An alternative derivation is presented of the Feynman-Hellwarth-Iddings-Platzman (FHIP) result for the impedance function of a polaron. In the present calculation only simple operator techniques are used. The FHIP result is worked out in such a way that we are able to identify which scattering processes are incorporated. The inverse problem is also solved.
An alternative derivation is presented of the Feynman-Hellwarth-Iddings-Platzman (FHIP) result for the impedance function of a polaron. In the present calculation only simple operator techniques are used. The FHIP result is worked out in such a way that we are able to identify which scattering processes are incorporated. The inverse problem is also solved.
The optical absorption of polarons at rest at zero temperature is calculated starting from the Feynman-Hellwarth-Iddings-Platzman (FHIP) theory of the impedance. The results are compared with the results of theories whose physical interpretation is clearer [weak-coupling theory of Gurevich, Lang, and Firsov (GLF) and product-ansatz strong-coupling theory of Kartheuser, Evrard, and Devreese (KED)] in order to obtain a better understanding of the FHIP approximation. We are particularly interested in the possible role of lattice relaxation [leading to relaxed excited states (RES)] in the optical absorption process. If the FHIP perturbation method were used to expand the conductivity (this would be the normal procedure), essentially Franck-Condon transitions would be found in the spectrum, and lattice relaxation would be absent. In this case the results do not fit with the product ansatz and provide merely the asymptotic limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, where $\ensuremath{\alpha}$ is the electron-phonon coupling constant. If, however, the impedance function rather than the conductivity is expanded (as preferred by FHIP for intuitive reasons, without further justification) more reliable results for the optical absorption appear. For $\ensuremath{\alpha}>5$, intense absorption peaks now occur, which presumably correspond to transitions to RES, and the results are in qualitative agreement with the predictions of the product-ansatz treatment in this coupling region. Also in the limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ the correct behavior is found. For $3\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\lesssim}5$, the interpretation of the results is somewhat delicate but the possibility that RES contribute to the oscillator strength as soon as $\ensuremath{\alpha}>3$ should be considered. The results so obtained for the optical absorption seem reliable at all $\ensuremath{\alpha}$. This provides an indirect justification for the expansion of $Z(\ensuremath{\Omega})$ rather than $\frac{1}{Z(\ensuremath{\Omega})}$ in FHIP theory and a confirmation of the qualitative strong-coupling predictions of KED. The present study indicates that optical absorption peaks due to free polarons should be observable experimentally in crystals for which $\ensuremath{\alpha}>1$.
The Feynman path-integral variational approach to the polaron problem, along with the associated Feynman-Hellwarth-Iddings-Platzman (FHIP) linear-response mobility theory, provides a computationally amenable method to predict the frequency-resolved temperature-dependent charge-carrier mobility, and other experimental observables in polar semiconductors. We show that the FHIP mobility theory predicts non-Drude transport behavior, and shows remarkably good agreement with the recent diagrammatic Monte Carlo mobility simulations of Mishchenko et al. [Phys. Rev. Lett. 123, 076601 (2019)] for the abstract Fr\"ohlich Hamiltonian. We extend this method to multiple phonon modes in the Fr\"ohlich model action. This enables a slightly better variational solution, as inferred from the resulting energy. We carry forward this extra complexity into the mobility theory, which shows a richer structure in the frequency and temperature-dependent mobility, due to the different phonon modes activating at different energies. The method provides a computationally efficient and fully quantitative method of predicting polaron mobility and response in real materials.
We have obtained an approximate expression for the impedance function at all frequencies, temperatures, and coupling strengths of an electron coupled to a polar lattice (a system commonly called a polaron). The starting point for the calculation is the quantum mechanical expression for the expected current. The phonon coordinates are eliminated from this expression by well-known field-theory techniques. The resulting exact "influence functional" is then approximated by a corresponding quadratic "influence functional" which, it is hoped, imitates the real polaron. Correction terms are computed to account for the difference between the approximate impedance and the exact polaron impedance in a manner closely analogous to Feynman's treatment of the polaron self-energy. In fact, the analytic evaluation of the expression for the impedance obtained here is carried out using the approximate "influence functional" that was successfully employed in minimizing the binding (and free) energy of the polaron in earlier calculations. However, the accuracy obtained using this approximation, for the impedance calculation, is less satisfactory and its limitations are discussed. Nevertheless, beginning at intermediate coupling strengths, the approximate impedance produces a level structure of increasing complexity and narrowing resonances as the coupling strengthens. This suggests that further refinements may be fruitful. Methods for finding a better quadratic influence functional for use in our impedance expression as well as ways of improving the expression further are suggested. A comparison of our results with those of the Boltzmann equation points up interesting differences which arise from reversing the order of taking limits of zero frequency and coupling.
A method is presented to obtain the linear response coefficients of a system coupled to a bath. The method is based on a systematic truncation of the Liouville equation for the reduced distribution function. The first-order truncation results are expected to be accurate in the low-temperature and weak-coupling regime. Explicit expressions for the conductivity of the Fröhlich polaron are obtained, and the discrepancy between the Kadanoff and the Feynman-Hellwarth-Iddings-Platzman mobility is elucidated.
An variational expression for the zero temperature polaron impedance is obtained by minimizing the free energy in a generalized quadratic Feynman model. The impedance function of the quadratic model serves as the variational parameter. It is shown that a very small change in the energy can be accompanied by a large change in the optical conductivity. This is related to the insensitivity of the Jensen-Feynman free energy to the UV properties of the model. Analytic and numeric results are derived for the Fröhlich polaron in weak and strong coupling. Standard results are recovered at weak coupling but, more importantly, strong coupling inconsistencies are removed.
Direct evaluation of Feynman's functional integral for the polaron maps it into a standard type of many-body theory. The perturbation expansion is shown to be well controlled and does not require any renormalization. This perturbation expansion has the pleasing feature that nth-order terms in the polaron coupling constant are obtained from nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth-order perturbation calculation to generate the same results. In contrast to previous beliefs, the method is shown to enable the calculation of the polaron effective mass and relaxation time directly from the functional-integral expression. The ground-state energy is computed to second order in the polaron coupling constant to illustrate the use of the method and to establish agreement with previous perturbation treatments. In addition, we provide the first perturbation calculation of the polaron effective mass through second order in the coupling constant. Pad\'e approximants are shown to provide a vehicle for analytically extending the perturbation results into the region of intermediate and strong couplings, and the procedure yields reasonable agreement with variational calculations for values of the polaron coupling constant up to about $\ensuremath{\alpha}=10$. In addition, conditions are derived under which the original polaron action, used in the functional integral, reduces to some of the best variational trial actions utilized in the literature.