论文标题
磁场中三维理想费米气体的接地状态的纠缠熵
Entanglement entropy of ground states of the three-dimensional ideal Fermi gas in a magnetic field
论文作者
论文摘要
我们研究了$ \ Mathbb r^3 $在$ \ Mathbb r^3 $中的纠缠核熵的渐近生长，受到垂直于平面的非零的恒定磁场。至于没有磁场的情况，我们发现的是$ l^2 \ ln（l）$，这是该熵的对数增强区域定律，用于有界的，分段的Lipschitz区域$Lλ\ subset \ subset \ Mathbb r^3 $作为缩放参数$ l $ Infinity to Infinity to Infinity。这与二维情况相反，因为粒子现在可以自由地沿磁场的方向移动，这会导致额外的$ \ ln（l）$ kinter。领先顺序系数的显式表达包含类似于非磁性情况下的宽度公式的表面积分。但是，它的不同是，对边界的依赖不仅仅是其面积，而是在“垂直于磁场方向”的区域。在途中，我们证明了具有不连续符号的一维wiener--hopf操作员的某些痕迹的两项渐近扩展（直到命令1的误差项）。这是独立的兴趣，并导致了相关跟踪的$ l^2 $的改进误差项，用于分段$ \ mathsf {c}^{1，α} $ smooth $ \partialλ$。
We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in $\mathbb R^3$ subject to a non-zero, constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order $L^2\ln(L)$, a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region $LΛ\subset \mathbb R^3$ as the scaling parameter $L$ tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra $\ln(L)$ factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom formula in the non-magnetic case. It differs however in the sense that the dependence on the boundary is not solely on its area but on the "area perpendicular to the direction of the magnetic field". On the way we prove an improved two-term asymptotic expansion (up to an error term of order one) of certain traces of one-dimensional Wiener--Hopf operators with a discontinuous symbol. This is of independent interest and leads to an improved error term of the order $L^2$ of the relevant trace for piecewise $\mathsf{C}^{1,α}$ smooth surfaces $\partial Λ$.
