  For every conserved quantity written as a sum of local terms, there exists a
corresponding current operator that satisfies the continuity equation. The
expectation values of current operators at equilibrium define the persistent
currents that characterize spontaneous flows in the system. In this work, we
consider quantum many-body systems on a finite one-dimensional lattice and
discuss the scaling of the persistent currents as a function of the system
size. We show that, when the conserved quantities are given as the Noether
charges associated with internal symmetries or the Hamiltonian itself, the
corresponding persistent currents can be bounded by a correlation function of
two operators at a distance proportional to the system size, implying that they
decay at least algebraically as the system size increases. In contrast, the
persistent currents of accidentally conserved quantities can be nonzero even in
the thermodynamic limit and even in the presence of the time-reversal symmetry.
We discuss `the current of energy current' in $S=1/2$ XXZ spin chain as an
example and obtain an analytic expression of the persistent current.
