Extremal Horizons with Reduced Symmetry: Hyperscaling Violation, Stripes, and a Classification for the Homogeneous Case

TL;DR

This paper classifies extremal black brane horizons with reduced symmetry, incorporating hyperscaling violation and striped phases, and extends the mathematical framework to include real four-algebras for a richer set of homogeneous geometries.

Contribution

It introduces a comprehensive classification of homogeneous extremal horizons using real four-algebras, including new phases with hyperscaling violation and striped horizons, advancing holographic ground state analysis.

Findings

Hyperscaling violation naturally incorporated into Bianchi horizons.

Analytical examples of striped horizons provided.

New homogeneous geometries classified via real four-algebras.

Abstract

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.