Q-tensor continuum energies as limits of head-to-tail-symmetric spin systems

TL;DR

This paper studies the continuum limit of head-to-tail symmetric spin systems, showing that the limit can be expressed as a Q-tensor functional and characterizing the energy density in various dimensions.

Contribution

It introduces a framework for deriving continuum energies from discrete spin systems with head-to-tail symmetry, including second-order theories and gradient models.

Findings

Limit energies can be expressed as Q-tensor functionals.

Characterization of energy density in 2D and 3D cases.

Development of second-order and gradient models in the planar case.

Abstract

We consider a class of spin-type discrete systems and analyze their continuum limit as the lattice spacing goes to zero. Under standard coerciveness and growth assumptions together with an additional head-to-tail symmetry condition, we observe that this limit can be conveniently written as a functional in the space of $Q$-tensors. We further characterize the limit energy density in several cases (both in 2 and 3 dimensions). In the planar case we also develop a second-order theory and we derive gradient or concentration-type models according to the chosen scaling.