硬壁势中非相互作用费米子的Wigner函数
Wigner function for noninteracting fermions in hard wall potentials
摘要：
Wigner函数$W_N({\bf x}, {\bf p})$是表征$N$ -体系统在相空间中量子涨落的一个有用的量。本文研究了温度为$T=0$，半径为$R$的$d$维球形硬盒中的无相互作用的无自旋费米子的$W_N({\bf x}, {\bf p})$.在大的$N$极限中，局域密度近似（LDA）预测$W_N({\bf x}, {\bf p}) \approx 1/(2 π\hbar)^d$在$({\bf x}, {\bf p})$平面的有限区域内，即对于$|{\bf x}| < R$和$|{\bf p}| < k_F$，其中$k_F$是费米动量，而$W_N({\bf x}, {\bf p})$在由量子涨落确定的尺度上消失在该区域之外或“液滴”。在本文中，我们系统地调查，在这个量子区域，沿着这个液滴的边缘，称为费米面的维格纳函数的结构。在一维，我们发现，有三个不同的边缘区域沿费米冲浪，我们精确计算相关的非平凡的标度函数在每个政权。我们还研究了动量分布$\hat ρ_N(p)$，并发现一个惊人的代数尾非常大的动量$\hat ρ_N(p) \propto 1/p^4$，远远超过$k_F$，让人想起一个类似的尾巴中发现的相互作用的量子系统（讨论的背景下谭的关系）。然后，我们将这些结果推广到更高的$d$，并发现，值得注意的是，接近盒子边缘的尺度函数是通用的，即，独立于维度$d$。
英文摘要：
The Wigner function $W_N({\bf x}, {\bf p})$ is a useful quantity to characterize the quantum fluctuations of an $N$-body system in its phase space. Here we study $W_N({\bf x}, {\bf p})$ for $N$ noninteracting spinless fermions in a $d$-dimensional spherical hard box of radius $R$ at temperature $T=0$. In the large $N$ limit, the local density approximation (LDA) predicts that $W_N({\bf x}, {\bf p}) \approx 1/(2 π\hbar)^d$ inside a finite region of the $({\bf x}, {\bf p})$ plane, namely for $|{\bf x}| < R$ and $|{\bf p}| < k_F$ where $k_F$ is the Fermi momentum, while $W_N({\bf x}, {\bf p})$ vanishes outside this region, or "droplet", on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution $\hat ρ_N(p)$ and find a striking algebraic tail for very large momenta $\hat ρ_N(p) \propto 1/p^4$, well beyond $k_F$, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher $d$ and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension~$d$.
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