具有随机能谱的非相互作用费米子的基态能量
Ground state energy of noninteracting fermions with a random energy spectrum
摘要：
我们用解析方法导出了无序环境中$K$非相互作用费米子的基态能量的全分布，用一个谱由$N$ i.i.d.~具有分布$p(\varepsilon)$（具有$\varepsilon \geq 0$）的随机能级，其精神与“随机能量模型”相同。我们证明了对于每个固定的$K$，基态能量$E_0$的分布$P_{K,N}(E_0)$在大$N$的极限下具有普遍的标度形式。我们计算这个通用的尺度函数，并表明，它只依赖于$K$和指数$α$表征小$\varepsilon$行为的$p(\varepsilon) \sim \varepsilon^α$。我们比较了分析预测与数值模拟的结果。为此，我们采用了一个复杂的重要性抽样算法，使我们能够获得分布在一个大范围的支持下的概率小到$10^{-160}$。我们发现渐近的分析预测和数值结果之间的一个非常好的协议。
英文摘要：
We derive analytically the full distribution of the ground-state energy of $K$ non-interacting fermions in a disordered environment, modelled by a Hamiltonian whose spectrum consists of $N$ i.i.d.~random energy levels with distribution $p(\varepsilon)$ (with $\varepsilon \geq 0$), in the same spirit as the `Random Energy Model'. We show that for each fixed $K$, the distribution $P_{K,N}(E_0)$ of the ground-state energy $E_0$ has a universal scaling form in the limit of large $N$. We compute this universal scaling function and show that it depends only on $K$ and the exponent $α$ characterizing the small $\varepsilon$ behaviour of $p(\varepsilon) \sim \varepsilon^α$. We compared the analytical predictions with results from numerical simulations. For this purpose we employed a sophisticated importance-sampling algorithm that allowed us to obtain the distributions over a large range of the support down to probabilities as small as $10^{-160}$. We found asymptotically a very good agreement between analytical predictions and numerical results.
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