  We argue, at a very basic effective field theory level, that higher dimension
operators in scalar theories that break symmetries at scales close to their
ultraviolet completion cutoff, include terms that favour the breaking of
translation (Lorentz) invariance, potentially resulting in striped, chequered
board or general crystal-like phases. Such descriptions can be thought of as
the effective low energy description of QCD-like gauge theories near their
strong coupling scale where terms involving higher dimension operators are
generated. Our low energy theory consists of scalar fields describing operators
such as $\bar{q} q$ and $\bar{q} F^{(2n)} q$. Such scalars can have kinetic
mixing terms that generate effective momentum dependent contributions to the
mass matrix. We show that these can destabilize the translationally invariant
vacuum. It is possible that in some real gauge theory such operators could
become sufficiently dominant to realize such phases and it would be interesting
to look for them in lattice simulations. We present a holographic model of the
same phenomena which includes RG running. A key phenomenological motive to look
at such states is recent work that shows that the non-linear response in $R^2$
gravity to such short range fluctuations can mimic a cosmological constant.
Intriguingly in a cosmology with such a Starobinsky inflation term, to generate
the observed value of the present day acceleration would require stripes at the
electroweak scale. Unfortunately, low energy phenomenological constraints on
Lorentz violation in the electron-photon system appear to strongly rule out any
such possibility outside of a disconnected dark sector.

High Energy Physics - Phenomenology;General Relativity and Quantum Cosmology;High Energy Physics - Theory