磁场中三维理想费米气体基态的纠缠熵
Entanglement entropy of ground states of the three-dimensional ideal Fermi gas in a magnetic field
摘要：
研究了在垂直于平面的非零恒定磁场作用下，$\mathbb R^3$中非相互作用（无自旋）费米子基态纠缠熵的渐近增长。对于没有磁场的情形，我们发现，对于首阶$L^2\ln(L)$，当标度参数$L$趋于无穷大时，对于有界的分段Lipschitz区域$LΛ\subset \mathbb R^3$，这种熵的一个数学上增强的面积律.这与二维情况相反，因为粒子现在可以在磁场的方向上自由移动，这导致额外的$\ln(L)$因子。在非磁性情况下，首阶系数的显式表达式包含类似于Widom公式的表面积分。然而，它的不同之处在于，对边界的依赖性不仅取决于其面积，而且取决于“垂直于磁场方向的面积“。在此基础上，我们证明了具有间断符号的一维Wiener-Hopf算子的某些迹的改进的两项渐近展开式（直到一阶误差项）。这是独立的利益，并导致改进的误差项的顺序$L^2$的相关跟踪分段$\mathsf{C}^{1,α}$光滑表面$\partial Λ$。
英文摘要：
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 $\partial Λ$.
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