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arXiv:cond-mat/0601548 · Physical Review B

Variational path-integral treatment of a translation invariant many-polaron system

F. Brosens, Serghei N. Klimin, J. T. Devreese

Published Jun 6, 2005• 14 citations
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Abstract

A translation invariant $N$-polaron system is investigated at arbitrary electron-phonon coupling strength, using a variational principle for path integrals for identical particles. An upper bound for the ground-state energy is found as a function of the number of spin up and spin down polarons, taking the electron-electron interaction and the Fermi statistics into account. The resulting addition energies and the criteria for multipolaron formation are discussed.

DOI
10.1103/physrevb.71.214301

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A translation invariant NNN-polaron system is investigated at arbitrary electron-phonon coupling strength, using a variational principle for path integrals for identical particles. An upper bound for the ground-state energy is found as a function of the number of spin up and spin down polarons, taking the electron-electron interaction and the Fermi statistics into account. The resulting addition energies and the criteria for multipolaron formation are discussed.
The ground-state energy of an NNN-polaron system confined to a quantum dot with a neutralizing background charge is investigated within an all-coupling many-body path-integral variational principle taking into account both Fermi statistics of polarons and the electron-electron interaction. The treatment of the ground-state energy is performed for both closed- and open-shell systems. The average fermion density in the neutral spherical dot is characterized by the Wigner-Seitz parameter rs{r}_{s}rs​. For a sufficiently large but finite number of polarons, the dependency of the ground-state energy on rs{r}_{s}rs​ is very similar to that for a polaron gas in bulk. From this, we can conclude that the ground-state energy of a polaron gas in bulk can be qualitatively described using a model of a finite number of polarons in a confinement potential provided by a neutralizing background charge.
The ground-state energy of an N-polaron system, confined to a spherical quantum dot with a neutralizing background charge, is investigated within an all-coupling many-body path-integral variational principle, taking into account both the Fermi statistics of the polarons and the electron-electron interaction. The treatment of the ground-state energy is performed for both closed-shell and open-shell systems. The av. fermion d. in the neutral spherical dot is characterized by the Wigner-Seitz parameter rs. For a sufficiently large but finite no. of polarons, the dependence of the ground-state energy on rs is very similar to that for a polarons, gas in bulk. Hence, we can conclude that the ground-state properties of a polaron gas in bulk can be qual. described using a model of a finite no. of polarons in a confinement potential provided by a neutralizing background charge. [on SciFinder (R)]
The polaron features due to electron-phonon interactions with different coupling ranges are investigated by adopting a variational approach. The ground-state energy, the spectral weight, the average kinetic energy, the mean number of phonons, and the electron-lattice correlation function are discussed for the system with coupling to local and nearest-neighbor lattice displacements comparing the results with the long-range case. For large values of the coupling with nearest-neighbor sites, most physical quantities show a strong resemblance with those obtained for the long-range electron-phonon interaction. Moreover, for intermediate values of interaction strength, the correlation function between electron and nearest-neighbor lattice displacements is characterized by an upturn as function of the electron-phonon coupling constant.
The polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.
A novel variational approach is presented for the calculation of the ground-state energy of the polaron in arbitrary N dimensions in the strong-coupling limit. By using the phonon coherent state to represent the wavefunction of phonons, a self-consistent integro-differential equation for the electron wavefunction is derived. The calculated results of the ground-state energy for N = 1, 2 and 3 agree well with the best results in the literature. It is also found that, for arbitrary N, the present results are less than the Feynman path integral ones by small percentages. It is proposed that this approach should be universal for systems involving polarons in the strong-coupling regime.
The free energy of an ideal polaron gas in a static magnetic field is calculated using Feynman's path-integral formalism. The trial action, used in Feynman's polaron theory, is extended to take into account the anisotropy of the effective electron-phonon interaction. This results in a free-energy expression with four variational parameters. According to Feynman's conjecture, the resulting free energy is an upper bound to the exact result. The approximate free energy is expected to provide accurate results for arbitrary electron-phonon coupling strength ($\ensuremath{\alpha}$), temperature (TTT), and magnetic field strength (H\mathcal{H}H). The free energy per polaron is evaluated for limiting values of $\ensuremath{\alpha}$, TTT, and H\mathcal{H}H.