  The symmetry and the locality are the two major sources of various nontrivial
statements 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. The argument is based on the exponential decay of various types of
equal-time correlation functions.

Statistical Mechanics;Strongly Correlated Electrons