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Physical review. B, Condensed matter

Ground-state symmetry of a generalized polaron

Matthew P. A. Fisher, W. Zwerger

Published Oct 15, 1986• 25 citations
Publisher

Abstract

We study the structure of the ground-state wave function for a general model of a continuum polaron in d dimensions, coupled to either optical or acoustical phonons. Using a path-integral approach we derive a simple criterion which is sufficient to guarantee that the exact ground state is delocalized. This criterion applies to most polaron models studied so far and thus implies that interaction with phonons generally does not lead to a breaking of the translational symmetry, even for arbitrary strong coupling. We discuss briefly the distinction between self-trapping and localization.

DOI
10.1103/physrevb.34.5912

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Journal of the Physical Society of Japan•Nov 1, 1975•36 citations
We study the structure of the ground-state wave function for a general model of a continuum polaron in d dimensions, coupled to either optical or acoustical phonons. Using a path-integral approach we derive a simple criterion which is sufficient to guarantee that the exact ground state is delocalized. This criterion applies to most polaron models studied so far and thus implies that interaction with phonons generally does not lead to a breaking of the translational symmetry, even for arbitrary strong coupling. We discuss briefly the distinction between self-trapping and localization.
A novel variational approach is presented for the calculation of the ground-state energy of the polaron in arbitrary N dimensions in the strong-coupling limit. By using the phonon coherent state to represent the wavefunction of phonons, a self-consistent integro-differential equation for the electron wavefunction is derived. The calculated results of the ground-state energy for N = 1, 2 and 3 agree well with the best results in the literature. It is also found that, for arbitrary N, the present results are less than the Feynman path integral ones by small percentages. It is proposed that this approach should be universal for systems involving polarons in the strong-coupling regime.
We calculate the low-temperature properties of an acoustic polaron in three dimensions in thermal equilibrium at a given temperature using a specialized path-integral Monte Carlo method. In particular we find numerical evidence that the chosen Hamiltonian for the acoustic polaron describes a phase transition from a localized state to an unlocalized state for the electron as the phonon-electron coupling constant decreases. The phase transition manifests itself with a jump discontinuity in the potential energy as a function of the coupling constant. In the weak coupling regime the electron is in an extended state whereas in the strong coupling regime it is found in a self-trapped state.
Self-trapping of polarons is studied with use of path-integral techniques within a generalized Fr\"ohlich Hamiltonian. The effects of dimension and interaction range on the self-trapping transition are considered quantitatively. Results show conclusively that dimension by itself has no qualitative effect on the polaron localization transition. Numerical values for the polaron mass and energy discontinuity at the transition are obtained for a wide class of electron-phonon interaction in all dimensions.
The polaron path integral introduced by Feynman is generalized to the case of an electron moving in d space dimensions. A scheme of systematic calculations of the obtained path integral is developed to estimate the ground-state energy of the polaron in the same way for different values of the electron-phonon coupling constant \ensuremath{\alpha}. The leading-order term in this approach yields an upper bound to the polaron self-energy and improves Feynman's variational estimates for d=2 and d=3. A scaling relation between polaron self-energies for different space dimensions is obtained for this term. The next correction to this estimation is calculated by numerical integration for d=2 and d=3.
Based on the unitary transformation, a general treatment of the bound polaron is proposed. In distinction from earlier calculations, the present method gives explicit expressions for not only the energy but also the wave functions both in the ground state and in the excited states such as the relaxed excited state and the phonon scattering state. These expressions involve a summation on the complete set of eigenstates of a trial Hamiltonian, and in actual calculations the Coulomb Hamiltonian and the harmonic oscillator Hamiltonian are chosen as trial Hamiltonians. For the ground state, the energy expression is the same as that obtained by Matsuura with the path integral method. The present method is valid in the whole region of the electron-phonon coupling constant α and Coulomb force β except for β→0 and α>6.