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arXiv:cond-mat/9909305 · Physical review. B, Condensed matter

Variational approach for the Holstein molecular-crystal model

V. Cataudella, G. De Filippis, Giuseppe Iadonisi

Published Dec 1, 1999• 42 citations
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Abstract

A variational technique is developed to investigate the polaronic features of the Holstein molecular-crystal model. It is based on a linear superposition of Bloch states that describe large and small polaron wave functions. It is shown that this method provides a very good description of the regime characterized by intermediate values of the electron-phonon coupling constant (the so-called intermediate polaron) for any value of the adiabatic parameter ${\ensuremath{\omega}}_{0}/t.$ The polaron ground state energy in one and two dimensions is calculated and successfully compared with the best estimates available providing a clear physical interpretation of the intermediate polaron. The band structure, the spectral weight of the ground state and the lattice displacement associated to the polaron are also calculated and discussed. The method has the advantage of requiring very little computational effort.

DOI
10.1103/physrevb.60.15163

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A variational technique is developed to investigate the polaronic features of the Holstein molecular-crystal model. It is based on a linear superposition of Bloch states that describe large and small polaron wave functions. It is shown that this method provides a very good description of the regime characterized by intermediate values of the electron-phonon coupling constant (the so-called intermediate polaron) for any value of the adiabatic parameter \ensuremathω0/t.{\ensuremath{\omega}}_{0}/t.\ensuremathω0​/t. The polaron ground state energy in one and two dimensions is calculated and successfully compared with the best estimates available providing a clear physical interpretation of the intermediate polaron. The band structure, the spectral weight of the ground state and the lattice displacement associated to the polaron are also calculated and discussed. The method has the advantage of requiring very little computational effort.
The polaron features of the one-dimensional Holstein model are investigated by improving a variational method introduced recently and based on a linear superposition of Bloch states that describe large and small polaron wave functions. The mean number of phonons, the polaron kinetic energy, the electron-phonon local correlation function, and the ground-state spectral weight are calculated and discussed. A crossover regime between large and small polarons for any value of the adiabatic parameter ${\ensuremath{\omega}}_{0}/t$ is found and a polaron phase diagram is proposed.
We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
The polaron features for long-range electron–phonon interaction are investigated by extending a variational approach previously proposed for the study of systems with local coupling. The ground state spectral weight, the average kinetic energy, the mean number of phonons, and the electron–lattice correlation function are discussed for a wide range of model parameters focusing on the adiabatic regime and comparing the results with the short-range case (Holstein model). A strong mixing of electronic and phononic degrees of freedom for small values of the electron–phonon coupling constant is found in the adiabatic case due to the long-range interaction. Finally a polaron ‘phase diagram’ is proposed.
We present selected results for the Holstein molecular crystal model in one space dimension as determined by the Global–Local variational method, including complete polaron energy bands, ground state energies, and effective masses. We juxtapose our results with specific comparable results of numerous other methodologies of current interest, including quantum Monte Carlo, cluster diagonalization, dynamical mean field theory, density matrix renormalization group, semiclassical analysis, weak-coupling perturbation theory, and strong-coupling perturbation theory. Taken as a whole, these methodologies are mutually confirming and provide a comprehensive and quantitatively accurate description of polaron properties in essentially any regime. In particular, this comparison confirms the Global–Local variational method as being highly accurate over a wide range of the polaron parameter space, from the nonadiabatic limit to the extremes of high adiabaticity, from weak coupling through intermediate coupling to strong coupling.
The ground state of the Holstein molecular crystal model of a single electron in one dimension is studied using the Global-Local variational method, analyzing in particular the total energy, kinetic energy, phonon energy, and interaction energy over a broad region of the polaron parameter space. Through the application of objective criteria, a curve is identified that simply, accurately, and robustly locates the self-trapping transition separating small polaron and large polaron behavior. Particular attention is given to the kinetic energy, which is compared quantitatively with other perturbative and nonperturbative methods.
We present selected results for the Holstein molecular crystal model in one space dimension as determined by the Global-Local variational method, including complete polaron energy bands, ground state energies, and effective masses. We juxtapose our results with specific comparable results of numerous other methodologies of current interest, including quantum Monte Carlo, cluster diagonalization, dynamical mean field theory, density matrix renormalization group, semiclassical analysis, weak-coupling perturbation theory, and strong-coupling perturbation theory. This comparison confirms the Global-Local variational method as being highly accurate over a wide range of the polaron parameter space, from the non-adiabatic limit to the extremes of high adiabaticity, from weak coupling through intermediate coupling to strong coupling.