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EXACT SOLUTION OF A PERIODIC-CLUSTER ELECTRON-PHONON SYSTEM

César R. Proetto, L. M. Falicov

Published Aug 1, 1988• 0 citations
OpenAlex

Abstract

The ground state of a one-dimensional electron-phonon system is studied by means of a small-crystal approach: an exact solution of a two-site cluster with two electrans and periodic boundary conditions.A single electron band is considered in the tight-binding approximation, together with two-electron interactions between electrons in the same site (Hubbard model); the electrons are coupled to longitudinal acoustic phonons through a bilinear interaction.The problem is isomorphic with that of the homopolar diatomic molecule with a vibronic degree of freedom and coupling between the electronic and vibronic modes.In the adiabatic (large ionic mass, small vibrational frequency) limit there is an analytical solution which indicates that the transition between the non-distorted (weak coupling) and the distorted (strong coupling) phases can be either continuous or discontinuous, depending on whether the value of the electron-electron interaction is larger or smaller than a critical value.In the extremely high-frequency limit (ionic mass M ~ 0 ) it is also possible to find an analytical solution (for M = 0) that shows that the system remains undistorted for any value of the electron-phonon coupling and the electron-electron interaction.In the intermediate case (finite, nonvanishing M) numerical solutions exhibit a continuous transition from the non-distorted to the distorted phase.

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Phase diagram of one-dimensional electron-phonon systems. II. The molecular-crystal model

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We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin ($n$). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.

Physical review. B, Condensed matter•Apr 1, 1983•236 citations
The ground state of a one-dimensional electron-phonon system is studied by means of a small-crystal approach: an exact solution of a two-site cluster with two electrans and periodic boundary conditions.A single electron band is considered in the tight-binding approximation, together with two-electron interactions between electrons in the same site (Hubbard model); the electrons are coupled to longitudinal acoustic phonons through a bilinear interaction.The problem is isomorphic with that of the homopolar diatomic molecule with a vibronic degree of freedom and coupling between the electronic and vibronic modes.In the adiabatic (large ionic mass, small vibrational frequency) limit there is an analytical solution which indicates that the transition between the non-distorted (weak coupling) and the distorted (strong coupling) phases can be either continuous or discontinuous, depending on whether the value of the electron-electron interaction is larger or smaller than a critical value.In the extremely high-frequency limit (ionic mass M ~ 0 ) it is also possible to find an analytical solution (for M = 0) that shows that the system remains undistorted for any value of the electron-phonon coupling and the electron-electron interaction.In the intermediate case (finite, nonvanishing M) numerical solutions exhibit a continuous transition from the non-distorted to the distorted phase.
The ground state of a one-dimensional electron-phonon system is studied by means of a small-crystal approach: an exact solution of a two-site cluster with two electrons and periodic boundary conditions. A single electron band is considered in the tight-binding approximation, together with two-electron interactions between electrons in the same site (Hubbard model); the electrons are coupled to longitudinal-acoustic phonons through a bilinear interaction. The problem is isomorphic with that of the homopolar diatomic molecule with a vibronic degree of freedom and coupling between the electronic and vibronic modes. In the adiabatic (large-ionic-mass, small-vibrational-frequency) limit there is an analytical solution which indicates that the transition between the nondistorted (weak-coupling) and the distorted (strong-coupling) phases can be either continuous or discontinuous, depending on whether the value of electron-electron interaction is smaller or larger than a critical value. In the extremely-high-frequency limit (ionic mass M\ensuremath{\rightarrow}0) it is also possible to find an analytical solution (for M=0) that shows that the system remains undistorted for any value of the electron-phonon coupling and the electron-electron interaction. In the intermediate case (finite, nonvanishing M), numerical solutions exhibit a continuous transition from the nondistorted to the distorted phase.
We investigate polaron formation in a many-electron system in the presence of a local repulsion sufficiently strong to prevent local-bipolaron formation. Specifically, we consider a Hubbard-Holstein model of interacting electrons coupled to dispersionless phonons of frequency ω 0 . Numerically solving the model in a small one-dimensional cluster, we find that in the nearly adiabatic case ω 0 < t , the necessary and sufficient condition for the polaronic regime to occur is that the energy gain in the atomic ( i.e. , extremely localized) regime ε pol overcomes the energy of the purely electronic system ε el . In the antiadiabatic case, ω 0 > t , polaron formation is instead driven by the condition of a large ionic displacement g / ω 0 < 1 ( g being the electron-phonon coupling). Dynamical properties of the model in the weak and moderately strong coupling regimes are also analyzed.
We study the nature of the ground state of the Su-Schrieffer-Heeger model for electron-phonon interactions in one dimension in the half-filled-band case. We consider the cases of spinless electrons (n=1n=1n=1) and spin-\textonehalf{} electrons (n=2n=2n=2), and discuss the stability of the Peierls-dimerized ground state as a function of the ionic mass and electron-phonon coupling constant. We first consider the zero-mass limit of the theory and extend our results to finite mass using renormalization-group arguments. For spinless electrons, it is found that quantum fluctuations destroy the long-range dimerization order for the small electron-phonon coupling constant if the ionic mass is finite. For spin-\textonehalf{} electrons, the system is dimerized for an arbitrary coupling constant and phonon frequency. Renormalization-group trajectories show that the low-energy behavior of the system is governed by the zero-mass limit of the theory, an nnn-component Gross-Neveu model. Monte Carlo simulations are performed for the model at finite phonon frequencies, and the results are compared with the static limit result. We study in particular the set of parameters appropriate for polyacetylene and find a 15% reduction in the phonon order parameter due to fluctuations of the phonon field. A finite-size scaling analysis of the numerical data for the cases n=1and2n=1 \mathrm{and} 2n=1and2 is performed, which confirms the results obtained from the renormalization-group analysis.
The exact eigenvalues for the ground state and the excitation spectra were obtained for a system with two electrons on two sites, taking into account the transfer energy, the Coulomb repulsion on the same site, and the intersite correlation energy. The vibrational modes couple linearly to the electronic density. We discuss the interplay of these interactions resulting in different phases of distorted and undistorted eigenstates with corresponding localized and delocalized electronic wave functions and accompanying charge transfer. Several simple approximation schemes generalizable to larger systems are evaluated. The comparison with the exact results shows that their validity depends strongly on the system parameters. The according limits of applicability are discussed.
We study the small-polaron problem of a single electron interacting with the lattice for the Holstein model in the adiabatic limit on a comb lattice hen the electron-phonon interaction acts only on the base sites. The ground state properties can be easily deduced from the ones of a linear chain with a appropriate rescaling of the coupling constant. On the other hand, the dynamical properties, that involve the complete spectrum of the system, present an "exotic" behavior. In the weak coupling limit the Drude weight (zero-frequency-conductivity) is enhanced with respect to its free-case value, contrary to the linear chain case, where for every finite value one has suppression of the Drude peak. More interestingly, the loss of coherent electron motion and the polaronic localization of the carrier occurs for different coupling values. Thus for intermediate coupling, a different phase appears with large kinetic energy and no coherent motion. RI Capone, Massimo/A-7762-2008
We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin (nnn). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.