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arXiv:1409.5415 · Journal of Mathematical Physics

Ground state energy of large polaron systems

Rafael D. Benguria, Rupert L. Frank, Élliott H. Lieb

Published Feb 1, 2015• 0 citations
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Abstract

The last unsolved problem about the many-polaron system, in the Pekar–Tomasevich approximation, is the case of bosons with the electron-electron Coulomb repulsion of strength exactly 1 (the “neutral case”). We prove that the ground state energy, for large N, goes exactly as −N7/5, and we give upper and lower bounds on the asymptotic coefficient that agree to within a factor of 22/5.

DOI
10.1063/1.4908125

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The last unsolved problem about the many-polaron system, in the Pekar–Tomasevich approximation, is the case of bosons with the electron-electron Coulomb repulsion of strength exactly 1 (the “neutral case”). We prove that the ground state energy, for large N, goes exactly as −N7/5, and we give upper and lower bounds on the asymptotic coefficient that agree to within a factor of 22/5.
The binding of polarons, or its absence, is an old and subtle topic. After defining the model we state some recent theorems of ours. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at U=2α, where U is the electronic Coulomb repulsion and αis the polaron coupling constant. Second, if U is large enough, there is no multi-polaron binding of any kind. We also discuss the Pekar-Tomasevich approximation to the ground state energy, which is valid for large α. Finally, we derive exact results, not reported before, about the one-dimensional toy model introduced by E. P. Gross.
The polaron Hamiltonian would be easily soluble were it not for the quartic term appearing therein. It is proposed to substitute for the quartic term a quadratic term having roughly the same properties, and in such a way that the ground-state energy of the new Hamiltonian is rigorously a lower bound for the true energy. With a very small amount of work one can obtain a lower bound as a continuous function of $\ensuremath{\alpha}$ for all values of $\ensuremath{\alpha}$. The result agrees fairly well with the results obtained by other methods. Using the equivalent Hamiltonian one can also obtain an analytic expression for the effective mass, although one cannot say it is a bound for the true effective mass. Futhermore, once one has obtained a lower bound for the energy as a continuous function of the parameters of the Hamiltonian, one can rigorously derive upper and lower bounds for the ground-state expectation values of various operators. For example, it can be shown that for large $\ensuremath{\alpha}$ and large k, $〈{{a}_{\mathrm{k}}}^{*}{a}_{\mathrm{k}}〉\ensuremath{\sim}{\mathrm{k}}^{\ensuremath{-}6}$ and not ${\mathbf{k}}^{\ensuremath{-}2}$ exp(-${\mathbf{k}}^{\ensuremath{-}2}$) as in Pekar's solution. Because of its simplicity, it is possible that this method may have application to other ground-state problems.
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
Abstract The properties of the ground state of the bound large polaron within the Fock approximation are investigated by using a new model spectrum which is known to describe better the polaron at low external fields. The results are compared with those of Matsuura who used the same approximation, obtained from a path integral formulation, with the usual Gaussian and Coulombic trial spectra, Both formulations are variational. It is found that the new spectrum gives better results (lower variational energies) than those of Matsuura for intermediate or large values of electron–phonon coupling and small values of the electrostatic coupling of the electron to the defect. This is related to that fact that the polaron seems to prefer a cylindrical symmetry to a spherical one in the absence of external fields.
The ground-state energy of the optical polaron is calculated by the generalized path-integral formulation. The ground-state approximation is made to simplify the complicated expression, and the Ritz variational and direct integration methods are used to obtain the ground-state energy in the whole range of coupling strength. The results agree with previous work, and it is found that there is a transition for coupling constant ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}9.2$. The ground-state energy obtained by harmonic approximation, which is equivalert to taking Gaussian-like trial wave functions, is compared with those obtained by Pekar's trial wave function.