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Physical review. B, Condensed matter

One particle interacting with the acoustical phonons in a discrete chain

A. La Magna, R. Pucci

Published Mar 1, 1997• 11 citations
Publisher

Abstract

We have applied a variational method to study both the Davydov model and the Su, Schrieffer, and Heeger model for the interaction between an electron and the acoustical phonons of a discrete chain. A class of localized solutions has been found for both of these models. These solutions appear in the intermediate values of the adiabaticity \ensuremath{\gamma}\ensuremath{\sim}1 when the particle-phonon coupling constant \ensuremath{\lambda} is near the critical point for the delocalization-localization transition. We have widely discussed the limits of the continuum approximation, stressing the importance of the application of the method to a discrete system if we want to explore the transition from a large mobile polaron to a strong localized heavy polaron. Furthermore, we have considered the connection between localization and transport in the low-density limit and the possible applications of the method to understanding of the physical properties of some systems with low-dimensionality.

DOI
10.1103/physrevb.55.6296

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•Jan 1, 2011•1 citation
We have applied a variational method to study both the Davydov model and the Su, Schrieffer, and Heeger model for the interaction between an electron and the acoustical phonons of a discrete chain. A class of localized solutions has been found for both of these models. These solutions appear in the intermediate values of the adiabaticity \ensuremath{\gamma}\ensuremath{\sim}1 when the particle-phonon coupling constant \ensuremath{\lambda} is near the critical point for the delocalization-localization transition. We have widely discussed the limits of the continuum approximation, stressing the importance of the application of the method to a discrete system if we want to explore the transition from a large mobile polaron to a strong localized heavy polaron. Furthermore, we have considered the connection between localization and transport in the low-density limit and the possible applications of the method to understanding of the physical properties of some systems with low-dimensionality.
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We consider a polaron localized on a trap in a one-dimensional lattice with a harmonic potential of nearest-neighbor interaction. We study the lattice oscillations in the framework of quantum mechanics and regard the polaron energy as a functional of the wave function. We take the electron—phonon interaction into account in the framework of the linear Su—Schrieffer—Heeger approximation. We show that the results do not differ from the adiabatic consideration if the lattice oscillations are described classically. For the polaron ground state, we obtain approximate analytic expressions that agree well with the results of numerical simulation.
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 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.
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.
Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.