Effect of Dimensionality and Anisotropy on the Holstein Polaron
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ABSTRACT We apply weak-coupling perturbation theory and strong-coupling perturbation theory to the Holstein molecular crystal model in order to elucidate the effects of anisotropy on polaron properties in D dimensions. The ground state energy is considered as a primary criterion through which to study the effects of anisotropy on the self-trapping transition, with particular attention given to shifting of the self-trapping line and the adiabatic critical point. The effects of dimensionality and anisotropy on electron-phonon correlations and polaronic mass enhancement are studied, with particular attention given to the polaron radius and the characteristics of quasi-l-D and quasi-2-D structures. Perturbative results are confirmed by selected comparisons with variational calculations and quantum Monte Carlo data.
INTRODUCTION As quantum quasiparticles, polarons describe the states of excitations correlated with the deformation or polarization quanta of a host medium. Often characterized in terms such as "large" and "small" and "free" and "self-trapped", the properties of polarons have long been expected to depend on the effective dimensionality of the host system. Well-known and widely-invoked results tied to the adiabatic approximation suggest that polarons in 2-D and 3-D should be qualitatively distinct from those found in 1-D, and that even the notion of selftrapping should take on different meaning in low and high dimensions (1-16]. The root of this lies in stability arguments suggesting that in 1-D all polaron states should be characterized by finite radii, while in 2-D and 3-D polaron states may have either infinite radii ("free") or finite radii ("self-trapped"), with the self-trapping transition taken to mean the abrupt transition from "free" states characterized by the free electron mass to "self-trapped" states characterized by strongly-enhanced effective masses. The self-trapping transition is not, in fact, an abrupt phenomenon except in the adiabatic limit; at finite parameter values the physically-meaningful transition is more in keeping with a smooth, if rapid, "crossover" from polaron structures characteristic of the weak-coupling regime to structures characteristic of the strong-coupling regime. The "self-trapping line" describing this transition can be located by criteria sensitive to changes in polaron structure; these may involve physical observables such as the polaron ground state energy and effective mass, or may rely upon more formal properties less accessible to direct physical measurement. Here, we will consider both physical observables and formal criteria drawn from a variety of methods at both finite parameters and in asymptotic regimes. Through these, we are able to characterize self-trapping in one, two, and three dimensions for any degree of anisotropy. For the explicit calculations to follow, we use the Holstein Hamiltonian [3,17] on a D-dimensional
Euclidean lattice Ht
=
Hkin±. + Hp +
,
(1)
D
fJikn
=
-ZZEJiaX(anj+j- ±ag...j) , A i=1
ftph = hw :bnbq, -gw I
aa,=
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