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A proof of crystallization in two dimensions
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UNSPECIFIED (2006) A proof of crystallization in two dimensions. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 262 (1). pp. 209-236. doi:10.1007/s00220-005-1458-7 ISSN 0010-3616.
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Official URL: http://dx.doi.org/10.1007/s00220-005-1458-7
Abstract
Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type E({y}) = (1 <= x < x'<= N)Sigma V(vertical bar y(x) - y(x')vertical bar), where y(x) is an element of R-2 is the position of particle x and V(r) is an element of R is the pair-interaction energy of two particles which are placed at distance r. Due to the Mermin-Wagner theorem it can't be expected that at finite temperature this system exhibits long-range ordering. We focus on the zero temperature case and show rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E-*:
(N ->infinity)lim (1)/(N) (y:{1...N}-> R2)min E({y}) = E-*,
where E-* is an element of R is the minimum of a simple function on [0, infinity). Furthermore, if suitable Dirichlet- or periodic boundary conditions are used, then the minimizers form a triangular lattice. To the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the Euclidean symmetry group consisting of rotations and translations.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QC Physics | ||||
Journal or Publication Title: | COMMUNICATIONS IN MATHEMATICAL PHYSICS | ||||
Publisher: | SPRINGER | ||||
ISSN: | 0010-3616 | ||||
Official Date: | February 2006 | ||||
Dates: |
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Volume: | 262 | ||||
Number: | 1 | ||||
Number of Pages: | 28 | ||||
Page Range: | pp. 209-236 | ||||
DOI: | 10.1007/s00220-005-1458-7 | ||||
Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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