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Physical review. B, Condensed matter

Hole-polaron formation in the two-dimensional Holsteint-Jmodel: A variational Lanczos study

Holger Fehske, Heinrich Röder, Gerhard Wellein, Antonis D Mistriotis

Published Jun 15, 1995• 94 citations
Publisher

Abstract

Polaronic features of dopant-induced charge carriers are manifest in recent experiments on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4+\mathit{y}}$ and ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ni}}_{\mathit{x}}$${\mathrm{CuO}}_{4+\mathit{y}}$. To study the formation of hole (bi)polarons in such systems exhibiting strong Coulomb interactions, the Holstein t-J model is examined by means of a variational diagonalization technqiue for a wide range of phonon frequencies and electron-phonon (EP) couplings on finite square lattices, up to 18 effective sites in size. Including static displacement field, polaron, and squeezing effects, our approach allows for the description of nearly free polarons in the weak-coupling limit as well as for adiabatic Holstein polarons and nonadiabatic Lang-Firsov polarons in the case of strong EP interactions. At low doping levels and low phonon frequencies we demonstrate that antiferromagnetic spin correlations and EP interactions reinforce each other to the effect of lowering the threshold for polaronic self-localization in a strongly distorted lattice. This is contrasted with the nonadiabatic regime, where we observe the formation of Lang-Firsov polarons with moderate polaronic mass enhancement in a nearly undistorted lattice. In the case of two doped holes, the hole binding energy is analyzed in detail and we find that hole binding is enhanced as a dynamical (static) effect of a rather weak (extremely strong) EP interaction for delocalized (self-trapped) polarons. At quarter-filling we notice, in the adiabatic regime, a sequence of transitions as the EP coupling increases from nearly free mobile polarons to a polaronic superlattice and finally to a charge-separated state. The ordering of polarons disappears at higher phonon frequencies.

DOI
10.1103/physrevb.51.16582

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Polaronic features of dopant-induced charge carriers are manifest in recent experiments on {\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4+\mathit{y}} and {\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ni}}_{\mathit{x}}$${\mathrm{CuO}}_{4+\mathit{y}}. To study the formation of hole (bi)polarons in such systems exhibiting strong Coulomb interactions, the Holstein t-J model is examined by means of a variational diagonalization technqiue for a wide range of phonon frequencies and electron-phonon (EP) couplings on finite square lattices, up to 18 effective sites in size. Including static displacement field, polaron, and squeezing effects, our approach allows for the description of nearly free polarons in the weak-coupling limit as well as for adiabatic Holstein polarons and nonadiabatic Lang-Firsov polarons in the case of strong EP interactions. At low doping levels and low phonon frequencies we demonstrate that antiferromagnetic spin correlations and EP interactions reinforce each other to the effect of lowering the threshold for polaronic self-localization in a strongly distorted lattice. This is contrasted with the nonadiabatic regime, where we observe the formation of Lang-Firsov polarons with moderate polaronic mass enhancement in a nearly undistorted lattice. In the case of two doped holes, the hole binding energy is analyzed in detail and we find that hole binding is enhanced as a dynamical (static) effect of a rather weak (extremely strong) EP interaction for delocalized (self-trapped) polarons. At quarter-filling we notice, in the adiabatic regime, a sequence of transitions as the EP coupling increases from nearly free mobile polarons to a polaronic superlattice and finally to a charge-separated state. The ordering of polarons disappears at higher phonon frequencies.
We investigate the interplay of electron–phonon (EP) coupling and strong electronic correlations in the frame of the two–dimensional (2D) Holstein t–J model (HtJM), focusing on polaronic ordering phenomena for the quarter–filled band case. The use of direct Lanczos diagonalization on finite lattices allows us to include the effects of quantum phonon fluctuations in the calculation of spin/charge structure factors and hole–phonon correlation functions. In the adiabatic strong coupling regime we found evidence for “self-localization ” of polaronic carriers in a (π, π) charge-modulated structure, a type of superlattice solidification reminiscent of those observed in the nickel perovskites La2−xSrxNiO4+y. Electron diffraction measurements on the nickel oxide analogue of the layered high–Tc 214 copper oxide compound, La2−xNixCuO4, reveal a (π, π)–superstructure spot at quarter filling, i.e. for x = 0.5, which has been interpreted as sign of a truly 2D polaron ordered phase [1]. To study charge – and spin–ordering phenomena in such systems, exhibiting besides a substantial EP interaction strong Coulomb correlations, let us consider the planar t-J model with an additional on-site Holstein hole-phonon coupling:
The Holstein Hubbard and Holstein t-J models are studied for a wide range of phonon frequencies, electron-electron, and electron-phonon interaction strengths on finite lattices with up to ten sites by means of direct Lanczos diagonalization. Previously the necessary truncation of the phononic Hilbert space caused serious limitations to either very small systems (four or even two sites) or to weak electron-phonon coupling, in particular in the adiabatic regime. Using parallel computers we were able to investigate the transition from ``large'' to ``small'' polarons in detail. By resolving the low-lying eigenstates of the Hamiltonian and by calculating the spectral function, we can identify a polaron band in the strong-coupling case, whose dispersion deviates from the free-particle dispersion at low and intermediate phonon frequencies. For two electrons (holes) we establish the existence of bipolaronic states and discuss the formation of a bipolaron band. For the two-dimensional Holstein t-J model, we demonstrate that the formation of hole polarons is favored by strong Coulomb correlations. Analyzing hole-hole correlation functions, we find that hole binding is enhanced as a dynamical effect of the electron-phonon interaction. \textcopyright{} 1996 The American Physical Society.
missing to date is an application of the powerful ED technique to the calculation of dynamical properties of the Holstein t--J model (HtJM), including the full quantum nature of phonons. In this contribution, we employ the Lanczos algorithm in combination with a kernel polynomial moment expansion and the Maximum Entropy method [5] to investigate the quasiparticle spectrum of a single hole--polaron in the 2D HtJM on a ten--site square lattice. Moreover, we compute different hole--hole/phonon correlation functions at higher doping level in order to comment on hole--binding effects and charge-densitywave (CDW) formation. The HtJM is described by the Hamiltonian [4] H = H ph + H t\\GammaJ \\Gamma p " p ¯h! X i \\Gamma b y i + b i \\Delta ~ h i<F25.
We study the tunneling dynamics of dopant-induced hole polarons that are self-localized by electron-phonon coupling in a two-dimensional antiferromagnet. Our treatment is based on a path-integral formulation of the adiabatic (Born-Oppenheimer) approximation, combined with many-body tight-binding, instanton, constrained lattice dynamics, and many-body exact diagonalization techniques. The applicability and limitations of the adiabatic approximation in polaron tunneling problems are discussed in detail and adiabatic results are compared to exact numerical results for a two-site polaron problem. Our results are mainly based on the Holstein-tJ\mathrm{tJ}tJ and, for comparison, on the Holstein-Hubbard model. We also study the effects of second-neighbor hopping and long-range electron-electron Coulomb repulsion. The polaron tunneling dynamics is mapped onto an effective low-energy Hamiltonian that takes the form of a fermion tight-binding model with occupancy-dependent, predominantly second- and third-neighbor tunneling matrix elements, excluded double occupancy, and effective intersite charge interactions. Antiferromagnetic spin correlations in the original many-electron Hamiltonian are reflected by an attractive contribution to the first-neighbor charge interaction and by Berry phase factors that determine the signs of effective polaron tunneling matrix elements. In the two-polaron case, these phase factors lead to polaron-pair wave functions of either ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave symmetry or p-wave symmetry with zero and nonzero total pair momentum, respectively. Implications for the doping-dependent isotope effect, pseudogap, and Tc{T}_{c}Tc​ of a superconducting polaron-pair condensate are discussed and compared to observed properties of the cuprate high-Tc{T}_{c}Tc​ materials.
We present exact results for the ground-state properties of the two-dimensional single-hole Peierls t-J model on finite square lattices, up to 18 effective sites in size. Using a Lanczos technique for unrestricted energy minimization with respect to local lattice distortions, we study the competition between intrasite electron-phonon and intersite antiferromagnetic (AFM) exchange coupling on the hole state over a wide range of interaction strengths. The structure of the doping state is examined by calculating local expectation values of spin and electron densities as well as magnetic and charge correlation functions. As the local electron-phonon interaction increases, we find a self-trapping transition from a delocalized hole (polaronlike) state to a mainly localized hole state. Despite a weak lattice distortion the delocalized case has similar properties to the ground state of the pure t-J model, whereas the localized hole state exhibits strong local lattice distortion and a quenched magnetic moment coupled ferromagnetically to an otherwise unperturbed AFM background. Aspects of the phase diagram are discussed.
We present a detailed numerical study of the one-dimensional Holstein model with a view to understanding the self-trapping process of electrons or excitons in crystals with short-range particle-lattice interactions. Applying a very efficient variational Lanczos method, we are able to analyze the ground-state properties of the system in the weak- and strong-coupling, adiabatic and nonadiabatic regimes on lattices large enough to eliminate finite-size effects. In particular, we obtain the complete phase diagram and comment on the existence of a critical length for self-trapping in finite (closed) one-dimensional systems. In order to characterize large and small polaron states we calculate self-consistently the lattice distortions and the particle-phonon correlation functions. In the strong-coupling case, two distinct types of small polaron states are shown to be possible according to the relative importance of static displacement field and dynamic polaron effects. Special emphasis is on the intermediate-coupling regime, which we also study by means of direct diagonalization, preserving the full dynamics and quantum nature of phonons. The crossover from large to small polarons shows up in a strong decrease of the kinetic energy accompanied by a substantial change in the optical absorption spectra. We show that our numerical results in all important limiting cases reveal excellent agreement with both analytical perturbation theory predictions and very recent density matrix renormalization group data.