Abstract In this paper, we study two-loop contribution to the effective action of a two-dimensional non-linear sigma model. We derive new formula, which can be applicable regularization general type. As examples, obtain known results for dimensional and investigate types cutoff one. Also, discuss non-local contributions restrictions on regularization.

Abstract Using the recently developed theory of rigorously validated numerics, we address phase-field-crystal model at microscopic (atomistic) level. We show existence critical points and local minimizers associated with ‘classical’ candidates, grain boundaries, localized patterns. further dynamical relationships between observed patterns for fixed parameters across parameter space, then formul...