Design of lattice surface energies

Abstract

We provide a general framework for the design of surface energies on lattices. We prove sharp bounds for the homogenization of discrete systems describing mixtures of ferromagnetic interactions by constructing optimal microgeometries, and we prove a localization principle which allows to reduce to the periodic setting in the general nonperiodic case.

Crystallinity of the homogenized energy density of periodic lattice systems

“…Indeed, example 2.8 shows that for infinite range interactions this is in general not true. In [15,16] it is shown that, as the periodicity T of the interactions tends to +∞, it is possible to approximate any norm as surface energy density satisfying suitable growth conditions. We refer to [2] for a random setting where it is shown that an isotropic energy density (and thus non-crystalline) can be obtained in the limit.…”

The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

“…where Q ν is the square with one side orthogonal to ν, u pos ε and u neg ε are the ground states depicted in Figure 1, and ∂ ± ε Q ν are a discrete version of the top/bottom parts of ∂Q ν . Asymptotic formulas like (1.3) are common in discrete-to-continuum variational analyses and are often used to represent Γ-limits of discrete energies [6,10,13,12,8,29]. However, proving an asymptotic lower bound with the density (1.3) for this model requires additional care and is the technically most demanding contribution of this paper.…”

Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

“…For N fixed, the physical system is expected to behave like a classical Ising-type system with N phases. (See also [16,1,3,15,2,19,11,13,18,14] for the analysis of spin systems in the surface scaling.) According to the results proven for the Ising system, we expect the limit energy to be finite on functions of bounded variation with values in the finite set S N .…”