Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138, USA.
Phys Rev Lett. 2013 Jun 28;110(26):260601. doi: 10.1103/PhysRevLett.110.260601.
Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.
巴达森、波尔曼和摩尔最近的数值研究揭示了在所谓的多体局域相中，初始乘积态的纠缠熵随时间呈缓慢的对数增长。我们表明，这种令人惊讶的现象是由于不同态的本征能的相位退相干引起的，这种相位退相干是由于相互作用引起的小指数修正。对于弱相互作用，我们发现纠缠熵的增长与 ξln(Vt/ℏ)成正比，其中 V 是相互作用强度，ξ是单粒子局域长度。长时间的纠缠熵饱和值由初始态的参与率决定，参与率是初始态相对于子系统本征态的占比。我们的工作表明，对数纠缠增长是多体局域相中普遍存在的现象，并且揭示了这种相中的广泛的相位退相干时间尺度层次结构。
