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arXiv:cond-mat/0509059 · Physical Review B

Effects of lattice geometry and interaction range on polaron dynamics

J. P. Hague, P. E. Kornilovitch, A. S. Alexandrov, John H. Samson

Published Feb 16, 2006• 47 citations
arXivHTMLPDF

Abstract

We study the effects of lattice type on polaron dynamics using a continuous-time quantum Monte Carlo approach. Holstein and screened Fr\"ohlich polarons are simulated on a number of different Bravais lattices. The effective mass, isotope coefficients, ground-state energy and energy spectra, phonon numbers, and density of states are calculated. In addition, the results are compared with weak- and strong-coupling perturbation theory. For the Holstein polaron, it is found that the crossover between weak- and strong-coupling results becomes sharper as the coordination number is increased. In higher dimensions, polarons are much less mobile at strong coupling, with more phonons contributing to the polaron. The total energy decreases monotonically with coupling. Spectral properties of the polaron depend on the lattice type considered, with the dimensionality contributing to the shape and the coordination number to the bandwidth. As the range of the electron-phonon interaction is increased, the coordination number becomes less important, with the dimensionality taking the leading role.

DOI
10.1103/physrevb.73.054303

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We study the effects of lattice type on polaron dynamics using a continuous-time quantum Monte Carlo approach. Holstein and screened Fr"ohlich polarons are simulated on a number of different Bravais lattices. The effective mass, isotope coefficients, ground-state energy and energy spectra, phonon numbers, and density of states are calculated. In addition, the results are compared with weak- and strong-coupling perturbation theory. For the Holstein polaron, it is found that the crossover between weak- and strong-coupling results becomes sharper as the coordination number is increased. In higher dimensions, polarons are much less mobile at strong coupling, with more phonons contributing to the polaron. The total energy decreases monotonically with coupling. Spectral properties of the polaron depend on the lattice type considered, with the dimensionality contributing to the shape and the coordination number to the bandwidth. As the range of the electron-phonon interaction is increased, the coordination number becomes less important, with the dimensionality taking the leading role.
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.
Based on a recently developed variational method, we explore the properties of the Holstein polaron on an infinite lattice in D dimensions, where 1< D< 4.1<~D<~4.1< D< 4. The computational method converges as a power law, so that highly accurate results can be achieved with modest resources. We present the most accurate ground state energy (with no small parameter) to date for polaron problems, 21 digits for the one-dimensional (1D) polaron at intermediate coupling. The dimensionality effects on polaron band dispersion, effective mass, and electron-phonon (el-ph) correlation functions are investigated in all coupling regimes. It is found that the crossover to large effective mass of the higher-dimensional polaron is much sharper than the 1D polaron. The correlation length between the electron and phonons decreases significantly as the dimension increases. Our results compare favorably with those of the quantum Monte Carlo, dynamical mean-field theory, density-matrix renormalization-group, and Toyozawa variational methods. We demonstrate that the Toyozawa wave function is qualitatively correct for the ground-state energy and the two-point electron-phonon correlation functions, but fails for the three-point functions. Based on this finding, we propose an improved Toyozawa variational wave function.
We provide a comprehensive solution to the lattice dynamics problem in the two dimensional Holstein model at finite electron density and finite temperature. We work in the physically relevant adiabatic regime and vary the electron-phonon interaction from the weak coupling perturbative window to the strong coupling polaronic regime. We explore three typical electron densities, dilute - where spatial correlations between polarons is weak, intermediate - where correlations are significant, and half-filling - where there is long range checkerboard order at low temperature. We use two methods both of which exploit the "slowness" of the phonons to handle the problem. These are (i)~a standard random phase approximation (RPA), adapted to capture small quantum fluctuations on Monte Carlo generated classical thermal backgrounds, and (ii)~a Langevin dynamics scheme, with a simplified ``thermal noise'', that can address large amplitude dynamical fluctuations. The Langevin scheme, as we argue in the paper, is the superior method in the strong coupling part of the phase diagram, where lattice distortions are large. It reveals a non trivial multi-peak momentum resolved spectrum with a high energy part, on the scale of the bare phonon frequency ΩΩΩ, and a low energy peak at ω≪Ωω\ll Ωω≪Ω. Below the polaronic threshold, the high energy dispersion changes only modestly with temperature TTT, while the broadening, arising from mode coupling, increases linearly with TTT at low temperature. The low energy peak shows up at strong coupling and finite temperature and arises from the slow tunneling of polarons. The tunneling events become spatially correlated as electron density increases towards half-filling, and the weight becomes strongly momentum and temperature dependent. We suggest the analytic basis of these results.
We present details of a continuous-time quantum Monte Carlo algorithm for the screened Hubbard-Fr\"ohlich bipolaron. We simulate the bipolaron in one dimension with arbitrary interaction range in the presence of Coulomb repulsion, computing the effective mass, binding energy, total number of phonons associated with the bipolaron, mass isotope exponent, and bipolaron radius in a comprehensive survey of the parameter space. We discuss the role of the range of the electron-phonon interaction, demonstrating the evolution from Holstein to Fr\"ohlich bipolarons and we compare the properties of bipolarons with singlet and triplet pairing. Finally, we present simulations of the bipolaron dispersion. The bandwidth of the Fr\"ohlich bipolaron is found to be broad and the decrease in bandwidth as the two polarons bind into a bipolaron is found to be far less rapid than in the case of the Holstein interaction. The properties of bipolarons formed from long-range electron-phonon interactions, such as light strongly bound bipolarons and intersite pairing when Coulomb repulsion is large, are found to be robust against screening, with qualitative differences between Holstein and screened Fr\"ohlich bipolarons found even for interactions screened within a single lattice site.
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.
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^{2}2, the mass m∗{\mathit{m}}^{\mathrm{*}}m∗/mb{\mathit{m}}_{\mathit{b}}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^{2}2, and in (b) the strong-coupling limit E=-0.333 088\ensuremath{\alpha}' 2^{2}2,m∗{\mathit{m}}^{\mathrm{*}}m∗/mb{\mathit{m}}_{\mathit{b}}mb​=2.1254\ensuremath{\alpha}' 4^{4}4, and N=0.666 176\ensuremath{\alpha}' 2^{2}2. Here, \ensuremath{\alpha}' is the normalized electron-phonon coupling constant, which was chosen such that E=-\ensuremath{\alpha}' within second-order perturbation theory.