Lifshitz Topological Black Holes

TL;DR

The paper addresses holographic descriptions of (2+1)-D Lifshitz critical systems by constructing and analyzing Lifshitz black holes with topologically nontrivial horizons in a (3+1)-D gravity theory with gauge fields and a topological coupling. It develops Lifshitz asymptotics for $z=2$, provides an exact topological Lifshitz black-hole solution, and builds a numerical framework (with large-$r$ and near-horizon series plus shooting) to map the horizon data to asymptotic charges. Key results include genus-insensitive thermodynamics for large black holes, genus-dependent thermodynamics for small black holes, and boundary-screening behavior via Wilson-loop analysis, highlighting how horizon topology influences holographic transport. The findings advance Lifshitz holography by incorporating topology, offering insight into the dual boundary theory and outlining future avenues for higher dimensions, other $z$, and holographic renormalization in nonrelativistic settings.

Abstract

I find a class of black hole solutions to a (3+1) dimensional theory gravity coupled to abelian gauge fields with negative cosmological constant that has been proposed as the dual theory to a Lifshitz theory describing critical phenomena in (2+1) dimensions. These black holes are all asymptotic to a Lifshitz fixed point geometry and depend on a single parameter that determines both their area (or size) and their charge. Most of the solutions are obtained numerically, but an exact solution is also obtained for a particular value of this parameter. The thermodynamic behaviour of large black holes is almost the same regardless of genus, but differs considerably for small black holes. Screening behaviour is exhibited in the dual theory for any genus, but the critical length at which it sets in is genus-dependent for small black holes.

Figures (9)

• Figure 1: $h_0$ vs. $r_0$ on a log-log scale for spherical ($k=1$, solid), toroidal ($k=0$, dot) and higher-genus ($k=-1$, dot-dash) black holes. The limit given in eq. (\ref{['eq23']}) for $k=1$ is the dashed line. The intersection point of the $k=-1$ curve with the $r_0$ axis corresponds to the exact solution (\ref{['eq20']}).
• Figure 2: A plot of the metric and gauge functions for $r_0=20$ for all three values of $k$. The three curves overlap within the plotting resolution.
• Figure 3: A plot of the metric and gauge functions for $r_0=1.6$ for $k=1$ (solid), $k=0$ (dot), and $k=-1$ (dot-dash).
• Figure 4: A plot of the metric functions for $r_0=0.4$ for $k=1$ (solid), $k=0$ (dot), and $k=-1$ (dot-dash).
• Figure 5: A plot of the gauge functions for $r_0=0.4$ for $k=1$ (solid), $k=0$ (dot), and $k=-1$ (dot-dash).