Timelike Entanglement Entropy and Phase Transitions in non-Conformal Theories

TL;DR

This work defines a holographic timelike entanglement entropy (tEE) for non-conformal theories by combining spacelike and timelike extremal surfaces with a dilaton-including action, yielding a complex pseudoentropy. The authors establish a merging prescription at the IR tip to fix the boundary subregion length, and demonstrate that in confining backgrounds the tEE exhibits a nonzero imaginary part and a maximal connected-length threshold, making it a potential order parameter for confinement and related phase transitions. They develop both analytic (deep IR and near-boundary) and numerical methods to compute the tEE, and compare Lorentzian results with Euclidean analytic continuations, highlighting qualitative similarities but quantitative differences. The results provide a framework to probe confinement and phase structure in non-conformal holographic theories and open avenues for extensions to black-hole spacetimes and RG flows. Overall, the tEE offers a novel, bulk-determined observable that captures Lorentzian information and nonconformal dynamics beyond standard spacelike entanglement entropy.

Abstract

We propose a holographic formalism for a timelike entanglement entropy in non-conformal theories. This pseudoentropy is a complex-valued measure of information, which, in holographic non-conformal theories, receives contributions from a set of spacelike surfaces and a finite timelike bulk surface with mirror symmetry. We suggest a method of merging the surfaces so that the boundary length of the subregion is exclusively specified by holography. We show that in confining theories, the surfaces can be merged in the bulk at the infrared tip of the geometry and are homologous to the boundary region. The timelike entanglement entropy receives its imaginary and real contributions from the timelike and the spacelike surfaces, respectively. Additionally, we demonstrate that in confining theories, there exists a critical length within which a connected non-trivial surface can exist, and the imaginary part of the timelike entanglement entropy is non-zero. Therefore, the timelike entanglement entropy exhibits unique behavior in confining theories, making it a probe of confinement and phase transitions. Finally, we discuss the entanglement entropy in Euclidean spacetime in confining theories and the effect of a simple analytical continuation from a spacelike subsystem to a timelike one.

Figures (5)

• Figure 1: The different ways to merge the timelike and spacelike surfaces obtained by solving \ref{['eqtt1', 'eqtt2']}. Following our prescription we choose the blue colored surfaces over the green ones, since green surfaces violate the prescription. Moreover, the green surfaces intersect with each other when $u_0$ is near the tip of the geometry $u_k$. The blue and red surfaces are smoothly patched as depicted also in \ref{['surface2']}.
• Figure 2: The surfaces comprising the tEE. The red-colored curve is a solution of the equations of motion \ref{['eqtt1']} and extends from the turning point $u_0$ to $u_k$. The blue-colored curve consists of the solutions of \ref{['eqtt2']} and extends from the boundary to $u_k$. In the left figure, we plot only three representative solutions with a focus on the regime of $u_0$ around $u_k$. The right figure contains the line of the natural IR cut-off $u=u_k$ that is the tip of the cigar geometry and contains the curves of a wide range of turning points of $u_0$. As the turning point distances itself from the tip of the geometry, the surface increases its boundary length. For all our plots we fix $u_k=1$ and we normalize all the parameters with it, or equivalently with the fixed radius $R_0$.
• Figure 3: The subsystem lengths as a function of $u_0$. (a) The subsystem lengths $T_{Im}$ and $T_{Re}$ corresponding to the solutions described in \ref{['eqtt1', 'eqtt2']} are plotted with increasing $u_0$. Notice the maximum value that both lengths have which is in exact agreement with the analytic computations (\ref{['u0_T_tot']}). Both the subsystem lengths increase faster when the turning point is deep inside the bulk. (b) The ratio $\frac{T_{Re}}{T_{Im}}$ versus $u_0$. In the region where $u_0$ is close to $u_k$, it is above the unit. With increasing $u_0$ the ratio drops below the unit indicating $T_{Im}>T_{Re}$. In $u_0\gg u_k$, we observe $T_{Im}\approx T_{Re}$.
• Figure 4: Analysis of $\mathcal{\hat{S}}^T$ scaled with $\frac{{\cal V}^{d_0-2}}{4G_N^{d+1}}$ as a function of $u_0$ and $T$. (a) The real and the imaginary parts of $\mathcal{\hat{S}}^T$ are plotted as a function of $T$ where $T=T_{Re}+T_{Im}$. For small subsystem sizes, both $|\mathcal{\hat{S}}^T_{Re}|$ and $\mathcal{\hat{S}}^T_{Im}$ are small whereas they obtain large values where the turning point $u_0$ is far away from $u_k$. In the embedded plot notice the existence of the $T_{crit}$ beyond which a connected non-trivial surface seizes to exist and a discontinuous phase transition occurs. (b) The ratio of the real and the imaginary parts of tEE in \ref{['tgen']}. In regions $u_0\approx u_k$ and $u_0\gg u_k$ the real part is larger than the imaginary part. However, there is an intermediate region where the imaginary part dominates. In the near boundary regime $|\mathcal{\hat{S}}^T_{Re}|\approx 1.05~\mathcal{\hat{S}}^T_{Im}$.
• Figure 5: Plot of the EE $\mathcal{\hat{S}}$$vs$$T$ in Euclidean scenario. Here, $\mathcal{\hat{S}}$ is the difference between the areas of the real part of connected and the disconnected straight surfaces which are two plausible solutions of \ref{['gentprime']}. The generic expression of $\mathcal{\hat{S}}$ is given in \ref{['genSrenorm']}. Note that the plot shows two branches where the lower branch represents the favorable solution. Following this solution, we observe a change in dominance between the two solutions of \ref{['gentprime']} at $T=T^{(E)}_{crit}$ where $\mathcal{\hat{S}}=0$. For regions $T<T^{(E)}_{crit}$ and $T>T^{(E)}_{crit}$ the dominant contributions are received from the connected and the disconnected surfaces respectively. Besides, we observe that the subsystem size obtains a bound which is indicated by $T_{max}$. Beyond this particular value, there is no connected solution of \ref{['gentprime']}.