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Physical review. B, Condensed matter

Ground-state energy of the optical polaron

Chih-Yuan Lu, Chi-Kuang Shen

Published Oct 15, 1982• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.26.4707

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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 \ensuremathαc\ensuremath≃9.2{\ensuremath{\alpha}}_{c}\ensuremath{\simeq}9.2\ensuremathαc​\ensuremath≃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.
Based on the unitary transformation, a general treatment of the bound polaron is proposed. In distinction from earlier calculations, the present method gives explicit expressions for not only the energy but also the wave functions both in the ground state and in the excited states such as the relaxed excited state and the phonon scattering state. These expressions involve a summation on the complete set of eigenstates of a trial Hamiltonian, and in actual calculations the Coulomb Hamiltonian and the harmonic oscillator Hamiltonian are chosen as trial Hamiltonians. For the ground state, the energy expression is the same as that obtained by Matsuura with the path integral method. The present method is valid in the whole region of the electron-phonon coupling constant α and Coulomb force β except for β→0 and α>6.
Recently Feynman's path-integral formalism of the Fröhlich optical polaron problem is generalized, by which it is easy and natural to get the second-order perturbation result in the weak-coupling case and the Pekar's result in the strongcoupling case, even in the crudest ground-state approximation. By using numerical method to the equations which are given by the ground-state approximation, the ground-state energy of the electron-optical phonon system can obtained for the overall range of coupling strength. In addition, a self-consistent set of equations is derived for a central optimized potential without using the ground-state approximation. This effective local, central potential is the solution of a linear integral equation which was similar to the extensive work of optimized-potential model (OPM) in atomic physics. By using OPM formalism, no particular choice of interaction form is taken, therefore, except the approximation by Jensen's inequality, the ground-state energy of the electron-optical phonon interacting system can be obtained without any approximation.
Taking the same trial wave function, the ground-state energy of Fröhlich polaron is investigated by variational method and coherent-state expansion (CSE) one, respectively. Within the accuracy to α2α^{2}α2(the electron-phonon coupling constant), both methods can give the same analytic expressions of polaron ground-state energy as the function of electron-phonon coupling constant. We find that the CSE method can much simplify the calculation and shows more advantages in higher order approximations.
The ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation E2D{E}_{2\mathrm{D}}E2D​(\ensuremath{\alpha})=(2/3)E3D{E}_{3\mathrm{D}}E3D​(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where E2D{E}_{2\mathrm{D}}E2D​ (E3D{E}_{3\mathrm{D}}E3D​) is the ground-state energy of the 2D (3D) polaron.
The system of a charged impurity in an interacting Bose gas has gained significant attention due to the long-range ion-atom interactions and the study of transport properties. Here, the ground-state energy of a charged Bose polaron is calculated within the Bogoliubov approximation using a generalized Feynman variational path-integral approach, which obtained accurate results for other polaron problems. We find that compared to other existing approaches, our approach gives similar and improved results while being numerically less intensive. Optical absorption, which is a standard tool to study the solid-state polaron, is introduced here as a probing method for the Bose polaron. It is calculated within the Bogoliubov approximation for weak ion-atom interactions, and the effect of finite temperature is studied. We show that the coupling of the ion to an oscillating external electric field offers a straightforward experimental probe for the charged polaron in a Bose gas, different from but complementary to existing spectroscopic techniques.