# Construction of Hamiltonians by supervised learning of energy and entanglement spectra

## TL;DR

This paper introduces a machine learning-based method to construct effective Hamiltonians from energy and entanglement spectra, demonstrated on Hubbard and Heisenberg ladder models, aiding the study of strongly-correlated systems.

## Contribution

It presents a novel scheme for deriving Hamiltonians from spectra using supervised learning, applicable to both energy and entanglement spectra in quantum many-body systems.

## Key findings

- Successfully constructed low-energy spin models for the Hubbard model.
- Revealed non-locality of entanglement Hamiltonian in the Haldane phase.
- Restored locality by tuning anisotropy in the entanglement Hamiltonian.

## Abstract

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.