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arXiv:1803.09608 · Physical review. B./Physical review. B

Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions

Thomas Hahn, Sergei N. Klimin, Jacques Tempere, Jozef T. Devreese, Cesare Franchini

Published Apr 20, 2018• 38 citations
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Abstract

We present results for the solution of the large polaron Fr\"ohlich Hamiltonian in 3 dimensions (3D) and 2 dimensions (2D) obtained via the diagrammatic Monte Carlo (DMC) method. Our implementation is based on the approach by Mishchenko [A. S. Mishchenko et al., Phys. Rev. B 62, 6317 (2000)]. Polaron ground state energies and effective polaron masses are successfully benchmarked with data obtained using Feynman's path integral formalism. By comparing 3D and 2D data, we verify the analytically exact scaling relations for energies and effective masses from $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$, which provides a stringent test for the quality of DMC predictions. The accuracy of our results is further proven by providing values for the exactly known coefficients in weak- and strong-coupling expansions. Moreover, we compute polaron dispersion curves which are validated with analytically known lower and upper limits in the small-coupling regime and verify the first-order expansion results for larger couplings, thus disproving previous critiques on the apparent incompatibility of DMC with analytical results and furnishing useful reference for a wide range of coupling strengths.

DOI
10.1103/physrevb.97.134305

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We present results for the solution of the large polaron Fr"ohlich Hamiltonian in 3 dimensions (3D) and 2 dimensions (2D) obtained via the diagrammatic Monte Carlo (DMC) method. Our implementation is based on the approach by Mishchenko [A. S. Mishchenko et al., Phys. Rev. B 62, 6317 (2000)]. Polaron ground state energies and effective polaron masses are successfully benchmarked with data obtained using Feynman's path integral formalism. By comparing 3D and 2D data, we verify the analytically exact scaling relations for energies and effective masses from 3D\ensuremath→2D3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}3D\ensuremath→2D, which provides a stringent test for the quality of DMC predictions. The accuracy of our results is further proven by providing values for the exactly known coefficients in weak- and strong-coupling expansions. Moreover, we compute polaron dispersion curves which are validated with analytically known lower and upper limits in the small-coupling regime and verify the first-order expansion results for larger couplings, thus disproving previous critiques on the apparent incompatibility of DMC with analytical results and furnishing useful reference for a wide range of coupling strengths.
The effective mass of the Fröhlich polaron model in d spatial dimensions is obtained within the Gaussian-equivalent representation, that is, within a suitably rearranged perturbative path-integral approach, whose leading order already takes care of all Gaussian fluctuations. In doing so, the authors extend their earlier investigations ( Phys. Rev. B 50 3733 (1994)) of the polaron ground-state energy. Analytical expressions for the effective mass are derived in the weak- and strong-coupling limit. In the intermediate-coupling regime, the effective mass is calculated numerically for the cases of two and three dimensions. The new results are compared with previously known data.
A detailed study of the Fr\"ohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method [N. V. Prokof'ev and B. V. Svistunov, Phys. Rev. Lett. 81,81,81, 2514 (1998)]. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green's function, within an approach developed in the present paper. We present up to date results for the binding energy, and make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals nontrivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.
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.
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.
We study the properties of the two-dimensional Fermi polaron model in which an impurity attractively interacts with a Fermi sea of particles in the zero-range limit. We use a diagrammatic Monte Carlo (DiagMC) method which allows us to sample a Feynman diagrammatic series to very high order. The convergence properties of the series and the role of multiple particle-hole excitations are discussed. We study the polaron and molecule energy as a function of the coupling strength, revealing a transition from a polaron to a molecule in the ground state. We find a value for the critical interaction strength which complies with the experimentally measured one and predictions from variational methods. For all considered interaction strengths, the polaron ZZZ factor from the full diagrammatic series almost coincides with the one-particle-hole result. We also formally link the DiagMC and the variational approaches for the polaron problem at hand.
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.