Pastur Leonid, Slavin Victor
Department of Mathematics, King's College, London WC2R 2LS, UK.
B. Verkin Institute for Low Temperature Physics and Engineering, 61103 Kharkiv, Ukraine.
Entropy (Basel). 2024 Jun 30;26(7):564. doi: 10.3390/e26070564.
We consider a quantum system of large size and its subsystem of size , assuming that is much larger than , which can also be sufficiently large, i.e., 1≪L≲N. A widely accepted mathematical version of this inequality is the asymptotic regime of successive limits: first the macroscopic limit N→∞, then an asymptotic analysis of the entanglement entropy as L→∞. In this paper, we consider another version of the above inequality: the regime of asymptotically proportional and , i.e., the simultaneous limits L→∞,N→∞,L/N→λ>0. Specifically, we consider a system of free fermions that is in its ground state, and such that its one-body Hamiltonian is a large random matrix, which is often used to model long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-range hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give streamlined proof of Page's formula for the entanglement entropy of black hole radiation for a wide class of typical ground states, thereby proving the universality and the typicality of the formula.
我们考虑一个大尺寸的量子系统及其尺寸为(L)的子系统，假设(L)远大于(l)，(l)也可以足够大，即(1\ll L\lesssim N)。这个不等式的一个被广泛接受的数学版本是连续极限的渐近区域：首先是宏观极限(N\rightarrow\infty)，然后是当(L\rightarrow\infty)时对纠缠熵的渐近分析。在本文中，我们考虑上述不等式的另一个版本：(L)与(N)渐近成比例的区域，即同时极限(L\rightarrow\infty)，(N\rightarrow\infty)，(L/N\rightarrow\lambda>0)。具体来说，我们考虑一个处于基态的自由费米子系统，其单体哈密顿量是一个大随机矩阵，该矩阵常用于模拟长程跳跃。通过使用随机矩阵理论，我们表明在这种情况下，纠缠熵遵循具有短程跳跃的系统所熟知的体积定律，但该系统由哈密顿量的混合态或纯强激发态描述。我们还给出了一类广泛的典型基态下黑洞辐射纠缠熵的佩奇公式的简化证明，从而证明了该公式的普遍性和典型性。
