Introduction to statistical field-theory: from a toy model to a one-component plasma

Working with a toy model whose partition function consists of a discrete summation, we introduce the statistical field-theory methodology by transforming a partition function via a formal Gaussian integral relation (the Hubbard-Stratonovich transformation). We then consider Gaussian type of approximations, wherein correlational contributions enter as harmonic fluctuations around the saddle-point solution. The work focuses on how to construct a self-consistent, non-perturbative approximation without recourse to a variational construction based on the Gibbs-Bogolyubov-Feynman inequality that is inapplicable to a complex action. To address this problem, we propose a construction based on a selective satisfaction of a set of exact relations generated by considering a dual representation of a partition function, in its original and transformed form.