# Area Law Micro-State Entropy from Criticality and Spherical Symmetry. (arXiv:1712.02233v1 [hep-th])

It is often assumed that the area law of micro-state entropy and the
holography are intrinsic properties exclusively of the gravitational systems,
such as black holes. We construct a non-gravitational model that exhibits an
entropy that scales as area of a sphere of one dimension less. It is
represented by a non-relativistic bosonic field living on a d-dimensional
sphere of radius R and experiencing an angular-momentum-dependent attractive
interaction. We show that the system possesses a quantum critical point with
the emergent gapless modes. Their number is equal to the area of a
(d-1)-dimensional sphere of the same radius R. These gapless modes create an
exponentially large number of degenerate micro-states with the corresponding
micro-state entropy given by the area of the same (d-1)-dimensional sphere.
Thanks to a double-scaling limit, the counting of the entropy and of the number
of the gapless modes is made exact. The phenomenon takes place for arbitrary
number of dimensions and can be viewed as a version of holography.