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Physical review. B, Condensed matter

Variational study of the discrete Holstein model

A. La Magna, R. Pucci

Published Apr 1, 1996• 42 citations
Publisher

Abstract

The Holstein model for the interaction between one particle and a large (\ensuremath{\sim}100 sites) chain of oscillators has been treated through a numerical self-consistent procedure. The adopted variational state is a generalization of the most commonly used trial wave functions and the results are correct both in the weak and the strong-coupling limits. The comparison of our variational ground-state energy with the exact calculations on small clusters also supports the validity of the approach. Moreover, all the relevant physical quantities are analytical functions of the parameters. The appearance of a self-trapped state is discussed critically in connection with the changes of the dynamical condition of the system. The role of the quantum fluctuations of the lattice in the intermediate-coupling regime is emphasized by calculating the phonons wave functions. The application of the results to some physical systems is presented in the conclusion. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.8449

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The Holstein model for the interaction between one particle and a large (\ensuremath{\sim}100 sites) chain of oscillators has been treated through a numerical self-consistent procedure. The adopted variational state is a generalization of the most commonly used trial wave functions and the results are correct both in the weak and the strong-coupling limits. The comparison of our variational ground-state energy with the exact calculations on small clusters also supports the validity of the approach. Moreover, all the relevant physical quantities are analytical functions of the parameters. The appearance of a self-trapped state is discussed critically in connection with the changes of the dynamical condition of the system. The role of the quantum fluctuations of the lattice in the intermediate-coupling regime is emphasized by calculating the phonons wave functions. The application of the results to some physical systems is presented in the conclusion. \textcopyright{} 1996 The American Physical Society.
A concise variational approach is to propose to calculate the ground-state properties of none-dimensional Holstein model. The results in the weak and strong coupling limit can be easily recovered analytically. It is shown that, in the nontrivial intermediate electron-phonon coupling regime, the present results are in good agreement with those by density-matrix renormalization group and numerical exact diagonalizations. The present approach is more concise than any other analytical ones in this field, and can be easily generalized to Holstein models in higher dimensions and with more electrons.
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.
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.
The paper deals with the ground and the first excited state of the polaron in the one-dimensional Holstein model. Various variational methods are used to investigate both the weak-coupling and strong-coupling case, as well as the crossover regime between them. Two of the methods, presented here, introduce interesting elements to the understanding of the nature of the polaron. Reliable numerical evidence is found that, in the strong-coupling regime, the ground and the first excited state of the self-trapped polaron are well described within the adiabatic limit. The lattice vibration modes associated with the self-trapped polarons are analyzed in detail, and the frequency softening of the vibration mode at the central site of the small polaron is estimated. It is shown that the first excited state of the system in the strong-coupling regime corresponds to the excitation of the soft phonon mode within the polaron. In the crossover regime, the ground and the first excited state of the system can be approximated by the anticrossing of the self-trapped and the delocalized polaron state. In this way, the connection between the behavior of the ground and the first excited state is qualitatively explained.
We adapt a variational procedure to calculate ground-state properties of the Holstein model in the adiabatic limit. At strong coupling, this adaption leads to rapid convergence of results. The intermediate coupling regime is further handled with an adaptive algorithm. We also use semiclassically derived results for the adiabatic end point, along with weak coupling perturbation theory. These establish weak and strong coupling (or large and small polaron, respectively) regimes in two dimensions or higher. As is well known, these are connected smoothly, but the crossover becomes increasingly abrupt as the phonon frequency decreases.
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.