Dissipation in Open Holography

TL;DR

The paper studies open holography by modeling a CFT$_d$ coupled to a bath through a marginal double-trace deformation implemented via transparent boundary conditions between two AdS bulks. It computes the energy transmission coefficient across the boundary, uncovers a strong/weak coupling duality that maps the coupling $g$ to $-1/g$ with swapped quantization, and quantifies dissipation through quasinormal modes. Analytical results show that transmission coefficients depend only on boundary data and obey energy conservation, with a conformal-flat-space check at the conformally coupled point. Numerical explorations with Schwarzschild BTZ and mixed-boundary setups illustrate the duality and dissipation, suggesting broader applicability to open holographic systems and potential Lindblad-type dynamics in holography.

Abstract

We exploit the holographic realization of a conformal theory coupled to an external bath realized via a double trace deformation and its gravity dual in terms of transparent boundary conditions in order to map out some basic dissipative properties of this simple open holographic system. In particular, we determine the energy transmission coefficient across the boundary, discover a novel duality relating weak and strong coupling to the external bath, and quantify the dissipation in the system by working out the quasi normal modes.

Figures (11)

• Figure 1: Transmission coefficient across the transparent boundary conditions.
• Figure 2: On the left a black hole coupled to empty AdS and on the right we couple two black holes.
• Figure 3: The parameters for this plot are: $\Delta= 1.8$, $M=.8$. The colors correspond to 61 different values of $g$: the purple correspond to $g=0$ the red correspond to $g=3$ and in between we increase the value of g in increments of $.05$. The duality between $g$ and $1/g$ is manifest: the bluer the dots are, the smaller g is and the closer we get to the modes associated with the $\alpha =0$ for standard quantization for the black hole and $\beta=0$ for standard quantization for empty AdS. The larger g gets, the redder the dots are and the closer we get to the modes associated with $\beta=0$ for standard quantization for the BH and $\alpha=0$ for standard quantization for empty AdS.
• Figure 4: Phase diagram for the strong/week duality. We show that we don't need to cover the whole range of what could $\Delta_L$ be as long as we are running the coupling over large values above 1. The modes of the left side of the diagram covers the whole range of possible modes. The quasinormal modes of the bottom left side is the same as those of the top right and those of the bottom right is the same as those of the top left.
• Figure 5: Left: Imaginary component of the modes vs $\Delta$ for a Schwarzschild BTZ black hole with no coupling. Middle: Imaginary part of the modes vs $\Delta$ for a Schwarzschild black hole in AdS coupled to an empty AdS with coupling constant $h=.5$. Right: AdS Schwarzschild coupled to empty AdS with coupling constant $h=5$. Due to the coupling, there are extra set of modes picked up on the plots of finite $h$. For small h such that $h(2\Delta-2)=g <1$, the imaginary component of the modes has a similar behavior to the no coupling case. However for values of h such that $h(2\Delta-2)=g >1$ we have the imaginary component of the modes becoming less negative. The transition from decreasing into increasing happening approximately at $\Delta = \frac{1}{2h}+1$.