The ground-state energy of a two-dimensional (2D) Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and gives E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-(\ensuremath{\pi}/2)\ensuremath{\alpha}-0.063 97${\ensuremath{\alpha}}^{2}$ with \ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$ the surface optical-phonon energy. In the strong-coupling limit the adiabatic approximation is used and E/\ensuremath{\Elzxh}${\ensuremath{\omega}}_{s}$=-0.4047${\ensuremath{\alpha}}^{2}$ is found to leading order in \ensuremath{\alpha}. The Feynman path-integral approximation, the Gaussian approximation, and the modified Lee-Low-Pines unitary transformation approximation to the polaron ground-state energy all satisfy the scaling relation
E2D(\ensuremath{\alpha})=(2/3)
E3D(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where
E2D (
E3D) is the ground-state energy of the 2D (3D) polaron.
According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupling term in the Hamiltonian into the form of an effective hopping integral and derive the single-particle Green's function describing propagation of the small Frohlich polaron. One and two dimensional spectral functions are studied by expanding the Green's function perturbatively. Numerical calculations of the spectral functions are produced. Remarkably, the coherent spectral weight (Z) and effective mass (Z') renormalisation exponents are found to be different with Z'>>Z, which can explain a small coherent spectral weight and a relatively moderate mass enhancement in oxides.
The polaron mass of a two-dimensional Fr\"ohlich polaron is calculated to second order in the coupling constant (\ensuremath{\alpha}) and is found to be equal to
m∗/
mb=1+(\ensuremath{\pi}/8)\ensuremath{\alpha}+0.127 23${\mathrm{\ensuremath{\alpha}}}^{2}$, with
mb the band mass of the electron. In the strong-coupling limit we found the exact result
m∗/
mb=0.733${\ensuremath{\alpha}}^{4}$. A comparison is made with the polaron mass as obtained from the Feynman polaron theory, and the validity of a scaling relation is discussed.
The Fr\"ohlich polaron confined to move in one space dimension and interacting with a three-dimensional lattice is studied. We calculate exact and approximate results for the ground-state energy and the effective mass. The following exact results were obtained. In (a) the weak-coupling limit for the energy E=-\ensuremath{\alpha}'-[(3/2 \ensuremath{\surd}2 )-1]\ensuremath{\alpha}'
2, the mass
m∗/
mb=1/2\ensuremath{\alpha}'+[(5/2 \ensuremath{\surd}2 )-1]/4, and the average number of virtual phonons N=1/2\ensuremath{\alpha}'+[(3/ \ensuremath{\surd}2 )-2]\ensuremath{\alpha}'
2, and in (b) the strong-coupling limit E=-0.333 088\ensuremath{\alpha}'
2,
m∗/
mb=2.1254\ensuremath{\alpha}'
4, and N=0.666 176\ensuremath{\alpha}'
2. Here, \ensuremath{\alpha}' is the normalized electron-phonon coupling constant, which was chosen such that E=-\ensuremath{\alpha}' within second-order perturbation theory.
We compute the Holstein polaron spectral function on a one-dimensional ring using the finite-temperature
T Lanczos method. With increasing
T additional features in the spectral function emerge even at temperatures below the phonon frequency. We observe a substantial spread of the spectral weight towards lower frequencies and the broadening of the quasiparticle (QP) peak. In the weak-coupling regime the QP peak merges with the continuum in the high-
T limit. In the strong-coupling regime the main features of the low-
T spectral function remain detectable up to the highest
T used in our calculations. The effective polaron mass shows a nonmonotonic behavior as a function of
T at small phonon frequency but increases with
T at larger frequencies. The self-energy remains
k independent even at elevated
T in the frequency range corresponding to the polaron band, while at higher frequencies it develops a distinguishable
k dependence. Analytical expressions for the first few frequency moments are derived, and they agree well with those extracted from numerical calculations in a wide-
T regime.
According to recent quantum Monte Carlo simulations, the small polaron theory is practically exact in a wide range of the long-range (Fröhlich) electron-phonon coupling and adiabatic ratio. Applying Lang-Firsov transformation, we derive a single-particle Green's function describing the propagation of small Fröhlich polaron. One- and two-dimensional spectral functions are studied by expanding Green's function perturbatively. Remarkably, the coherent spectral weight ( Z ) and effective mass ( Z ′) renormalisation exponents are found to be different with Z ′ ≫ Z , which can explain a small coherent spectral weight and a relatively moderate mass enhancement in oxides.