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Physical review. B, Condensed matter

Energy spectrum of the optical polaron at finite total momentum

B. Gerlach, F. Kalina

Published Oct 15, 1999• 9 citations
Publisher

Abstract

In the following discussion we are concerned with the standard Fr\"ohlich model for an optical polaron. We clarify the qualitative properties of the energy spectrum for arbitrary total momentum Q. Concerning the ground-state energy, we establish an effective lower bound. Until now, we have to assume that the electron-phonon coupling parameter $\ensuremath{\alpha}$ does not exceed a specified positive value. Using this bound, we demonstrate that the ground-state energy coincides with the continuum edge for $|Q|>~|{Q}_{C}|,$ ${Q}_{C}$ being finite. Consequently, it is only for $|Q|<|{Q}_{C}|$ that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and $\ensuremath{\alpha},$ we find an increase (strict decrease) with increasing $|Q|(\ensuremath{\alpha}).$ In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.

DOI
10.1103/physrevb.60.10886

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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.

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In the following discussion we are concerned with the standard Fr"ohlich model for an optical polaron. We clarify the qualitative properties of the energy spectrum for arbitrary total momentum Q. Concerning the ground-state energy, we establish an effective lower bound. Until now, we have to assume that the electron-phonon coupling parameter \ensuremathα\ensuremath{\alpha}\ensuremathα does not exceed a specified positive value. Using this bound, we demonstrate that the ground-state energy coincides with the continuum edge for ∣Q∣> ∣QC∣,|Q|>~|{Q}_{C}|,∣Q∣> ∣QC​∣, QC{Q}_{C}QC​ being finite. Consequently, it is only for ∣Q∣<∣QC∣|Q|<|{Q}_{C}|∣Q∣<∣QC​∣ that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and \ensuremathα,\ensuremath{\alpha},\ensuremathα, we find an increase (strict decrease) with increasing ∣Q∣(\ensuremathα).|Q|(\ensuremath{\alpha}).∣Q∣(\ensuremathα). In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.
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.
Method of intermediate problems in the theory of linear semi-bounded self-adjoint operators on rigged Hilbert space was applied to the investigation of the ground state energy of the Fr öhlich polaron model.It was shown that various infinite sequences of non-decreasing improvable lower bound estimates for the polaron ground state energy can be derived for arbitrary values of the electron-phonon interaction constant.The proposed approach allows for explicit numerical evaluation of the thus obtained lower bound estimates at all orders and can be straightforwardly generalized for investigation of the low-lying branch of the slow-moving polaron excitation energy spectral curve adjacent to the ground state energy of the polaron at rest.In conjunction with numerous, already derived by multitudinous methods, well-known upper bound estimates for the energy spectral curve of the Fr öhlich polaron as a function of the electron-phonon 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.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.
Recently Feynman's path-integral formalism of the Fröhlich optical polaron problem is generalized, by which it is easy and natural to get the second-order perturbation result in the weak-coupling case and the Pekar's result in the strongcoupling case, even in the crudest ground-state approximation. By using numerical method to the equations which are given by the ground-state approximation, the ground-state energy of the electron-optical phonon system can obtained for the overall range of coupling strength. In addition, a self-consistent set of equations is derived for a central optimized potential without using the ground-state approximation. This effective local, central potential is the solution of a linear integral equation which was similar to the extensive work of optimized-potential model (OPM) in atomic physics. By using OPM formalism, no particular choice of interaction form is taken, therefore, except the approximation by Jensen's inequality, the ground-state energy of the electron-optical phonon interacting system can be obtained without any approximation.
The next lowest order term of the ground state energy has its origin in lowering of frequencies of phonons localized around the polaron. An eigenvalue equation is derived, the solution of which gives phonon eigenmodes and frequencies in the presence of the polaron. Several eigenvalues are obtained by numerical integration. From the results, the ground state energy is estimated as E0/ℏω0=−0.108513α2−2.836+O(1/α2)E_{0}/\hbar\omega_{0}{=}-0.108513\alpha^{2}-2.836+O(1/\alpha^{2})E0​/ℏω0​=−0.108513α2−2.836+O(1/α2), where ω 0 is the frequency of the optical phonon and α is the electron-phonon coupling constant. The numerical value of the effective mass is 0.0227019α 4 .
The one-dimensional Fr\"ohlich model describing the motion of a single electron interacting with optical phonons is a paradigmatic model of quantum many-body physics. We predict the existence of an arbitrarily large number of bound excited states in the strong-coupling limit and calculate their excitation energies. Numerical simulations of a discretized model demonstrate the complete amelioration of the projector Monte Carlo sign problem by walker annihilation in an infinite Hilbert space. They reveal the threshold for the occurrence of the first bound excited states at a value of $\ensuremath{\alpha}\ensuremath{\approx}1.73$ for the dimensionless coupling constant. This puts the threshold into the regime of intermediate interaction strength. We find a significant spectral weight and increased phonon number of the bound excited state at threshold.