Energy–momentum relations in polaron problems with arbitrary coupling

Abstract

A theory is derived suitable to obtain t h e energy-momentum dependence for an arbitrarily coupling polaron. Using a variational principle general expressions are given which describe this dependence when a n arbitrary shape of the electron surface of constant energy is considered and the polarization field is approximated by any number of fictitious particles. This method is able t o yield the lowest band of electrons interacting with any mode of lattice vibrations. I n particular, numerical results for the energy, depending on the total momentum, are obtained for isotropic optical and piezoelectric polarons in a wide range of a dimensionless coupling constant. Comparison of the results with known ones (only weak and strong-coupling limits were investigated earlier) shows, that the energy values presented lie lower, though they represent themselves an upper bound for the exact polaron energymomentum relation.

Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?

“…We mention a particularly simple example, which was extensively studied in the literature: putting ^(k)^ |l?k| _1 , where B is a real, symmetrical matrix with strictly positive eigenvalues, one can qualitatively describe optical polarons in anisotropic crystals. As for variational and Monte Carlo calculations, we refer to Kahn (1968); Pekar (1969); Pekar, Sheka, and Dmitrenko (1973); Okamoto (1974); Pekar, Khazan, and Sheka (1974); Pokatilov and Tarakanova (1974); Hattori (1975a); ; Sheka, Khazan, and Mozdor (1975); and Gerlach and Schliffke(1984).…”

Note on the Path-Integral Variational Approach in Many-Body Theory

“…For the model system (14) we choose a Lagrangian L M , in which each fermion harmonically interacts with one fictitious particle. We do this uncritically, deliberately overlooking the problem of the required symmetry of the state space of the model action and choose…”

“…The free energy F v from this inequality is calculated analytically. The parameters M and k of the model "Lagrangian" (14) are then found by minimizing the value of the supposed upper bound F v . This calculation has shown that the many-body variational principle for path integrals (26) is violated for this model system, as clearly illustrated in Fig.…”

“…A discussion of this problem lies beyond the scope of the present paper. For more details on the status of this problem, see the literature 12,13,14,15,16,17,18 . The Jensen-Feynman inequality is reminiscent of the Bogolubov inequality 19,20 , which provides the following upper bound to the free energy F of a system described by the Hamiltonian H…”

“…A discussion of this problem lies beyond the scope of the present paper. For more details on the status of this problem, see the literature 12,13,14,15,16,17,18 .…”