Vidmar Lev, Hackl Lucas, Bianchi Eugenio, Rigol Marcos
Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA.
Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, Pennsylvania 16802, USA.
Phys Rev Lett. 2017 Jul 14;119(2):020601. doi: 10.1103/PhysRevLett.119.020601. Epub 2017 Jul 11.
In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)PRLTAO0031-900710.1103/PhysRevLett.71.1291], Page proved that the average entanglement entropy of subsystems of random pure states is S_{ave}≃lnD_{A}-(1/2)D_{A}^{2}/D for 1≪D_{A}≤sqrt[D], where D_{A} and D are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy ⟨S⟩ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models lnD_{A}-(lnD_{A})^{2}/lnD≤⟨S⟩≤lnD_{A}-1/(2ln2)^{2}/lnD. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then ⟨S⟩<lnD_{A} in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.
在一篇具有开创性的论文[D. N. 佩奇，《物理评论快报》71, 1291 (1993)PRLTAO0031 - 900710.1103/PhysRevLett.71.1291]中，佩奇证明了对于(1\ll D_{A}\leq\sqrt{D})，随机纯态子系统的平均纠缠熵为(S_{ave}\simeq\ln D_{A}-(1/2)D_{A}^{2}/D)，其中(D_{A})和(D)分别是子系统和系统的希尔伯特空间维度。因此，典型的纯态是（近乎）最大纠缠的。我们开发了工具来计算二次费米子哈密顿量所有本征态的平均纠缠熵(\langle S\rangle)。特别地，我们为最一般的平移不变模型推导出了精确的界(\ln D_{A}-(\ln D_{A})^{2}/\ln D\leq\langle S\rangle\leq\ln D_{A}-[1/(2\ln2)](\ln D_{A})^{2}/\ln D)。因此，我们证明了：(i) 如果子系统大小是系统大小的有限分数，那么在热力学极限下(\langle S\rangle<\ln D_{A})；即，哈密顿量本征态的平均值偏离了典型纯态的结果；(ii) 在子系统大小是系统大小的可忽略分数的极限情况下，平均纠缠熵最大；即，此类哈密顿量的典型本征态表现出本征态热化。
