The application of numerical techniques to the study of energy landscapes oflarge systems relies on sufficient sampling of the stationary points. Since thenumber of stationary points is believed to grow exponentially with system size,we can only sample a small fraction. We investigate the interplay between thisrestricted sample size and the physical features of the potential energylandscape for the two-dimensional $XY$ model in the absence of disorder with upto $N=100$ spins. Using an eigenvector-following technique, we numericallycompute stationary points with a given Hessian index $I$ for all possiblevalues of $I$. We investigate the number of stationary points, their energy andindex distributions, and other related quantities, with particular focus on thescaling with $N$. The results are used to test a number of conjectures andapproximate analytic results for the general properties of energy landscapes.
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机译：数值技术在大型系统能量分布图研究中的应用依赖于对固定点的充分采样。由于固定点的数量被认为随系统大小成指数增长，因此我们只能采样一小部分。我们研究了在没有$ N = 100 $旋转的无序情况下，二维$ XY $模型的受限样本量与势能景观的物理特征之间的相互作用。使用特征向量跟随技术，我们对给定的Hessian索引$ I $的所有可能值$ I $进行数值计算。我们研究固定点的数量，它们的能量和指数分布以及其他相关数量，尤其关注$ N $的缩放。该结果用于检验许多猜想和近似的能量格局一般性质分析结果。
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