  Let $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus
q_d\mathbb{Z}$, where $q_l\in \mathbb{Z}_+$, $l=1,2,\cdots,d$. Let $\Delta+V$
be the discrete Schr\"odinger operator, where $\Delta$ is the discrete
Laplacian on $\mathbb{Z}^d$ and the potential $V:\mathbb{Z}^d\to \mathbb{R}$ is
$\Gamma$-periodic. We prove three rigidity theorems for discrete periodic
Schr\"odinger operators in any dimension $d\geq 3$:
  (1) if at some energy level, Fermi varieties of the $\Gamma$-periodic
potential $V$ and the $\Gamma$-periodic potential $Y$ are the same (this
feature is referred to as {\it Fermi isospectrality} of $V$ and $Y$), and $Y $
is a separable function, then $V$ is separable;
  (2) if potentials $V$ and $Y$ are Fermi isospectral and both
$V=\bigoplus_{j=1}^rV_j$ and $Y=\bigoplus_{j=1}^r Y_j$ are separable functions,
then, up to a constant, lower dimensional decompositions $V_j$ and $Y_j$ are
Floquet isospectral, $j=1,2,\cdots,r$;
  (3) if a potential $V$ and the zero potential are Fermi isospectral, then $V$
is zero.
  In particular, all conclusions in (1), (2) and (3) hold if we replace the
assumption "Fermi isospectrality" with a stronger assumption "Floquet
isospectrality".
