A free-energy theorem for polarons is derived as an extension to arbitrary lattice temperature of the ground-state theorem derived by Lemmens, deSitter, and and Devreese for zero lattice temperature. These theorems relate the exact free energy of the polaron to the exact absorption spectrum integrated over frequency. With approximate solutions in general, and with approximate solutions to many-body problems in particular, it is of interest to know how closely such sum rules are satisfied. We prove that the free-energy theorem is valid for any Feynman-approximation solution to the polaron problem. Using this analytic result, we are able to determine how sensitively satisfying the free-energy theorem depends upon the details of the absorption spectrum. Considering examples of two quite different spectra at zero temperature, we conclude that the ground-state theorem is of little value in discriminating between different calculated absorption spectra. A more sensitive alternative is discussed which makes use of the fluctuation-dissipation theorem for comparing the calculated absorption with the fluctuation energy at each frequency. Since this alternative method requires equality between two functions for each value of their argument (frequency), rather than merely between two numbers as with sum rules, the method easily distinguishes between spectra, each of which may satisfy the ground-state rule very closely. This method is closely related to a self-consistent procedure developed previously. The latter procedure removed the ambiguity of which approximate influence functional to use in path-integral treatments of polaronlike problems.
A free-energy theorem for polarons is derived as an extension to arbitrary lattice temperature of the ground-state theorem derived by Lemmens, deSitter, and and Devreese for zero lattice temperature. These theorems relate the exact free energy of the polaron to the exact absorption spectrum integrated over frequency. With approximate solutions in general, and with approximate solutions to many-body problems in particular, it is of interest to know how closely such sum rules are satisfied. We prove that the free-energy theorem is valid for any Feynman-approximation solution to the polaron problem. Using this analytic result, we are able to determine how sensitively satisfying the free-energy theorem depends upon the details of the absorption spectrum. Considering examples of two quite different spectra at zero temperature, we conclude that the ground-state theorem is of little value in discriminating between different calculated absorption spectra. A more sensitive alternative is discussed which makes use of the fluctuation-dissipation theorem for comparing the calculated absorption with the fluctuation energy at each frequency. Since this alternative method requires equality between two functions for each value of their argument (frequency), rather than merely between two numbers as with sum rules, the method easily distinguishes between spectra, each of which may satisfy the ground-state rule very closely. This method is closely related to a self-consistent procedure developed previously. The latter procedure removed the ambiguity of which approximate influence functional to use in path-integral treatments of polaronlike problems.
A method previously derived for the approximate construction of the nucleon Green's function is here applied to that of the polaron. After making an arbitrary translation of the phonon variables, a linear integral equation for the Green's function is derived by means of a symmetrical treatment and a noncorrelation assumption, in complete analogy to the nucleon problem. This equation is solved through the introduction of a spectral representation in the special case of zero total momentum. The lowest energy state of the system is calculated in terms of the arbitrary translations and then minimized with respect to them. Using the simplest nontrivial cutoff procedure to obtain the variational equation and its solution, results are obtained for values of the coupling parameter $\ensuremath{\alpha}<~3$, and are compared with those of Feynman.
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.
We develop and compare several analytical approximations for the ground state and zero-temperature polaron dispersion in finite-width, nonparabolic conduction bands. The main focus of the work is an extension of the Feynman variational method to a tight-binding lattice, where the effective-mass approximation is no longer applicable. The resulting variational formulation is not restricted to a specific phonon dispersion or electron-phonon interaction and provides a uniform description across weak-, intermediate-, and strong-coupling regimes. In addition, we revisit and generalize other analytical approaches traditionally formulated for continuum polarons, including canonical transformations and self-consistent Wigner-Brillouin-type approximations. For lattice polarons, these methods exhibit qualitative features absent in the continuum case, such as a nontrivial connection between weak- and strong-coupling limits. We show that an improved Wigner-Brillouin scheme yields a momentum-dependent polaron self-energy free of resonances and in good agreement with numerically exact results. All methods are applied to the Holstein model on a tight-binding lattice and are benchmarked against numerically exact calculations, including diagrammatic Monte Carlo (both our calculations and preceding works) and exact diagonalization results. Furthermore, the analytical approaches are extended to polarons with Rashba-type spin-orbit coupling, providing a stringent test of their applicability in systems with nontrivial band structure. Our results demonstrate that the modified Feynman variational method yields ground-state energies and dispersions with accuracy comparable to, and in many cases exceeding, that of other established analytical approaches. The developed framework offers a versatile and reliable analytical description of lattice polarons beyond the continuum approximation.
A ground-state theorem for the Fr\"ohlich polaron is derived. We show that the zeroth moment of the current-current correlation function is proportional to the kinetic energy of the polaron. While the electron-phonon interaction does not depend on the electron mass, the coupling constant $\ensuremath{\alpha}$ does; this allows us to relate the kinetic energy with the ground-state energy using the Feynman-Helmann theorem. As an example we compare the polaron ground-state energy on variational grounds with the ground-state energy obtained from the function $\ensuremath{\chi}(\ensuremath{\omega})$ calculated by Feynman, Hellwarth, Iddings, and Platzman. For $\ensuremath{\alpha}<5$ both ground-state energies turn out to be identical, as can be concluded from numerical calculations.
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 $\ensuremath{\alpha}>1$.
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.