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The quantum variance of an observable is a fundamental quantity in quantummechanics, and the variance provides additional information other than theaverage itself. By examining the relation between the particle-current variance$(\delta J)^2$ and the average current $J$ in both closed and open interactingfermionic systems, we show the emergence of a multi-valued Lissajous curvebetween $\delta J$ and $J$ due to interactions. As a closed system weconsidered the persistent current in a benzene-like lattice enclosing aneffective magnetic flux and solved it by exact diagonalization. For the opensystem, the steady-state current flowing through a few lattice sites coupled totwo particle reservoirs was investigated using a Lindblad equation. In bothcases, 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 ultracoldatoms, thus offering an alternative way of probing the dynamics of many-bodysystems.

Many-body multi-valuedness of particle-current variance in closed and open cold-atom systems