If time translation invariance is spontaneously broken to a discrete subgroup (much like a crystal lattice in time), then we can have interesting Early Universe behaviour while still recovering approximate Poincare invariance at very late times. 1512.02304 recently studied a scalar field moving periodically in an expanding Universe (a non-canonical k-essence theory), and identified a phase of cosmological matter with this property. Here it is shown that the existence of such a periodic trajectory necessarily requires (periodic) NEC violation, which cannot happen dynamically in k-essence models (without introducing ghosts and gradient instabilities). This is argued both at the level of the classical Hamiltonian phase space (from the Bendixson-Dulac theorem), and also at the level of quantum fluctuations (which become strongly coupled at NEC violation).

Spontaneously breaking Poincare invariance relaxes a number of consistency conditions which otherwise forbid certain novel dynamics and phenomenology. Of course, the Universe today appears locally Poincare invariant to a high degree, so any cosmologically viable theory must allow some kind of (at least approximate) restoration of Poincare invariance. One example of this is to break continuous time-translations, t -> t +c, down to a discrete subgroup, t-> t + nT, where n is an integer and T is some fixed period. Then at late times, t >> T, we recover approximate time translation invariance. Such a model can be useful in modelling early stages of inflation or late stages of dark energy, and an explicit realisation of such a symmetry breaking phase was recently outlined in 1512.02304. However, it now seems that this phase must contain pathological instabilities.

In graphene, each atom contributes three bound electrons to the lattice and one free electron to the conduction band. Ignoring the free electrons, the pairs of bound electrons in the lattice behave like scalar fields on an M0-brane, enjoying a high degree of symmetry. The free electrons break this symmetry and create two gauge fields, which play the role of gravitons. One gauge field (the `gravity') is produced by anti-parallel electron spins, while the other (the `anti-gravity') is produced by parallel electron spins. In the absence of defects, these exactly cancel and overall the system has zero curvature. However, the presence of suitable defects can leads to a nonzero curvature, and the two gauge fields are then described by a modified theory of gravity, which can be used to describe the current density in the graphene.

While the study of Mp-branes in M-theory was originally from the point of view of fundamental physics, it is now finding interesting applications in describing condensed matter systems such as graphene. Many experiments have recently been carried out on graphene samples, not least because the symmetries of its honeycomb lattice give a low mass energy excitation for fermions, and accurate measurements of its surface free energy, density of states, free current density and the effect of defects can be made.