Physical Mathematics

Puiseux Series Expansions for the Eigenvalues of Transfer Matrices and Partition Functions from the Newton Polygon Method for Nanotubes and Ribbons

Abstract

Jeffrey R Schmidt and Dileep Karanth

For certain classes of lattice models of nanosystems the eigenvalues of the row-to-row transfer matrix and the components of the corner transfer matrix truncations are algebraic functions of the fugacity and of Boltzmann weights. Such functions can be expanded in Puiseux series using techniques from algebraic geometry. Each successive term in the expansions in powers a Boltzmann weight is obtained exactly without modifying previous terms. We are able to obtain useful analytical expressions for any thermodynamic function for these systems from the series in circumstances in which no exact solutions can be found.