Quantum lattice dynamical effects on the single-particle excitations in 1D Mott and Peierls

a r X i v :c o n d -m a t /0312426v 2 [c o n d -m a t .s t r -e l ] 31 M a r 2004Quantum lattice dynamical effects on the single-particle excitationsin 1D Mott and Peierls insulatorsH.Fehske,1G.Wellein,2G.Hager,2A.Weiße,3and A.R.Bishop 41Institut f¨u r Physik,Ernst-Moritz-Arndt Universit¨a t Greifswald,D-17487Greifswald,Germany 2Regionales Rechenzentrum Erlangen,Universit¨a t Erlangen,D-91058Erlangen,Germany 3School of Physics,The University of New South Wales,Sydney,NSW 2052,Australia and4Theoretical Division and Center for Nonlinear Studies,Los Alamos National Laboratory,Los Alamos,New Mexico 87545(Dated:February 2,2008)As a generic model describing quasi-one-dimensional Mott and Peierls insulators,we investigate the Holstein-Hubbard model for half-filled bands using numerical bining Lanczos diagonalization with Chebyshev moment expansion we calculate exactly the photoemission and inverse photoemission spectra and use these to establish the phase diagram of the model.While polaronic features emerge only at strong electron-phonon couplings,pronounced phonon signatures,such as multi-quanta band states,can be found in the Mott insulating regime as well.In order to corroborate the Mott to Peierls transition scenario,we determine the spin and charge excitation gaps by a finite-size scaling analysis based on density-matrix renormalization group calculations.PACS numbers:71.27.+a,71.30.+h,71.45.Lr,71.38.+i,63.20.Kr,71.10.FdThe one-dimensional (1D)Holstein-Hubbard model (HHM)has been used extensively to describe for novel low-dimensional materials, e.g.,conjugated polymers,organic charge transfer salts or halogen-bridged tran-sition metal complexes 1,and the associated metal-insulator 2,3and insulator-insulator transitions 4,5.6The HHM accounts for a tight-binding electron band,intra-site Coulomb repulsion between electrons of opposite spin,and a local coupling of the charge carriers to the phonon system:H =−ti,σ(c †iσc i +1σ+H.c.)+U in i ↑n i ↓−√2finite periodic chains,the MI-PI quantum phase transi-tion could be identified by a ground-state level crossingassociated with a change in the parity eigenvalue P.4Note that this scenario differs from the(weakly interact-ing)HHM with frozen phonons10,where there is strongevidence in favor of two quantum critical points,as inthe ionic Hubbard model.11,12The aim of this work is to establish the physical picturedeveloped to date for the interplay of spin,charge andlattice degrees of freedom in the1D Holstein Hubbardmodel.In particular we attempt to verify the proposedphase diagram by examining the single-particle excita-tions.For these purposes we employ Lanczos exact di-agonalization(ED)13,kernel polynomial14,and density-matrix renormalization group(DMRG)15,16methods todetermine the(inverse)photoemission spectra as well asthe spin and charge excitation gaps.These quasi-exactnumerical techniques allow us to obtain reliable resultsfor all interaction strengths with the full quantum dy-namics of phonons taken into account.Exact diagonal-izations are seriously limited in achievable system sizesN,but have the advantage that spectral quantities areeasily plementary,the DMRG algorithmyields specific eigenstates of large systems by implement-ing a renormalization scheme and an optimal truncationof the Hilbert space.Thus it permits for afinite-size anal-ysis of the ground-state energies in different particle andspin sectors,which is required to determine the behav-ior of the various excitation gaps in the thermodynamiclimit.We begin by studying the spectral density of single-particle excitations associated with the injection of aspin-σelectron with wave number K,A+Kσ(ω)(inversephotoemission(IPE)),and the corresponding quantityfor the emission of an electron,A−Kσ(ω)(photoemission(PE)),whereA±Kσ(ω)= m| ψ(N el±1)m|c±Kσ|ψ(N el)0 |2×δ[ω∓(E(N el±1)m−E(N el))](2)with c+Kσ=c†Kσand c−Kσ=c Kσ.|ψ(N el)0 is the groundstate of the system with N el electrons and|ψ(N el±1)m are eigenstates of the(N el±1)-particle system.E(N el)andE(N el±1)m are the corresponding energies.Adding the spectral densities of(photo-)emission and absorption we obtain the spectral function A Kσ(ω)=A+Kσ(ω)+A−Kσ(ω),which obeys various sum rules and allows for aconnection to angle-resolved photoemission spectroscopy (ARPES).The simplest sum rule, ∞−∞A Kσ(ω)dω=1, reflects the normalization of A Kσ(ω)but is not useful for ARPES since it involves both occupied and unoccupied states. σ ∞−∞n F(ω)A Kσ(ω)dω=n(K)(where n F(ω) is the Fermi function)is more important,since it relates the ARPES intensity to the number of electrons in a mo-mentum state K:n(K)= σ c†Kσc Kσ .The ED results presented for A±Kσ(ω)in the following were obtained for an eight-site system with periodic boundary conditions.17 Let usfirst consider the MI regime.Figure2dis-plays the IPE and PE spectra for the HHM at the allowed wave numbers of ourfinite system:K= 0,±π/4,±π/2,±3π/4,andπ.To reliably monitor a possible band splitting induced by the Hubbard and EP couplings at half-filling it is necessary to guarantee that the Fermi momenta K F=±π/2are occupied,which is the case for N=4l(l integer,periodic boundary con-ditions).The most prominent feature we observe in the MI regime is the opening of a gap at K=±π/2,in-dicating massive charge excitations.A comparison with the results obtained for the pure Hubbard model classi-fies this gap as the Mott-Hubbard correlation gap.Its value∆/t≃3.25almost coincides with the optical gap ∆opt we determined by evaluating the regular part of the optical conductivity for the same parameters.The dispersion of the lower(upper)Hubbard band can be de-rived tracing the uppermost(lowest)excitations in each K sector.Due to thefiniteness of our system and the rather moderate value u=1.5,PE(IPE)excitations with K=±3π/4andπ(K=±π/4and0)have stillfi-nite spectral weight.This can be seen from the integrated spectral densities S±Kσ(ω)= ω∓∞dω′A±Kσ(ω′),which,in addition to the sum rule S−Kσ(−∞)+S+Kσ(∞)=1,sat-isfy the relations S±K,σ(±∞)+S±π−K,σ(±∞)=1(K≥0). Since the spectral weight of the PE excitations with K>π/2is expected to vanish as N goes to infinity for u≫1,the lower Hubbard band will be completelyfilled ( |K|≤K F,σ −∞∞dωA−Kσ(ω)≃N el),and consequently the system behaves as an insulator at T=0.20As a result of the coupling to the phonon system the elec-tronic levels in each K sector split,creating phonon side bands.The distinct peaks are separated by multiples of the bare phonon frequency and can be assigned to re-laxation processes of the Q=0phonon modes.21The number of phonons involved is controlled by g2.S±Kσ(ω) shows clearly that the total spectral weight of the result-ing excitation bands equals the weight of the respective electronic excitations in the pure Hubbard model.In-terestingly,mediated by Q=0phonons,there appear “shadows”of the band belonging to a dominant elec-tronic excitation in a certain K sector in other K sectors, giving rise to a weak“breather-like”excitation22,which is almost dispersionsless in the Brillouin zone.If we decrease the Hubbard interaction atfixed EP coupling strength the Mott-Hubbard gap weakens andfi-nally closes at about(u/λ)c≃1,which marks the MI-PI crossover.This is the situation shown in Fig.3.Ap-proaching the critical point from above and below,the ground state and thefirst excited state become degener-ate.These states have different eigenvalues P of the site-inversion operator P c†iσP†=c†N−iσ(i=0,...,N−1) and we have verified that the ground-state site parity is P=+1in the MI and P=−1in the PI.Obviously the critical point is characterized by gapless charge excita-tions at the Fermi momenta but should not be consideredFIG.3:PE(red lines)and IPE(red lines)spectra near the Mott insulator Peierls insulator transition point(u≃λ).as metallic because the Drude weight is ill-defined.11,23 If the Hubbard interaction is further reduced,i.e.the EP coupling overcomes the on-site Coulomb repulsion,a CDW accompanied by a dimerization of the lattice de-velops.As a result the electronic band structure be-comes gapped again(see Fig.4(upper panel)).The form of the spectra,however,is quite different from MI case.While in the MI regime the lowest peak in each K sector is clearly the dominant one,in the BI phase distributed,which is a clear signature of multi-phononabsorption and emission processes that accompany every single-particle excitations in the PI.The lineshape thenreflects the(Poisson-like)distribution of the phonons in the ground state.Again low-intensity“shadow bands”become visible.Remarkably,now the cumulative spec-tral weight |K|≤π/2S−Kσ(−∞)gives nearly the total number N el of electrons,i.e.,thefinite-size effect men-tioned above for the MI are much less pronounced inthe BI state.The situation changes radically if the in-sulating behavior is associated with localized bipolarons forming a CDW state(see Fig.4,lower panel).Due to4citation gaps in the HHM with dynamical phonons(λ=0.35,α=0.1).Note the different scale of∆c in the MIphase.Open andfilled symbols denote DMRG results forperiodic(PBC)and open(OPC)boundary conditions,re-spectively.The accessible system sizes are smaller at largerλ/u,where an increasing number of(phononic)pseudosites isrequired to reach convergence with respect to the phonons.Stars represent the ED results for the eight-site system.The arrow marks the value of the optical gap∆opt for theBethe-ansatz solvable1D Hubbard model,which is given by∆opt/4t=u−1+ln(2)/2u in the limit of large u>1.24strong polaronic effects an almostflat band dispersionresults with exponentially small(electronic)quasiparti-cle weight.Now the dominant peaks in the incoherentpart of the(I)PE spectra are related to multiples of the(large)bare phonon frequency broadened by electronicexcitations.Since many-body gaps to excited states form the basisfor making contact with experimentally measurable exci-tation gaps and can also be used to characterize differentphases of the HHM,wefinally determine the charge andspin gaps,∆c=E(N+1)0(12)−2E(N)(0)(3)∆s=E(N)0(1)−E(N)(0),(4)DMRG.18Here E(M)(S z)is the ground-state en-of the HHM with M partcles in the sector with totalz component S z.thefinite-size scaling presented in Fig.5c/ssubstantiates our introductory discussion of thediagram(cf.Fig.1).∆c and∆s arefinite in thewill converge further as N→∞.Both gaps seemat the quantum phase transition point of thewithfinite-frequency phonons,but in the criticalthefinite-size scaling is extremely delicate.In thefound afinite charge excitation gap,which in theu/λ≫1scales to the optical gap of the Hubbardwhereas the extrapolated spin gap remains zero.19summary,we have presented a comprehensive pic-ture of the physical properties of the1D half-filledfinite-phonon frequency Holstein-Hubbard model.With re-spect to the metal the electron-electron coupling favorsthe Mott insulating state whereas the electron-phononinteraction is responsible for the Peierls insulator to oc-cur.The PI typifies a band insulator in the adiabaticweak-to-intermediate coupling range or a bipolaronic in-sulator for non-to-antiadiabatic strong-coupling.Our re-sults for the single-particle spectra and spin/charge ex-citation gaps give clear indication of a Mott-to Peierls-insulator quantum phase transition at u/λ≃1.Quan-tum phonon dynamics yields pronounced effects in the(I)PE spectra,which might be of great importance for in-terpreting photoemission experiments of low-dimensionalstrongly correlated electron-phonon systems such as MX-chain compounds.1We gratefully acknowledge stimulating discussionswith F.G¨o hmann,E.Jeckelmann,and A.P.Kampf.Work in Greifswald,Erlangen and Los Alamos was sup-ported by Deutsche Forschungsgemeinschaft(Focus pro-gramme SPP1073),Bavarian Competence Network forHigh Performance Computing,and US DOE,respec-tively.Numerical calculations have been performed atthe HLRN Berlin-Hannover,RRZ Erlangen and LRZM¨u nchen.1N.Tsuda,K.Nasu,A.Yanese,K.Siratori,Electronic Con-duction in 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