LG (Landau-Ginzburg) in GL (Gregory-Laflamme)

TL;DR

This work addresses the Gregory-Laflamme instability of black strings by introducing a Landau-Ginzburg (LG) framework to determine the order of the GL transition and by extending the analysis from a single circle to arbitrary torus compactifications ${\bf T}^p$. The LG approach expands the free energy around the critical GL point in powers of the order parameters, showing that the sign of the quartic coefficient ${\cal C}$ (and its tensor generalization for multiple tachyons) fixes whether the transition is first or second order, while avoiding some of the full metric-order computations. For ${\bf T}^1$, the method reproduces known results and yields canonical and microcanonical coefficients that align with prior analyses, including a canonical critical dimension $D^*_{can}=12.5$ and microcanonical indicators. Extending to square tori ${\bf T}^p$, the authors demonstrate that the transition order depends only on the number of extended dimensions $d$ and is robust across $p$, with spontaneous symmetry-breaking tendencies favoring single-tachyon directions. The findings provide a more efficient framework for mapping the phase structure of black strings in higher-dimensional compactifications and constrain how toroidal geometry can influence GL transitions.

Abstract

This paper continues the study of the Gregory-Laflamme instability of black strings, or more precisely of the order of the transition, being either first or second order, and the critical dimension which separates the two cases. First, we describe a novel method based on the Landau-Ginzburg perspective for the thermodynamics that somewhat improves the existing techniques. Second, we generalize the computation from a circle compactification to an arbitrary torus compactifications. We explain that the critical dimension cannot be lowered in this way, and moreover in all cases studied the transition order depends only on the number of extended dimensions. We discuss the richer phase structure that appears in the torus case.

Figures (4)

• Figure 1: The uniform black string together with the definition of the cylindrical coordinates $(r,z)$. $r_0$ is the Schwarzschild radius.
• Figure 2: An illustration of a first order phase transition. A condensed plot showing two kinds of graphs. The thin solid lines show the free energy as a function of $\tilde{\lambda}$ for a sequence of $\mu$ values, while for the thick lines the vertical axis becomes $\mu$ (the horizontal remains $\tilde{\lambda}$) and they designate the various phases corresponding to the extremum of the free energy. The free energy has a minimum for $\mu>\mu_c$ that corresponds to the stable symmetric phase (thick solid line) which becomes unstable below $\mu_c$ (thick dashed line). It follows that the asymmetric phase branch emergent from the critical point (thick dotted line) is unstable since the free energy has a negative direction for $\mu \ge \mu_c$ at $\tilde{\lambda}=\tilde{\lambda}_b$. Note that the free energy in this phase is higher relative to that of the critical state.
• Figure 3: An illustration of a second order transition from a symmetric to an asymmetric phase, with the same condensed conventions as in figure \ref{['fig_O1F']}. The free energy (designated by the thin solid line) has a minimum for $\mu>\mu_c$ corresponding to the stable symmetric phase (thick solid line) which becomes unstable below $\mu_c$ (thick dashed line). The free energy is at minimum also for $\mu= \mu_c$ and the minimum continuously moves away from $\tilde{\lambda}=0$, indicating that the emergent asymmetric phase is stable and has the asymmetry developing smoothly.
• Figure 4: The potential (solid line) and the wave-function (dashed line) for the negative mode, obeying the equation $-d^2\psi/d\rho^2 + V\, \psi = -k^2\psi$, for $d=5$.