Non-uniform black strings and the critical dimension in the $1/D$ expansion

TL;DR

This paper uses a large D effective theory approach to analyze non-uniform black strings, deriving an effective equation and thermodynamic properties up to NNLO, and identifies a critical dimension where phase transition order changes.

Contribution

The study provides the first NNLO analytical results for NUBS thermodynamics and confirms the critical dimension for phase transition change using the $1/D$ expansion.

Findings

Critical dimension D* ≈ 13.5 for phase transition change

Analytical NNLO thermodynamics matches numerical results

Effective equations derived up to NNLO in 1/D expansion

Abstract

Non-uniform black strings (NUBS) are studied by the large $D$ effective theory approach. By solving the near-horizon geometry in the $1/D$ expansion, we obtain the effective equation for the deformed horizon up to the next-to-next-to-leading order (NNLO) in $1/D$. We also solve the far-zone geometry by the Newtonian approximation. Matching the near and far zones, the thermodynamic variables are computed in the $1/D$ expansion. As the result, the large $D$ analysis gives a critical dimension $D_*\simeq13.5$ at which the translation-symmetry-breaking phase transition changes between first and second order. This value of $D_*$ agrees perfectly, within the precision of the $1/D$ expansion, with the result previously obtained by E. Sorkin through the numerical resolution. We also compare our NNLO results for the thermodynamics of NUBS to earlier numerical calculations, and find good agreement within the expected precision.