EPJ B Colloquium - Finite size and boundary effects in critical two-dimensional free-fermion models

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Published on 28 August 2017

In the study of phase transitions and critical phenomena, it is important to understand finite size corrections to thermodynamic quantities. Finite-size scaling concerns the critical behavior of systems in which one or more directions are finite. It is valuable in the analysis of experimental and numerical data in many situations, for example for films of finite thickness.
As soon as one has a finite system one must consider the question of boundary conditions on the outer surfaces or “walls” of the system because the critical behavior near boundaries normally differs from the bulk behavior.
The author of this EPJ B Colloquium investigates the effects of boundary conditions on finite-size corrections through the study of model systems, especially those which have exact results and can be analysed without numerical errors, such as the Ising model, the dimer model, the resistor network and the spanning tree model.

Two-dimensional models of statistical mechanics have long served as a proving ground when trying to understand critical behavior and to test the general ideas of finite-size scaling.

This Colloquium tries to unify the different results on exact finite-size corrections to a common framework. The author considers the finite-size scaling, finite-size corrections and boundary effects for the critical two-dimensional free-fermion models. It shows that the partition functions of the three models used can be written in terms of the only object, namely, the partition function with twisted boundary conditions.