Invariant Hamiltonian Bootstrap for Quantum Many-body Ground States

Hamiltonian bootstrap methods use semidefinite relaxation to find certifiable lower bounds to the ground state energy of a Hamiltonian, along with approximations of ground state correlation functions. We give a thorough treatment of the role of symmetry in finite-dimensional Hamiltonian bootstrap, and show that symmetry can be used to significantly reduce the required computational resources. We additionally incorporate arbitrary Hermitian linear constraints in our analysis, which allows one to find properties of the ground state within a specified subspace. We demonstrate our approach using the 1D Hubbard model. We find quantitative agreement with exact diagonalization on 10 sites at both half-filling and quarter-filling. Additionally, we apply our method on 100 sites at half-filling and find quantitative agreement with the exact ground state energy density in the thermodynamic limit derived from the Bethe ansatz. This talk is based on arXiv:2410.00810.