We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane $\mathbb R^2$ perpendicular to an external constant magnetic field of strength $B>0$. We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential $\mu\ge B$ (in suitable physical units). For this (pure) state we define its local entropy $S(\Lambda)$ associated with a bounded (sub)region $\Lambda\subset \mathbb R^2$ as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $\Lambda$ of finite area $|\Lambda|$. In this setting we prove that the leading asymptotic growth of $S(L\Lambda)$, as the dimensionless scaling parameter $L>0$ tends to infinity, has the form $L\sqrt{B}|\partial\Lambda|$ up to a precisely given (positive multiplicative) coefficient which is independent of $\Lambda$ and dependent on $B$ and $\mu$ only through the integer part of $(\mu/B-1)/2$. Here we have assumed the boundary curve $\partial\Lambda$ of $\Lambda$ to be sufficiently smooth which, in particular, ensures that its arc length $|\partial\Lambda|$ is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case $B=0$, where an additional logarithmic factor $\ln(L)$ is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space $\text L^2(\mathbb R^2)$ to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Renyi entropies.

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理想费米气体在恒定磁场中局部基态熵的渐近增长

我们考虑没有自旋但带电荷的不可区分粒子的理想费米气体，限制在欧几里得平面 $\mathbb R^2$ 垂直于强度 $B>0$ 的外部恒定磁场。我们假设这种（无限）量子气体在零温度下处于热平衡状态，即处于具有化学势 $\mu\ge B$（以合适的物理单位表示）的基态。对于这个（纯）状态，我们将其与有界（子）区域 $\Lambda\subset \mathbb R^2$ 相关联的局部熵 $S(\Lambda)$ 定义为获得的（混合）局部子状态的冯诺依曼熵通过将无限区域基态减少到有限区域 $|\Lambda|$ 的这个区域 $\Lambda$。在这个设置中，我们证明了 $S(L\Lambda)$ 的领先渐近增长，因为无量纲缩放参数 $L>0$ 趋于无穷大，具有形式 $L\sqrt{B}|\partial\Lambda|$ 直到一个精确给定的（正乘法）系数，该系数独立于 $\Lambda$ 并且仅通过整数依赖于 $B$ 和 $\mu$ $(\mu/B-1)/2$ 的一部分。这里我们假设 $\Lambda$ 的边界曲线 $\partial\Lambda$ 足够平滑，特别是确保其弧长 $|\partial\Lambda|$ 是明确定义的。该结果与所谓的面积律缩放（对于两个空间维度）一致。它与零场情况 $B=0$ 形成对比，其中已知存在额外的对数因子 $\ln(L)$。对于更简单的情况，即潜在的单粒子哈密顿量（称为朗道哈密顿量），我们也有类似的结果，但系数稍微更明确，受限于其自然希尔伯特空间 $\text L^2(\mathbb R^2)$ 到单个但任意朗道级别的特征空间。这两个结果都扩展到了量子人一熵的整个单参数族。