Optical conductivity in the t-J-Holstein Model
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Optical conductivity in the t - J -Holstein Model L. Vidmar, J. Bonˇca,
2, 1 and S. Maekawa
3, 4 J. Stefan Institute, 1000 Ljubljana, Slovenia Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan (Dated: October 28, 2018)Using recently developed numerical method we compute charge stiffness and optical conductivity of the t - J model coupled to optical phonons. Coherent hole motion is most strongly influenced by the electron-phononcoupling within the physically relevant regime of the exchange interaction. We find unusual non-monotonousdependence of the charge stiffness as a function of the exchange coupling near the crossover to the strongelectron-phonon coupling regime. Optical conductivity in this regime shows a two-peak structure. The low-frequency peak represents local magnetic excitation, attached to the hole, while the higher-frequency peakcorresponds to the mid infrared band that originates from coupling to spin-wave excitations, broadened andrenormalized by phonon excitations. We observe no separate peak at or slightly above the phonon frequency.This finding suggests that the two peak structure seen in recent optical measurements is due to magnetic excita-tions coupled to lattice degrees of freedom via doped charge carriers. PACS numbers: 71.10.Pm,71.27.+a,78.67.-n,71.38.-k
I. INTRODUCTION
Despite many years of intensive research of transport prop-erties of a hole doped in an antiferromagnetic background theproper description of this system remains a challenging the-oretical problem. The transport of a doped hole leaves inits wake locally distorted, slowly relaxing spin background,leading to the formation of a dressed quasiparticle with anenhanced effective mass and renormalized charge stiffness -a measure of a coherent, free particle like transport. Additionof lattice degrees of freedom to this already elaborate problemreflects the current scientific interest in the field of correlatedelectron systems.Long after the pioneering work , the enhanced interest incorrelated models, coupled to lattice degrees of freedom is pri-marily fuelled by experimental evidence given in part by an-gular resolved photoemission data demonstrating that strongelectron-phonon (EP) interaction plays an important role inlow-energy physics of high- T c materials . Moreover, re-cent estimates of transport based on the pure the t − J model yield substantially smaller resistivity in comparison to exper-iments in the low-doping regime of La − x Sr x CuO , that canbe explained as a lack of additional lattice degrees of freedom.Recent numerical methods investigating optical conductiv-ity (OC) in correlated electron systems and systems, whereelectrons are coupled to bosonic degrees of freedom, havebeen focused on the generalized t − J model , Holstein andgeneralized electron-boson model , while investigations ofthe t − J -Holstein model have been until recently limitedto small clusters with 10 sites . Diagrammatic QuantumMonte Carlo (DMC) method has been applied to resolve OCof the Fr¨ohlich polaron and recently also the t − J -Holsteinmodel . In the later work authors report on a two-peak struc-ture in the optical response where the low- ω peak is due topolaronic effects while the peak at higher ω is due to magneticexcitations, renormalized by lattice degrees of freedom. DMCmethod does not reproduce the well known string states seen in the spectral function neither the signature of local mag-netic excitations in optical conductivity since authors use theself consistent Born approximation (SCBA) without magnon-magnon vertex corrections for treatig spin degrees of freedom.Their result seems to contradict calculations based on the dy-namical mean-field theory (DMFT) where a two peak struc-ture is seen in OC only at small J/t . . . In this workauthors also show that the low- ω peak is of the magnetic ori-gin while the higher- ω peak represents the broad polaronicband. II. MODEL AND NUMERICAL METHOD
The main goal of this work is to investigate in depth opticalproperties of the t - J -Holstein model for the case of a singlehole in the antiferromagnetic background. We first define the t - J -Holstein model on a square lattice H = − t X h i,j i ,s ˜ c † i,s ˜ c j,s + J X h i,j i S i S j , + g X i (1 − n i )( a + i + a i ) + ω X i a + i a i , (1)where ˜ c i,s = c i,s (1 − n i, − s ) is a fermion operator, projectedonto a space of no double occupancy, t represents nearest-neighbor overlap integral, the sum h i, j i runs over pairs ofnearest neighbors, a i are phonon annihilation operators and n i = P s n i,s . The third term represents EP coupling g = √ λω t , where λ is the dimensionless EP coupling constant,and the last term represents the energy of Einstein phonons ω .We use recently developed method based on the ex-act diagonalization within the limited functional space(EDLFS). Since details of the method have been pub-lished elsewhere, we now briefly discuss only the mainsteps of the method. We first construct the limited func-tional space by starting from a N´eel state with one hole witha given momentum k and zero phonon degrees of freedom | φ (0 , k i = c k | Neel; 0 i , and applying the generator of states {| φ ( N h ,M ) k l i} = ( H kin + H Mg ) N h | φ (0 , k i , where H kin and H g represent the first and the third term respectively of Eq. 1.This procedure generates exponentially growing basis spaceof states, consisting of different shapes of strings in the vicin-ity of the hole with maximum lengths given by N h as well asphonon quanta that are as well located in the vicinity of thehole, at a maximal distance N h . Parameter M provides gen-eration of additional phonon quanta leading to a maximumnumber N max ph = M N h . Full Hamiltonian given by Eq. 1 isdiagonalized within this limited functional space taking intoaccount the translational symmetry while the continued frac-tion expansion is used to obtain dynamical properties of themodel. The method treats spin, charge as well as lattice de-grees of freedom on equal footing.We define OC per doped hole σ ( ω ) = iω + ( h τ i − χ ( ω )) (2) χ ( ω ) = i Z ∞ e iω + t h [ j ( t ) , j (0)] i d t (3)where τ = P h i,j i ,s t ij ( R ij ⊗ R ij )˜ c † j,s ˜ c i,s represents the stresstensor, j = i P h i,j i ,s t ij R ij ˜ c † j,s ˜ c i,s is the current operator, t ij = − t for next nearest neighbors only and zero otherwise,and R ij = R j − R i . We also note that in the case of next-neighbor tight binding models, h τ i is related to the kineticenergy, h τ µ,µ i = −h H kin i / . III. CHARGE STIFFNESS AND SUM-RULES
Charge stiffness per doped hole can be on a square lat-tice for the t - J -Holstein model computed via its spectralrepresentation D µ,µ = − h | H k in | i + X n h | j µ | n ih n | j µ | i ( E − E n ) , (4) D µ,µ = S tot − S regµ,µ , (5)where S tot represents normalized optical sum-rule R ∞−∞ σ ′ µ,µ ( ω )d ω = 2 πS tot , while S r eg is defined by R ∞ + σ ′ µ,µ ( ω )d ω = πS reg µ,µ and σ ′ ( ω ) represents the realpart of the optical conductivity tensor in Eq.2. We havecomputed D µ,µ in the single-hole ground-state, i.e. at k = ( ± π/ , ± π/ . It is well know that the dispersion E ( k ) is highly anisotropic around its single-hole minimum, whichis in turn reflected in the anisotropy of the effective masstensor . It is thus instructive to compute tensors repre-senting the charge stiffness as well as the OC in the directionof their eigen-axis, i.e. along the nodal ( ( π/ , π/ → (0 , )direction that gives D k , σ k ( ω ) , and along the anti-nodal( ( π/ , π/ → ( π, ) direction that leads to D ⊥ , and σ ⊥ ( ω ) . In Fig. 1(a) we present the charge stiffness vs. J/t for vari-ous values of EP coupling strength. To obtain accurate resultsin the strong EP coupling (SC) limit, we had to rely on only N st = 9786 different combinations of spin-flip states, whilethe total number of states, including phonon degrees of free-dom, was N st = 9 × . To test the quality of λ = 0 results,we show with the dashed line D k computed with zero phonondegrees of freedom using N st = 5 × . Agreement withthe λ = 0 case, obtained with N st = 9786 is rather surpris-ing, given the fact that results were computed using Hilbertspaces that differ nearly three orders of magnitude. This fastconvergence is in contrast to calculations on finite-size clus-ters where due to the existence of persistent currents D variesrather uncontrollably between different system sizes . S t o t D || D ⊥ < N ph > J/t a) b) ω /t= ω /t= J/t λ =0,0.1,0.2,0.25,0.3 λ =0,0.1,0.2,0.25,0.3 J/t ω /t= ω /t= λ=0.3λ=0.25λ=0.2 Figure 1: (Color online) a) Charge stiffness D k ( D ⊥ in the inset),b) optical sum-rule S tot ( h N ph i = P i a † i a i in the insert) vs. J/t at ω /t = 0 . (full lines) and ω /t = 0 . (dashed lines). In thisand all subsequent figures (except for the dotted line in a) or elseotherwise indicated) we used: N h = 8 , M = 7 , and N st ∼ × .Dotted line in a) was for λ = 0 obtained with states with zero phonondegrees of freedom and the following set of parameters: N h = 14 , M = 0 , and N st ∼ × . Exploring further λ = 0 results we observe D k ∼ at J/t ∼ indicating strong scattering on spin degrees of free-dom. With increasing J/t D k steeply increases and around J/t ∼ reaches at D k ∼ . a broad maximum that as wellcoincides with the maximum of the bandwidth W . In con-trast, the optical sum rule S tot = −h H kin i / monotonicallydecreases in the range . J/t . . with increasing J/t ,Fig. 1(b). Due to strong anisotropy in E ( k ) , D ⊥ remainsnearly an order of magnitude smaller than D k for J/t & . (see the inset of Fig. 1(a)).Turning to finite λ , D k expectedly decreases, due to addi-tional scattering on lattice degrees of freedom. The effect of λ on the value of D k however varies with J/t . This is bestseen in the case of λ = 0 . and ω /t = 0 . where D k isapproximately equal to its λ = 0 value for J/t . . , it thendecreases with increasing J/t , reaching its minimum valuearound
J/t ∼ . and finally, for larger values of J/t & . ,steeply increases. This non-monotonous begavior is as wellreflected in the bell shaped average phonon number h N ph i vs. J/t , presented in the insert of Fig. 1(b). This behavior is alsoconsistent with the non-monotonous functional dependence of λ c ( J/t ) , representing the crossover EP coupling strength tothe SC regime, Refs. .We now make some general comments about the effect ofthe EP interaction on the correlated system at the onset of theSC regime. At small values of J/t . . EP coupling is lesseffective, which seems to be in contrast to naive expectations.We attribute this disentanglement from lattice degrees of free-dom to the increase of the kinetic energy and the vicinity ofthe Nagaoka regime. This effect particularly evident fromthe
J/t -dependence of the average phonon number h N ph i at λ = 0 . (see the insert of Fig. 1(b)) where an increase fol-lowed by a sharp drop of h N ph i is seen with lowering of J/t .At the onset of the SC regime, i.e. at λ ∼ . , EP couplingis most effective in the physically relevant J/t ∼ . − . regime, where there is a strong competition between kineticenergy and magnetic excitations. The critical λ c as wellreaches its minimum around J/t ∼ . as shown in Ref. .At larger J/t ∼ EP coupling becomes again less effectivedue to more coherent quasiparticle motion as reflected in theenhanced charge stiffess, quasiparticle weight, as well as thebandwidth .The optical sum-rule S tot , presented in Fig. 1(b), as welldecreases with increasing λ . It however remains finite evendeep in the SC regime where D k ∼ since S tot includesboth coherent as well as incoherent transport. The latter re-mains finite due to processes, where the hole hops back andforth between neighboring sites while leaving lattice deforma-tion unchanged. Despite charge localization we thus expectnonzero optical response σ ( ω ) even deep in the SC regime,with its spectral weight shifted towards larger ω and zero con-tribution at ω = 0 . Due to localization we also expect OC tobe isotropic in the SC regime, i.e. σ k ( ω ) ∼ σ ⊥ ( ω ) . < j ⊥ > < j || > S || r e g S ⊥ r e g J/t a) b) J/t λ =0.3,0.25,0.2,0.1,0 ω /t= J/t ω /t= Figure 2: (Color online) a) Expectation value of the square of theelectrical current along the nodal direction h j k i ( S reg k in the inset),b) expectation value of the square of the electrical current along theanti-nodal direction h j ⊥ i ( S reg ⊥ in the inset) vs. J/t at ω /t = 0 . (full lines) and ω /t = 0 . (dashed lines). In the insets of Fig. 2 we show S reg k and S reg ⊥ representingthe integrated regular part of the OC. S reg k steeply decreaseswith increasing J/t due to the simultaneous increase of co-herent transport, captured by D k , as well as due to decreaseof S tot , see also Eq. 5. We observe, that EP coupling havelittle effect on S reg k for J/t . . , since its value is rather in-dependent on λ except deep in the SC regime, i.e. at λ = 0 . in this particular case. Due to small values of D ⊥ we find S reg ⊥ ∼ S tot , see Figs. 1 and 2 .In Fig. 2 we present the average of the square ofthe electrical current defining the following sum-rule R ∞ + ωσ ′ µ,µ ( ω )d ω = π h j µ,µ i , that furthermore represents thefluctuation of the current operator. In the ground state there isno persistent currents that usually appear on finite-size clus-ters, since our method is defined on an infinite lattice. Thisenables more reliable calculation of the charge stiffness. At λ = 0 h j k i (in Fig. 2(a)) and h j ⊥ i (in Fig. 2(b)) displayrather distinctive J/t dependence. While h j k i shows weaknon-monotonous dependence on J/t , h j ⊥ i shows a substan-tial increase. With increasing λ current fluctuations as wellincrease in both directions even though the increase is morepronounced in the case of h j k i . In the SC regime we obtain h j k i ∼ h j ⊥ i as a consequence of localization due to latticedegrees of freedom. ω /t ω /t J/t=1.0 J/t=0.4
J/t=0.05
J/t=1.0 J/t=0.4 J/t=0.05 N st =9786 a) b) c) d) e) f) σ || /π σ ⊥ /πσ xx /π × × N st =9786 J/t ω I /t ω I Figure 3: (Color online) σ k in a), b), and c), and σ ⊥ in d), e), and f)for three different values of J/t as indicated in the figures for the t - J model for a single doped hole at λ = 0 and k = ( π/ , π/ . Hilbertspace with no phonon degrees of freedom and N st = 5 × wasused in all cases except in b) and e) where for comparison we inaddition present calculations with N st = 9786 . In a), b), and d)we also show σ xx = ( σ k + σ ⊥ ) / using turquoise (dark grey) fill.Arrows in a) and b) indicate positions of lowest-energy peaks, ω I .Dashed lines in c) and f) are given by σ µ,µ ( ω ) = π/ ( zω ) . Insert in c)represents scaling of ω I vs. J/t . In this and in the subsequent figures,the Drude peak is not shown. Artificial broadening ǫ = 0 . t wasused. Dashed areas in c) and f) delineate small frequency regimes( ω/t . . ) where at J/t = 0 . EDLFS does not lead accurateresults due to the vicinity of the Nagoka regime.
IV. DYNAMIC PROPERTIES
Turning to dynamic properties we first establish numericalefficiency of our method by presenting optical properties ofthe t - J model. In Fig. 3 we display different components ofthe conductivity tensor σ µ,µ ( ω ) in the single-hole minimum k = ( π/ , π/ , computed using EDLFS. At physically rel-evant value J/t = 0 . we reproduce well known features,characteristic of σ xx ( ω ) : a) in the regime . J . ω . t we find peaks forming a rather broad band, appearing withinthe well known mid-infrared (MIR) frequency regime, sepa-rated from the Drude peak (not shown) by a gap of the orderof J and b) there is a broad featureless tail, extending to largefrequencies, ω & t . MIR peaks for J/t & . scale withthe exchange coupling ( J/t ) η , where η ∼ . We stress thatsuch scaling is consistent with local magnetic excitations aswell as spin waves. Obtained scaling is however not consis-tent with the string picture where η = 2 / (see also Ref. ).At J/t = 0 . as well as at J/t = 1 , the lowest peak appearsat
J/t ∼ . . Location of the lowest-frequency peak (indi-cated by arrows in Figs 3(a) and (b)) is surprisingly close tothe location of the peak in OC of the t - J z model in the limit J z /t → ∞ , given by σ ( ω ) ∼ t /J z δ ( ω − J z ) . In thistrivial case the peak appears at the frequency that correspondsto the energy (measured from the N´eel state) of a single spin-flip, attached to the hole, created as the hole hops one latticesite from its origin in the undisturbed N´eel background. It issomewhat surprising that such a naive interpretation seems tosurvive even in the (spin) isotropic t - J model and at rathersmall value of J/t = 0 . . The scaling of the position ofthe low-frequency peak closely follows the following expres-sion ω I = 1 . J/t ) . , indicated by a dashed line, connect-ing the circles shown the insert of Fig. 3(c). Our results ofOC qualitatively agree with those, obtained on small latticesystems .When conductivity tensor σ ( ω ) is computed in its eigen di-rections, distinct (incoherent) finite- ω peaks are obtained inthe case of σ k ( ω ) and σ ⊥ ( ω ) , as best seen at J/t = 1 and
J/t = 0 . . For comparison we present in Figa. 3 (a),(b), and(d) σ xx ( ω ) , that consists of all the peaks characteristic for both σ k ( ω ) as well as σ ⊥ ( ω ) . The reason is, that the ground stateat k = ( π/ , π/ or Σ - point belongs to an irreducible repre-sentation Σ of the small group of k , i.e. C . Current opera-tors j k and j ⊥ , defining σ k ( ω ) and σ ⊥ ( ω ) through Eqs.2, and3 transform as distinct irreducible representations Σ and Σ .Selection rules allow only transitions into states that transformaccording to a direct product of irreducible representations ofthe group C . Since j x does not transform according to irre-ducible representations of C , the above mentioned selectionrules do not apply.In Figs. 3 (b) and (e) we present as well results, computedon a much smaller set of states, i.e. with N st = 9786 . Apartfor a small shift of one of the MIR peaks at larger ω , theagreement with results, obtained with more than three ordersof magnitude larger systems ( N st = 5 × ) underlines theefficiency of our method. Obtaining relevant results for thepure t - J model at moderate number of states is of crucial im-portance for successful implementation of additional lattice ω/ t ω / t= ω / t= a) b) λ=0.00λ=0.10λ=0.20λ=0.25λ=0.30λ=0.35 c) σ xx λ=0.20λ=0.21λ=0.22λ=0.23λ=0.24λ=0.25λ=0.00λ=0.10λ=0.20λ=0.25λ=0.30λ=0.35 J/t
J/t= ω / t= Figure 4: (Color online) σ xx for ω /t = 0 . in a), ω /t = 0 . in b and c) at J/t = 0 . , and at k = ( π/ , π/ . Total numberof functions was N st = 9 × . Up to 56 phonon quanta wasused to obtain accurate results for λ & . . Arrows in b) indicate ω II = 16 λ ˜ t where ˜ t = 0 . t . Units of σ are arbitrary, yet chosenidentical in a) and b); a different scale was used for c), neverthelessidentical among different plots in c). Artificial broadening was set to ǫ/t = 0 . . degrees of freedom. Last, we present in Figs. 3(c) and (f)results at small J/t = 0 . . Dashed lines represent knownanalytical estimate σ ( ω ) = π/ ( zω ) , where z = 4 , Ref. .This result is characteristic for systems with a nearly constantdensity of states and diffusive hole motion where current ma-trix elements |h | j µ | n i| are roughly independent of n . Goodagreement with the analytical results in the small J/t limitis of particular importance since our method is by construc-tion, based on the existence of the long-range N´eel order, tar-geted to be valid predominantly in the regime of intermediateto large values of the exchange constant
J/t . We also notethat in the limit of small-
J/t optical properties for ω/t & J/t become isotropic.We now focus on the influence of increasing EP cou-pling λ on optical properties of the t - J Holstein model inthe adiabatic regime, i.e. for ω /t = 0 . , Fig. 4(a) and ω /t = 0 . , Fig. 4(b) and (c) the latter value being relevantfor cuprates. Increasing EP coupling λ leads to three maineffects: a) the spectra progressively shift towards higher fre-quencies while the total spectral weight decreases (see also theinset of Fig. 3(b)), b) magnetic excitations that form a band inthe MIR regime broaden and diminish with increasing λ ; theyfinally disappear in the SC regime where they are replacedby a broad polaron-like band that clearly originated from therenormalized MIR peaks. The peak of the well formed broadband at ω /t = 0 . in the regime . . λ . . roughlyscales with ω II ∼ λ ˜ t where ˜ t = 0 . t represents renormal-ized hopping due to EP interaction. At ω /t = 0 . a broader,featureless band is formed, and c) a large gap opens in the SCregime.In contrast to numerical results of Ref. , we observe no ω/ t λ =0.3, ω / t= λ =0.3, ω / t= λ =0.0 a) b) J/t=
J/t= c) σ xx Figure 5: (Color online) σ xx at λ = 0 . ; a) ω /t = 0 . , b) ω /t = 0 . , and λ = 0 . in c) at k = ( π/ , π/ . Arrows in b)and c) indicate positions of the lowest-frequency state as it appearsat respective values of J/t at λ = 0 . Artificial broadening was set to ǫ/t = 0 . . separate peak at or slightly above the phonon frequency. Thisis more clearly seen in Fig. 4(c) where σ xx ( ω ) is shownin an expanded frequency range. This result is consistentwith DMFT calculations of Ref. . Nevertheless, we findquantitative agreement at λ ∼ . with measurements on(Eu − x Ca x )Ba Cu O in the low hole-doping regime pub-lished in Ref. . In our calculation λ ∼ . represents themaximum EP coupling constant where the low- ω peak, lo-cated at ω I ∼ . J ∼ meV ( choosing t = 400 meV and J/t = 0 . ), is just barely visible. This peak is, as dis-cussed above, due to the local magnetic excitation and re-mains separated from the continuum forming the rest of theMIR band. Experimental value of the corresponding peak is ω expI = 174 meV . The higher ω − peak at ω II ∼ . t =560 meV (experimental value is ω expII = 590 meV ) corre-sponds to MIR band, slightly broadened and renormalized byphonon excitations. This part of OC is in agreement with cal-culations in Ref. .Our explanation of the experimental results relies on theconjecture that lightly doped (Eu − x Ca x )Ba Cu O com-pound lies in the crossover from from weak to strong couplingelectron-phonon regime where physical properties (quasipar-ticle weight, charge stiffness and dynamic properties) are ex-tremely sensitive to small changes of λ . This is evident fromFig. 4 and from results, published in Ref. . MIR peak in OCis at λ = 0 centered around ω II = 2 J = 240 meV . This valuecorresponds to the peak of the magnon density of states ithowever underestimates the position of the main peak, seen inthe experiment of Ref. . Increasing λ beyond the weak cou-pling regime λ > λ c , the center of MIR peaks starts movingtowards higher frequencies and broadens as it transforms intoa wide polaron band, thus approaching the experimental value.Simultaneously the peak due to the local magnetic excitation at ω I as well broadens and disappears above λ & . .The lack of a peak at ω ∼ ω in OC can be explained insimple terms in the large- J z /t limit of the simplified t - J z -Holstein model. Starting from a hole in the N´eel background,the lowest energy contribution to σ xx ( ω ) comes from the hopof the hole to the neighboring site. This move generates a sin-gle spin-flip with the energy E = 3 J z / above the groundstate. The contribution to OC that would include a singlephonon excitation would thus be located at ω & J z / ω .In order to explore the interplay of magnetic and polaronicdegrees of freedom in the structure of σ xx ( ω ) in more detail,we present in Figs. 5(a) and (b) comparison of optical spectraat fixed λ = 0 . and different values of the exchange inter-action J/t . Decreasing
J/t leads to a shift of the broad pola-ronic peak towards smaller values of ω . At smaller ω /t = 0 . more pronounced structure abruptly appears at low ω/t . . at small J/t = 0 . , Fig. 5(a). At larger value of ω /t = 0 . ,Fig. 5(b), a shoulder starts appearing at J/t = 0 . in the low- ω regime that corresponds to the onset of the respective mag-netic peaks (as indicated by arrows in Fig. 5(c)) of the pure t - J model. Below J/t . . well formed peaks emerge be-ing clearly of the magnetic origin. The disentanglement oflattice degrees of freedom, clearly seen in Fig. 5(b), is consis-tent with DMFT calcutions . V. SUMMARY
In summary, we have explored effects of magnetic as wellas lattice degrees o freedom on optical properties of the t - J -Holstein model. EDLFS captures well optical properties of asingle hole in the t - J -Holstein model in the range of physi-cally relevant parameters of the model since it treats spin andlattice degrees of freedom on equal footing. Competition be-tween kinetic energy and spin degrees of freedom strongly in-fluences the coherent hole motion as measured by charge stiff-ness near the crossover to SC polaron regime. In the adiabaticregime increasing EP coupling leads to the shift of the OCspectra towards higher frequencies and broadening of peaksthat in the pure t - J model originate in magnetic excitations.As an important as well as unusual finding we report a lack ofa peak in the OC spectra at or slightly above the phonon fre-quency that we attribute to the inherently strong correlationsthat are present in the t - J model. This finding suggests thatthe two peak structure seen in recent optical measurements isentirely due to magnetic excitations. Based on our calcula-tions, the two peak structure can be explained with the obser-vation of local magnetic excitations, created by the hole mo-tion at lower frequencies and the contribution of spin waves,coupled via doped hole to lattice degrees of freedom at higherfrequencies. Acknowledgments
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