Construction of Hamiltonians by supervised learning of energy and entanglement spectra

Correlated many-body problems ubiquitously appear in various fields of physics such as condensed matter, nuclear, 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 supervised learning algorithm for constructing Hamiltonians from given energy or entanglement spectra. We apply the proposed scheme to the Hubbard model at the half-filling, and compare the obtained effective low-energy spin 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 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 nonlocal by nature, and the locality can be restored by introducing the anisotropy and turning the ground state into the large-D phase. Possible applications to the model construction from experimental data and to various problems of strongly correlated systems are discussed.