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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. 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