The symmetry and the locality are the two major ingredients leading to various nontrivial statements in quantum many-body systems. In this talk I will show 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 twisted angle of the boundary condition in the limit of the large system size. The argument is based on the exponential decay of several types of equal-time correlation functions.

Remark: joint with IPMU Mathematics-String Theory seminar