The inverse problem of 'eigenstates-to-Hamiltonian' is considered for an open chain of $N$ quantum spins in the context of Many-Body-Localization. We first construct the simplest basis of the Hilbert space made of $2^N$ orthonormal Matrix-Product-States (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding $N$ Local Integrals of Motions (LIOMs) that can be considered as the local building blocks of these $2^N$ MPS, in order to construct the parent Hamiltonians that have these $2^N$ MPS as eigenstates. Finally we study the Matrix-Product-Operator form of the Diagonal Ensemble Density Matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states.

中文翻译：

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构建多体局部模型，其中所有本征态都是矩阵乘积态

在多体定位的背景下，对于 $N$ 量子自旋的开放链，考虑了“本征态到汉密尔顿”的逆问题。我们首先构造由 $2^N$ 正交矩阵乘积状态 (MPS) 组成的希尔伯特空间的最简单基，因此将自动满足纠缠面积定律。然后我们分析相应的 $N$ 局部运动积分 (LIOM)，它们可以被视为这些 $2^N$ MPS 的局部构建块，以构建具有这些 $2^N$ MPS 作为本征态的父哈密顿量。最后，我们研究了对角集合密度矩阵的矩阵-乘积-运算符形式，它允许计算幺正动力学的长时间平均可观察量。给出了局部可观察值的记忆和算子空间中的纠缠性质的明确结果，