Puiseux series expansions for the eigenvalues of transfer matrices and partition functions from the Newton polygon method for nanotubes and ribbons. (English) Zbl 1334.82089

Karanth, Dileep; Schmidt, Jeffrey R.

J. Phys. Math. 5, Article ID 1000125, 13 p. (2014).

The Newton polygon algorithm offers at least three advantages for the study of nanotubes and ribbons. The first one – series expansions for all eigenvalues of the transfer matrix can be found by symbolic-numerical methods as functions of some quantity such as fugacity or a Boltzmann weight. Both very large and very low fugacity regimes, and the cases of ferromagnetic or anti-ferromagnetic systems, are equally approachable. The second one – the technique is not an approximation, i.e., each iteration gives the next order in the series expansion exactly. Besides, if a root truncates to a finite sum of terms, the Newton polygon process itself simply halts, no new segments are produced once the last term in the series is reached. The third one – the degeneracy of a root presents none of the problems normally encountered by numerical root extraction methods in this case. The level of degeneracy of a root has tell-tale signs in the calculation. It is noted that the technique proposed is not limited to a single expansion variable. There is the example to use an extended Hensel construction based upon the Newton polygon algorithm to factor multivariate polynomials.

Reviewer: Yulianna Perepelkina (Moskva)

MSC:

82D80	Statistical mechanics of nanostructures and nanoparticles
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
15A18	Eigenvalues, singular values, and eigenvectors

Keywords:

nanostructures; nanoparticles; lattice systems; eigenvalues