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arXiv:1005.0257 · Journal of the Physical Society of Japan

First Order Bipolaronic Transition at Finite Temperature in the Holstein Model

貴裕 布施, Yoshiaki Ōno

Published Aug 25, 2010• 16 citations
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Abstract

We investigate the Holstein model by using the dynamical mean-field theory combined with the exact diagonalization method. Below a critical temperature Tcr, a coexistence of the polaronic and the bipolaronic solutions is found for the same value of the electron-phonon coupling $ in the range gc1(T)
DOI
10.1143/jpsj.79.093702

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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.
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.
At large electron-phonon coupling, the phonon of the Holstein-Hubbard model can be (generally) treated classically. Exact results can be obtained for this model at any band filling and any finite dimension where the Hubbard term with coefficient U is treated rigorously. The existence of chaotic bipolaronic states is proven when U is smaller than a certain value Uc and for an electron-phonon coupling k sufficiently large. The existence of magnetic polaronic states is also proven for positive U and k sufficiently large. For 0<U<Uc, there exists an overlap region with chaotic mixed polaronic-bipolaronic states. There is a region for U=Uc/2 and large k of special interest, where the bipolarons in their ground-state become singlet pairs of polarons bounded by a magnetic resonance. Then, the Peierls-Nabarro barrier which usually pins polarons and bipolarons to the lattice depresses to almost zero and the validity of the classical lattice approximation breaks down so that the initial exact results do not apply in that specific region. The formation of a superconducting state with possibly a high critical temperature, is then conjectured at a large coupling k in a narrow domain in-between the two insulating domains corresponding to Bipolaronic Charge Density Waves and to magnetic Polaronic Spin Density Waves respectively.
We investigate metal-insulator transitions in the half-filled Holstein-Hubbard model as a function of the on-site electron-electron interaction U and the electron-phonon coupling g . We use several different numerical methods to calculate the phase diagram, the results of which are in excellent agreement. When the electron-electron interaction U is dominant, the transition is to a Mott insulator; when the electron-phonon interaction dominates, the transition is to a localized bipolaronic state. In the former case, the transition is always found to be second order. This is in contrast to the transition to the bipolaronic state, which is clearly first order for larger values of U . We also present results for the quasiparticle weight and the double occupancy as functions of U and g .
In this paper we have studied the properties of a two-fermion system interacting with the phonon field in the framework of the Holstein Hubbard model in one, two and three dimensions. We have chosen the modified Lang-Firsov variational wave function with on-site and nearest-neighbour lattice distortion for the phonon subsystem to obtain an effective electronic Hamiltonian. This effective electronic Hamiltonian is then solved exactly to obtain the binding energy and effective mass of the bipolaron. We investigated the stability of the bipolaronic phase by comparing the ground-state energy of a bipolaron and two free polarons. We observed a critical repulsive on-site Hubbard interaction U c below which the bipolaronic phases are stable for a fixed electron-phonon coupling g . In the absence of on-site repulsion, bipolaronic phases are stable over the entire range of electron-phonon coupling for one dimension, whereas there is a critical electron-phonon coupling g c for formation of a stable bipolaron in two and three dimensions.