论文标题
恒定磁场中理想费米气体的局部地面熵的渐近生长
Asymptotic growth of the local ground-state entropy of the ideal Fermi gas in a constant magnetic field
论文作者
论文摘要
我们认为没有旋转但电荷的不可区分的颗粒的理想费米气体局限于欧几里得平面$ \ mathbb r^2 $垂直于外部恒定磁场的强度$ b> 0 $。我们假设（无限）量子气在零温度下处于热平衡状态，即处于其基态，具有化学势$μ\ ge b $（以合适的物理单位为单位）。为此，我们将其本地熵$ s（λ）$定义与有限的（子）区域$λ\ subset \ mathbb r^2 $相关联，为（混合）本地替代的von Neumann熵，通过减少无限区域的基层状态，以降低该区域$λ$λ$λ$λ$λ$λ$。在这种情况下，我们证明了$ s（lλ）$的主要渐近增长，因为无尺寸的缩放参数$ l> 0 $倾向于无限，具有$ l \ sqrt {b} | \partialλ| $的形式，最高为$λ$和$λ$的$λ$和$λ$的$λ$和$ b $ n独立于$λ$和$λ$的元素$（μ/b-1）/2 $。在这里，我们假设$λ$的边界曲线$ \partialλ$要变得足够光滑，特别是确保其弧长$ | \partialλ| $定义明确。该结果与所谓的区域法缩放（对于两个空间维度）一致。它与零场案例$ b = 0 $进行了对比，其中已知存在附加的对数因子$ \ ln（l）$。我们也有相似的结果，并且在更简单的情况下，基础单粒子哈密顿量（称为Landau Hamiltonian）受其天然希尔伯特（Hilbert Space）$ \ text l^2（\ mathbb r^2）$限制到单个但任意的landau级别的特征。这两个结果都扩展到整个量子式熵家族。
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 $μ\ge B$ (in suitable physical units). For this (pure) state we define its local entropy $S(Λ)$ associated with a bounded (sub)region $Λ\subset \mathbb R^2$ as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $Λ$ of finite area $|Λ|$. In this setting we prove that the leading asymptotic growth of $S(LΛ)$, as the dimensionless scaling parameter $L>0$ tends to infinity, has the form $L\sqrt{B}|\partialΛ|$ up to a precisely given (positive multiplicative) coefficient which is independent of $Λ$ and dependent on $B$ and $μ$ only through the integer part of $(μ/B-1)/2$. Here we have assumed the boundary curve $\partialΛ$ of $Λ$ to be sufficiently smooth which, in particular, ensures that its arc length $|\partialΛ|$ 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 Rényi entropies.
