  We define a distinct phase of matter, a "pair density wave" (PDW), in which
the superconducting order parameter $\phi$ varies periodically as a function of
position such that when averaged over the center of mass position, all
components of $\phi$ vanish identically. Specifically, we study the simplest,
unidirectional PDW, the "striped superconductor," which we argue may be at the
heart of a number of spectacular experimental anomalies that have been observed
in the failed high temperature superconductor, La$_{2-x}$ Ba$_x$CuO$_4$. We
present a solvable microscopic model with strong electron-electron interactions
which supports a PDW groundstate. We also discuss, at the level of Landau
theory, the nature of the coupling between the PDW and other order parameters,
and the origins and some consequences of the unusual sensitivity of this state
to quenched disorder.
