Exact Ground State Energy of the Strong-Coupling Polaron

Abstract

Dedicated to the memory of R.L. Dobrushin and P.W. Kasteleyn, two founders of modern statistical mechanics. AbstractThe polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant, α. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result, E 0 ∼ −Cα 2 , and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons.The polaron Hamiltonian of Fröhlich [1] is a model for the Coulomb interaction of one or more electrons with the quantized phonons of an ionic crystal. In the course of time it was also seen to be an interesting field theory model of non-relativistic particles interacting with a scalar boson field, and it was widely studied [2,3,4,5,6,7,8,9] in both contexts.

Symmetry of Bipolaron Bound States for Small Coulomb Repulsion

“…This energy is more than just an approximation, for it is asymptotically exact as α and U tend to infinity with U/α fixed. This is stated in [17], following the technique of [16]; see also [2]. A benchmark for the bipolaron problem is the energy of a single polaron.…”

Method of intermediate problems in the Fröhlich polaron model

“…To our knowledge, lower bounds to the polaron energy spectrum have been receiving much less attention than the upper bounds throughout very long history of polaron studies. Among the most remarkable contributions several works by E.H. Lieb should be mentioned first of all [11,12] as well as the succeeding work by D.M. Larsen [13], who improved the result of [11], though neither of the lower bounds to the polaron ground state energy presented in these papers comes close to the best lowest upper bounds available so far.…”

“…In other words, in the partitioning (21), (22) of the original Hamiltonian (3) the Hamiltonian (22) is "too positive" whilst the Hamiltonian (21) is, in some sense, "not positive enough", which makes the whole partitioning too unbalanced for successful application of the method in the limiting case k D → ∞. Fortunately, much more balanced, albeit not so simple, partitioning schemes exist for the Fröhlich polaron model, for example, the partitionings derived in [11,12], which surely allow for treatment of the "field-theoretical" polaron model. The analysis of this and other balanced partitioning schemes with respect to the method of intermediate problems will be presented somewhere else later.…”

Estimates on Functional Integrals of Non-Relativistic Quantum Field Theory, with Applications to the Nelson and Polaron Models

“…In the case N = 1, for instance, f (t) = αe −t and θ = 1, and inequality (3.2) yields the lower bound −α − α 2 /4, sharp for small α [13] but off by a large factor of about 2.5 for large α [9,33]. In any case, this is an explicit lower bound valid for all α ≥ 0, and is an improvement over Lieb and Yamazaki's bound [34], −α − α 2 /3.…”

“…Its Hamiltonian is of the form (1.1) but its exact shape is irrelevant at this time, and will be studied in some detail below. Many results have been published about the model since Fröhlich's paper, including estimates on the ground state energy [34,13,18,28,29,16], the effective mass problem [13,42,43,44,31], asymptotics for large coupling [39,9,33], stability and absence of binding for multiparticle polarons [14], and many others. In Chapter 4 we give an explicit lower bound on the N -polaron (N electrons interacting with a polar crystal) without interelectronic repulsion, E ≥ −αN − α 2 N 3 /4, where α is the polaron coupling constant.…”