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arXiv 2008.01045cond-mat.stat-mechcond-mat.quant-gasmath-phmath.MP
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$d$维势中非相互作用费米子的计数统计

Counting statistics for non-interacting fermions in a $d$-dimensional potential

Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar, Gregory Schehr

备注：
Main text: 8 pages, 1 figure. Supplemental material: 22 pages, 6 figures
发表信息：
Phys. Rev. E 103, 030105 (2021)
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摘要：

我们发展了一种第一性原理方法来计算任意维$d$（$d>1$的中心）中一般势中$N$非相互作用无自旋费米子基态的计数统计。在限制势中，费米气体被支撑在有界区域上。在$d=1$中，对于特定电位，该系统与标准随机矩阵系综相关。我们研究的量子涨落的费米子${\cal N}_{\cal D}$的数量在一个域$\cal{D}$的宏观尺寸的大部分的支持。我们发现${\cal N}_{\cal D}$的方差随着大$N$的$N^{(d-1)/d} (A_d \log N + B_d)$而增长，并得到了$A_d, B_d$对势和${\cal D}$的大小的显式依赖性（对于$d>1$中的球形域）。这推广了$d=1$中由随机矩阵理论的Dyson-Mehta渐近性给出的微观区域的自由费米子结果。这使我们猜想类似的渐近子系统$\cal{D}$的纠缠熵，在任何维度上，支持$d=1$的精确结果。

英文摘要：

We develop a first-principle approach to compute the counting statistics in the ground-state of $N$ noninteracting spinless fermions in a general potential in arbitrary dimensions $d$ (central for $d>1$). In a confining potential, the Fermi gas is supported over a bounded domain. In $d=1$, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the support. We show that the variance of ${\cal N}_{\cal D}$ grows as $N^{(d-1)/d} (A_d \log N + B_d)$ for large $N$, and obtain the explicit dependence of $A_d, B_d$ on the potential and on the size of ${\cal D}$ (for a spherical domain in $d>1$). This generalizes the free-fermion results for microscopic domains, given in $d=1$ by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem $\cal{D}$, in any dimension, supported by exact results for $d=1$.

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