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Handling the large number of degrees of freedom with proper approximations, namely the construction of the effective Hamiltonian is at the heart of the condensed matter physics. Here we propose a simple scheme of constructing Hamiltonians from given energy spectrum using the supervised learning technique. Taking the Hubbard model at the half-filling as an example, we show that we can find the reduced description, namely the effective spin model, of the Hubbard model in a way that the estimation bias and error are well controlled. We reproduce the effective model at (t/U)^6 obtained previously by arduous perturbative calculations, just by minimizing the error in the spectrum (semi-)automatically using the supervised learning algorithms. We also show that the same approach is useful to construct the entanglement Hamiltonian of a quantum many-body state from its entanglement spectrum, taking the ground states of the S=1/2 two-leg Heisenberg ladders, as an example. We find the qualitative difference of the entanglement Hamiltonian in the two different phases of that model which has not been known previously. Compared to the known approach based on the full diagonalization of the reduced density matrix, our approach is computationally much cheeper thus offering a way of studying the entanglement nature of large (sub)systems.

[1] H. Fujita, Y. O. Nakagawa, S. Sugiura, and M. Oshikawa, Phys. Rev. B 97, 075114 (2018)

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