  We consider a class of parabolic stochastic PDEs on bounded domains
$D\subseteq\mathbb{R}^d$ that includes the stochastic heat equation, but with a
fractional power $\gamma$ of the Laplacian. Viewing the solution as a process
with values in a scale of fractional Sobolev spaces $H_r$, with $r < \gamma -
d/2$, we study its power variations in $H_r$ along regular partitions of the
time-axis. As the mesh size tends to zero, we find a phase transition at
$r=-d/2$: the solutions have a nontrivial quadratic variation when $r<-d/2$ and
a nontrivial $p$th order variation for $p= 2\gamma/(\gamma-d/2-r)>2$ when
$r>-d/2$. More generally, suitably normalized power variations of any order
satisfy a genuine law of large numbers in the first case and a degenerate limit
theorem in the second case. When $r<-d/2$, the quadratic variation is given
explicitly via an expression that involves the spectral zeta function, which
reduces to the Riemann zeta function when $d=1$ and $D$ is an interval.
