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Stripes: Why hole rich lines are antiphase domain walls? Oron Zachar ICTP, 11 strada Costiera, Trieste 34100, Italia (February 1, 2008) For stripes of hole rich lines in doped antiferromagnets, we investigate the competition between 0 anti-phaseandin-phasedomainwall groundstateconfigurations. Wearguethataphasetransition 0 mustoccureasafunctionoftheelectron/holefillingfraction ofthedomain wall. Duetotransverse 0 kinetic hole fluctuations, empty domain walls are always anti-phase. At arbitrary electron filling 2 fraction (δ) of the domain wall (and in particular for δ ≈1/4 as in LaNdSrCuO), it is essential to n account also for the transverse magnetic interactions of the electrons and their mobility along the a domain wall. J Wefindthatthetransitionfromanti-phasetoin-phasestripedomainwalloccursatacriticalfilling 6 fraction 0.28 < δc < 0.30, for any value of Jt < 31. We further use our model to estimate the 1 spin-wave velocity in a stripe system. Finally, we relate the results of our microscopic model to previous Landau theory approach to stripes. ] l I. INTRODUCTION (below half-filling) per site along the domain wall; e - r nh =1 2δ. (1) t Ananti-phasedomainwallinstripesisthestatewhere − s . the localantiferromagnetic(AFM) spin order parameter Experimentally [1], anti-phase domain wall stripes in t a undergo a π phase shift across a hole rich line. Such Nickel-Oxides have nh 1 (corresponding to one hole m periodic stripe structures were experimentally found in per site along the doma≈in wall, or equivalently an elec- - doped Copper oxides and Nickel oxides [1]. Historically, tronemptydomainwall,δ =0),whileanti-phasedomain d thiswellacceptedfeaturewasfirstconsideredtobeanat- wall stripes in Copper-Oxides have nh 1 (correspond- n uraloutcomeofmean-fieldtheoryFermisurfaceinstabil- ing to one hole per two sites along the≈do2main wall, or o ity[2,3]. Yet,nosimilarrigorousmicroscopicexplanation electron1/4filleddomainwall,δ 0.25). These domain c ≈ [ was given within the frustrated phase separationpicture wallfillingfractionsremainroughlyconstantoverawide [4] which is currently regarded as underlying the micro- range of dopings in the respective materials. 1 scopic originof stripes in the relevant materials [5]. One Wefindthatthetransitionfromanti-phasetoin-phase v of our goals here is to close this gap in the theory. stripe domain wall occurs at critical filling fraction 7 In contrast with common folklore, we show that hole 1 0.28<δ <0.30 (2) 2 rich lines are not necessarily anti-phase domain walls of c 1 AFMspindomains. First,onsimplegeneralgrounds,we dependingonthevalueof J . Inotherwords,forexam- 0 argue that there must be a phase transition from anti- t ple, we predictthatstripe domainwalls with ahole den- 0 phasetoin-phasedomainwallconfigurationasafunction (cid:0) (cid:1) sity of one hole per three sites are always in-phase and 0 ofincreasedelectronfillingfractionδ ofthedomainwall. / not anti-phase. Appropriate numerical simulations can t We then proceed to construct microscopic t-J models of a be constructed to get exact numbers beyond the limits the local electronic dynamics which determines the re- m of our analyticalapproximations. Yet, since our analysis sulting spin order across a hole line. indicatesthatthesensitivityto J isratherweak,wedo d- It is frequently argued theoretically [4,5] and exempli- not expect the exact numerical rtesults to deviate much n fied experimentally [1] that the charge segregation into (cid:0) (cid:1) from our predictions. o hole rich stripe lines is prior to the establishment an- We further apply our model to evaluate the of spin- c tiphasespindomains. Therefore,themicroscopicmecha- : wave velocity in a stripe systems, and compare with ex- v nismbywhichtheholelinesenslavethespinordershould periments. In addition, we relate the competing interac- i be distinguished and considered separately. X tionsinourmicroscopicmodeltopreviousLandautheory Wedevelopeaqualitativeandquantitativemicroscopic approach to stripes order [5,6]. Thus, we advance to- r understanding of the domain wall magnetic order. Our a wards a coherent microscopic and phenomenological un- analysisaccountsfortheelectronicdynamicsbothtrans- derstandingofthestripesstructurewithinthefrustrated verse and along a pre-established hole rich line between phase-separationpicture. two AFM domains. Thus we focus solely on the mechanism which give rise to the spin antiphase domain wall feature, by examining the competition between anti- A. Overview of the model construction, analysis and phaseandin-phaseconfigurationsatgivenelectronfilling main results fraction δ of the hole rich line. In the stripes literature it is more commonto describe Sincethepaperisquitelong,weheresupplythereader the domain wall filling in terms of the number of holes with an overview of the gist of our model development 1 and main results. wall configuration, while magnetic interaction between The general intuitive argument for the domain wall electrons favors an in-phase domain configuration (see transition is the following: In one extreme case, nh = 1, figure-1below). These competinginteractionsdetermine where there is a hole on each site along the domain wall the preferred local spin configuration. All calculations (i.e., itisempty ofelectrons,δ =0,asinNickel-Oxides), are done to the first significant order in J . Our anal- t it is clear that an anti-phase domain wall configuration ysis surprisingly shows that if dynamics along the do- (cid:0) (cid:1) is favoredby transversehole fluctuations. Now, consider main wall is ignored then transition to an in-phase do- the state of domain walls with increasing electron filling main wall would have occurred already at nh 2/3 ≤ fraction. In the opposite extreme casewhere the domain (i.e., at a density of two holes per three sites), in con- wall is half-filled with electrons (i.e., nh =0, δ = 1), we flictwith experimentalobservationsofanti-phasestripes 2 should recover the undoped ordered AFM state. There- in (LaNd)SrCuO materials with nh 1/2. Hence, it ≈ fore, there must be some intermediate critical electron is essential to investigate the effect of kinetic dynamics filling fraction δ of the domain wall at which there is a alongthedomainwall,whichweundertakeinsection-IV. c transition from an anti-phase toin-phase local AFMspin In section-IV, we model the kinetic dynamics of the configuration across a hole rich line. electrons moving along a stripe domain wall in an effec- The mainquantitativeobjective ofthis paper is to de- tive external magnetic mean field due to its AFM envi- termine the critical domain wall filling fraction δ as a ronment. Along an in-phase domain wall there is an ef- c function ofthe t-J model parameters. In the process,we fectivenetstaggeredexternalmagneticfield, whilealong develope an advanced qualitative and quantitative un- an anti-phase domain wall the net external field is zero derstanding of the various competing local interactions on each site. We analyze two extreme limits: (a) Non- in the single stripe physics. Additional applications are interacting electrons moving along the domain wall, and discussed in sections-V and VI. (b) Large U t limit for the interaction of electrons ≫ Intuitively, increase of electron filling of the domain alongthedomainwall. Inbothcases,the essentialresult wall amounts to increase of magnetic interactions across isthatkineticfluctuationsalongthedomainwallweaken thedomainwalluntiltheyarestrongenoughtodominate the average magnetic interaction energy which favors an overthechargefluctuationdynamics(duetoholes). The in-phasedomainwall,andthusextendthestabilityofan possibility of a transition from topological (anti-phase) anti-phase domain wall configuration to higher electron to non-topological (in-phase) stripes, due to increase of filling fractions (i.e., lower hole densities). AFM interactions, was first speculated by Neto & Hone In section-V, we relate the competing interactions in [7,8](though,onnotquiterigorousgrounds,itwassome- our microscopic model to previous Landau theory ap- how related to the spin correlation length). proach to stripes order [5,6]. Wefindthatthetopological/non-topologicalnatureof Insection-VI,weapplyourmodeltoevaluatethespin- the stripe charge wall can be completely determined by wavevelocityv inastripestatecomparedwithv inthe ⊥ 0 the local dynamics, which in turn is determined by the parentAFMmaterial(wherev isvelocityperpendicular ⊥ electron filling fraction of the wall (assuming fixed given to the stripes). t-J model parameters). We do not see a way by which We argue that our results are quantitatively accurate interaction between domain walls, the width of the spin well beyond the seemingly rough approximations of our domainsoranysimilarlonglengthscalearesignificantly simple models. The heart of the matter is the fact that relevant. our quantitative results are sensitive only to the ener- In section-II, we list some preliminary assumptions of getic difference between an anti-phase and in-phase do- our model of the stripe domain wall: (a) The effective mainwallconfiguration. Processeswhichareneglectedin electron dynamics is captured by a one band t-J model our treatment (e.g., deeper penetration of hole hopping forelectronshoppingbetweenCoppersites(i.e.,inwhich into the AFM environment) have the same contribution the Oxygen sites are integrated out). (b) It is assumed in both domain wall configurations and thus only very thattheholeline,itsmeanposition,andholedensityare weakly affect the energetic difference between them. already pre-established by some higher energy processes (presumably phase separationand coulomb frustration). We hence focus solely on determining the preferred spin II. PRELIMINARY ASSUMPTIONS configuration across a pre-established hole line. (c) The spin order is determined by comparing the energetics of The stripes characteristics can vary to include both anti-phaseandin-phase domainwallconfigurations(and diagonal and vertical stripes, which may be insulating notwith the motionofdilute holes inauniformAFM as or conducting. Therefore, the anti-phase domain wall was done in most previous work [9,10]). mechanism should be rather simple, robust, and not too Insection-III,weintroduceourmicroscopicmodeland sensitive to the above mentioned variations. start with the consideration of only transverse interactions and dynamics, (i.e., ignoring the effects of dynam- 1. The most microscopic Hubbard type model of the ics along the domain wall). The kinetic energy due to CuO orNiO planesincludesdistinctOxygenand 2 2 transverse hopping of holes favors an anti-phase domain 2 Copper (or Nickel) orbitals bands. Yet, as com- etc. can be incorporated into a renormalizedvalue monly argued, we assume that the effective dy- of ε without changing our results. namic is capturedby a one-bandt-J model [11](in 7. Our quantitative results are sensitive only to the whichtheoxygensitesdegreesoffreedomhavebeen energetic difference between an anti-phase and in- integrated out) where electrons hop directly be- phase domain wall configuration. Therefore, many tweenCopperlatticesites. Inthecontextofstripes processes which are neglected in our treatment theory, the above assumption is supported by the (e.g., deeper penetration of hole hopping into the factthatcorrectdomainwallconfigurationsturned AFM environment, magnetic interaction between out in numerical simulations [12] of one band t-J electrons on the wall) have the same contribution models. in both domain wall configurations and thus only 2. Experimental evidence [1] and theoretical consid- very weakly affect the energetic difference between erations [5] suggest that stripe formation is com- them. monly charge driven, i.e., that periodic hole line stripes form first and enslave the formation of the AFM spin domain. Therefore, for our purpose in III. THE EFFECT OF TRANSVERSE this paper we take the hole lines to be pre-formed. INTERACTION 3. StripeswerefoundinbothSpin-1 andSpin-1doped 2 Our analysis proceed in two stages; First, in this sec- antiferromagnets[1]. Adopedholecanthuscorre- spondtoaspin-0oraspin-1 siterespectively. Yet, tion, we consider solely the transverse fluctuations of a 2 hole and magnetic interaction of electrons in the stripe, inboth casesthe hole andits dynamics are carried whilethe holes/electronsconfigurationalongthe domain on only within the dx2−y2 orbital band. Indeed, wallistakenasstatic. Inasecondstage(section-IV),we the model which we construct and analyze works equally well for both doped Spin-1 and Spin-1 an- consider the implications of electron mobility along the 2 stripe. tiferromagnets.We chosetopresentouranalysisin terms of a doped Spin-1 AFM (i.e., corresponding 2 to stripes in a CuO plane). 2 A. Model of site-centered stripe 4. Aswithsuperexchangemechanismofantiferromag- netism in the undoped parent system, we argue The average hole line position is fixed by introducing that the domain wall spin order is determined by a chemical potential shift (µ) on particular sites [7,8]. local interactions across the hole rich line. Hence, (In Figure-1,hole line µ-shifted sites are representedby itis sufficienttoconsiderasinglestripesegmentin darkcircles).Inthegroundstate,holesarepreferentially isolation (see figure-1). situated on the µ shifted sites. Hence, the effective local chemical potential shift, µ, incorporates the net effect of 5. There’s a non-trivial distinction between site- thehighenergydynamics(e.g.,coulombfrustratedphase centered and bond-centered domain walls [13], separation [4]) which lead to the stripe formation. The in the sense that the spin alignment across an magnitude of µ determines the stripe stability, and can anti-phase domain wall is antiferromagnetic for be associated with the stripe creation temperature for site-centered stripe and ferromagnetic for bond- the purpose of extracting an experimental estimate of centered stripe. The presentation in this paper it’s magnitude. Thus, it is assumed that µ<J <t. is conducted in terms of site-centered stripes, and An antiferromagnetic spin exchange interaction J ex- elsewhere[14]wewillshowthatthesameprinciples ists betweenelectronspins onnearestneighborssites. In apply for bond-centered stripes as well. order to fix the spin order, it is enough to determine the 6. Probably the main quantitative approximation in relative orientation of the spins immediately to the left our model is that we treat the AFM regions be- and to the rightof the hole line. Therefore,we include a tween the hole lines as if they were antiferromag- transverse hoping interaction, t, only between a domain neticallyordered. Weneglectspinexchangefluctu- wall site (indicated by a dark circle in figure-1a) and its ations and the quantum nature of the AFM corre- right and left nearest neighbors in the AFM regions. As lationsinthe rathernarrowladdergeometryofthe notedbefore,wearguethatprocessesoffurther penetra- stripes. Inotherwords,ourquantitativeresultsare tion of a hole into the AFM regions are not significantly rigorously valid for an Ising model approximation affecting our results. Hopping interaction t along the k of the AFM regions. Yet, in all of our calculations domain wall will be considered in section-IV. we use a parameter ε which is defined to be the Intheabsenceofkineticmotionalongthedomainwall, energydifferencebetweenparallelandanti-parallel the Hamiltoniandescribingasinglesite centereddomain near-neighbor spin states (see equation (5)). Only wall is in the end we substitute ε = J for intuitive con- creteness. Inprinciple,alltheeffectsoffluctuations 3 H = H (3) The Hamiltonian (4) is thus given by the matrix i i X 0 t t 1 Hi =µn0,i+J2 jjX′ ii′ Sj,i·Sj′,i′ (4) Hα = tt E01α E01α  (6) h ih i   t c† c +c† c +h.c. (α=anti/in)whichcanbediagonalizedexactly,resulting − 1,i,s 0,i,s −1,i,s 0,i,s Xi,s h(cid:16) (cid:17) i with the ground state energy Egα where Sj,i is the spin ofanelectronatcolumnj andline Eα = 1 Eα (Eα)2+8t2 (7) i (where j = 0 is the domain wall position). n0,i is the g 2 1 − 1 occupation number of a domain wall site at position i (cid:18) q (cid:19) alongthe wall. c†1,i,s andc†−1,i,s are electroncreationop- 12 E1α−√8t 1+ (E161αt2)2 for J ≪t erators respectively to the right and left of the domain ≈( h2t2 (cid:16) (cid:17)i for J t wall site n . (t-J model projection operator which ex- −E1 ≫ 0,i cludes double occupancy is implicitly assumed through The difference in kinetic energy gain due to transverse out the paper). hoping, between an anti-phase and an in-phase domain wall configurations, is given by B. Kinetic transverse hole fluctuations ∆E1kin =Eganti−Egin (8) ε + ε2 for J t In the absence of hoping interaction, t, the anti-phase −2 O t ≪ (9) stripe groundstate (a) and the in-phase stripe ground- ≈(−tε2 (cid:16) (cid:17) for J ≫t state (a’) are degenerate. But, the energy of the corre- Our result agrees with a previous large-d calculation sponding excited states (b) and (b’) differ. As a result, (valid only in the limit J t) [16]. But, in the ex- transverse hole hoping interaction removes the ground perimental systems of intere≫st J 1, and thus they are state degeneracy in favor of anti-phase domain wall [15]. t ≈ 3 better approximated by calculations in the limit J t ≪ which will be assumed for the rest of our discussion. Anti-phase stripe In-phase stripe From equation (8) we conclude that, in a site-centered stripe geometry, the zero point transverse kinetic fluctu- ation of the holes favor an antiferromagnetic alignment of the neighboring spins. Notethattheaboveresultisinstarkcontrastwiththe conventional wisdom that hole’s zero-point kinetic fluctuations always favors ferromagnetic alignment of their environment [17,18], which is based mostly on isolated (a) (a') (b) (b') hole models. It is a rather simple demonstration of the differencebetweencollectiveandsingleholepropertiesin FIG.1. Thedomainwallsitesareindicatedbycoloredcir- doped antiferromagnets. cles. The static ground state configuration of anti-phase and in-phase domain walls are depicted in figures (a) and (b) respectively. The corresponding excited states (a’) and (b’) re- C. Competing magnetic interactions at electron sultfromtransversehopingdynamicsofholes. Notethe”bad filling δ6=0 bond” created by the hoping in (b’). The depicted domain wall groundstate segments have one electron per 5 sites on the domain wall (i.e., a filling fraction δe = 0.1). Note the Fromtheanalysisoftheabovemodel,weconcludethat the transverse kinetic fluctuations of holes are sufficient difference in the magnetic bonds of the domain wall electron to induce an anti-phase domain wall ground state in an between (a) and (b). empty domain wall δ = 0 (i.e., one hole per site along To be general, we define ε to be the energy difference the wall, nh =1) of stripes. between an antiferromagnetic ”good bond” and a ferro- Foradomainwallwithelectronfillingfractionofδ,2δ magnetic ”bad bond” of two neighboring spins. Obvi- isthe numberofelectronsper sitealongthe domainwall ously, ε is proportional to J (and ε = J in the case of (inthelargeUlimit),andnh =(1 2δ)isthenumberof − Ising model of the AFM spin domains). We label by holes in the lower Hubbard band. Each hole contributes E1anti and E1in the bare excited states energy in the case akineticenergydifference∆E1kin. Hence,atanarbitrary of an anti-phase and in-phase domain walls as depicted electronfilling fractionδ of the domainwall,the average in figure (1b) and (1b′) respectively. Clearly, transverse kinetic energy gain per site ∆E⊥kin(δ) of an anti-phase domain wall in comparison with an in-phase Ein Eanti =ε (5) domain wall is 1 − 1 4 ∆Ekin ε where σ = 1 for spin , respectively, and ∆E⊥kin(δ)= N 1 =−2(1−2δ) (10) ± ↑ ↓ site 0 for anti-phase B = (16) For each electron on the domain wall there is an energy J for in-phase difference ∆Emag = +ε in favor of an in-phase domain (cid:26) ≈ 4 1 wall,due to spinexchangeinteractionwith the AFM en- When considering the competition of stripes versus vironment. At electron filling δ, the average magnetic droplets forms of local phase separation, Nayak&Wilzek energy difference (of an anti-phase domain wall in com- [9] proposed that a significant energy is gained by the parison with an in-phase domain) per site along the do- increasedmobilityalong ananti-phasedomainwall. But main wall is we note that the kinetic energy gain in an in-phase do- mainwallispracticallythesameasinanti-phasedomain ∆Emag ∆Emag(δ)= 1 =+ε(2δ). (11) wallstate. Therefore,itisessentialtomakeamorecare- ⊥ Nsite ful analysis of the electronic dynamics along the domain wall in both cases. The competition of these transverse interaction alone Below, we extract quantitative results in two limits: would predict a transition as a function of domain wall (a) For non-interacting electrons moving along the do- filling from anti-phase to in-phase domain wall when main wall, and (b) In the large U t limit for the in- ≫ ∆E (δ)=∆Ekin+∆Emag >0 (12) teraction of electrons along the domain wall. In both ⊥ ⊥ ⊥ cases, the essential result is that kinetic fluctuations at along the domain wall weakens the magnetic interaction energy gain which favors an in-phase domain wall en- 1 vironment, and thus extends the stability of anti-phase δ (13) ≥ 6 domain wall configuration to higher electron filling fractions (i.e., lower hole densities). In particular, we con- (corresponding to nh 2/3, i.e., at a density of two ≤ clude that dynamics fluctuations along the domain wall holes per three sites), in conflict with experimental ob- allowforthe establishmentofanti-phasestripeswithdo- servations of anti-phase stripes in (LaNd)SrCuO mate- main wall filling δ 1 (as in Copper-Oxides systems). rials with δ 1/4. This conflict is resolved in the next ≈ 4 ≈ section, where we account for effect of kinetic dynamics along the domain wall. A. Effect of motion along the domain wall for non-interacting electrons at arbitrary filling IV. IMPLICATIONS OF ELECTRON DYNAMICS We here model the dynamics along the hole rich do- ALONG THE DOMAIN WALL main wall by an effective one dimensional lattice model (14) of non-interacting electrons (U = 0). For an in- Inthis sectionweinvestigatethe consequencesofelec- phasedomainwall,theexchangecouplingtothespinsin trondynamicsalongthedomainwallforthecompetition the AFM environment result with an effective external between anti-phase and in-phase domain wall configura- staggered magnetic field on the electrons moving along tions. As a first order approximation, we assume static thedomainwall(seefigure-1b). Ourmodelofkineticmo- spin configuration of the antiferromagnetic domains on tion along the domainwall included only band structure the left and right of the domain wall. Thus, electrons effectsduetoastaticspinconfigurationoftheAFMenvi- moving along the domain wallare effectively modeled as ronment,butnodynamicscatteringinteractions(leading a one-dimensional electron gas (1DEG) in a static ex- to finite resistivity) [19]. ternal magnetic field. At this mean-field level, electrons The anti-phase domain wall spectrum (B =0) is moving in an anti-phase domain wall experience a net zero external field on each site, while electrons moving E(anti) = 2t cos(k a) µ(anti) (17) on an in-phase domain wall experience a staggered ex- n − k n − F 2π N N ternal magnetic field of magnitude proportional to the k a= n ( n ) (18) n spin interaction strength J. Hence, the effective domain N − 2 ≤ ≤ 2 wall 1DEG Hamiltonian has the form µ(anti) = 2t cos 2πn (19) F − k N F H =H (B)+U n n (14) (cid:18) (cid:19) 0 j↑ j↓ j where N is the number of sites. X For an in-phase domain wall (B = 0), the staggered H0(B) =tk c†jσcj+1,σ +h.c. −µF c†jσcjσ (15) field doubles the unit cell. The non-i6nteracting Hamilto- Xjσ (cid:16) (cid:17) Xjσ nian (15) H0(B =0) is diagonalized by the appropriate +2B σ( 1)jc† c Bloch states; 6 − jσ jσ jσ X 5 N/2 B 1+sin(πδ) ψk†,σ =rN2 Xj=1e+ik(2ja)Wj,σ(k) (20) =2JB(cid:20)J2(cid:18)tk(cid:19)1l+n(cid:18)sin1(−πδs)in(πδ)(cid:19)(cid:21) (27) 1 ln (28) W (k) = c† +f e−ikac† ≈ 4 t 1 sin(πδ) j,σ 1+ f 2 2j,σ k,σ 2j−1,σ (cid:20) k (cid:18) − (cid:19)(cid:21) k,σ h i | | (Notethatthelimitt 0oftheprevioussectioncannot qσ B B 2+cos2(k) berecoveredsinceweku→sedapproximationsbasedontk fk,σ = − (cid:0)t(cid:1)∓qco(cid:0)s(tk(cid:1)) BW≈eJa/r4e).now in a positionto give a firstanalyticales≫ti- mate of the critical transitionpoint filling δ determined The resulting energy spectrum for the in-phase domain c given by (using (10) and (28)) wall is En(in) =±2tks(cid:18)Btk(cid:19)2+cos2(cid:18)2Nπn(cid:19)−µ(Fin) (21) 0=∆EJ⊥(1(δc)2≈δc)E+⊥kinJ+J∆Elnma1g +sin(πδc) ((3209)) ≈−2 − 4 t 1 sin(πδ ) N N (cid:20) (cid:18) − c (cid:19)(cid:21) ( n ) (22) − 4 ≤ ≤ 4 The geometrical solution for J = 1,1,1,1, 1 is plotted t 3 4 6 8 10 in figure-2 2n F δ = (23) N (cid:13) InthiscontextwecommentthatforaHubbardmodelin an external staggered magnetic field, unlike the case of staggeredchargepotential/interaction,thereisnocharge 2 gap opening at 1/4 filling [14]. For simplicity, the rest of the formulas are written for the case where the odd- 2(1-d c ) sitesarewiththe magneticfieldanti-paralleltothe spin. J 1 = Moreover, we remind the reader that we assume B t 3 ∼ J t . Well below half filling, (i.e., t cos(k)>0), k k ≪ 1 B+ B2+t 2cos2(k) k f = − k,↑ t cos(k) pk 2 B B 1 + (24) ≈ − t cos(k) O t cos(k) k (cid:18) k (cid:19) d 0 c The gain in magnetic energy per k-state is due to the differencebetweentheoccupationprobabilityofoddand even sites for a given electron spin, (this is the main FIG.2. geometrical solution for Jt = 31,14,61,18,110 difference between 1DEG in an in-phase vs. anti-phase domain wall), Themainqualitativeresultisthatelectronhopingfluc- tuations along a partially filled domain wall inhibit the c† c 2 c† c 2 (25) effective magnetic interaction energy across the domain 2j,σ 2j,σ k − 2j−1,σ 2j−1,σ k wall. As a result, an anti-phase domain wall ground- (cid:12)(cid:12)D E (cid:12)(cid:12) (cid:12)(cid:12)ND/2 2 1 E (cid:12)(cid:12) B state configuration can be obtained beyond δ = 16, de- (cid:12) (cid:12) (cid:12) (cid:12)2 pending on the value of J/t. For 1 J 1 we find ≈Xj=1 N 1+|fk,σ|2 (cid:18) tkcos(k)(cid:19) 0.3<δc <0.4. Note that the critica3lfi≤llintg≤fra1c0tionδc is 2 not very sensitive to the value of J/t. Mostimportantly, B B + it is above the value δ 0.25 of anti-phase stripes in ≈ t cos(k) O t cos(k) ≈ (cid:18) k (cid:19) (cid:18) k (cid:19) (LaNd)SrCuO systems [1]. Therefore, for filling up to k , the magnetic interaction F energy gain for an in-phase domain wall in comparison B. Large U ≫t model, and a second estimate of the with an anti-phase domain wall is given approximately critical domain wall filling δc by, πδ B Inthissubsectionweexaminethecompetitionbetween ∆Emag(δ) 2B 2 dk (26) anti-phase and in-phase domain wall configurations for ≈ " Z−πδ tkcos(k) # 6 strongly interacting electrons, U t, along the domain As shown in figure-3, in the U = limit, there are ≫ ∞ wall at arbitrary filling fraction. onlythreepossibleelectronoccupationnumbersofaunit The exact ground-state energy of the Hubbard model cell: one, two or zero electrons on the domain wall sites. (14) is well known for the case B =0 from Bethe ansatz Inourmodelapproximation,weignorethehoppingfrom solution. Unfortunately, we do not know of an estab- one unit cell to another. As we shall later explain, for lished solution for the groundstate energy of a one di- filling fractions δ 1 it will suffice to consider only unit ≥ 4 mensional Hubbard model in a staggered magnetic field cellswith oneandtwoelectronsonthe domainwallsites B = 0. In particular, when B = 0 the Hubbard inter- (i.e., as in figure-3(a1,a2and b1,b2)). 6 actionin momentum space takes the form Un n , and k↑ k↓ the U = limit is effectively captured by imposing a Anti-phase stripe In-phase stripe ∞ restriction of one electron per k state. But such a sim- − a1 a1’ b1 b1’ ple representationdoesnotexistintermsofthe W (k) t j,σ basis (20) in a staggeredexternal magnetic field. There- fore, we here develope a way to approximate the two competing wall configuration energies at the same level a2 b2 of approximation. Wewishtotressthatthequalitativeeffectoflongitudi- nalhopingfluctuations-toinhibittheeffectivemagnetic interaction energy across the domain wall - was already a3 b3 established in the preceding subsection. The purpose of the current subsection is to investigate the quantitative effect of including large Hubbard interactions between FIG. 3. Characteristic unit cells of a stripe domain wall. the electrons. Though we indeed resort to approxima- Any domain wall groundstate can be constructed by prop- tions at severalstages of our model and calculations, we erlyassemblingtheseunitcells. Fordomainwallgroundstate do capture the substantial effect. Once the core physics with filling fraction δe >1/4, unit cells (a3) and (b3) can be established, the degree to which our final numerical val- neglected. uesareexactcanbechecked(andimproved)innumerical simulations on finite systems. We are interested only in the effect of hopping dynam- Our main result is that the large Hubbard interaction, ics t of electron along the domain wall. Therefore we k U, delimitate the critical filling fraction to be in the nar- examinefirsttheunitcellswithoneholeandoneelectron row range on the domain wall sites. There’s spin exchange interactions of strength J between nearest-neighbor electrons. 0.28<δ <0.30 (31) c Yet, to zeroth level of approximation, the spin configuration is assumed to be static (apart from t electron for any value of J < 1. k t 3 hoping). Thus, we consider only the electron configu- Below,weexplainourmodeling approachandcalcula- rations of an anti-phase (a1,a1’) and in-phase (b1,b1’) tion. We first apply the model to examine the simplest domain wall unit cells. case of 1/4 filled domain wall, and show that always an Consider the case of unit cells with a single hole and anti-phase is obtained. Then, we apply the model to re- spin- electron hopping between the domain wall sites. estimate the critical filling fractionfor a transition to an ↓ We observe that while the anti-phase domainwall states in-phasedomainwall,andthusestablishtheresultnoted (a1,a1’) are degenerate, the in-phase domain wall states above in equation (31). (b1 and b1’) have an energy difference of magnitude 2ε (where ε J is the energy difference between paral- ∼ lel and anti-parallel neighboring spin states). Thus, the 1. Model of characteristic unit cells Hamiltonianofasingle electroninananti-phasedomain wall unit cell is The unit cell of a stripe domain wall comprise of two 0 t lines as depicted in figure-3. Our modeling idea in this Hanti = − k (32) hole-cell t 0 subsection is to solve exactly the Hamiltonian of iso- (cid:18)− k (cid:19) lated prototypical unit cells, and then approximate the withgroundstateenergy t,whileforanin-phasedomain full stripe as assembled of a collection of such unit cells. wall − Such anapproximationis building onour findings in the ε t previous subsection, where we argued that it is not the Hin = − − k (33) hole-cell t +ε global longitudinal mobility, but only the local kinetic (cid:18)− k (cid:19) fluctuations (in the occupation of odd/even numbered with groundstate energy sitesalongthewall)whichdistinguishanti-phaseandin- ε2+t2 t + 1 ε ε . (Note that the mag- phase domain walls. − k ≈ − k 2 tk r netic(cid:16)interac(cid:17)tion enhergy is(cid:16)alre(cid:17)adiy fully accounted for in 7 the eigenvalues, so it shouldn’t be counted again). Thus anti-phaseandin-phase wall,andsince the net magnetic we find that, for a stripe unit cell with one hole and one fieldaveragesto zeroalongbothanti-phaseandin-phase electronon the domain wallsites, the groundstate mag- wall, it is intuitively expected that the significant dif- neticinteractionenergydifferencebetweenananti-phase ference is not in the net electron mobility but mostly in and in-phase domain wall is the local kinetic fluctuations (Though magnetic scattering from AFM environment, which we neglect, may also ∆Emag =Eanti Ein (34) differ). hole-cell hole-cell− hole-cell Let us recall first the result of section-III. In the ab- =−tk+ ε2+t2k senceofhoppingalongthewall,theaveragetransverseki- r(cid:16) (cid:17) neticenergypersiteofananti-phasedomainwallincom- +1 ε ε for t J parisonwithanin-phasedomainwallwas(usingequation 2 tk k ≫ ≈( +ε (cid:16) (cid:17) for tk 0 (10)) → 1 Ekin ε ε The above needs to be divided by 2 in order to get the ∆Ekin δ = = ⊥ = (1 2δ)= ⊥ 4 N −2 − −4 energydifferencepersite,sincetheunitcellhastwosites (cid:18) (cid:19) along the domain wall. In the static limit tk 0, we re- and the competing transverse magnetic energy due to → cover equation (11) of section-III, ∆Emag(δ=1/4) +ε. Heisenberg interaction with the AFM environment was N ≈ 2 Butfortherestofthepaper,wefocusontheexperimen- (using equation (11)) talWlyemnoorweprreolecveaendttloimdeitmtokn≫strJat.efirsttheessenceofour ∆Emag δ = 1 = E⊥mag =+ε(2δ)=+ε. approach for the simplest case of δ = 1 filling. ⊥ (cid:18) 4(cid:19) N 2 4 Hence, it was the in-phase domain wall configuration which had the lowest energy ground state. Now, with 2. The special case of 1/4 filling theinclusionoftheeffectofthekineticfluctuationsalong thedomainwall,theaveragemagneticinteractionenergy In section-III, we have shown that dynamics of only is modified and (using (34) instead of (11)) the average transverse interactions (hole hopping and Heisenberg in- energy difference per site is teraction) will not suffice to stabilize an anti-phase do- ∆E 1 main wall for δ > 1/6. Therefore, by investigating the =∆Ekin+ ∆Emag (35) N ⊥ 2 hole-cell case of δ = 1/4 we can already answer the question of ∆E(δ =1/4) ε ε ε principle about the effect of electron dynamics along a + <0 (36) stripe domain wall. Moreover, it is a case of particular N ≈−4 (cid:18)tk(cid:19)4 interestforstripes inHTc,whicharefoundwith afilling where N is the number of sites along the domain wall, fraction very near δ =1/4. (The factor1/2in(35)isdue to havingtwodomainwall In the limit of large on-site interaction U t, at ≫ sites per unit cell). Therefore, we find that due to the δ =1/4thereiseffectivelyoneelectronforeverytwosites kinetic fluctuations along the domain wall, the magnetic along the wall. Moreover, to mimic the expected effect interaction energy gain of an in-phase domain wall con- of coulomb interactions, it makes sense to supplement figuration is reduced enough to make the anti-phase do- themodelwitharepulsiveinteractionalsobetweennear- main wall more favorable at 1/4 filling for any value of neighborholes (i.e., a t U V modelalongthe domain − − J ε<tk. wall)[20]. Thus,intheabsenceofkinetichopingdynam- ∼ ics along the domain wall,the groundstateofthe one dimensional electron gas is a periodic charge density wave 3. Transition at domain wall filling δc (CDW) with one electron per every second site. Even withtheinclusionofkineticfluctuations,thesameCDW We now attempt to determine the contribution of the still dominate the electron correlations at low tempera- effectdescribedaboveatarbitraryelectronfilling2δ > 1, tures. As a result, we argue that the occurrence of unit 2 and thus estimate the transition point. cells with an occupation of zero or two electrons on the As argued before, the 1DEG correlations on the do- domain wall sites (as in figure-3(a2,a3)) are rare events main wall are dominated by a CDW correlations where, and thus their contribution may be safely neglected. at electron filling 2δ > 1/2, the occurrence of two near- Therefore,atwo-linestripessegmentwithoneholeand neighbor holes is statistically negligible compared with one electrononthe domainwallsites can be regardedas other configurations in the ground state. Therefore, the the characteristicunit cellalong a1/4filled domainwall two-sites unit cells along the domain wall are basically (as in figure-3(a1,b1)). By solving such unit cell model only of two kinds: (a) There are (1 2δ) unit cells con- exactly, we are able to account for the effect of local − taining one hole,which arethe type analyzedinthe pre- kinetic fluctuations along the domain wall. Moreover, vious subsection. (b) There are 1 (1 2δ)= 2δ 1 since we are interested only in the difference between 2 − − − 2 unit cells containing no holes. (cid:0) (cid:1) 8 To estimate the average energy contribution per site, =r S 2+r S 2+r ρ 2 (44) we can use the result of the previous subsection for the Fq,k 1s | q~| 2s q~+~k c ~k hole cells, only with the added factor (1−2δ); +γ Sq∗~Sq~+~kρ~∗k(cid:12)(cid:12)(cid:12)+c.c(cid:12)(cid:12)(cid:12). +...(cid:12)(cid:12), (cid:12)(cid:12) 1 h i ∆Ea(δ) =∆E⊥kin+ 2∆Ehmoaleg-cell (37) where ρ~k is the complex-valued CDW amplitude with wave vector ~k, ρ∗ ρ , and similarly S is a complex ε ε ~k ≡ −~k q~ = 1 (1 2δ) (38) vector amplitude of the SDW. The quartic (and higher −4 − t − (cid:20) (cid:18) k(cid:19)(cid:21) order) terms required for stability are omitted. Zachar et al. [5] have considered the phase transition and add the contribution of magnetic energy difference between a stripe phase and a high-temperature disor- per site due to unit cells with no holes, dered state, as involving only a single SDW wave vector ∆Eb(δ)= 1Emag =+1 2δ 1 2ε. (39) (i.e., Sq∗~ = Sq~+~k). The existence of the cubic γ term, 2 2e-cell 2 − 2 coupling these two order parameters in the Landau free (cid:18) (cid:19) energy,drives the period of the SDW to be twice that of Thepreferredgroundstateconfigurationisdeterminedby the CDW, and the absence of any net AFM ordering is equivalent to the statement that the stripes are topolog- ∆E(δ) =∆Ea(δ)+∆Eb(δ) (40) ical. N In contrast, it was shown by Pryadko et al. [6], that ε ε 1 the same sort of cubic term in a Landau theory of the = 1 (1 2δ)+ 2δ ε. (41) −4 − t − − 2 transition from a homogeneous ordered antiferromag- (cid:20) (cid:18) k(cid:19)(cid:21) (cid:18) (cid:19) netic phase to a stripe ordered phase produces a state Atransitionfromanti-phasetoin-phasedomainwalloc- in which the antiferromagnetic magnetization does not curs when change its sign between the domains, i.e. the stripes are non-topological. ∆E(δ) > 0 When investigating the transition from a well- developedantiferromagnetwithamodulationvector~π = Therefore, the critical filling fraction δc is given by (π,π) to an incommensurate modulated phase, we must account for both the original AFM order parameter S ~π ε ε 1 0= 1 (1 2δ )+2 2δ ε (42) (which cannot be assumed small), and the spin density c c −2(cid:20) −(cid:18)tk(cid:19)(cid:21) − (cid:18) − 2(cid:19) wave S~π+~k. The most relevant terms in the Landau expansion of the effective free energy are then 3 1 ε δc = 25−−2(cid:16)tεktk(cid:17) (43) F =r1s |S~π|2+r2s S~π+~k 2+rc ρ~k 2 (45) The predicte(cid:16)d δc(cid:17)for several values of εt is given below, +γ S~π∗ S~π+~kρ~∗k(cid:12)(cid:12)(cid:12)+c.c(cid:12)(cid:12)(cid:12). +...(cid:12)(cid:12), (cid:12)(cid:12) h i where a finite r < 0 is assumed as given. This expres- ε 1 1s δc = =0.28571 sion implies that an instability in either spin or charge t 3 (cid:18) (cid:19) sector generates both spin- and charge-density waves at δ ε = 1 =0.2931 the wavevectors ~q =~π+~k and~k, respectively. Near the c t 6 transition the magnitude of the incommensurate peak is (cid:18) (cid:19) ε necessarily much smaller then the commensurate AFM δ 0 /0.3 c t → modulation S S . It is easy to see that this (cid:16) (cid:17) | ~π+~k| ≪ | ~π| corresponds to in-phase domain walls; The derived rela- We find that 0.28 < δc < 0.3, (for Jt < 13), and is only tionshipbetween~q and~k impliesthattheperiodsofspin weakly sensitive to the value of J. t and charge modulation are equal. Our analysis in this paper supply microscopic insight to the Landau theory results. First, we find that the V. CONNECTION WITH A LANDAU THEORY possibilityofbothtopologicalandnon-topologicalstripe OF STRIPES ORDER phasescanindeedberealizedwithinasinglemicroscopic model (as anti-phase and in-phase domain wall ground- The properties of a general Ginzburg-Landau free en- states respectively). ergy (44) of stripes were previously investigated in [5,6]. Second, we find that non-topological stripes (corre- Specifically, an ordered stripe phase is a unidirectional sponding to in-phase domain wall state) are indeed es- densitywavewhichconsistsofcoupledspin-densitywave tablishedduetoenhancedantiferromagneticinteractions (SDW) and charge-density wave (CDW) order parame- (asfirstspeculatedbyCastro-Neto[8]). Yet,thusfarthe ters. connection is more heuristic than rigorous. 9 Furthermore, after analyzing the effect higher order Neto&Hone [7] proposed an effective anisotropic derivativetermsinagradientexpansionoftheGinzburg- Heisenberg model for describing the low temperature Landaufree energy,Pryadkoetal. arguethat[6]: When spin dynamics in a striped Cu-O plane. Within the there is no frustration, topological stripes are not estab- anisotropic Heisenberg model [7], they derived the spin- lished in the ground state. However they speculate that wave velocity anisotropy frustration, such as competing first and second neighbor interactions, or opposite sign terms in the gradient αJ v v (48) ⊥ 0. expansion of the Ginzburg-Landau model (i.e. below a ≈ 2 r Lifshitzpoint),canstabilizetopologicalcollineardomain i.e., α = αJ. By fitting other consequences of their walls. In other words, topological stripes are a conse- v 2 modeltoexperimentalresults,Neto&Honeextractedthe quence of physics on short length scale, and they do not p value appear in a theory that considers only long-distance or low-energy physics. α 0.01. Pryadko et al. speculate that the missing frustrating J ≈ ingredient may be due to inter-stripe interactions such The fitted value of α , together with (48) ~v J ⊥ as long range coulomb interactions which are expected 50meV ˚A. As discussed by Tranquada et al. [21], th≈is to exist betweenthe chargeddomain walls. Our analysis value is−much too small to be compatible with inelastic in this paper demonstrate that such interactions are not neutron scattering experiments. It is further incompati- required. Wehaveshownthatlocalsinglestripedynamic ble with the range of energies over which stripes incom- interactions, (albeit involving finite strength competing mensurability is observed. interactions,i.e.,expectantlybeyondthereachofpertur- Aiming specifically to model the interaction between bative gradient expansion) are sufficient to realize both narrow stripe AFM regions, Tworzydlo et al. [22] made alternativepossibletopologicalmagneticorderacrossthe predictions based on models of coupled narrow spin lad- charge line. ders (e.g., 2-leg or 3-leg ladders). The resulting expres- sions for the spin wave velocity anisotropy were VI. ESTIMATE OF SPIN-WAVE VELOCITY 2αJ for 2-leg ladder α = 1+αJ (49) v  q 3αJ for 3-leg ladder In this section, we use our microscopic domain wall  1+2αJ model to estimate spin-wave velocity reduction q In terms of2-leg ladder model, Tranquada et al. pro- v⊥ claimed to fit the experiments well with a value of α = , (46) v v 0 α 0.35. J ≈ where v is the spin-wave velocity perpendicular to the ⊥ giving stripes (i.e., along the modulation direction), and v is 0 the spin-wave velocity in the undoped parent antiferro- α 0.72 magneticmaterial. Equivalently,itisthe predictedspin- v ≈ wavevelocityanisotropyin stripes (wherev0 is the same In principle, the ladder model results are quite sensitive asthevelocityalongthestripes). Castro-Neto&Hone[7] to the width of the stripes (which affects the ladder spin were the firstto point outthat within the stripes model, gap magnitude). Yet accidentally, the above value of one of the main effects of the hole rich lines would be to α 0.35 would give α 0.78 for 3-leg ladder, i.e., J v weaken the effective exchange interaction between spins prac≈tically the same as for≈2-leg. on either side across the domain wall. Thus, a cardi- Our microscopic model is based solely on the local nal parameter in the model is the effective weakening physics in the neighborhood of the hole line. As such, factor of the magnetic interaction JW across the stripes it is insensitive to details of the width of the AFM spin domainwallascomparedwiththeHeisenberginteraction domains. We arguethat, bydefinition, the energydiffer- J withintheholefreeAFMregions(andhencealongthe ence between alternative spin configurations across the stripes). domainwallis ameasureofthe effectivespininteraction acrossthedomainwallinanyappropriatelowenergythe- J α = W (47) ory. Therefore, using our microscopic model calculation, J J we suggest that The key new ingredient is our analytical estimate of αJ. J = ∆E(δ)=Ein Eanti (50) Consequently,while previous model estimates alwaysin- W − − volvedfitting of free parameters,we are able to estimate were ∆E(δ) is given in equation (40) analytically the spin-wave velocity anisotropy (without any free parameters). ε ε 1 ∆E(δ) 1 (1 2δ)+2 2δ ε ≈−2 − t − − 2 h (cid:16) (cid:17)i (cid:18) (cid:19) 10