The Holstein
Molecular Crystal Model is investigated by a
strong coupling perturbative method which,
unlike the standard Lang-Firsov approach,
accounts for retardation effects due to the
spreading of the polaron size. The effective
mass is calculated to the second perturbative
order in any lattice dimensionality for a broad
range of (anti)adiabatic regimes and
electron-phonon couplings. The crossover from a
large to a small polaron state is found in all
dimensionalities for adiabatic and intermediate
adiabatic regimes. The phonon dispersion largely
smoothes such crossover which
is signalled by polaron mass enhancement and on-site localization
of the correlation function. The notion of self-trapping together
with the conditions for the existence of light polarons, mainly in
two- and three-dimensions, is discussed. By the imaginary time
path integral formalism I show how nonlocal electron-phonon
correlations, due to dispersive phonons, renormalize downwards the

The interest for phonons and lattice distortions in High-Temperature Superconductors (HTSc) is today more than alive [

With regard to small polarons, path integral investigations started with the groundbreaking numerical work by De Raedt and Lagendijk [

It has been questioned whether small (bi)polarons could indeed account for high

In view of the relevance of the polaron mass issue, I review in this article some work [

It is worth emphasizing that our previous investigations on the Holstein polaron and the present paper assume dimensional effects as driven by the lattice while the electron transfer integral is a scalar quantity. In a different picture, first proposed by Emin [

Being central to polaron studies, the effective mass behavior ultimately reflects the abovementioned property of the polaron problem: when the electron moves through the lattice, it induces and drags a lattice deformation which however does not follow instantaneously, the retardation causing a spread in the size of the quasiparticle. The electron-phonon correlation function provides a measure of the deformation around the instantaneous position of the electron and may be used to quantify the polaron size. Several refined theoretical tools, including quantum Monte Carlo calculations [

Extremely interesting is that range of

The retardation effect and its consequences are here treated by means of a variational analytical method based on the Modified Lang Firsov (MLF) transformation [

The Holstein diatomic molecular model was originally cast [

In second quantization the dimension dependent Holstein Hamiltonian with dispersive harmonic optical phonons reads

The LF transformation uses a phonon basis of fixed displacements (at the electron residing site) which diagonalizes the Hamiltonian in (

The idea underlying the MLF transformation [

Explicitly, the MLF transformed Holstein Hamiltonian in (

The coordination number

Looking at (

The energy eigenstates of

The second-order correction to the ground-state energy of the polaron with momentum

By minimizing the zone center ground state energy, the variational parameters

Given the formal background, the polaron mass is calculated both for the Lang-Firsov and for the Modified Lang-Firsov method to the second order in SCPT. Generally the second order correction (i) makes a relevant contribution to the ground state energy, (ii) does not affect the bandwidth, (iii) introduces the mass dependence on the adiabatic parameter which would be absent in the first order. Altogether the polaron landscape introduced by the second order SCPT is much more articulated than the simple picture suggested by the first order of SCPT which nonetheless retains its validity towards the antiadiabatic limit. In first order SCPT, ground state energy, bandwidth, and effective mass appear as equivalent, interchangeable properties to describe the polaron state while band narrowing and abrupt mass enhancement are

As emphasized in Section

Hereafter I take

Figure

Ratio of the one-dimensional polaron mass to the bare band mass versus

Figure

Figure

It should be reminded that the concept of

However, these conclusions have been critically reexamined in the recent polaron literature and the same notion of

As fluctuations in the lattice displacements around the electron site are included in the MLF variational wavefunction, the calculated polaron mass should not display discontinuities by varying the Hamiltonian parameters through the crossover [

Ratio of the Modified Lang-Firsov polaron mass to the bare band mass versus

In 1D, see Figure

Some significant results are found in 2D as shown in Figure

This effect is even more evident in 3D, see Figure

The self trapping transition appears in Figure

Nonetheless, the findings displayed in Figure

Within the MLF formalism one may also compute the electron-phonon correlation functions in the polaron ground state. This offers a measure of the polaron size as electron and phonons displacements can be taken at different neighbors sites. The on-site

Electron-phonon correlation functions: on-site

Altogether the diamond marked loci displayed in Figure

The results obtained so far can be put on sound physical bases by applying the space-time path integral method to the dispersive Holstein Hamiltonian. The method permits to incorporate the effect of the electron-phonon correlations in a momentum dependent effective

The phonons operators in (

Then, the

The sum over

While, in principle, the sum over

I apply to the Holstein Hamiltonian space-time mapping techniques [

I introduce

Consider now the

Averaging (

Then, on the base of (

Note that the Holstein source current does not depend on the electron path coordinates. The time dependence is incorporated only in the atomic displacements. This property will allow us to disentangle phonon and electron degrees of freedom in the path integral and in the total partition function.

After these premises, one can proceed to write the general path integral for an Holstein electron in a bath of dispersive phonons. Assuming a mixed representation, the electron paths are taken in real space while the phonon paths are in momentum space. Thus, the electron path integral reads

The total partition function can be derived from (

The phonon degrees of freedom in (

Equation (

The time (temperature) averaged

In Figure

(a) Time averaged ^{-1} versus wave vector for a linear chain.

The projections of the two-dimensional

Then, an increased range for the

According to the traditional notion of small polaron, the strong electron coupling to the lattice deformation implies a polaron collapse of the electron bandwidth (

In all dimensions, there is room for non trapped intermediate polarons in the region of the moderate couplings. Such polarons spreads over a few lattice sites and their real space extension grows with dimensionality. Thus, analytical methods can suitably describe some polaron properties also in that interesting intermediate window of

The physical origins for the possibility of light polarons have been analysed treating the dispersive Holstein Hamiltonian model by the imaginary time path integral method. The perturbing source current is the time averaged

Joyous and fruitful collaboration with Professor A. N. Das is acknowledged.