Phase transitions in 3D gravity and fractal dimension

TL;DR

The paper identifies a second-order phase transition in 3D AdS gravity with higher-genus boundaries, driven by the condensation of a sufficiently light scalar with dimension $\Delta<\Delta_c$, and shows that $\Delta_c$ equals the Hausdorff dimension $\delta$ of the limit set of the Schottky group $\Gamma$. It develops a dual CFT picture using higher-genus conformal blocks and a generalized free field to bound $\Delta_c$, and exposes a bulk instability via zero modes in locally hyperbolic handlebodies, connecting spectral theory to fractal geometry. Analytically, $\Delta_c$ is computed near the moduli-space boundary via McMullen’s algorithm and numerically across moduli and genus, with results in close agreement with CFT bounds (e.g. $\Delta_c\approx 0.18912$ for genus-2). The work implies new phase structure for holographic CFTs at higher genus, affecting Rényi entropies and illuminating deep links between gravity, spectral theory, and fractal geometry.

Abstract

We show that for three dimensional gravity with higher genus boundary conditions, if the theory possesses a sufficiently light scalar, there is a second order phase transition where the scalar field condenses. This three dimensional version of the holographic superconducting phase transition occurs even though the pure gravity solutions are locally AdS$_3$. This is in addition to the first order Hawking-Page-like phase transitions between different locally AdS$_3$ handlebodies. This implies that the Rényi entropies of holographic CFTs will undergo phase transitions as the Rényi parameter is varied, as long as the theory possesses a scalar operator which is lighter than a certain critical dimension. We show that this critical dimension has an elegant mathematical interpretation as the Hausdorff dimension of the limit set of a quotient group of AdS$_3$, and use this to compute it, analytically near the boundary of moduli space and numerically in the interior of moduli space. We compare this to a CFT computation generalizing recent work of Belin, Keller and Zadeh, bounding the critical dimension using higher genus conformal blocks, and find a surprisingly good match.

Figures (6)

• Figure 1: A genus 2 surface, which is cut into two pairs of pants glued together along the three black circles. Along each of the three circles we can insert a projection onto the descendants of a primary of dimension $\Delta_i$ ($i=1,2,3$) to obtain the block $\mathcal{F}(\{\Delta_i\},c;\tau)$. The dual handlebody is found by 'filling in' the surface, as indicated by the shaded disks. The block $\mathcal{F}(\{\Delta_i\},c;\tau)$ can be computed in the bulk in a semi-classical approximation, valid in the limit $1\ll\Delta_i\ll c$, by computing the action of the network of bulk geodesics indicated in red.
• Figure 2: The limit sets for two of the ${\mathbb{Z}}_3$ symmetric genus two Schottky groups that arise when investigating $n=3$ Rényi entropies. The parameter $q$ defining the groups is an eigenvalue of one of the generators as specified in \ref{['exampleSurfaces']}. We give the value of the Hausdorff dimension $\delta$ for these two limit sets, computed using the methods of \ref{['criticalDimSec']}.
• Figure 3: An example of computing the transition matrix for McMullen's algorithm, in the case of a Kleinian group freely generated by two loxodromic elements $g,h$, so that ${\mathbb{C}}^*/\Gamma$ is a genus two surface. In the figure, we have drawn a fundamental domain for $\Gamma$, the exterior of the four outermost circles (those corresponding to $g^{-1}$, $h^{-1}$ are not shown in their entirety). Break the limit set into the four pieces $E_\gamma$ contained in each of these circles, labelled by $\gamma=g,h,g^{-1},h^{-1}$ corresponding to the element of the group that maps the fundamental domain to the interior of the circle, and choose points $w_\gamma\in E_\gamma$, for example the attractive fixed point of $\gamma$. The piece of the limit set $E_g$ can be broken up into three disjoint pieces, inside the circles labelled $g^2$, $gh$ and $gh^{-1}$, which are the images under $g$ of $E_g$, $E_h$ and $E_{h^{-1}}$ respectively. The scalings of these limit sets under the action of $g$ go into the top row of the transition matrix: $T = |g'(w_g)||g'(w_h)|0|g'(w_{h^{-1}})||h'(w_g)||h'(w_h)||h'(w_{g^{-1}})|00|(g^{-1})'(w_h)||(g^{-1})'(w_{g^{-1}})||(g^{-1})'(w_{h^{-1}})||(h^{-1})'(w_g)|0|(h^{-1})'(w_{g^{-1}})||(h^{-1})'(w_{h^{-1}})|$The other three rows repeat the same exercise for the other three regions, and finding $\delta$ such that the spectral radius of $T^\delta$ is unity gives an approximation for the Hausdorff dimension. This can be refined by breaking the limit set up into the $3 \times 4^{n-1}$ regions $E_\gamma$ labelled by words of length $n$ in $g,h,g^{-1},h^{-1}$, and applying the $\delta$-invariance imposed by considering the preimage of $E_\gamma$ under the first element ($g,h,g^{-1}$, or $h^{-1}$) appearing in the word $\gamma$. Then $T$ will be a sparse matrix, with three nonzero elements in each row and column, and the algorithm has error decreasing exponentially with $n$. The figure includes labels for words of length two, but also shows the images of circles under words of length three (unlabelled).
• Figure 4: The critical dimension $\Delta_c=\delta$ as a function of cross-ratio $x$ for the handlebodies corresponding to the Rényi entropies of a pair of intervals. From top to bottom, the curves correspond to genus $2,3,4,5,6$, and finally the $n\to\infty$ result in black. The shading visible on the right side of the plot indicates the bounds achieved by applying McMullen's algorithm at the crudest level of approximation.
• Figure 5: The critical dimension for the $x=\frac{1}{2}$ Rényi surface as a function of replica number $n$. The asymptote is the computed limit as $n\to\infty$. On the right is a log-log plot showing convergence to this value.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2: Patterson-Sullivan
• Theorem 3: Prime geodesic theorem