量子格点系统的平衡时间尺度

The equilibration time scales of the quantum lattice model

平衡时间尺度问题是理解量子系统平衡化过程的一个重要的问题,此问题尚无一个广泛适用而又准确的解答.本文研究了量子格点系统的局域平衡时间尺度问题,通过平衡与纠缠熵的关系,给出一种新的平衡的判据,相对其他判据, Rényi熵的计算更简单,能与信息的传播相联系.对平衡时间上界,我们沿用观测上时间平均平衡这个判据.由于这个判据与观测对应,因此定义准确,但计算比较复杂.通过适当地假设系统的初始状态,我们给出了一个新的结果.由于对系统哈密顿量没有限制,此结果适用范围很广.此结果给出的平衡时间与具体观测相关,正比于能隙坐标下观测算符与初始密度矩阵乘积分布的二阶Rényi熵.当观测限于一个小区域时,平衡时间上界可被限制得很小.对于平衡时间下界,通过计算短程相互作用、指数衰减相互作用、长程相互作用系统的局域二阶Rényi熵变化率,我们给出了这些系统平衡时间新的下界.本文得到的平衡时间上下界对于理解量子系统平衡化具有重要意义.

Finding the equilibration time scale is an important open question in studying the equilibration of quantum systems. There are many kinds of systems that are unable to achieve equilibration, such as Anderson insulator, many-body localization systems and some integrable systems. For those systems that can reach equilibration, it was proved that there exists a general equilibration time scale. But these upper bounds are unrealistically long, and the lower bounds are also unrealistically short. How to get an accurate and general equilibration time scale is still unclear. In this paper, we study local equilibration time scales in quantum lattice systems. With the relation between equilibration and entanglement enU＇opy, we define a new criterion for equilibration. This criterion is based on Rrnyi entropy, which is simpler for calculation. Moreover, the production of Rrnyi entropy is highly dependent on the spreading of information, and hence the tools developed in quantum information theory can help a lot. For the upper bound of equilibration time, we use the normal criterion of the time average ofthe fluctuation of the observation. This equilibration criterion is assessed with concrete observable operator, hence it is more accurate than the criterion of R6nyi entropy. But this criterion is more complicated for calculation. With an appropriate assumption about the initial state, we present a new upper bound of equilibration time. Since the results are not constrained by the Hamiltonian of the system, this bound can be applied to various situations. If we are concerned with the local equilibration, we can limit the observation to a small region. If the whole system is big enough, the local observation would always find that the rest part is staying at the canonical ensemble, so that the local equilibration time will not increase with the size of whole system. When the local region is small enough, the upper bound of equilibration time can be much shorter. The local Rrnyi entropy has close relation to the propagation of information. The limitation of the speed of information propagation in a quantum lattice system can be given by the Lieb-Robinson bound, with which we evaluate the production rate of local 2-R6nyi entropy. With these, we give a new lower bound of equilibration time for systems whose interactions are respectively short-range, exponentially-decaying and long-range. In earlier works, the lower bound of equilibration time is equal to the Lieb-Robinson time of the local system. In our rigorous proof, however, the lower bound is related to the strength of hopping, which also decides the Lieb-Robinson velocity. This means the equilibration time is related to the Lieb-Robinson time indeed. But the strictly equal relation may not hold. These new bounds are important to understanding the process of quantum equilibration.