Bootstrap bounds for Quantum Spin Systems using String Operators

Hamiltonian bootstrap has emerged as a numerical method to provide rigorous bounds on ground-state observables by imposing constraints on local correlators. The quality of the bounds is sensitive to the choice of operators in the constraints. Particularly, tight bounds require the inclusion of operators that annihilate excitations. This is a problem for spontaneous symmetry-breaking phases due to the existence of low-energy domain wall excitations created by non-local operators with unbounded support. In this work, we provide a general framework for including string-like extended operators within the bootstrap program to obtain rigorous, tight bounds in the thermodynamic limit. We illustrate our approach using the 1D transverse field Ising model and the 3-state quantum Potts model, and find that the inclusion of these strings substantially tightens the bounds on the energy density and static correlation functions in both models. Our results extend the classes of constraints that can be imposed and open the door to bootstrapping symmetry-broken and topological phases, where important low-energy excitations are non-local in nature.