Insensitivity of bulk properties to the twisted boundary condition

The symmetry and the locality are the two major sources of various general theorems in quantum many-body systems. We demonstrate that, in gapped phases of a U(1) symmetric Hamiltonian with finite-range interactions, the bulk properties such as the expectation value of local operators, the ground-state energy and the excitation gap, and the static and low-frequency dynamical responses in general do not depend on the U(1) phase of the twisted boundary condition in the limit of the large system size. Specifically, their dependence on the twisted angle is exponentially suppressed with the linear dimension of the system. Our argument is solely based on the exponential decay of various types of equal-time correlation functions and does not assume any details of the Hamiltonian, meaning that the statement applies quite generally regardless of the dimensionality or the interaction strength of the system.