  We find novel perturbative fixed points by introducing mildly
spacetime-dependent couplings into otherwise marginal terms. In
four-dimensional QFT, these are physical analogues of the small-$\epsilon$
Wilson-Fisher fixed point. Rather than considering $4-\epsilon$ dimensions, we
stay in four dimensions but introduce couplings whose leading spacetime
dependence is of the form $\lambda x^\kappa \mu^\kappa$, with a small parameter
$\kappa$ playing a role analogous to $\epsilon$. We show, in $\phi^4$ theory
and in QED and QCD with massless flavors, that this leads to a critical theory
under perturbative control over an exponentially wide window of spacetime
positions $x$. The exact fixed point coupling $\lambda_*(x)$ in our theory is
identical to the running coupling of the translationally invariant theory, with
the scale replaced by $1/x$. Similar statements hold for three-dimensional
$\phi^6$ theories and two-dimensional sigma models with curved target spaces.
We also describe strongly coupled examples using conformal perturbation theory.
