Abstract：A complete set of local integrals of motion (LIOM) is a key concept for describing many-body localization (MBL), which explains a variety of intriguing phenomena in MBL systems. For example, LIOM constrain the dynamics and result in ergodicity violation and breakdown of the eigenstate thermalization hypothesis. However, it is difficult to find a complete set of LIOM explicitly and accurately in practice, which impedes some quantitative structural characterizations of MBL systems. Here we propose an accurate numerical method for constructing LIOM, discover through the LIOM an interaction-induced characteristic length $\xi_+$, and prove a 'quasi-product-state' structure of the eigenstates with that characteristic length $\xi_+$ for MBL systems. More specifically, we find that there are two characteristic lengths in the LIOM. The first one is governed by disorder and is of Anderson-localization nature. The second one is induced by interaction but shows a discontinuity at zero interaction, showing a nonperturbative nature. We prove that the entanglement and correlation in any eigenstate extend not longer than twice the second length and thus the eigenstates of the system are the quasi-product states with such a localization length.
${\boldsymbol R} = f(g({\boldsymbol E}))$. Every element of ${\boldsymbol E}$ is 0 or 1. It is 1 only when it is permuted via ${\boldsymbol P}^? $ to a diagonal element of $[{\boldsymbol U}^{\rm d} | {\boldsymbol E}] {\boldsymbol P}^? $. Function $g({\boldsymbol E})$ orthonomalizes every column of ${\boldsymbol E}$ to all columns of ${\boldsymbol U}^{\rm d}$. Function $f({\boldsymbol S})$ unitarizes ${\boldsymbol S}$ in the space spanned by the columns of ${\boldsymbol S}$ via repeated applications of ${\boldsymbol S} \leftarrow \frac 32 {\boldsymbol S} - \frac 12 {\boldsymbol S} {\boldsymbol S}^? {\boldsymbol S}$ until convergence, i. e., ${\boldsymbol S} = \frac 32 {\boldsymbol S} - \frac 12 {\boldsymbol S} {\boldsymbol S}^? {\boldsymbol S}$. In other words, function $f({\boldsymbol S})$ produces a matrix close to ${\boldsymbol S}$ and so that $[{\boldsymbol U}^{\rm d} | {\boldsymbol R}]$ is unitary
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The number of steps in this minimization is 1 in this work, which is found to be the most efficient for the whole LIOM construction. The step size $|\delta {\boldsymbol X}|$ is set to $|\delta {\boldsymbol X}| / |{\boldsymbol I}| \sim 0.08$, to which the minimization efficiency is not very sensitive
2018, Vol. 35 
