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Physical Review

Density Matrix Formulation of Small-Polaron Motion

Lionel R. Friedman

Published Jul 6, 1964• 77 citations
Publisher

Abstract

A density-matrix treatment of small-polaron motion is presented for the case in which the electronic overlap term of the total Hamiltonian is a small perturbation. The principal result of the density matrix formalism is that total small-polaron mobility can be expressed as the sum of a band part $〈{{v}_{o}}^{2}\ensuremath{\tau}〉$, characteristic of the low-temperature regime ($T<{T}_{t}$), plus a part (${W}_{T}{a}^{2}$) describing the hopping motion dominant at high temperatures ($T>{T}_{t}$). This verifies the separation of the above two types of motion made on the basis of physical arguments. In addition, the present treatment avoids certain formal divergences in the integrals for the jump probabilities. Furthermore, the hopping contribution does not require localization of the polaron at a particular site, but follows from a translationally invariant formulation. These results are obtained, in part, from lowest order Boltzmann equations which are derived both in the local-site and polaron-band representations. The principal contributions to the scattering terms of the Boltzmann equations are determined by interference effects between the matrix elements, which are examined in some detail.

DOI
10.1103/physrev.135.a233

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Theory of Polaron Mobility

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The polaron mobility is calculated by making use of the general theory of electrical conductivity. We take the states determined by Feynman's trial action as the unperturbed states and treat the difference between the true action and the trial action as a perturbation. Numerical values of the polaron mobility at very .low temperatures are given and are dis· cussed in comparison with the results obtained by Shultz and .by Morita. § I. Introduction We have investigated the static properties of polaron at finite temperatures in the previous paper1> (hereafter referred to as I). We shall investigB;te the polaron mobility in the present paper. In the same way as in I, we calculate the polaron mobility by using Feynman's path-integral method. In the polaron problem, the interaction between an electron and lattice vibrations is so large that the usual perturbation theoretic treatment fails. Therefore the calculation of the mobility in the present paper will be based on the general theory of electrical conductivity recently developed by many authors,2 > which is applicable to a system which does not allow us to set up the Boltzmann equation. We take the states determined by Feynman's trial action as unperturbed states and treat the difference between the trial action and the true action as a perturba­ tion which yields a decay of electronic current correlation. The following assumptions will be made in the present paper. i) The electrical conductivity is determined by the asymptotic form of a correlation function of electronic current. ii) This asymptotic form has the property of exponential decay in time. These assumptions are valid in the case of weak interaction. It is not sure whether or not they are applicable to the present problem which includes the case of strong interaction. In the present treat­ ment, however, ~he states determined by Feynman's trial action instead of the states of a free electron are chosen as the unperturbed states. It is expected that the perturbation in the former case may be made weaker than in the latter case and that the above mentioned assumptions may be valid. In fact the self­ energy of polaron at 0°K calculated by using these assumptions agrees with that obtained by Feynman3> as will be shown _in § 3. Therefore, the approxi

Progress of Theoretical Physics•Apr 1, 1961•5 citations

Polaron conductivity in the narrow-band Hubbard model

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We consider a Hubbard model in which electrons are locally coupled to the lattice and form polarons. The frequency-dependent polaron conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{pot}}$(\ensuremath{\omega}) is calculated in the narrow-band limit as a function of electron density. There are two contributions: a band term that arises from coherent motion of dressed electrons and a hopping term that describes the incoherent process by which electrons ``shake off'' their polaron distortion. The possible relevance of these results to the structure in \ensuremath{\sigma}(\ensuremath{\omega}) observed in high-${\mathit{T}}_{\mathit{c}}$ superconductors is briefly discussed.

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Polarons by translationally invariant diagrammatic perturbation theory

O. S. Barišić, S. Barišić

The structure of the translationally-invariant diagrammatic perturbation theory for one polaron is examined on the 1D discrete lattice described by the Holstein Hamiltonian. The latter is characterized by the electron hopping t, the phonon frequency ω_0 and the electron-phonon coupling g. It is shown that the polaron localization (and translation) properties are contained in the electron propagator of one electron, intermittently added to the lattice, and/or in the phonon correlation function with one electron permanently present in the lattice. The order by order analysis in g/ω_0 shows that the expansion of the irreducible electron self-energy corresponds to the expansion of the phonon correlation function, rather than of the irreducible phonon self-energy. The range of polaronic correlations is determined in this way. For small t/ω_0 and already to the second order g/ω_0 small, the electron-lattice correlation becomes very short ranged, i.e. the polaron is already localized to one site, although the overall translational symmetry remains unbroken. For large t/ω0, the second order result is meaningful up to large g/ω_0 ≈ (t/ω_0)^\frac14, where it becomes degenerate with the results for the large adiabatic Holstein polaron. This suggests that the translationally invariant perturbation theory crosses then over smoothly, without symmetry breaking, into the adiabatic, continuous quantum limit, as rigorously demonstrated in the companion paper. Thus the quantum theory of the large adiabatic Holstein polaron provides a simple, instructive example of the quantum crossover which replaces the behavior in the quantum critical point.

FizA•Jan 1, 2005•0 citations

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We have developed a new equation of motion Green's function theory for the large polaron, that is completely nonperturbative in origin, and therefore gives a unified treatment of the polaron ground state energy for all interaction strengths. The theory also incorporates translational invariance rigorously, a priori. We find the following new results. (a) Previous theories that are valid for all coupling strengths are based on a nonperturbative summation of infinite subsets of entire diagrams (equations of motion), rather than on recoil parts of all diagrams. We can thus recover lowest order results of Feynman for example, via a translationally noninvariant form of the Fock approximation, which resembles that reported in a previous work. (b) The total ground state energy we obtain in the strong coupling limit is E0 = −0.426α2 to lowest order in the coupling constant α, a result substantially below that of previous calculations, E0 = −0.106α2 (using gaussian trial functions in both cases). Our result is also below the lower bound of Lieb and Yamazaki, although we show in a following paper that our result is an upper bound. (c) The origin of the difference of our results compared to that of other strong coupling theories is several fold, and will be presented in detail in subsequent papers. However, the essence of the difference is that the phonons in the field, via their interaction with the electron, assume the role of independent degrees of freedom, in the limit of an infinite number of phonons. We then have a field theoretic problem with an infinite number of degrees of freedom which can exhibit a totally different physical behavior compared to one with a finite number of degrees of freedom. This aspect of the problem has not been given explicit recognition in all other theories, including that of Lieb and Yamazaki.

Canadian Journal of Physics•Jun 1, 1973•3 citations

Solitons, generalised master equations and small-polaron propagation

Vladislav Čápek

For an electron interacting with harmonic dispersive phonons in a periodic lattice, the possibility of wave (soliton)-like solutions of the generalised master equations (GME) for the electron site occupation probabilities is investigated. For the current form of the small-polaron Hamiltonian, the long-time asymptotics of the memory functions (kernels of the GME) in the small-polaron basis are discussed to all orders in the electron hopping integrals.Using this result, one obtains (for the initially localised electron) the t 2 -behaviour of the mean-square electron displacement at zero temperature. This corresponds to existing soliton solutions of the Schrodinger equation. At non-zero temperatures, the wave solution is damped.

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Direct path-integral treatment of the polaron problem

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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 $n\mathrm{th}$-order terms in the polaron coupling constant are obtained from $n\mathrm{th}$-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a $2n\mathrm{th}$-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.

Physical review. B, Condensed matter•Apr 15, 1983•23 citations
A density-matrix treatment of small-polaron motion is presented for the case in which the electronic overlap term of the total Hamiltonian is a small perturbation. The principal result of the density matrix formalism is that total small-polaron mobility can be expressed as the sum of a band part 〈vo2\ensuremathτ〉〈{{v}_{o}}^{2}\ensuremath{\tau}〉〈vo​2\ensuremathτ〉, characteristic of the low-temperature regime (T<TtT<{T}_{t}T<Tt​), plus a part (WTa2{W}_{T}{a}^{2}WT​a2) describing the hopping motion dominant at high temperatures (T>TtT>{T}_{t}T>Tt​). This verifies the separation of the above two types of motion made on the basis of physical arguments. In addition, the present treatment avoids certain formal divergences in the integrals for the jump probabilities. Furthermore, the hopping contribution does not require localization of the polaron at a particular site, but follows from a translationally invariant formulation. These results are obtained, in part, from lowest order Boltzmann equations which are derived both in the local-site and polaron-band representations. The principal contributions to the scattering terms of the Boltzmann equations are determined by interference effects between the matrix elements, which are examined in some detail.
The polaron mobility is calculated by making use of the general theory of electrical conductivity. We take the states determined by Feynman's trial action as the unperturbed states and treat the difference between the true action and the trial action as a perturbation. Numerical values of the polaron mobility at very .low temperatures are given and are dis· cussed in comparison with the results obtained by Shultz and .by Morita. § I. Introduction We have investigated the static properties of polaron at finite temperatures in the previous paper1> (hereafter referred to as I). We shall investigB;te the polaron mobility in the present paper. In the same way as in I, we calculate the polaron mobility by using Feynman's path-integral method. In the polaron problem, the interaction between an electron and lattice vibrations is so large that the usual perturbation theoretic treatment fails. Therefore the calculation of the mobility in the present paper will be based on the general theory of electrical conductivity recently developed by many authors,2 > which is applicable to a system which does not allow us to set up the Boltzmann equation. We take the states determined by Feynman's trial action as unperturbed states and treat the difference between the trial action and the true action as a perturba­ tion which yields a decay of electronic current correlation. The following assumptions will be made in the present paper. i) The electrical conductivity is determined by the asymptotic form of a correlation function of electronic current. ii) This asymptotic form has the property of exponential decay in time. These assumptions are valid in the case of weak interaction. It is not sure whether or not they are applicable to the present problem which includes the case of strong interaction. In the present treat­ ment, however, ~he states determined by Feynman's trial action instead of the states of a free electron are chosen as the unperturbed states. It is expected that the perturbation in the former case may be made weaker than in the latter case and that the above mentioned assumptions may be valid. In fact the self­ energy of polaron at 0°K calculated by using these assumptions agrees with that obtained by Feynman3> as will be shown _in § 3. Therefore, the approxi
We consider a Hubbard model in which electrons are locally coupled to the lattice and form polarons. The frequency-dependent polaron conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{pot}}$(\ensuremath{\omega}) is calculated in the narrow-band limit as a function of electron density. There are two contributions: a band term that arises from coherent motion of dressed electrons and a hopping term that describes the incoherent process by which electrons ``shake off'' their polaron distortion. The possible relevance of these results to the structure in \ensuremath{\sigma}(\ensuremath{\omega}) observed in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors is briefly discussed.
The structure of the translationally-invariant diagrammatic perturbation theory for one polaron is examined on the 1D discrete lattice described by the Holstein Hamiltonian. The latter is characterized by the electron hopping t, the phonon frequency ω_0 and the electron-phonon coupling g. It is shown that the polaron localization (and translation) properties are contained in the electron propagator of one electron, intermittently added to the lattice, and/or in the phonon correlation function with one electron permanently present in the lattice. The order by order analysis in g/ω_0 shows that the expansion of the irreducible electron self-energy corresponds to the expansion of the phonon correlation function, rather than of the irreducible phonon self-energy. The range of polaronic correlations is determined in this way. For small t/ω_0 and already to the second order g/ω_0 small, the electron-lattice correlation becomes very short ranged, i.e. the polaron is already localized to one site, although the overall translational symmetry remains unbroken. For large t/ω0, the second order result is meaningful up to large g/ω_0 ≈ (t/ω_0)^\frac14, where it becomes degenerate with the results for the large adiabatic Holstein polaron. This suggests that the translationally invariant perturbation theory crosses then over smoothly, without symmetry breaking, into the adiabatic, continuous quantum limit, as rigorously demonstrated in the companion paper. Thus the quantum theory of the large adiabatic Holstein polaron provides a simple, instructive example of the quantum crossover which replaces the behavior in the quantum critical point.
We have developed a new equation of motion Green's function theory for the large polaron, that is completely nonperturbative in origin, and therefore gives a unified treatment of the polaron ground state energy for all interaction strengths. The theory also incorporates translational invariance rigorously, a priori. We find the following new results. (a) Previous theories that are valid for all coupling strengths are based on a nonperturbative summation of infinite subsets of entire diagrams (equations of motion), rather than on recoil parts of all diagrams. We can thus recover lowest order results of Feynman for example, via a translationally noninvariant form of the Fock approximation, which resembles that reported in a previous work. (b) The total ground state energy we obtain in the strong coupling limit is E0 = −0.426α2 to lowest order in the coupling constant α, a result substantially below that of previous calculations, E0 = −0.106α2 (using gaussian trial functions in both cases). Our result is also below the lower bound of Lieb and Yamazaki, although we show in a following paper that our result is an upper bound. (c) The origin of the difference of our results compared to that of other strong coupling theories is several fold, and will be presented in detail in subsequent papers. However, the essence of the difference is that the phonons in the field, via their interaction with the electron, assume the role of independent degrees of freedom, in the limit of an infinite number of phonons. We then have a field theoretic problem with an infinite number of degrees of freedom which can exhibit a totally different physical behavior compared to one with a finite number of degrees of freedom. This aspect of the problem has not been given explicit recognition in all other theories, including that of Lieb and Yamazaki.
For an electron interacting with harmonic dispersive phonons in a periodic lattice, the possibility of wave (soliton)-like solutions of the generalised master equations (GME) for the electron site occupation probabilities is investigated. For the current form of the small-polaron Hamiltonian, the long-time asymptotics of the memory functions (kernels of the GME) in the small-polaron basis are discussed to all orders in the electron hopping integrals.Using this result, one obtains (for the initially localised electron) the t 2 -behaviour of the mean-square electron displacement at zero temperature. This corresponds to existing soliton solutions of the Schrodinger equation. At non-zero temperatures, the wave solution is damped.
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 nthn\mathrm{th}nth-order terms in the polaron coupling constant are obtained from nthn\mathrm{th}nth-order perturbation theory, whereas the traditional perturbation formulation in terms of creation and annihilation operators requires a 2nth2n\mathrm{th}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.