  Classifying the zero-temperature ground states of quantum field theories with
finite charge density is a very interesting problem. Via holography, this
problem is mapped to the classification of extremal charged black brane
geometries with anti-de Sitter asymptotics. In a recent paper [1], we proposed
a Bianchi classification of the extremal near-horizon geometries in five
dimensions, in the case where they are homogeneous but, in general,
anisotropic. Here, we extend our study in two directions: we show that Bianchi
attractors can lead to new phases, and generalize the classification of
homogeneous phases in a way suggested by holography. In the first direction, we
show that hyperscaling violation can naturally be incorporated into the Bianchi
horizons. We also find analytical examples of "striped" horizons. In the second
direction, we propose a more complete classification of homogeneous horizon
geometries where the natural mathematics involves real four-algebras with three
dimensional sub-algebras. This gives rise to a richer set of possible
near-horizon geometries, where the holographic radial direction is
non-trivially intertwined with field theory spatial coordinates. We find
examples of several of the new types in systems consisting of reasonably simple
matter sectors coupled to gravity, while arguing that others are forbidden by
the Null Energy Condition. Extremal horizons in four dimensions governed by
three-algebras or four-algebras are also discussed.
