  The quantum variance of an observable is a fundamental quantity in quantum
mechanics, and the variance provides additional information other than the
average itself. By examining the relation between the particle-current variance
$(\delta J)^2$ and the average current $J$ in both closed and open interacting
fermionic systems, we show the emergence of a multi-valued Lissajous curve
between $\delta J$ and $J$ due to interactions. As a closed system we
considered the persistent current in a benzene-like lattice enclosing an
effective magnetic flux and solved it by exact diagonalization. For the open
system, the steady-state current flowing through a few lattice sites coupled to
two particle reservoirs was investigated using a Lindblad equation. In both
cases, interactions open a loop and change the topology of the corresponding
$\delta J$-$J$ Lissajous curve, showing that this effect is model-independent.
We finally discuss how the predicted phenomena can be observed in ultracold
atoms, thus offering an alternative way of probing the dynamics of many-body
systems.
