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arXiv:1510.06912 · Condensed Matter Physics

Renormalized energy of ground and first excited state of Fröhlich polaron in the range of weak coupling

N. V. Tkach, Julia O. Seti, Voitsekhivska, Olesya Pytiuk

Published Sep 1, 2015• 7 citations
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Abstract

Partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Fröhlich Hamiltonian is performed using the Feynman-Pines diagram technique.Renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for a weak electron-phonon coupling at T = 0 K.It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra.The ground state stays stationary and the excited one decays at a bigger coupling constant.

DOI
10.5488/cmp.18.33707

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Partial summing of infinite range of diagrams for the two-phonon mass operator of polaron described by Fröhlich Hamiltonian is performed using the Feynman-Pines diagram technique.Renormalized spectral parameters of ground and first excited (phonon repeat) polaron state are accurately calculated for a weak electron-phonon coupling at T = 0 K.It is shown that the stronger electron-phonon interaction shifts the energy of both states into low-energy region of the spectra.The ground state stays stationary and the excited one decays at a bigger coupling constant.
After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.
Method of intermediate problems was applied to investigation of the energy spectrum of the Fröhlich polaron model. It was shown that various infinite sequences of non-decreasing improvable lower bound estimates for the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy can be derived for arbitrary values of the electron-phonon interaction constant. These bound estimates allow for explicit numerical evaluation at all orders. In conjunction with numerous well-known upper bound estimates for the energy spectral curve of the Fröhlich polaron as a function of the interaction constant and the polaron total momentum, the aforesaid improvable lower bound estimates might provide one with virtually precise magnitude for the energy of the slow-moving polaron. Possible generalization to the case of the Fröhlich polaron in external magnetic field is outlined.
An upper bound to the ground-state energy of a tight-binding polaron (a polaron in a narrow band material) in a polar crystal is obtained, in the nonadiabatic limit, from a Hamiltonian that includes two contributions to the electron–phonon coupling: a short-range deformation potential interaction and a long-range dipolar interaction. A Debye cutoff on the phonon wave vectors is also assumed. A variational ground-state energy is deduced, using a modification of a tight-binding type Ansatz used in small-polaron theory, in the nonadiabatic limit. In the weak-coupling limit the ground state corresponds to that of a large polaron, while the strong-coupling limit gives a small-polaron state. The bandwidth greatly influences the state of the system and phase diagrams are presented that show the regions where the small and the large polarons are energetically favored. We find, in this variational context, that for a narrow band, the transition between the self-trapped state and the band state is continuous and that an intermediate-polaron state is involved. For a larger but moderate bandwidth, the system is found to switch abruptly from one state to the other, with a coexistence of both states around the transition region. Each of the two types of coupling is found to have an equivalent role in increasing the trapping energy of the electron and in reducing the bandwidth exponentially. Furthermore, their effect is cumulative.
A ground-state theorem for the Fr\"ohlich polaron is derived. We show that the zeroth moment of the current-current correlation function is proportional to the kinetic energy of the polaron. While the electron-phonon interaction does not depend on the electron mass, the coupling constant $\ensuremath{\alpha}$ does; this allows us to relate the kinetic energy with the ground-state energy using the Feynman-Helmann theorem. As an example we compare the polaron ground-state energy on variational grounds with the ground-state energy obtained from the function $\ensuremath{\chi}(\ensuremath{\omega})$ calculated by Feynman, Hellwarth, Iddings, and Platzman. For $\ensuremath{\alpha}<5$ both ground-state energies turn out to be identical, as can be concluded from numerical calculations.
The Fr\"ohlich model describes the interaction of a mobile impurity with a surrounding bath of phonons which leads to the formation of a quasiparticle, the polaron. In this paper an efficient renormalization group approach is presented which provides a description of Fr\"ohlich polarons in all regimes ranging from weak to strong coupling. The extended renormalization group approach introduced here is capable of predicting ground state properties for arbitrarily small impurity masses. This allows us to obtain the full phase diagram of the Fr\"ohlich Hamiltonian, which we present concretely for the Bogoliubov-Fr\"ohlich model originally introduced to describe ultracold impurities in a Bose-Einstein condensate. For this model, whose regime of validity in ultracold quantum gases is the subject of debate, we benchmark our method by comparison of the ground state energy to recent diagrammatic quantum Monte Carlo calculations.
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{E}_{2\mathrm{D}}E2D​(\ensuremath{\alpha})=(2/3)E3D{E}_{3\mathrm{D}}E3D​(( 3\ensuremath{\pi}/4)\ensuremath{\alpha}), where E2D{E}_{2\mathrm{D}}E2D​ (E3D{E}_{3\mathrm{D}}E3D​) is the ground-state energy of the 2D (3D) polaron.