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Physics of Particles and Nuclei

On lower bound estimates for the energy spectrum of the Fröhlich polaron model

Andrey Vladimirovich Soldatov

Published Nov 25, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s1063779610070221

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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.
Method of intermediate problems in the theory of linear semi-bounded self-adjoint operators on rigged Hilbert space was applied to the investigation of the ground state energy of the Fr öhlich polaron model.It was shown that various infinite sequences of non-decreasing improvable lower bound estimates for the polaron ground state energy can be derived for arbitrary values of the electron-phonon interaction constant.The proposed approach allows for explicit numerical evaluation of the thus obtained lower bound estimates at all orders and can be straightforwardly generalized for investigation of the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy of the polaron at rest.In conjunction with numerous, already derived by multitudinous methods, well-known upper bound estimates for the energy spectral curve of the Fr öhlich polaron as a function of the electron-phonon 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.
An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slow-moving "physical" Fröhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them.
An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slowmoving "physical" Fr öhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method.The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them.
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.
Partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Fröhlich Hamiltonian is performed using the Feynman-Pines diagram technique.Renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for a weak electron-phonon coupling at T = 0 K.It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra.The ground state stays stationary and the excited one decays at a bigger coupling constant.