Archive for Rational Mechanics and Analysis ( IF 2.6 ) Pub Date : 2021-03-11 , DOI: 10.1007/s00205-021-01627-6 Laurent Bétermin , Lucia De Luca , Mircea Petrache
We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form
$$\begin{aligned} \mathcal {E}[V](X):=\sum _{1\leqq i<j\leqq N}V(|X(i)-X(j)|), \end{aligned}$$where \(X(j)\in \mathbb R^2\) represents the position of the particle j and \(V(r)\in \mathbb R\) 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 \(\overline{\mathcal E}_{\mathrm {sq}}[V]\), which is the same as the energy per particle in the square lattice infinite configuration. We thus have
$$\begin{aligned} N{\overline{\mathcal E}_{\mathrm {sq}}[V]}\leqq \min _{X:\{1,\ldots ,N\}\rightarrow \mathbb R^2}\mathcal E[V](X)\leqq N{\overline{\mathcal E}_{\mathrm {sq}}[V]}+O(N^{\frac{1}{2}}). \end{aligned}$$Moreover \(\overline{\mathcal E}_{\mathrm {sq}}[V]\) is also re-expressed as the minimizer of a four point energy. In particular, this happens if the potential V is such that \(V(r)=+\infty \) for\(r<1\), \(V(r)=-1\) for \(r\in [1,\sqrt{2}]\), \(V(r)=0\) if \(r>\sqrt{2}\), in which case \({\overline{\mathcal E}_{\mathrm {sq}}[V]}=-4\). To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.
中文翻译：
两体势的方晶格结晶
我们考虑N粒子的二维零温度系统，我们将形式能量与之关联
$$ \ begin {aligned} \ mathcal {E} [V]（X）：= \ sum _ {1 \ leqq i <j \ leqq N} V（| X（i）-X（j）|），\结束{aligned} $$其中\（X（J）\在\ mathbb R ^ 2 \）表示颗粒的位置Ĵ和\（V（R）\在\ mathbb r \）被成对相互作用能量两个粒子的放置在距离潜在ř。我们表明，在单阱电势V的适当假设下，每个粒子的基态能量收敛到一个显式常数\（\ overline {\ mathcal E} _ {\ mathrm {sq}} [V] \），即等于方格无限配置中每个粒子的能量。因此，我们有
$$ \ begin {aligned} N {\ overline {\ mathcal E} _ {\ mathrm {sq}} [V]} \ leqq \ min _ {X：\ {1，\ ldots，N \} \ rightarrow \ mathbb R ^ 2} \ mathcal E [V]（X）\ leqq N {\ overline {\ mathcal E} _ {\ mathrm {sq}} [V]} + O（N ^ {\ frac {1} {2} }）。\ end {aligned} $$此外，\（\ overline {\ mathcal E} _ {\ mathrm {sq}} [V] \）也被重新表示为四点能量的最小化器。特别地，这种情况如果电位V是这样的：\（V（R）= + \ infty \）为\（R <1 \） ，\（V（R）= - 1 \）为\（R \在[1，\ SQRT {2}] \），\（V（R）= 0 \）如果\（R> \ SQRT {2} \），在这种情况下\（{\划线{\ mathcalë} _ { \ mathrm {sq}} [V]} =-4 \）。据我们所知，这是两体相互作用能结晶成方格的第一个证明。
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