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Bounds on the minimal energy of translation invariant n-polaron systems

Marcel Griesemer, Jacob Schach Møller

Published Aug 4, 2013• 21 citations
OpenAlex

Abstract

For systems of N charged fermions (e.g. electrons) interacting with longi-tudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U> 0 and α> 0 measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition 2α < U, which follows from the dependence of U and α on electrical properties of the crystal. We show that the large N asymptotic behavior of the minimal energy EN changes at 2α = U and that 2α ≤ U is necessary for ther-modynamic stability: for 2α> U the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and EN behaves like −N7/3. 2 M. Griesemer and J. S. Møller 1

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In the following discussion we are concerned with the standard Fr\"ohlich model for an optical polaron. We clarify the qualitative properties of the energy spectrum for arbitrary total momentum Q. Concerning the ground-state energy, we establish an effective lower bound. Until now, we have to assume that the electron-phonon coupling parameter $\ensuremath{\alpha}$ does not exceed a specified positive value. Using this bound, we demonstrate that the ground-state energy coincides with the continuum edge for $|Q|>~|{Q}_{C}|,$ ${Q}_{C}$ being finite. Consequently, it is only for $|Q|<|{Q}_{C}|$ that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and $\ensuremath{\alpha},$ we find an increase (strict decrease) with increasing $|Q|(\ensuremath{\alpha}).$ In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.

Physical review. B, Condensed matter•Oct 15, 1999•9 citations
For systems of N charged fermions (e.g. electrons) interacting with longi-tudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U> 0 and α> 0 measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition 2α < U, which follows from the dependence of U and α on electrical properties of the crystal. We show that the large N asymptotic behavior of the minimal energy EN changes at 2α = U and that 2α ≤ U is necessary for ther-modynamic stability: for 2α> U the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and EN behaves like −N7/3. 2 M. Griesemer and J. S. Møller 1
Method of intermediate problems was applied to investigation of the energy spectrum of the Fröhlich polaron model. It was shown that various infinite sequences of non-decreasing improvable lower bound estimates for the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy can be derived for arbitrary values of the electron-phonon interaction constant. These bound estimates allow for explicit numerical evaluation at all orders. In conjunction with numerous well-known upper bound estimates for the energy spectral curve of the Fröhlich polaron as a function of the interaction constant and the polaron total momentum, the aforesaid improvable lower bound estimates might provide one with virtually precise magnitude for the energy of the slow-moving polaron. Possible generalization to the case of the Fröhlich polaron in external magnetic field is outlined.
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.
We have developed a new equation of motion Green's function theory for the large polaron, that is completely nonperturbative in origin, and therefore gives a unified treatment of the polaron ground state energy for all interaction strengths. The theory also incorporates translational invariance rigorously, a priori. We find the following new results. (a) Previous theories that are valid for all coupling strengths are based on a nonperturbative summation of infinite subsets of entire diagrams (equations of motion), rather than on recoil parts of all diagrams. We can thus recover lowest order results of Feynman for example, via a translationally noninvariant form of the Fock approximation, which resembles that reported in a previous work. (b) The total ground state energy we obtain in the strong coupling limit is E0 = −0.426α2 to lowest order in the coupling constant α, a result substantially below that of previous calculations, E0 = −0.106α2 (using gaussian trial functions in both cases). Our result is also below the lower bound of Lieb and Yamazaki, although we show in a following paper that our result is an upper bound. (c) The origin of the difference of our results compared to that of other strong coupling theories is several fold, and will be presented in detail in subsequent papers. However, the essence of the difference is that the phonons in the field, via their interaction with the electron, assume the role of independent degrees of freedom, in the limit of an infinite number of phonons. We then have a field theoretic problem with an infinite number of degrees of freedom which can exhibit a totally different physical behavior compared to one with a finite number of degrees of freedom. This aspect of the problem has not been given explicit recognition in all other theories, including that of Lieb and Yamazaki.
In the following discussion we are concerned with the standard Fr\"ohlich model for an optical polaron. We clarify the qualitative properties of the energy spectrum for arbitrary total momentum Q. Concerning the ground-state energy, we establish an effective lower bound. Until now, we have to assume that the electron-phonon coupling parameter $\ensuremath{\alpha}$ does not exceed a specified positive value. Using this bound, we demonstrate that the ground-state energy coincides with the continuum edge for ∣Q∣> ∣QC∣,|Q|>~|{Q}_{C}|,∣Q∣> ∣QC​∣, QC{Q}_{C}QC​ being finite. Consequently, it is only for ∣Q∣<∣QC∣|Q|<|{Q}_{C}|∣Q∣<∣QC​∣ that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and $\ensuremath{\alpha},$ we find an increase (strict decrease) with increasing $|Q|(\ensuremath{\alpha}).$ In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.