Holographic Mutual Information at Finite Temperature

TL;DR

The paper studies how mutual information between two disjoint rectangular regions behaves in finite-temperature holographic theories using the Ryu-Takayanagi formula ${S_A = \frac{\mathrm{Area}(\gamma_A)}{4 G_N^{(d+1)}}}$ and the definition ${I(A,B) = S_A + S_B - S_{A\cup B}}$. It shows a disentangling transition: as separation ${x}$ or temperature ${T}$ increases, mutual information vanishes beyond a critical boundary, with universal qualitative features across relativistic CFTs in ${d}$ dimensions and non-relativistic backgrounds such as Lifshitz and hyperscaling-violating geometries. The authors obtain analytic results in several regimes (e.g., ${d=2}$, ${T \ll 1/l,1/x}$, ${1/l \ll T \ll 1/x}$, ${T \gg 1/x}$) and demonstrate that, at large temperature, ${I(A,B)}$ subtracts the thermal entropy and tracks the quantum entanglement part of the entanglement entropy. These findings yield phase diagrams in the ${(x/l, xT)}$-plane and indicate a robust, universal structure for disentanglement across holographic duals, with possible extensions to Rényi mutual information and more general backgrounds.

Abstract

Using the Ryu-Takayanagi conjectured formula for entanglement entropy in the context of gauge-gravity duality, we investigate properties of mutual information between two disjoint rectangular sub-systems in finite temperature relativistic conformal field theories in d-spacetime dimensions and non-relativistic scale-invariant theories in some generic examples. In all these cases mutual information undergoes a transition beyond which it is identically zero. We study this transition in detail and find universal qualitative features for the above class of theories which has holographic dual descriptions. We also obtain analytical results for mutual information in the specific regime of the parameter space. This demonstrates that mutual information contains the quantum entanglement part of the entanglement entropy, which is otherwise dominated by the thermal entropy at large temperatures.

Figures (5)

• Figure 1: The two disjoint sub-systems $A$ and $B$, each of length $l$ along $X$-direction and separated by a distance $x$. The schematic diagram on the right shows the possible candidates for minimal area surfaces which is relevant for computing $S_{A\cup B}$. The choice on top gives $S_{A\cup B} = S_A + S_B = 2 S(l)$; and the choice at the bottom gives $S_{A\cup B} = S(2l+x) + S(x)$. This is also summarized in (\ref{['twoch']}).
• Figure 2: 2-dimensional parameter space for the (1+1)-dimensional boundary theory. The mutual informational is non-zero only in the blue shaded region.
• Figure 3: The left panel: the dependence of $a_d$, as defined in (\ref{['transition']}), with respect to $d$. The right panel: the dependence of $b_d$, as defined in (\ref{['transitionT']}), with respect to $d$. The solid dots represent the corresponding value of $a_d$ or $b_d$ beyond which mutual information vanishes.
• Figure 4: 2-dimensional parameter space for the (3+1)-dimensional boundary theory. The mutual informational is non-zero only in the blue shaded region. The corresponding parameter space looks qualitatively similar for general $d$, thus we have showed one representative example here.
• Figure 5: $2$-dimensional parameter space for a scale-invariant $(2+1)$-dimensional field theory with Lifshitz scaling. The dynamical exponent is $z=2$. Mutual information is non-zero in the shaded region.