This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg-Landau framework. © 2014 Springer-Verlag Berlin Heidelberg

This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects of the displacement-gradient fields. Following the core radius approach, we introduce a parameter ε > 0 representing the lattice spacing of the crystal, we remove a disc of radius ε around each dislocation and compute the elastic energy stored outside the union of such discs, namely outside the core region. Then, we analyze the asymptotic behaviour of the elastic energy as ε → 0, in terms of Γ-convergence. We focus on the self energy regime of order log 1/ε; we show that configurations with logarithmic diverging energy converge, up to a subsequence, to a finite number of multiple dislocations and we compute the corresponding Γ-limit. © 2012 Springer-Verlag

Abstract. We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem [16], which concerns a system of N identical atoms in two dimensions interacting via the idealized pair potential V (r) = +∞ if r < 1, −1 if r = 1, 0 if r > 1. This is done by endowing the bond graph of a general particle configuration with a suitable notion of discrete curvature, and appealing to a discrete GaussBonnet theorem [18] which, as its continuous cousins, relates the sum/integral of the curvature to topological invariants. This leads to an exact geometric decomposition of the Heitmann-Radin energy into (i) a combinatorial bulk term, (ii) a combinatorial perimeter, (iii) a multiple of the Euler characteristic, and (iv) a natural topological energy contribution due to defects. An analogous exact geometric decomposition is also established for soft potentials such as the Lennard-Jones potential V (r) = r −6 − 2r −12 , where two additional contributions arise, (v) elastic energy and (vi) energy due to non-bonded interactions.

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Abstract. We consider a variational model for two interacting species (or phases), subject to cross and self attractive forces. We show existence and several qualitative properties of minimizers. Depending on the strengths of the forces, different behaviors are possible: phase mixing or phase separation with nested or disjoint phases. In the case of Coulomb interaction forces, we characterize the ground state configurations.

We propose and analyze a generalized two dimensional XY model, whose interaction potential has n weighted wells, describing corresponding symmetries of the system. As the lattice spacing vanishes, we derive by Γ-convergence the discrete-to-continuum limit of this model. In the energy regime we deal with, the asymptotic ground states exhibit fractional vortices, connected by string defects. The Γ-limit takes into account both contributions, through a renormalized energy, depending on the configuration of fractional vortices, and a surface energy, proportional to the length of the strings.Our model describes in a simple way several topological singularities arising in Physics and Materials Science. Among them, disclinations and string defects in liquid crystals, fractional vortices and domain walls in micromagnetics, partial dislocations and stacking faults in crystal plasticity.

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