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physica status solidi (b)

On the Strong Coupling Techniques of Polaron Theory

J. T. Devreese, R. P. Evrard

Published Jan 1, 1965• 7 citations
Publisher

Abstract

Abstract The validity of Pekar's polaron theory is investigated by using it to solve Gross's simplified model in which the electron interacts with only one lattice oscillator. The results are compared with the exact solutions. This shows as expected, that Pekar's approximation provides an accurate strong coupling limit. The accuracy decreases with the mean number of virtual phonons, which is used to define the coupling strength. For a certain value of this parameter, Pekar's method becomes less accurate than the Lee‐Low‐Pines method. At this point the relative error is about 15%. The present study makes it possible to introduce an “intermediate coupling region” in Gross's model. In this region the relative error of the Lee‐Low‐Pines approximation is about 10%.

DOI
10.1002/pssb.19650090213

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Journal of the Physical Society of Japan•Jan 15, 1975•105 citations
Abstract The validity of Pekar's polaron theory is investigated by using it to solve Gross's simplified model in which the electron interacts with only one lattice oscillator. The results are compared with the exact solutions. This shows as expected, that Pekar's approximation provides an accurate strong coupling limit. The accuracy decreases with the mean number of virtual phonons, which is used to define the coupling strength. For a certain value of this parameter, Pekar's method becomes less accurate than the Lee‐Low‐Pines method. At this point the relative error is about 15%. The present study makes it possible to introduce an “intermediate coupling region” in Gross's model. In this region the relative error of the Lee‐Low‐Pines approximation is about 10%.
This chapter describes that the methods of Gross and Huybrechts, because these methods, with some shortcomings notwithstanding, have gained wider acceptance and are still being used by many researchers. One of the early methods which attempted to deal with the entire coupling range was by Hohler. In the Hohler method, one chooses a wave function by taking a linear combination of Pekar-type wave functions such that the trial wave function is an eigenfunction of the total momentum. The Lee-Low-Pines-Huybrechts (LLPH) method is a modified version of the Low-Lee-Pines (LLP) method that was developed by Huybrechts to address small-coupling region, as well. The LLP method was also modified by Gross to make it valid for the entire range of the coupling constant. This method can be easily seen to give the weak- and the strong-coupling limits. For the polaron radius, Landau-Pekar method however gives an infinite value.
In this work we demonstrate that accurate ground state wave functions may be constructed for polarons in a fully ab initio setting across the wide range of couplings associated with both the large and small polaron limits. We present a single general unitary transformation approach which encompasses an ab initio version of the Lee-Low-Pines theory at weak coupling and the coherent state Landau-Pekar framework at strong coupling while interpolating between these limits in general cases. We show that perturbation theory around these limits may be performed in a facile manner to assess the accuracy of the approach, as well as provide an independent route to the ab initio properties of polarons. We test these ideas on the case of LiF, where the electron-polaron is expected to be large and relatively weakly coupled, while the hole-polaron is expected to be a strongly coupled small polaron.
In this work, we demonstrate that accurate ground-state wave functions may be constructed for polarons in a fully ab initio setting across the wide range of couplings associated with both the large and small polaron limits. We present a single general unitary transformation approach which encompasses an ab initio version of the Lee-Low-Pines theory at weak coupling and the coherent-state Landau-Pekar framework at strong coupling, while interpolating between these limits in general cases. We show that perturbation theory around these limits may be performed in a facile manner to assess the accuracy of the approach, as well as provide an independent route to the ab initio properties of polarons. We test these ideas on the case of LiF, where the electron polaron is expected to be large and relatively weakly coupled, while the hole polaron is expected to be a strongly coupled small polaron.
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.
A detailed consideration is given to the translation-invariant theory of Tulub polaron constructed without the use of Pekar ansatz.A fundamental result of the theory is that the value of the polaron energy is lower than that obtained on the basis of Pekar ansatz which was considered as an asymptotically exact solution in the strong coupling limit.In the case of bipolarons the theory yields the best values of the coupling energy and critical parameters of their stability.Numerous physical consequences of the existence of translation-invariant polarons and bipolarons are discussed.