硬边势中的非相互作用费米子
Non-interacting fermions in hard-edge potentials
摘要：
本文研究了非相互作用的$N$粒子的费米气体在$N$维非光滑势场中的空间量子涨落和热涨落。我们首先对球对称纯硬盒势进行了深入的研究，在$T=0$和$T>0$两种情况下，盒内势为零。我们发现，近壁的相关性描述的“硬边”内核，这取决于$d$和$T$，这是不同的“软边”艾里内核，其较高的$d$概括，发现光滑的潜力。我们将这些结果推广到势在盒内不均匀的情形，发现存在一类插值于上述“硬边”核和“软边”核之间的核.最后，我们考虑一维奇异势的形式$V(x)\sim |x|^{-γ}$与$γ>0$。我们发现，在$x=0$的奇异性接近的相关性描述的“硬边”内核为$1\leqγ<2$，而他们被称为贝塞尔内核$γ=2$，最后由艾里内核$γ>2$一个更广泛的家庭的“硬边”内核。这些一维内核也出现在随机矩阵理论，我们在这里提供的$1d$费米子模型和相应的随机矩阵合奏之间的映射。这些结果的一部分在最近的一封信中公布，EPL 120，10006（2017）。
英文摘要：
We consider the spatial quantum and thermal fluctuations of non-interacting Fermi gases of $N$ particles confined in $d$-dimensional non-smooth potentials. We first present a thorough study of the spherically symmetric pure hard-box potential, with vanishing potential inside the box, both at $T=0$ and $T>0$. We find that the correlations near the wall are described by a "hard edge" kernel, which depend both on $d$ and $T$, and which is different from the "soft edge" Airy kernel, and its higher $d$ generalizations, found for smooth potentials. We extend these results to the case where the potential is non-uniform inside the box, and find that there exists a family of kernels which interpolate between the above "hard edge" kernel and the "soft edge" kernels. Finally, we consider one-dimensional singular potentials of the form $V(x)\sim |x|^{-γ}$ with $γ>0$. We show that the correlations close to the singularity at $x=0$ are described by this "hard edge" kernel for $1\leqγ<2$ while they are described by a broader family of "hard edge" kernels known as the Bessel kernel for $γ=2$ and, finally by the Airy kernel for $γ>2$. These one-dimensional kernels also appear in random matrix theory, and we provide here the mapping between the $1d$ fermion models and the corresponding random matrix ensembles. Part of these results were announced in a recent Letter, EPL 120, 10006 (2017).
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