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

TL;DR

The paper broadens the holographic classification of zero-temperature ground states at finite density by extending Bianchi-type horizons to include hyperscaling violation and striped phases, and by generalizing the homogeneous classification to real four-algebras with three-dimensional subgroups. It demonstrates constructive pathways via Kaluza–Klein reduction and explicit matter couplings to realize HV geometries in 4d and 5d, including several striped and purely homogeneous solutions, while enforcing physical viability through NEC and Nernst-type conditions. It also develops a systematic 4d classification framework for homogeneous spaces using the 12 four-dimensional algebras $A_{4,k}$ and their 3d subalgebras, and analyzes the corresponding NEC constraints to delineate which algebras yield admissible static horizons. Collectively, the work highlights rich infrared phases of holographic field theories with reduced spatial symmetries and points to future investigations in stability, string-theory embeddings, and fully inhomogeneous horizon classifications.

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.

Figures (2)

• Figure 1: Experience, and wishes, of a typical graduate student at Stanford or TIFR.
• Figure 2: The darkest parameter region is the allowed region in the $(\eta_y , b)$ parameter space. We have set $\eta_x =1$ and $a=1$.