Adsorption of a macromolecule in an external field: An exactly solvable model with bicritical behavior

Abstract

We introduce a model of an adsorbing polymer chain in an external field, which admits exact analytical solution for the partition function for a finite number of the units, $N.\mathrm{}$ In the thermodynamic limit, the system has an isotropic and two ordered phases, exhibits continuous and discontinuous phase transitions, and has a bicritical point. We obtain exact expressions for the Landau free energy as a function of an order parameter in the vicinity of the first- and second-order phase transition lines, and compare them to the original theory. The Landau free energy is also calculated as a function of two independent order parameters in the vicinity of the bicritical point. The distribution of complex zeros of the partition function is found for both first- and second-order transitions. In the thermodynamic limit it is described by exact analytical expressions, allowing a comparison to the existing phenomenological results within the framework of the Yang-Lee-Fisher approach. An advantage of the model presented is that the partition function can be calculated analytically not only in the thermodynamic limit, but for finite $N$ as well. The ideas of finite-size scaling analysis are checked against the exact solution, in terms of both the functional form of the free energy and the $N$ dependence of the distribution of Fisher zeros.