Determining a local Hamiltonian from a single eigenstate
Determining a local Hamiltonian from a single eigenstate
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DOI：
10.22331/q-2019-07-08-159
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发表时间：
2019-07-08
期刊：
影响因子：
6.4
通讯作者：
Ranard, Daniel
中科院分区：
文献类型：
--
作者：
Qi, Xiao-Liang;Ranard, Daniel
我们询问对一个局域哈密顿量的单个本征态的了解是否足以唯一地确定该哈密顿量。我们给出证据表明，对于一般的局域哈密顿量，无论是基态还是激发态本征态，答案是“是”。实际上，一般来说，仅知道局域可观测量相对于本征态的两点等时关联函数就应该足以精确地恢复有限尺寸系统的哈密顿量，且数值算法的运行时间是系统尺寸的多项式。我们还研究了大系统极限、重构对误差的敏感性以及关联函数仅针对固定子区域上的可观测量已知的情况。数值演示支持了有限一维自旋链的结果（尽管在外推到高维的无限尺寸系统时必须谨慎）。为了我们的分析目的，我们定义了一个态的“k - 关联谱”，它揭示了态中局域关联的性质，并且可能具有独立的研究价值。
We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is "yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to in finite-size systems in higher dimensions). For the purpose of our analysis, we define the "k-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.