We present a numerically exact study of a polaron with quadratic coupling to the oscillator displacement, or ${X}^{2}$ polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes and the entire range of electron-phonon coupling ${g}_{2}$ , from the system's stability threshold at attractive ${g}_{2}=\ensuremath{-}1$ to arbitrary strong repulsion at ${g}_{2}\ensuremath{\gg}1$. The key properties of ${X}^{2}$ polarons prove dramatically different from their linear counterparts. They (i) are insensitive even to large quadratic coupling except in the antiadiabatic limit near the threshold of instability at attraction, (ii) depend only on the adiabatic ratio but are insensitive to the electron dispersion and dimension of space, and (iii) feature weak lattice deformations even at the instability point. Our results are of direct relevance to the properties of electrons at low densities in polar materials, including recent proposals for their superconducting states.We present a numerically exact study of a polaron with quadratic coupling to the oscillator displacement, or ${X}^{2}$ polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes and the entire range of electron-phonon coupling ${g}_{2}$, from the system's stability threshold at attractive ${g}_{2}=\ensuremath{-}1$ to arbitrary strong repulsion at ${g}_{2}\ensuremath{\gg}1$. The key properties of ${X}^{2}$ polarons prove dramatically different from their linear counterparts. They (i) are insensitive even to large quadratic coupling except in the antiadiabatic limit near the threshold of instability at attraction, (ii) depend only on the adiabatic ratio but are insensitive to the electron dispersion and dimension of space, and (iii) feature weak lattice deformations even at the instability point. Our results are of direct relevance to the properties of electrons at low densities in polar materials, including recent proposals for their superconducting states.

arXiv:2212.01475Physical review. B./Physical review. BMar 22, 202318 citations

Aspects of Polaron Theory

Nikolay N. Bogolubov, Nikolay N. Bogolubov

WORLD SCIENTIFIC eBooksSep 1, 200831 citations

A generalized functional approach to the polaron problem

Sergey Gorshkov, Victor Dmitrievich Lakhno, K. Rodriges +1 more

It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.

Soviet physics. DokladyOct 1, 19840 citations

The Fröhlich Hamiltonian

Ashok Chatterjee, Soma Mukhopadhyay

The picture of the polaron conceived by Frohlich is again that of a complex consisting of an electron and the lattice distortion induced by it, but unlike the case of Landau-Pekar polaron, the complex can be found anywhere in the lattice. The Frohlich model considers a low-lying or a slow electron so that the de Broglies wavelength is much larger than the lattice constant, and in this limit, the structure of the lattice becomes unimportant and lattice can be considered to be a continuum. In the Frohlich polaron model, one considers an extra slow electron in the conduction band of the crystal that interacts with the fields associated with the aforementioned polarization vectors to give rise to the electron-phonon Hamiltonian relevant for the polaron problem. Because the electron is low-lying, it will have a large de Broglie wave length associated with it.The picture of the polaron conceived by Frohlich is again that of a complex consisting of an electron and the lattice distortion induced by it, but unlike the case of Landau-Pekar polaron, the complex can be found anywhere in the lattice. The Frohlich model considers a low-lying or a slow electron so that the de Broglies wavelength is much larger than the lattice constant, and in this limit, the structure of the lattice becomes unimportant and lattice can be considered to be a continuum. In the Frohlich polaron model, one considers an extra slow electron in the conduction band of the crystal that interacts with the fields associated with the aforementioned polarization vectors to give rise to the electron-phonon Hamiltonian relevant for the polaron problem. Because the electron is low-lying, it will have a large de Broglie wave length associated with it.

CRC Press eBooksOct 9, 20180 citations

We present a numerically exact study of a polaron with quadratic coupling to the oscillator displacement, or

{X}^{2}

polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes and the entire range of electron-phonon coupling

{g}_{2}

, from the system's stability threshold at attractive ${g}_{2}=\ensuremath{-}1$ to arbitrary strong repulsion at ${g}_{2}\ensuremath{\gg}1$. The key properties of

{X}^{2}

polarons prove dramatically different from their linear counterparts. They (i) are insensitive even to large quadratic coupling except in the antiadiabatic limit near the threshold of instability at attraction, (ii) depend only on the adiabatic ratio but are insensitive to the electron dispersion and dimension of space, and (iii) feature weak lattice deformations even at the instability point. Our results are of direct relevance to the properties of electrons at low densities in polar materials, including recent proposals for their superconducting states.