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Journal of Physics Condensed Matter

Influence of quantum fluctuations on solitary-wave acoustic polaron motion

R. Ruckh, G. Seibold, E. Sigmund

Published Jun 28, 1993• 2 citations
Publisher

Abstract

The theory of Wilson's solitary-wave acoustic polaron motion is generalized to include the effects of quantum fluctuations and the non-adiabatic behaviour of the electron-phonon coupling. The polaron dynamics are described quantum mechanically in terms of master equations for the electronic variables. As a final result, it emerges that due to the considered quantum effects for v<<c (v=polaron velocity, c=sound velocity) the polaron behaviour is determined by the well-known semiclassical non-linear Schrodinger equation, whereas for v<c the polaron motion is overdamped but shows by no means a divergent behaviour as obtained in semiclassical theories. In addition, for v=c the phonon cloud around the electron disappears and the electron motion becomes diffusive.

DOI
10.1088/0953-8984/5/26/017

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The theory of Wilson's solitary-wave acoustic polaron motion is generalized to include the effects of quantum fluctuations and the non-adiabatic behaviour of the electron-phonon coupling. The polaron dynamics are described quantum mechanically in terms of master equations for the electronic variables. As a final result, it emerges that due to the considered quantum effects for v<<c (v=polaron velocity, c=sound velocity) the polaron behaviour is determined by the well-known semiclassical non-linear Schrodinger equation, whereas for v<c the polaron motion is overdamped but shows by no means a divergent behaviour as obtained in semiclassical theories. In addition, for v=c the phonon cloud around the electron disappears and the electron motion becomes diffusive.
A general class of acoustical-phonon mode induced-polaron systems is considered. The electron-phonon interaction for these systems is described by a Fr\"ohlich-interaction Hamiltonian involving Fourier components proportional to ${q}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}$ is an arbitrary real number. The energy versus total momentum ($E\ensuremath{-}P$) relations for these polarons are investigated. At weak coupling, in the context of the intermediate coupling theory, it is found that when $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}\frac{1}{2}$ a linear structure is present in the $E\ensuremath{-}P$ relations. As $\ensuremath{\gamma}$ decreases from -1/2 to -1, this linear structure becomes apparent only for larger and larger values of PPP. For $\ensuremath{\gamma}>\ensuremath{-}\frac{1}{2}$, this linear structure is not present, but rather the $E\ensuremath{-}P$ relations cut off at a critical value of PPP. When $\ensuremath{-}1<\ensuremath{\gamma}\ensuremath{\le}0$, the polaron mass for these systems grows large as the polaron velocity vvv approaches the speed of sound sss, but for $\ensuremath{\gamma}>0$ the polaron mass has a finite value as $v\ensuremath{\rightarrow}s$. As a function of $\ensuremath{\gamma}$, the self-energy reaches a maximum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.73$ and the $v\ensuremath{\rightarrow}0$ polaron mass reaches a minimum value for $\ensuremath{\gamma}\ensuremath{\approx}\ensuremath{-}0.25$, independent of coupling strength. From a strong-coupling theory appropriate for moving polarons it is found that a linear structure in the $E\ensuremath{-}P$ relations is obtained for all $\ensuremath{\gamma}>\ensuremath{-}1$. At both weak and strong coupling it is unclear that the $E\ensuremath{-}P$ relations are meaningful when $\ensuremath{\gamma}\ensuremath{\le}\ensuremath{-}1$.
Based on the Holstein model Hamiltonian of one-dimensional molecular crystals, by making use of the expansion approach of the correlated squeezed-coherent states of phonon instead of the two-phonon coherent state expansion scheme, the properties of the ground state and the anomalous quantum fluctuations are investigated in a strongly coupled electron—phonon system with special consideration of the electron-two-phonon interaction. The effective renormalization ( i ) of the displacement of the squeezed phonons with the effect of the squeezed-coherent states of phonon and both the electron-displaced phonon and the polaron-squeezed phonon correlations have been combined to obtain the anomalous quantum fluctuations for the corrections of the coherent state. Due to these non-adiabatic correlations, the effective displacement parameter i is larger than the ordinary parameter α i (0) . In comparison with the electron-one-phonon interaction ( g ) corrected as i g , we have found the electron-two-phonon interaction ( g 1 ) corrected as i 2 g 1 is enhanced significantly. For this reason, the ground state energy ( E 0 (2) ) contributed by the electron-two-phonon interaction is more negative than the single-phonon case ( E 0 (1) ) and the soliton solution is more stable. At the same time, the effects of the electron-two-phonon interaction greatly increase the polaron energy and the quantum fluctuations. Furthermore, in a deeper level, we have considered the effect of the polaron-squeezed phonon correlation ( f -correlation). Since this correlation parameter f > 1, this effect will strengthen the electron-one and two-phonon interactions by f i g and f 2 i 2 g 1 , respectively. The final results show that the ground state energy and the polaron energy will appear more negative further and the quantum fluctuations will gain further improvement.
The effects of the nonadiabatic phonon behaviours on the ground state of the system and the quantum modifications of the polaron characteristics and the modification form of the uncertainty relation between the phonon coordinate and momentum due to fluctuations of the electron density in the coupled electron-phonon systems have been studied by using a new variational ansatz for coherent phonons with correlated displacement and squeezing including internations with electrons. The correlated effect results in the following effect: (1) the energy of the ground state of the system is found to be lower than that for the phonons with uncorrelated displacement and squeezing; (2) the binding energy of the polaron was found to be increased; (3) the nonadiabatic coupling with electrons enhances the quantum uncertainty for the phonon coordinate and momentum when compared with the adiabatic and uncorrelated case.
For an electron interacting with harmonic dispersive phonons in a periodic lattice, the possibility of wave (soliton)-like solutions of the generalised master equations (GME) for the electron site occupation probabilities is investigated. For the current form of the small-polaron Hamiltonian, the long-time asymptotics of the memory functions (kernels of the GME) in the small-polaron basis are discussed to all orders in the electron hopping integrals.Using this result, one obtains (for the initially localised electron) the t 2 -behaviour of the mean-square electron displacement at zero temperature. This corresponds to existing soliton solutions of the Schrodinger equation. At non-zero temperatures, the wave solution is damped.
Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.