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Journal of the Physical Society of Japan

Intermediate Coupling Theory of a Polaron in External Fields

Kazuo Hattori

Published Jan 15, 1975• 9 citations
Publisher

Abstract

A new extension of the Lee, Low and Pines (L-L-P) method to the problem of a polaron weakly bound in a Coulomb potential and/or in a weak magnetic field is developed. The effective Hamiltonian of a polaron bound in a Coulomb potential is derived exactly to order P 4 . The validity of replacing P 2 by [\(P_{z}^{2}+eH{\hbar}(2n+1)/c\)] in the field-free polaron energy E ( P 2 ) within the framework of the L-L-P method is shown in the limit of a weak magnetic field H along the z axis in the intermediate coupling case. The energies of a polaron in a Coulomb potential and subject to a weak magnetic field are also discussed.

DOI
10.1143/jpsj.38.51

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Physical review. B, Solid state•Aug 15, 1971•39 citations
A new extension of the Lee, Low and Pines (L-L-P) method to the problem of a polaron weakly bound in a Coulomb potential and/or in a weak magnetic field is developed. The effective Hamiltonian of a polaron bound in a Coulomb potential is derived exactly to order P 4 . The validity of replacing P 2 by [(P_{z}^{2}+eH{\hbar}(2n+1)/c)] in the field-free polaron energy E ( P 2 ) within the framework of the L-L-P method is shown in the limit of a weak magnetic field H along the z axis in the intermediate coupling case. The energies of a polaron in a Coulomb potential and subject to a weak magnetic field are also discussed.
Energy levels of two-dimensional Fr\"ohlich polarons in a magnetic field are calculated variationally by use of a generalization of the Lee-Low-Pines ansatz for the variational wave function allowing the phonon distribution function to depend on the electronic coordinates. It is shown that this choice leads to the correct interpolation between the weak- and strong-field limits. Comparing our ground-state energies with those obtained within second-order perturbation theory and the path-integral method, we argue that neither of these methods gives a satisfactory description of the dynamics of the electron-phonon interaction under the influence of an external field.
After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.
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.
The spin polaron in the t-J model, i.e., a hole dressed by a cloud of virtual magnons of the antiferromagnetic spin background, is treated within the framework of intermediate-coupling theory. To this end the original t-J model is first reformulated in terms of spinless fermions and bosons by means of the generalized Dyson-Maleev representation (DMR). The latter may be regarded as the natural extension of the ordinary DMR of pure (undoped) spin systems to the case where holes are present, and is similar to the one originally proposed by Schmitt-Rink, Varma, and Ruckenstein. The reformulated t-J model, which is reminiscent of the Fr\"ohlich Hamiltonian, is then subjected to a series of unitary transformations, analogous to those employed by Lee, Low, and Pines in their treatment of the Fr\"ohlich polaron. Our approach yields an approximate quasiparticle energy E(k) as well as the corresponding eigenvector. To explore the range of validity of the theory presented here, the analytic expressions are then further analyzed for intermediate (J/t=0.4) and strong (J/t=0.08) coupling, where special attention is paid to the quasiparticle bandwidth W. The intermediate-coupling result for E(k) is in excellent agreement with the dispersion curve recently obtained by Dagotto and co-workers by means of a Green function Monte Carlo method. Surprisingly, even in the strong-coupling range the bandshape remains qualitatively correct.The bandwidth W is rather accurate for weak coupling (J/t\ensuremath{\gtrsim}3), as expected, and still reasonable in the intermediate range 0.4\ensuremath{\lesssim}J/t\ensuremath{\le}3, where it deviates from the correct values by some 10--20 %. Our theory fails, however, to describe the proper behavior of W in the strong-coupling regime. This shows that the limitations of our approach manifest themselves in the bandwidths rather than in the shapes of the dispersion curves. Our conclusion is that intermediate-coupling theory is appropriate for J/t\ensuremath{\gtrsim}0.4, whereas a genuine strong-coupling theory is required for all other cases. \textcopyright{} 1996 The American Physical Society.
The optical polaron bounded in a Coulomb potential is investigated within the modified variational scheme of the Lee-Low-Pines theory (1953) which is valid for the whole range of the electron-phonon coupling. The ground state energy, the mean number of phonons in the cloud around the electron and the spatial localisation of the electron for the optical polaron change abruptly at certain values of the electron-phonon coupling constant alpha and of the Coulomb potential beta . The polaron changes its state according to the values of the parameters alpha and beta . Such features are summarised in the phase diagram.
A general method is presented for obtaining an accurate intermediate-coupling theory from weak- and strong-coupling perturbation theory. The method uses two-point Pad\'e approximants to extrapolate (low-order) expansions about the weak- and the strong-coupling limits into the intermediate-coupling regime. The method is used to evaluate the ground-state energy and effective mass of the polaron with gratifying success. In addition, the weak-coupling perturbation theory of the polaron dispersion relation, ground-state energy, and effective mass are extended to fourth, sixth, and fourth order, respectively. A scheme based on two-point Pad\'e approximants is used to obtain an optimal polaron dispersion relation.