# Intermediate scalings for Solv, Nil and $SL_2({\cal R})$ black branes

## TL;DR

This paper explores Einstein-Maxwell black brane solutions with intermediate scaling regimes characterized by anisotropic scaling of spatial directions, analyzing their properties and DC conductivities.

## Contribution

It introduces new black brane solutions with non-translational invariance and studies their intermediate scaling behaviors and conductivities.

## Key findings

- Existence of Einstein-Maxwell solutions with Solv, Nil, and SL_2(R) geometries.
- Identification of anisotropic intermediate scaling regimes.
- Calculation of DC conductivities and their scaling properties.

## Abstract

In this work we will study black brane solutions that are not translationally invariant in the spatial directions along which it extends. Instead, we require homogeneity, which still allows points along the spatial directions to be related to each other by symmetries. We find Einstein-Maxwell black hole solutions whose near horizon geometry correspond to Solv (Bianchi $V1_{-1}$), Nil (Bianchi $II$) or $SL_2({\cal R})$ (Bianchi $VIII$). Interestingly we observe that at intermediate temperatures our solutions have an scaling regime where different spacetime directions scale differently. We also compute the DC conductivities for these charged solutions and study how they scale in this intermediate regime.