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Physical review. B, Condensed matter

Monte Carlo calculation of the ground-state energy of an optical polaron

W. Becker, B. Gerlach, H. Schliffke

Published Nov 15, 1983• 28 citations
Publisher

Abstract

We consider a (band) electron---LO-phonon system described by Fr\"ohlich's Hamiltonian. Making use of functional-integration methods, one can eliminate the phonon terms exactly leading to an effective one-particle system with the same spectral properties as the original one. The (formal) free energy of this effective system is calculated numerically by means of the Monte Carlo technique. We give a detailed description and first-error estimates of our numerical procedure. Furthermore, we draw a comparison with previous work. Our results compare favorably with all earlier results of variational type of which we are aware.

DOI
10.1103/physrevb.28.5735

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An approximate calculation is made of the polaron self-energy and effective mass as defined by Frohlich.The approach taken is a variational one based on an ex tended version of Hohler's ansatz in which the trial state function employed is an exact eigenfunction of the total wave vector of the polaron.The effective mass obtained simulates the Feynmann-Schultz result while the corresponding polaron self-energy is fairly accurate but inferior to the result of Feynman and Schultz.CONTENTS Chapters « -« T -| > £ ( l ^' (1'2) n in which n is the optical refractive index, k is the static dielectric constant, e is the electronic charge and 2 t\& is Plank's constant.The limit S->°° is to be taken with lim I = -~ S-*oo v 8tt d 3v .(1,3) A natural unit system is employed in which 2^=o)=2m=l.The first term in the Hamiltonian is the quantum mechanical operator corresponding to the kinetic energy of the electron.The second term represents the energy of the polarization field resulting from the long wave length longitudinal optical crystal modes in the absence of interaction with the electron.The last term represents the energy of interaction between the electron and the phonons.The name "polaron" has been given to the entity consisting of the Bloch conduction electron together with its accompanying nonradiative polarization field.Accordingly Frohlich defines the polaron self-energy, EQ (a), and the (ground-state) polaron effective mass, y(a), by the expansion EQ (a,ic) = Ec (a) + k 2/y(a) + 0(k4 ), (1,4)

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Comment on ‘‘Monte Carlo calculation of the ground-state energy of an optical polaron’’

François Maria Peeters, J. T. Devreese

Becker, Gerlach, and Schliffke [Phys. Rev. B 28, 5735 (1983)] calculated the free energy of a three-dimensional optical polaron at nonzero temperature (i.e., \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25) by the Monte Carlo method. In the present Comment (1) we show that the free energy of the Feynman polaron for \ensuremath{\alpha}\ensuremath{\lesssim}2 and \ensuremath{\alpha}>4 is lower than the corresponding Monte Carlo results of Becker, Gerlach, and Schliffke (but lies within the 1% error of the Monte Carlo results) and (2) we study (numerically) the polaron free energy as function of the temperature within the Feynman variational approach.

Physical review. B, Condensed matter•May 15, 1985•6 citations
We consider a (band) electron---LO-phonon system described by Fr"ohlich's Hamiltonian. Making use of functional-integration methods, one can eliminate the phonon terms exactly leading to an effective one-particle system with the same spectral properties as the original one. The (formal) free energy of this effective system is calculated numerically by means of the Monte Carlo technique. We give a detailed description and first-error estimates of our numerical procedure. Furthermore, we draw a comparison with previous work. Our results compare favorably with all earlier results of variational type of which we are aware.
We present the numerically exact ground-state energy, effective mass, and isotope exponents of a one-dimensional lattice polaron, valid for any range of electron-phonon interaction, applying a continuous-time quantum Monte Carlo (QMC) technique in a wide range of coupling strength and adiabatic ratio. The QMC method is free from any systematic finite-size and finite-time-step errors. We compare our numerically exact results with analytical weak-coupling theory and with the strong-coupling $1∕\ensuremath{\lambda}$ expansion. We show that the exact results agree well with the canonical Fr\"ohlich and Holstein-Lang-Firsov theories in the weak and strong coupling limits, respectively, for any range of interaction. We find a strong dependence of the polaron dynamics on the range of interaction. An increased range of interaction has a similar effect to an increased (less adiabatic) phonon frequency: specifically, a reduction in the effective mass.
We present a novel path integral Monte Carlo scheme to solve the Fröhlich polaron model. At intermediate and strong electron-phonon couplings, the polaron self-trapping is properly taken into account at the level of an effective action obtained by a preaveraging procedure with a retarded trial action. We compute the free energy at several couplings and temperatures in three and two dimensions. Our results show that the accuracy of the Feynman variational upper bound for the free energy is always within a few percent. The method can be generalized to the N-polarons case.
We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
An approximate calculation is made of the polaron self-energy and effective mass as defined by Frohlich.The approach taken is a variational one based on an ex tended version of Hohler's ansatz in which the trial state function employed is an exact eigenfunction of the total wave vector of the polaron.The effective mass obtained simulates the Feynmann-Schultz result while the corresponding polaron self-energy is fairly accurate but inferior to the result of Feynman and Schultz.CONTENTS Chapters « -« T -| > £ ( l ^' (1'2) n in which n is the optical refractive index, k is the static dielectric constant, e is the electronic charge and 2 t\& is Plank's constant.The limit S->°° is to be taken with lim I = -~ S-*oo v 8tt d 3v .(1,3) A natural unit system is employed in which 2^=o)=2m=l.The first term in the Hamiltonian is the quantum mechanical operator corresponding to the kinetic energy of the electron.The second term represents the energy of the polarization field resulting from the long wave length longitudinal optical crystal modes in the absence of interaction with the electron.The last term represents the energy of interaction between the electron and the phonons.The name "polaron" has been given to the entity consisting of the Bloch conduction electron together with its accompanying nonradiative polarization field.Accordingly Frohlich defines the polaron self-energy, EQ (a), and the (ground-state) polaron effective mass, y(a), by the expansion EQ (a,ic) = Ec (a) + k 2/y(a) + 0(k4 ), (1,4)
An efficient continuous-time path-integral quantum Monte Carlo algorithm for the lattice polaron is presented. It is based on Feynman's integration of phonons and subsequent simulation of the resulting single-particle self-interacting system. The method is free from the finite-size and finite-time-step errors and works in any dimensionality and for any range of electron-phonon interaction. The ground-state energy and effective mass of the polaron are calculated for several models. The polaron spectrum can be measured directly by Monte Carlo, which is of general interest.
Becker, Gerlach, and Schliffke [Phys. Rev. B 28, 5735 (1983)] calculated the free energy of a three-dimensional optical polaron at nonzero temperature (i.e., \ensuremath{\beta}\ensuremath{\Elzxh}\ensuremath{\omega}=25) by the Monte Carlo method. In the present Comment (1) we show that the free energy of the Feynman polaron for \ensuremath{\alpha}\ensuremath{\lesssim}2 and \ensuremath{\alpha}>4 is lower than the corresponding Monte Carlo results of Becker, Gerlach, and Schliffke (but lies within the 1% error of the Monte Carlo results) and (2) we study (numerically) the polaron free energy as function of the temperature within the Feynman variational approach.