Bootstrap bounds for Quantum Spin Systems using String Operators

Bootstrap is a numerical many-body method that provides rigorous bounds on ground-state observables by imposing a set of necessary constraints on the expectation values of operators. The quality of the resulting bounds is sensitive to the choice of operators entering the constraints. In particular, bounds on ground-state correlations are often loose in spontaneous symmetry-breaking (SSB) phases, since local operator sets cannot exclude domain-wall excitations. In this work, we introduce non-local, string-like operators into the bootstrap and show that the program can be formulated directly in thermodynamic limit. We then apply our construction to several 1D spin models. First, we obtain a significant tightening of the bounds in the SSB phase of the 1D transverse-field Ising model. Using the 1D axial next-nearest-neighbor Ising model, we further show that this tightening allows for a quantitative estimate of the locations of phase boundaries. Finally, we generalize the string operators to the 1D $\mathbb{Z}_3$ chiral clock model and track the behavior of the bounds across the phase diagram. Our results broaden the class of constraints available to the bootstrap and open a route toward bootstrapping more general symmetry-broken and topological phases, where the relevant constraints may involve non-local or extended operators.