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physica status solidi (b)

Microscopic Derivation of a Transport Equation for the Acoustic Polaron

S. Liu, Josef Sak

Published Oct 1, 1990• 1 citation
Publisher

Abstract

Abstract Starting with the Kubo formula, a transport equation for the acoustic polaron is derived. The collision integral in the transport equation contains not only irreducible vertex parts, but also “products” of them. These are products of irreducible verteces connected by non‐singular parts of Green's functions. They are important when one considers the second or higher order perturbations, as in the case of an acoustic polaron at very low temperatures.

DOI
10.1002/pssb.2221610222

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Abstract Starting with the Kubo formula, a transport equation for the acoustic polaron is derived. The collision integral in the transport equation contains not only irreducible vertex parts, but also “products” of them. These are products of irreducible verteces connected by non‐singular parts of Green's functions. They are important when one considers the second or higher order perturbations, as in the case of an acoustic polaron at very low temperatures.
The transport dynamics of a large polaron has been studied in a one-dimensional system whose underlying vibrational spectrum is acoustic in character. Such a polaron moves as a heavy quasiparticle with transport relaxation time, $\ensuremath{\tau}$, determined mainly by collisions with thermally ambient phonons. Concrete results show that $\ensuremath{\tau}$ decreases monotonically with increasing temperature, in sharp contrast to the previously treated problem of a large polaron in a one-dimensional Einstein lattice.
Abstract A general method for studying the properties of free and bound polarons at arbitrary strength of electron‐phonon coupling is given from a unified point of view by first performing a suitable canonical transformation of the electron and phonon field operators and subsequently applying the double‐time Green's function formalism to the transformed polaron‐phonon Hamiltonian. Calculating the one‐polaron Green's function in the lowest approximation and employing the variational principle for the free energy, a pair of coupled differential equations is derived for the effective wave function and a trial function as introduced by the canonical transformation. From these equations, the weakly excited polaron energy levels at low temperatures can be found in the entire range of coupling constants when external fields are present. In this paper, detailed calculations are confined to the optical‐polaron problem in the zero‐field case. Intermediate‐ and strong‐coupling formulas for temperature‐dependent polaron self‐energy and effective mass are given. The strong‐coupling regime appears to increase with temperature.
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<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.
Motivated by the recently observed failure of the kinetic theory for the bulk viscosity, we in turn revisit the shear viscosity and the thermal conductivity of two-component fermions with a zero-range interaction both in two and three dimensions. In particular, we show that their Kubo formula evaluated exactly in the high-temperature limit to the lowest order in fugacity is reduced to the linearized Boltzmann equation. Previously, such a microscopic derivation of the latter was achieved only incompletely corresponding to the relaxation-time approximation. Here, we complete it by resuming all contributions that are naively higher orders in fugacity but become comparable in the zero-frequency limit due to the pinch singularity, leading to a self-consistent equation for a vertex function identical to the linearized Boltzmann equation. We then compute the shear viscosity and the thermal conductivity in the high-temperature limit for an arbitrary scattering length and find that the Prandtl number exhibits a nonmonotonic behavior slightly below the constant value in the relaxation-time approximation.
We derive a Kubo-type formula that describes proper non-equilibrium stationary states for finite-size systems. We first argue that the usual Kubo formula considers only equilibrium states of the perturbed system, which are inappropriate to describe transport properties. Moreover, we show that the standard Kubo formula for the dc conductance is only appropriate in the thermodynamic limit. We demonstrate that taking into consideration explicitly the coupling to the leads/baths solves both problems. This approach results in well-behaved response functions, without the singular contributions from degenerate states encountered when Kubo formulae for infinite-size systems are inappropriately used for finite-size systems. We also derive a second, more efficient formulation which is valid only for a set of special physical quantities, which includes the charge current density operator.
Abstract An analytic theory for the spectral function for electrons coupled with phonons is formulated in the adiabatic limit. In the case when the chemical potential is large and negative μ → − ∞ the ground state does not have the adiabatic deformation and the spectral function is defined by the standard perturbation theory. In this limit we use the diagram technique in order to formulate an integral equation for the renormalized vertex. The spectral function was evaluated by solving the Dyson’s equation for the self-energy with the renormalized vertex. The moments of the spectral function satisfy the exact sum rules up to the 7th moment. In the case when the chemical potential is pinned at the polaron binding energy the spectral function is defined by the ground state with a nonzero adiabatic deformation. We calculate the spectral function with the finite polaron density in the adiabatic limit. We also demonstrate how the sum rules for higher moments may be evaluated in the adiabatic limit. Contrary to the case of zero polaron density the spectral function with the finite polaron concentration has some contributions which are characteristic for polarons.