Marginal Deformations and Rotating Horizons

TL;DR

The paper builds a holographically motivated connection between near-horizon AdS$_2$ physics and a disordered large-$N$ quantum mechanics with global $SU(2)$ symmetry, showing that an $SU(2)$-breaking marginal deformation can mimic rotation in Kerr–Newman black holes. It derives the low-energy effective action combining a Schwarzian sector with an $SU(2)$ (and later $U(1)$) coset sector, and demonstrates a gapless-to-gapped quantum phase transition controlled by a deformation parameter $z$ (analogous to the horizon angular velocity $\Omega$). The analysis yields exact saddle points, low-temperature thermodynamics with linear specific heat, and explicit phase-transition diagnostics, drawing qualitative parallels to rotating black holes’ thermodynamics and signaling potential extensions to higher dimensions and rotating de Sitter horizons. The work thus provides a concrete quantum-mechanical framework to explore holographic aspects of rotating horizons and superradiant phenomena in a controlled large-$N$ setting.

Abstract

Motivated by the near-horizon geometry of four-dimensional extremal black holes, we study a disordered quantum mechanical system invariant under a global $SU(2)$ symmetry. As in the Sachdev-Ye-Kitaev model, this system exhibits an approximate $SL(2,\mathbb{R})$ symmetry at low energies, but also allows for a continuous family of $SU(2)$ breaking marginal deformations. Beyond a certain critical value for the marginal coupling, the model exhibits a quantum phase transition from the gapless phase to a gapped one and we calculate the critical exponents of this transition. We also show that charged, rotating extremal black holes exhibit a transition when the angular velocity of the horizon is tuned to a certain critical value. Where possible we draw parallels between the disordered quantum mechanics and charged, rotating black holes.

Figures (6)

• Figure 1: Left: $J(\Omega)$ vs. $\Omega$. The blue curve represents the $J_+(\Omega)$ branch and the orange curve the $J_-(\Omega)$ one. Right: $J'(\Omega)$ displaying the divergent derivatives at $\Omega=\pm 1/\sqrt{8}$.
• Figure 2: The solid blue curve is the exact result for $S^a(u)$, while the dashed orange curve is the large time separation approximation $S_{\rm low}^a(u)= 1/\pi \gamma u$.
• Figure 3: Left: plot of $\delta E(\beta) = \langle E(\beta)\rangle-E_0$ and the $\zeta$-regularized value $E_{\rm low}$. We have performed the sum in (\ref{['eq:eexpect']}) numerically using the exact expression for $S^a(\omega_n)$ and cutoff our sum above and below at $n_c=\pm2.5\times 10^3$. As $n_c$ increases the plots become increasingly close. Right: Comparison between numerically computed $-\beta F/Nn$ obtained by evaluating the on-shell action (solid curve) and the low temperature approximation $-\beta E_0+C(\beta)/2$ (dashed curve). At low temperatures these curves align implying that the zero temperature entropy $S_0$ vanishes.
• Figure 4: The solid blue curves are the numerical Fourier transforms of (\ref{['eq:exactzdef']}), $S^1(u)$, $u\equiv \tau-\tau^\prime$, while the dashed orange curves represent the large time separation approximations $S_{\rm low}^1(u)$ in (\ref{['zcorrs']}). For $|z|<2\gamma$, $S_{\rm low}^1(u)$ matches the late time behavior of the numerical Fourier transform.
• Figure 5: Left: Comparison between $\delta J_z \equiv \left( \langle \hat{J}_z \rangle_\beta - \langle \hat{J}_z \rangle_0 \right)$ (solid-blue) and $\langle \hat{J}_z\rangle_{\rm low}$ (dashed-red) given in (\ref{['zetaJ']}), for $z=20$ (in units where $\beta=1$), with a cutoff on the sum given by $n_c=\pm2\times10^4$. Note the sharp transition at $\gamma=z/2$. Right: $\delta J_z$ vs. $z$ and $\gamma$. The red line indicates the locus $|z|=2\gamma$.