Holographic entanglement entropy and c-functions in conformal and confining backgrounds

Abstract

In this work we investigate holographic spacelike and timelike entanglement entropy using the Ryu-Takayanagi prescription, for slab-shaped and ball-shaped entangling regions. We work with an infinite family of 10-dimensional Type IIB supergravity solutions, which are gravity duals to an infinite set of linear quiver theories, with the backgrounds defined using the electrostatic potential formalism for brane configurations. The dual theories are 4-dimensional confining theories at low energy, but decompactify and flow to 5-dimensional SCFTs in the UV. We find that the entanglement entropy exhibits phase transition behaviour, and we use our results to investigate proposed c-functions constructed from the entanglement entropy. Comparing with the flow central charge, another proposed c-function, we find that each displays good behaviour, and reflects both UV and IR features of the dual theory.

Figures (12)

• Figure 1: An example rank function, which is continuous, convex, and piecewise linear as required by the quantisation of Page charges. The ranks of the associated gauge groups are encoded in the values of $\mathcal{R}(\eta)$ at the points where the gradient is discontinuous. Note that the rank function does not have to be symmetric.
• Figure 2: The Hanany-Witten brane setup is shown in (a), with the horizontal direction corresponding to $\eta$. Each stack of D5-branes is suspended between two NS5-branes, and there are transverse stacks of D7-branes providing flavour groups. In (b) is the corresponding quiver plot. Gauge nodes come from the stacks of D5-branes, and flavour nodes come from the stacks of D7-branes.
• Figure 3: The separation and entanglement entropy from equations \ref{['eqn: separation']} and \ref{['eqn: EE']}, with the integrals evaluated numerically. The separation increases from 0 at $r_0=r_*$ to some maximum value, then asymptotes back to 0 meaning every value of $T$ has two different associated turning points.
• Figure 4: Here we take three different values for $2\tilde{g}^2/9$ with $c_t=1$ and plot the values of $\mu$ and $c$ for which equation \ref{['eqn: turning point eqn']} has a single solution (the critical $\mu$ values mentioned above). The interior region (between each pair of lines) is the parameter space for which there are no real solutions to equation \ref{['eqn: turning point eqn']}. The shape of the plot matches our observations.
• Figure 5: The separation plotted against the EE. On the left, the approximate expressions are used, and we find a cusp indicating a phase transition. On the right, the analytic expression with integrals evaluated numerically is plotted. The red and orange are for different integration methods (Mathematica's "DoubleExponential" and "GlobalAdaptive" respectively). Red matches the approximate plot better, but since orange is still multi-valued it can still indicate a phase transition.