Inasimilarmanner, weshowthat
6 the Imry-Ma argument can be circumvented resulting in the formation of stable ordered states with discrete
1 brokensymmetriesindisorderedonedimensionalsystems. Specifically,wesimulateafamilyofHamiltonians
0 ofspinlessfermionswithcorrelateddisorderandinteractions,wherewefindthatachargedensitywaveisstable
2 uptoafinitecriticaldisorderstrength.HavingcircumventedtheImry-Mamechanism,wetheninvestigateother
n mechanismsbywhichdisordercandestroyanorderedstate.
a
J
Disorder can have drastic effects on electronic properties, casesn=1,2,3,4respectively.
2
1 especially in low dimensions. On the one hand, it lifts the Remarkably,materialswiththisformofdisorderhavebeen
degeneracybetweencompetingphasesthrough"orderbydis- identified; for example, it was found that the unusual trans-

n] order" mechanisms [1–3], and on the other it localizes clean portpropertiesofpolyanilinecanbeexplainedviaaneffective
n metallic states [4–8], and even creates unusual emergent ex- Hamiltonian approach that maps this molecule onto the ran-
- citations [9, 10]. In one dimension, the essential physics of dom dimer model [16]. Further interest in the n−mer mod-
s
i disorderiscapturedbyAndersonlocalization[4]fortransport els has been fueled by their possible relevance to describing
d
properties,andworkrelatedtotheseminalpaperofImryand transportin largeclasses ofbiomolecules suchas DNA[17–
.
t Ma[11–13]forunderstandingthedisorder-drivendestruction 20]. Most work on n-mer models has been devoted to the
a
m of ordered phases. The dimensionality dependence of both non-interacting case [14, 21], which is analytically and nu-
effectsweakenswithincreasingspatialdimension, thustheir mericallytractable,whiletheinteractingcasehasbeentreated
-
d strongesteffectsareseeninonedimension. onlyatthemeanfieldlevel[22,23],andwithexactdiagonal-
n An interesting exception to Anderson localization arises izationforsmallsystems[24].Thus,theeffectsofinteractions
o
due to correlations in the disorder. In particular, Refs. onthesephasesarestilllargelyunexplored.
c
[ [14, 15] discovered a class of non-interacting random n-mer The disorder-free system at small V/t is known to be
models where a band of single-particle states which exhibit a Tomonaga-Luttinger liquid [25] which at half-filling, and
1
nobackscatteringexist;aconditionkeyforcircumventinglo- a critical interaction strength (V/t = 2), forms a charge
v
1 calization. Thefocusofthisarticleistoshowthatcorrelated density wave (CDW) state that remains stable for all larger
4 disorder of this type can also avoid the Imry-Ma argument, V/t [25, 26]. However, according to the Imry-Ma argu-
0 andmayleadtothestabilizationoforderedphasesininteract- ment [11, 27], such a state should not exist upon the slight-
3 ing,disorderedchains. est introduction of disorder; here we show how the n−mer
0
Inalln-mermodelstherearetwotypesof‘atoms’, which modelsavoidthis.
.
1 wecallAandB,differingonlyintheiron-siteenergy,which
Given a pinned, commensurate CDW with every even site
0 are placed at random on a one-dimensional chain; with the
(mostly) occupied [28], as is depicted in Fig. 1, let us assess
6
conditionthatnB’s(an"n-mer")arealwaysplacedconsecu-
1 its stability to the introduction of weak disorder (for which
tively. Withtheinclusionofnearestneighborrepulsiveinter-
: wecloselyfollowRef.27). Considerasegmentoflength2L
v
actions, the Hamiltonian for spinless electrons that we study
thatispartofthefull1Dlatticewithalargenumberofsites,
i
X is:
and divide it into odd and even sublattices, to be labelled as
ar H =(cid:88)(cid:15)ini−t(cid:88)c†ici+1+H.c.+V (cid:88)nini+1 (1) 1theanedve2nressitpeescitsivgelrye.atIefrththeasnutmheosfuamlltohfethoen-osint-esietneeergnieersgioens
i i i
onalltheoddsites,thenitisenergeticallyfavorableforeach
where c† (c ) and n refer to the usual spinless electron cre- electron in the segment to shift by one site, despite the cost
i i i
ation(destruction)anddensityoperatorsrespectivelyonsitei of the repulsive interaction of neighboring electrons, hence
whichisoccupiedbyeitheranAorB atom,andcorrespond- formingadomainwall.
inglytheon-siteenergy(cid:15) iseither(cid:15) (settozerothroughout) Foruncorrelateddisorder,andforLsufficientlylargetoap-
i A
or (cid:15) , which will be referred to as the “disorder strength," plystatisticalarguments,thedifferencebetweenthesummed
B √
t is the nearest-neighbor hopping parameter, and V is the energies on the two sublattices is of the order ± L. Since
nearest-neighbor interaction strength. We use the terminol- forming a domain wall costs only an energy of order V, the
ogy-"monomer","dimer","trimer,"and"quadrumer"forthe former effect always wins for some large enough L; hence,
2
2L
400
1 2 1 2 1 2 1 2 1 2 1 2 CdpoDimnWna ewindi twChDa lolWlcal rgydifference233505000 Sublatticeenergydifference 23321100000000000000 MMMMMD2iooooom0nnnnneooooo0rmmmmm0eeeee0rrrrrsssssaaaaammmmm4ppppp0llllleeeee01234500 60000 80000 100000
e Uniform L
n200
e Gaussian
monomer e Monomer
A B B B A A B A B B A A ΔE ~ ±O(√L) tic150 Dimer
at Trimer
bl100 Quadrumer
dimer u
S
A B B B B A A B B A B B ΔE ~ 0 S 50
M
R 0
Figure 1. (Color online): The top panel shows a schematic of the 0 50 100 150 200 250 300
electronicdensity(proportionaltoareaofcircles)demonstratingfor- √L
mation of domain walls in a charge density wave in a generic dis-
orderedmodel. TheImry-Maargumentpredictsthatsuchdomains
are(typically)energeticallyfavorableevenforthesmallestnon-zero Figure2. (Coloronline): Rootmeansquaredvalueofthedifference
disorder. Thebottompanelshowsschematicsofadomainoflength insummedsublatticeenergies,∆E,versusthelengthofthesegment
2L in the random monomer and random dimer models. The red Lcomputedforsixtypesofdisorderdistributions. Foreachdisor-
andbluesitescorrespondtoAtype(monomerswithon-siteenergy dertype,2000realizations,eachcomprisingof105sites,wereused.
(cid:15) =0.0t)andBtype(monomersordimerswithon-siteenergy(cid:15) ) The uniform (box) distribution corresponds to maximum and min-
A B
sitesrespectively. Fortherandommonomercase,the√typicaldiffer- imum energies of 1 and −1 respectively, the Gaussian distribution
ence in summed sublattice energies of the order of L while it is has a mean of 0 and a spread (σ) of 1, and the monomer through
zerointherandomdimercase. quadrumermodelseachhave(cid:15)A = 0and(cid:15)B = 1. Inset: ∆E vsL
forseveralindividualdisorderrealizationsoftherandommonomer
the system acts to reduce its energy by the formation of do- anddimermodels.Theformershowslargefluctuationsin∆Ewhile
mains. Thus, there is no (quasi) long-range CDW order in thelatterhas∆E =0or±1(notvisualizedonthescaleoftheplot).
onedimensionupontheslightestintroductionofuncorrelated
disorder. henceEq.(3), holdsforany n-mermodelwith neven. This
However, thesituationismarkedlydifferentwhenthedis- is demonstrated in Figure 2 which shows the special cancel-
order is correlated. Let us define nα,j to be the number of lation(orlackthereof)ofthesublatticeenergyimbalancefor
siteswhereαisanindexforthedisordersite(AorB),andj theeven(odd)n−mermodelsforanensembleofdisorderre-
isthesublatticeindex(1or2). Then,foranydisorderrealiza- alizations.Ininstancesofsegmentswhereoneormorebound-
tionoftherandomdimermodel,andanyintervalof2Lsites, arycutsadimerinhalf,thereisanedgecorrectionofoneor
wehavetheconditions, two lattice sites, which is small on the scale of L and does
notaffectourconclusionsintheregimeofweak-to-moderate
n +n =L
A,1 B,1 disorder((cid:15)B <∼V).
n +n =L. (2)
A,2 B,2 We verify these arguments by performing numerically ac-
Since the instances of B occur only as dimers, n must be curate density matrix renormalization group (DMRG) [29]
B,1
equal to n (assuming the segment of length 2L does not calculationsofthen−mermodelsforn=1,2,3,4,discussed
B,2
contain any incomplete dimers) which, in turn, implies that further in the Methods section of the Supplemental Informa-
n = n , i.e., the number of A-type sites on each sub- tion. Resultsfromoursimulationsforindividualdisorderre-
A,1 A,2
latticearealsoexactlyequal. Forexample,the2L = 12site alizationsareshowninFig.3wherewehaveplottedtheelec-
segment of the random dimer disorder realization in Fig. 1 tronicdensityoneveryevensiteforV = 5tandthreedisor-
has n = 2 and n = 4, the latter forcing the condition der strengths. The boundary conditions have been chosen to
A,1 B,1
n = 4,andhencen = 2. Weemphasizethattherela- slightly favor the high occupation of the even sites, and thus
B,2 A,2
tionship(2)holdsgloballyand,moreimportantly,locallyfor anyrapiddecreasefromhightolowdensityisthesignatureof
anysubsetwithanevennumberoflatticesites. Thus,thedif- adomainwall.
ferencebetweensummedon-siteenergiesontheevenandodd At (cid:15) = 0.5t, the random monomer and trimer models
B
sublatticesiszero,i.e., show large but finite domains whose size decreases with in-
(cid:88) (cid:88) creasingdisorderstrength. Incomparison,therandomdimer
∆E ≡ (cid:15) − (cid:15) =0. (3)
i i andquadrumershownotendencytoformdomainwallsuptoa
i∈1 i∈2
critical(V-dependent)disorderstrength. Forexample,forall
This energy difference does not grow with L, and therefore oftheindividualrandomdimerandquadrumerrealizationsin
the Imry-Ma argument for the formation of domains is not Fig.3,thefirstdomainwallsareseenonlyaround(cid:15) = 2.5t
B
expected to apply. In fact, the condition n = n , and whenV =5t.
B,1 B,2
3
monomer dimer trimer quadrumer
1.0
0.8
y
t
† =0.5t 0.6 si
B n
0.4 e
D
0.2
0.0
1.0
0.8
y
t
† =2.0t 0.6 si
B n
0.4 e
D
0.2
0.0
1.0
0.8
y
t
† =2.5t 0.6 si
B n
0.4 e
D
0.2
0.0
0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000
Site Site Site Site
Figure3.(Coloronline):Fermionicdensityoneveryevensiteforindividualrealizationsoftherandommonomer,dimer,trimerandquadrumer
modelsatdisorderstrengths(cid:15) = 0.5t,2.0t,2.5tforV = 5t. Atsmalldisorder,theodd-nmodelsshowtheformationofdomainwallsin
B
agreementwiththeImry-Maargument,whiletheeven-nmodels,whichcircumventtheargument,donot(withinthesizeconsidered).Beyond
somecriticaldisorderstrength,domainwallformationisfavorableforallmodels,i.e.,CDWorderpersistsonlyinlocalpatches.
The eventual occurrence of domain walls in the random the Supplemental Information). In fact, this argument holds
dimer and quadrumer models can be explained as follows. for any even n-mer since favorable occupation on an even
First,forsufficientlylargedisorder(cid:15)B >∼ V,theeffectofthe numberofconsecutivesiteswillcauseaphaseslip.
heretoforeignorededgesinthen-merversionoftheImry-Ma Letusnowlookbeyondindividualrealizationsandperform
argumentnowstartstoplayanimportantrole. Theenergyof statistical analyses of our samples; Fig. 4 shows the average
the CDW is now reduced by order (cid:15)B, which is greater than size of the CDW domains as a function of disorder strength.
the price of forming a domain wall (order V). Second, any As is anticipated from the Imry-Ma argument, the random
B site would like to have lower density wherever possible, monomerandtrimermodelsshowdivergenceindomainsize
causingfluctuationsofthedensitythatgrowlargeenoughto around vanishing disorder for all V/t considered. This is in
destroy the ordered state. For example, for the realization in contrast to the random dimer and quadrumer models which
Fig. 3, for (cid:15)B = 0.5t, the density fluctuations are seen to be havenodomainwallsuntilacritical(cid:15)B∗(V)isreached.
small (∼ 0.03), compared to the maximum occupation of a We emphasize that to prevent the formation of domain
site (∼ 0.95), and eventually grow past 0.5, at which point walls,thecondition(3)mustbesatisfiedatallshortandlong
CDWorderislost. lengthscales. Forexample,toshowthatthelocalcancellation
Thissecondarymechanismcanalsobequalitativelyunder- isimportant,letusconcoctdisorderrealizationsofthefollow-
stood by considering just a single dimer of B sites placed at ingtype. TakearandomlygeneratedmonomerchainofL/2
the center of a 1D chain otherwise purely made of A sites. sitesanddefineits"complement":formarealizationoflength
When(cid:15) issmall,ournumericalcalculationsindicatethatthe L/2whereeveryAtypesiteisreplacedbyaB typesiteand
B
CDW is relatively robust with only a minor local rearrange- vice versa. Then place these two segments (sample and its
mentofelectronoccupations. Then,aboveafinite(non-zero) complement) side by side to form a L-site chain. Each such
critical(V dependent)(cid:15) ,itisenergeticallyfavorableforthe disorderrealizationhasn = n = n = n = L/4
B A,1 A,2 B,1 B,2
densityonboththeB sitestobesmall. Thiscreatesa"phase andthussatisfies(3),butonlyglobally. Interestingly,wefind
slip" on the dimer forcing the rest of the chain, made solely fromnumerics(shownintheSupplementalInformation)that
ofAsites,tomaintainaCDWwithoppositephasesoneither domain wall formation is still favorable, and the Imry-Ma
sideofthedimer. (Furtherdiscussionshavebeenpresentedin mechanismisstilleffective.Despitethisfailure,thisproposed
4
103 103 ported by the National Science Foundation Grant No. OCI-
n=1 n=2
D fi D fi 1053575, the Oak Ridge Leadership Computing Facility at
e › e › the Oak Ridge National Laboratory, which is supported by
n siz102 n siz102 theOfficeofScienceoftheU.S.DepartmentofEnergyunder
mai V=3.0t mai Contract No. DE-AC05-00OR22725 and the Taub campus
Do V=5.0t Do clusteratUIUC/NCSA.TLHissupportedbytheUSNational
V=7.0t
1001.0 0.2 0.4 0.6 0.8 1.0 1010 1 2 3 4 5 6 7 ScienceFoundationundergrantDMR1351895-CAR.
Disorder †B Disorder †B
103 103
n=3 n=4
D fi D fi
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