Determining a local Hamiltonian from a single eigenstate

Xiao-Liang Qi1,2 and Daniel Ranard1

1Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305 USA
2Institute for Advanced Study, Princeton NJ 08540 USA

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 infinite-size systems in higher dimensions). For the purpose of our analysis, we define the ``$\textit{k}$-correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

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