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arXiv:1010.0737 · Mathematical Results in Quantum Mechanics (QMath)

BINDING, STABILITY, AND NON-BINDING OF MULTI-POLARON SYSTEMS

Rupert L. Frank, Élliott H. Lieb, Robert Seiringer, Lawrence E. Thomas

Published May 1, 2011• 5 citations
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Abstract

The binding of polarons, or its absence, is an old and subtle topic. After defining the model we state some recent theorems of ours. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at U=2α, where U is the electronic Coulomb repulsion and αis the polaron coupling constant. Second, if U is large enough, there is no multi-polaron binding of any kind. We also discuss the Pekar-Tomasevich approximation to the ground state energy, which is valid for large α. Finally, we derive exact results, not reported before, about the one-dimensional toy model introduced by E. P. Gross.

DOI
10.1142/9789814350365_0002

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•Jan 1, 2006•0 citations
The binding of polarons, or its absence, is an old and subtle topic. After defining the model we state some recent theorems of ours. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at U=2α, where U is the electronic Coulomb repulsion and αis the polaron coupling constant. Second, if U is large enough, there is no multi-polaron binding of any kind. We also discuss the Pekar-Tomasevich approximation to the ground state energy, which is valid for large α. Finally, we derive exact results, not reported before, about the one-dimensional toy model introduced by E. P. Gross.
The binding of polarons, or its absence, is an old and subtle topic. Here we prove two things rigorously. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at U=2α, where U is the electronic Coulomb repulsion and α is the polaron coupling constant. Second, if U is large enough, there is no multipolaron binding of any kind. Considering the known fact that there is binding for some U>2α, these conclusions are not obvious and their proof has been an open problem for some time.
For systems of N charged fermions (e.g. electrons) interacting with longi-tudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U> 0 and α> 0 measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition 2α < U, which follows from the dependence of U and α on electrical properties of the crystal. We show that the large N asymptotic behavior of the minimal energy EN changes at 2α = U and that 2α ≤ U is necessary for ther-modynamic stability: for 2α> U the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and EN behaves like −N7/3. 2 M. Griesemer and J. S. Møller 1
In recent literature it was frequently claimed that free polarons should undergo a (formal) phase transition from a mobile to a localized state, if the electron-phonon coupling constant \ensuremath{\alpha} exceeds some critical value ${\ensuremath{\alpha}}_{c}$. For \ensuremath{\alpha}=${\ensuremath{\alpha}}_{c}$, the formal free energy should be nonanalytic in \ensuremath{\alpha}. In this paper we prove that for optical polarons no such transition exists for finite temperatures. To do so, we use functional-integration techniques and discuss perturbation theory in \ensuremath{\alpha} to infinite order.
We have developed a new equation of motion Green's function theory for the large polaron, that is completely nonperturbative in origin, and therefore gives a unified treatment of the polaron ground state energy for all interaction strengths. The theory also incorporates translational invariance rigorously, a priori. We find the following new results. (a) Previous theories that are valid for all coupling strengths are based on a nonperturbative summation of infinite subsets of entire diagrams (equations of motion), rather than on recoil parts of all diagrams. We can thus recover lowest order results of Feynman for example, via a translationally noninvariant form of the Fock approximation, which resembles that reported in a previous work. (b) The total ground state energy we obtain in the strong coupling limit is E0 = −0.426α2 to lowest order in the coupling constant α, a result substantially below that of previous calculations, E0 = −0.106α2 (using gaussian trial functions in both cases). Our result is also below the lower bound of Lieb and Yamazaki, although we show in a following paper that our result is an upper bound. (c) The origin of the difference of our results compared to that of other strong coupling theories is several fold, and will be presented in detail in subsequent papers. However, the essence of the difference is that the phonons in the field, via their interaction with the electron, assume the role of independent degrees of freedom, in the limit of an infinite number of phonons. We then have a field theoretic problem with an infinite number of degrees of freedom which can exhibit a totally different physical behavior compared to one with a finite number of degrees of freedom. This aspect of the problem has not been given explicit recognition in all other theories, including that of Lieb and Yamazaki.
Three different approaches to correlated electron systems are presented.(i)The single impurity Anderson model is solved by a semi-analytical approach based on the flow equation method.(ii)A numerically implementable Multi-scale Many-Body approach to strongly correlated electron systems is introduced.An extension to quantum cluster methods, it approximates correlations on any given length-scale commensurate with the strength of the correlations on the respective scale.(iii) The spectral properties of the one-dimensional Holstein and breathing polaron models using the self-consistent Born approximation are presented.4.10 a) Energy levels for zero coupling (g = 0).The solid line shows the non-interacting electron dispersion, the dashed ones are one phonon + one electron states and the dotted one two phonons + one electron, etc. k 1 is the momentum where the electron dispersion reaches ω 0 and has the most significant contribution to the scattering at the first phonon threshold.k 2 is the value of momentum with significant contribution to the scattering at the second phonon threshold.b) For the polaron with momentum k the scattering with momentum q has a significant contribution.Although the polaron momentum is close to k F , thus k a = k -k F is small, q = k a + k b is large because k b which connects two points on the Fermi surface can be large. . . . . . . . . . .