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arXiv:cond-mat/0510620

Polarons and Bipolarons in Holstein and Holstein $t-J$ models by Dynamical Mean Field Theory

Sergio Ciuchi, Massimo Capone, E. Cappelluti, Giorgio Sangiovanni

Published Oct 24, 2005• 0 citations
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Abstract

The results of a comparative study of electronic spectral properties as well as phonon Probability Distribution Function (PDF) are reported. The study is done using Dynamical Mean Field Theory for the half-filled spinless and spinful Holstein model and for a single hole in Holstein t-J model. Qualitative changes in phonon PDF signals a polaronic crossover while an electronic Metal Insulator Transition (MIT) due to pairing into a bipolaronic state can be seen from a gap in the electronic DOS or by the vanishing of the quasi particle spectral weight. MIT can be observed only at sufficiently high density provided the Coulomb repulsion is neglected. In the limit of low density the carrier show a polaronic behavior trough the development of a definite polarization shifting the average value of the phonon PDF. At large densities occupied and empty sites result an a bimodal the phonon PDF. At the polaron crossover fluctuation of phonon coordinates tends to be larger than those at any other coupling. In this regime Born-Oppenheimer approximation is shown to fail also when phonon frequency is much less then electron bandwidth. For a single hole in the t-J model a further source of localization is due to magnetic superexchange energy which tends to localize the spin-defect spreading due to the presence of the hole.

DOI
10.48550/arxiv.cond-mat/0510620

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Polarons and Bipolarons in Holstein and Holstein t−Jt-Jt−J models by Dynamical Mean Field Theory
The results of a comparative study of electronic spectral properties as well as phonon Probability Distribution Function (PDF) are reported. The study is done using Dynamical Mean Field Theory for the half-filled spinless and spinful Holstein model and for a single hole in Holstein t-J model. Qualitative changes in phonon PDF signals a polaronic crossover while an electronic Metal Insulator Transition (MIT) due to pairing into a bipolaronic state can be seen from a gap in the electronic DOS or by the vanishing of the quasi particle spectral weight. MIT can be observed only at sufficiently high density provided the Coulomb repulsion is neglected. In the limit of low density the carrier show a polaronic behavior trough the development of a definite polarization shifting the average value of the phonon PDF. At large densities occupied and empty sites result an a bimodal the phonon PDF. At the polaron crossover fluctuation of phonon coordinates tends to be larger than those at any other coupling. In this regime Born-Oppenheimer approximation is shown to fail also when phonon frequency is much less then electron bandwidth. For a single hole in the t-J model a further source of localization is due to magnetic superexchange energy which tends to localize the spin-defect spreading due to the presence of the hole.
The evolution of the properties of a finite density electronic system as a function of the electron-phonon coupling is investigated in the Holstein model using the dynamical mean-field theory (DMFT) that becomes exact in infinite dimensions. We compare the spinless fermion case, in which only isolated polarons can be formed, with the spinful model in which the polarons can bind and form bipolarons. In the latter case, the bipolaronic binding occurs within DMFT as a metal-insulator transition. In the adiabatic regime in which the phonon energy is small with respect to the electron hopping we compare numerically exact DMFT results with an analytical scheme inspired by the Born-Oppenheimer procedure. Within the latter approach, a truncation of the phononic Hilbert space leads to a mapping of the original model onto an Anderson spin-fermion model. In the anti-adiabatic regime (where the phonon energy exceeds the electronic scales) the standard treatment based on Lang-Firsov canonical transformation allows one to map the original model on to an attractive Hubbard model in the spinful case. The separate analysis of the two regimes supports the numerical evidence that the presence of well-defined polaronic lattice displacements is not necessarily associated to a metal-insulator transition, which is instead due to pairing between the carriers. The finite-dimensionality effects neglected in DMFT may lead to a finite conductivity in the bipolaronic state which is, however, not always associated with polaronic distortions. At the polaron crossover the Born-Oppenheimer approximation is shown to break down due to the entanglement of the electron-phonon state.
The formation of a finite-density polaronic state is analyzed in the context of the Holstein model using the dynamical mean-field theory. The spinless and spinful fermion cases are compared to disentangle the polaron crossover from the bipolaron formation. The exact solution of dynamical mean-field theory is compared with weak-coupling perturbation theory, noncrossing (Migdal), and vertex correction approximations. We show that polaron formation is not associated with a metal-insulator transition, which is instead due to bipolaron formation.
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.
We present results on the dynamical correlation functions of the particle-hole symmetric Holstein-Hubbard model at zero temperature, calculated using the dynamical mean-field theory which is solved by the numerical renormalization group method. We clarify the competing influences of the electron-electron and electron-phonon interactions particularity at the different metal to insulator transitions. The Coulomb repulsion is found to dominate the behavior in large parts of the metallic regime. By suppressing charge fluctuations, it effectively decouples electrons from phonons. The phonon propagator shows a characteristic softening near the metal to bipolaronic transition but there is very little softening on the approach to the Mott transition.
The spectral weight functions and the optical conductivity of the Holstein model are studied on a one-dimensional six-site lattice with periodic boundary conditions for three different electron concentrations: a single electron, two electrons of opposite spins, and half filling. A density matrix approach is used to obtain an optimal phonon basis and to truncate the phonon Hilbert space without significant loss of accuracy. This approach allows us to calculate spectral functions for electrons dressed locally by the optimal phonons as well as for bare electrons. We obtain evidence for a smooth crossover from quasifree electrons to a heavy itinerant small polaron (single-electron case) or bipolaron (two-electron case) as the electron-phonon coupling strength increases. At half filling, we observe a crossover from a quasifree-electron ground state to a quasidegenerate Peierls charge-density-wave ground state for a finite electron-phonon coupling. This crossover is marked by an abrupt drop of the Drude weight, which is vanishingly small in the Peierls phase.
We derive an exact solution for the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ of one hole in the Holstein $t\text{\ensuremath{-}}J$ model in the framework of dynamical mean-field theory. We investigate the magnetic and phonon features associated with polaron formation as a function of the exchange coupling JJJ, electron-phonon interaction $\ensuremath{\lambda}$, and temperature. Our solution directly relates the features of the optical conductivity to excitations in the single-particle spectral function, revealing two distinct mechanisms of closing and filling of the optical pseudogap that take place upon varying the microscopic parameters. We show that the optical absorption at the polaron crossover is characterized by a coexistence of a magnon peak at low frequency and a broad polaronic band at higher frequency. An analytical expression for $\ensuremath{\sigma}(\ensuremath{\omega})$ valid in the polaronic regime is presented.