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arXiv 2106.05014cond-mat.stat-mechcond-mat.quant-gasmath-phmath.MP
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相互作用囚禁费米子的全计数统计

Full counting statistics for interacting trapped fermions

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

备注：
61 pages, 6 figures
发表信息：
SciPost Phys. 11, 110 (2021)
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摘要：

我们研究了$N$无自旋费米子的基态，它在一维空间中受到一个外部势的限制，具有一般Calogero-Sutherland型的长程相互作用。对于某些潜在的选择，该系统映射到标准的随机矩阵合奏的戴森指数$β$的一般值。在费米子模型中，$β$控制着相互作用的强度，而$β=2$对应于非相互作用的情况。我们研究了费米气体中宏观尺寸的区域$\cal{D}$中费米子${\cal N}_{\cal D}$数的量子涨落。我们预测，对于一般的$β$，${\cal N}_{\cal D}$的方差随着$N \gg 1$的$A_β \log N + B_β$而增长，并且我们得到了$A_β$和$B_β$的公式。这是基于一个明确的计算$β\in\left\{ 1,2,4\right\} $和一个猜想，我们制定一般$β$。这一猜想进一步使我们能够得到一个通用的公式，更高的累积量的${\cal N}_{\cal D}$。我们的结果的方差在微观制度被发现是一致的Luttinger液体理论与参数$K = 2/β$的预测，并允许超越。此外，我们提出了家庭的相互作用费米子模型在一维，在其基态，可以映射到随机矩阵模型。我们得到的平均费米子密度为这些模型的一般相互作用参数$β$。在某些情况下，费米子密度表现出有趣的转变，例如，我们得到了一个非相互作用的费米子制定的Gross-Witten-Wadia模型。

英文摘要：

We study $N$ spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index $β$. In the fermion model $β$ controls the strength of the interaction, $β=2$ corresponding to the noninteracting case. 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 Fermi gas. We predict that for general $β$ the variance of ${\cal N}_{\cal D}$ grows as $A_β \log N + B_β$ for $N \gg 1$ and we obtain a formula for $A_β$ and $B_β$. This is based on an explicit calculation for $β\in\left\{ 1,2,4\right\} $ and on a conjecture that we formulate for general $β$. This conjecture further allows us to obtain a universal formula for the higher cumulants of ${\cal N}_{\cal D}$. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter $K = 2/β$, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter $β$. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

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