  The Bloch theorem is a powerful theorem stating that the expectation value of
the U(1) current operator averaged over the entire space vanishes in large
quantum systems. The theorem applies to the ground state and to the thermal
equilibrium at a finite temperature, irrespective of the details of the
Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this
work we present a simple yet rigorous proof for general lattice models. For
large but finite systems, we find that both the discussion and the conclusion
are sensitive to the boundary condition one assumes: under the periodic
boundary condition, one can only prove that the current expectation value is
inversely proportional to the linear dimension of the system, while the current
expectation value completely vanishes before taking the thermodynamic limit
when the open boundary condition is imposed. We also provide simple
tight-binding models that clarify the limitation of the theorem in dimensions
higher than one.
