Hamiltonian Bootstrap with gauged symmetries: Tight Bounds for Symmetry Broken Phases in Infinite Systems

Hamiltonian bootstrap is a recently developed numerical method capable of producing rigorous two-sided bounds on ground state correlation functions of local translation invariant Hamiltonians in the thermodynamic limit. While the method guarantees rigorous bounds, these bounds may be loose if the set of constraints is not properly chosen. For example, recent work on the transverse field Ising model (TFIM) in one dimension found upper bounds on ground state energy and lower bounds on spin correlations to be relatively loose in the small transverse field regime, in which the $\mathbb{Z}_2$ symmetry is spontaneously broken. In this work, we identify domain walls, which cannot be created or removed by any local operator, as the origin of the looseness of these bounds. We propose a general framework to implement constraints associated with these domain walls by gauging the $\mathbb{Z}_2$ symmetry. For the 1D TFIM, we show that we can numerically produce tight rigorous bounds on the energy density and spin correlations in all parameter regimes away from the phase transition. We show that we can also obtain tight bounds for the 2D TFIM deep inside the symmetric and symmetry broken phases. Our work establishes the gauging procedure as a general technique for treating nonlocal excitations within the bootstrap framework, yielding rigorous and tight bounds on important physical quantities.