Title: Emergent local integrals of motion without a complete set of localized eigenstates Show Chinese title

Title: 无完整局域本征态集合的涌现局部守恒量

Abstract: Systems where all energy eigenstates are localized are known to display an emergent local integrability, in the sense that one can construct an extensive number of operators that commute with the Hamiltonian and are localized in real space. Here we show that emergent local integrability does not require a complete set of localized eigenstates. Given a set of localized eigenstates comprising a nonzero fraction $(1-f)$ of the full many body spectrum, one can construct an extensive number of integrals of motion which are local in the sense that they have {\it nonzero weight} in a compact region of real space, in the thermodynamic limit. However, these modified integrals of motion have a `global dressing' whose weight vanishes as $\sim f$ as $f \rightarrow 0$. In this sense, the existence of a {\it non-zero fraction} of localized eigenstates is sufficient for emergent local integrability. We discuss the implications of our findings for systems where the spectrum contains delocalized states, for systems with projected Hilbert spaces, and for the robustness of quantum integrability. Show Chinese abstract

Abstract: 所有能级本征态都是局域化的系统已知会表现出一种涌现的局部可积性，即可以构造出与哈密顿量对易且在实空间中局域的大量算符。 在这里，我们表明，涌现的局部可积性并不要求存在完整的局域本征态集合。 给定一个包含全多体谱非零分数$(1-f)$的局域本征态集合，可以构造出大量守恒量，这些守恒量在热力学极限下，在实空间的紧致区域内具有{\it 非零权重}。 然而，这些修改后的守恒量具有一种“全局修饰”，其权重随着$\sim f$而消失，当$f \rightarrow 0$时。 从这个意义上说，局域本征态的{\it 非零分数}存在足以保证涌现的局部可积性。 我们讨论了这些发现对于谱中包含非局域态的系统、具有投影希尔伯特空间的系统以及量子可积性的鲁棒性的影响。