Crystallization to the Square Lattice for a Two-Body Potential
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Date
2021
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Abstract
We consider two-dimensional zero-temperature systems of N particles towhich we associate an energy of the form
E[V](X) := Sigma(1 <= i<) j(<= N) V(|X(i) - X(j)|),
where X(j) is an element of R-2 represents the position of the particle j and V(r) is an element of R is the pairwise interaction energy potential of two particles placed at distance r. We show that under suitable assumptions on the single-well potential V, the ground state energy per particle converges to an explicit constant <(E)over bar>(sq[V]), which is the same as the energy per particle in the square lattice infinite configuration. We thus have
N (E) over bar (sq[V]) <= (X:{1,...,N}-> R2) E[V](X) <= N (E) over bar (sq)[V] + O(N-1/2). Moreover (E) over bar (sq)[V] is also re-expressed as the minimizer of a four point energy. In particular, this happens if the potential V is such that V(r) = +infinity for r < 1, V(r) = -1 for r is an element of [1, root 2], V(r) = 0 if r > root 2, in which case (E) over bar (sq)[V] = -4. To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.
E[V](X) := Sigma(1 <= i<) j(<= N) V(|X(i) - X(j)|),
where X(j) is an element of R-2 represents the position of the particle j and V(r) is an element of R is the pairwise interaction energy potential of two particles placed at distance r. We show that under suitable assumptions on the single-well potential V, the ground state energy per particle converges to an explicit constant <(E)over bar>(sq[V]), which is the same as the energy per particle in the square lattice infinite configuration. We thus have
N (E) over bar (sq[V]) <= (X:{1,...,N}-> R2) E[V](X) <= N (E) over bar (sq)[V] + O(N-1/2). Moreover (E) over bar (sq)[V] is also re-expressed as the minimizer of a four point energy. In particular, this happens if the potential V is such that V(r) = +infinity for r < 1, V(r) = -1 for r is an element of [1, root 2], V(r) = 0 if r > root 2, in which case (E) over bar (sq)[V] = -4. To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.