  Correlated many-body problems ubiquitously appear in various fields of
physics such as condensed matter physics, nuclear physics, and statistical
physics. However, due to the interplay of the large number of degrees of
freedom, it is generically impossible to treat these problems from first
principles. Thus the construction of a proper model, namely effective
Hamiltonian, is essential. Here, we propose a simple scheme of constructing
Hamiltonians from given energy or entanglement spectra with machine learning.
We apply the proposed scheme to the Hubbard model at the half-filling, and
compare the obtained effective low-energy spin-1/2 model with several analytic
results based on the high order perturbation theory which have been
inconsistent with each other. We also show that our approach can be used to
construct the entanglement Hamiltonian of a quantum many-body state from its
entanglement spectrum as well. We exemplify this using the ground states of the
$S=1/2$ two-leg Heisenberg ladders. We observe a qualitative difference between
the entanglement Hamiltonians of the two phases (the Haldane phase and the Rung
Singlet phase) of the model due to the different origin of the entanglement. In
the Haldane phase, we find that the entanglement Hamiltonian is non-local by
nature, and the locality can be restored by introducing the anisotropy and
turning the system into the large-$D$ phase. Possible applications to the study
of strongly-correlated systems and the model construction from experimental
data are discussed.
