Holographic Timelike Entanglement Entropy in Non-relativistic Theories

TL;DR

The paper develops a holographic framework to study timelike entanglement entropy (tEE) and its Euclidean counterpart, temporal entanglement entropy, in non-relativistic theories with Lifshitz-like anisotropy and hyperscaling violation. By analyzing unions of spacelike and timelike extremal surfaces and their gradient-normal vectors, the authors connect the real and imaginary parts of tEE to stability conditions (NEC, thermodynamic stability) and to the presence of Fermi surfaces, with the imaginary part capturing Lifshitz-dependent features. They demonstrate that tEE encodes the theory’s stability and naturalness, and show that Fermi surfaces can be identified either via the logarithmic behavior of the real part or via a constant imaginary part, depending on the Lifshitz exponent and hyperscaling parameters. The work further provides a detailed classification of surface behaviors across directions (y- and x-localizations) and analyzes large-d limits, offering insights into non-relativistic quantum criticality and potential applications to quantum phase transitions in condensed matter systems.

Abstract

Timelike entanglement entropy is a complex measure of information that is holographically realized by an appropriate combination of spacelike and timelike extremal surfaces. This measure is highly sensitive to Lorentz invariance breaking. In this work, we study the timelike entanglement entropy in non-relativistic theories, focusing on theories with hyperscaling violation and Lifshitz-like spatial anisotropy. The properties of the extremal surfaces, as well as the timelike entanglement entropy itself, depend heavily on the symmetry-breaking parameters of the theory. Consequently, we show that timelike entanglement can encode, to a large extent, the stability and naturalness of the theory. Furthermore, we find that timelike entanglement entropy identifies Fermi surfaces either through the logarithmic behavior of its real part or, alternatively, via its constant imaginary part, with this constant value depending on the theory's Lifshitz exponent. This provides a novel interpretation for the imaginary component of this pseudoentropy. Additionally, we examine temporal entanglement entropy, an extension of timelike entanglement entropy to Euclidean space, and provide a comprehensive discussion of its properties in these theories.

Figures (11)

• Figure 1: The tEE surfaces for $z>1$ corresponding to $A_2$ of (\ref{['case2A']}). The boundary of the theory is at $r\rightarrow 0$. The equation of motion for the spacelike hypersurfaces $\Sigma_{Re}$ follows the boundary conditions $t^{\prime}_{Re}|_{r\rightarrow 0}=0$ and $t^{\prime}_{Re}|_{r\rightarrow \infty}=\infty$, whereas for the timelike surface $\Sigma_{Im}$, the boundary conditions are given by $t^{\prime}_{Im}|_{r\rightarrow r_0}=\infty$ and $t^{\prime}_{Im}|_{r\rightarrow \infty}=\infty$. We plot the timelike surfaces with red color and the spacelike ones with green. Here we choose $d=4$ and the turning point of the timelike surface at $r_0=5$. These choices are same for the neighboring plots of this section. Moreover, here $z=3/2$.
• Figure 2: The surfaces for $0<z<1$ which is discussed in $A_1$ of (\ref{['case2A']}). The boundary of the theory for the range of parameters $A_0$ and case $A_1$ is at $r=0$. The equation of motion for $\Sigma_{Re}$ follows the boundary conditions $t^{\prime}_{Re}|_{r\rightarrow 0}=0=t^{\prime}_{Re}|_{r\rightarrow \infty}$, whereas for $\Sigma_{Im}$ the boundary conditions are given by $t^{\prime}_{Im}|_{r\rightarrow r_0}=\infty$ and $t^{\prime}_{Im}|_{r\rightarrow \infty}=0$. Here we consider $z=3/10$.
• Figure 3: The surfaces for $-(d-2)^{-1}<z<0$ which is the $A_0$ of (\ref{['case2A']}). The $g_{xx}$ and $g_{yy}^{-1}$ diverge at $r\rightarrow \infty$. The equation of motion for $\Sigma_{Re}$ satisfies $t^{\prime}_{Re}|_{r\rightarrow 0}=\infty$ and $t^{\prime}_{Re}|_{r\rightarrow \infty}=0$, whereas for $\Sigma_{Im}$ we have $t^{\prime}_{Im}|_{r\rightarrow r_0}=\infty$ and $t^{\prime}_{Im}|_{r\rightarrow \infty}=0$. Here we have considered $z=-1/4$.
• Figure 4: The hypersurfaces for $z<-(d-2)^{-1}$ described by $B$ of (\ref{['case2B']}). $g_{xx}$ and $g_{yy}^{-1}$ diverge at $r\rightarrow \infty$. There are two discrete sets of surfaces for $z=-4/5$ with solid lines and and $z=-3/5$ plotted with dashed. For both the cases, $\Sigma_{Re}$ follow the same boundary conditions given by $t^{\prime}_{Re}|_{r\rightarrow 0}=\infty$ and $t^{\prime}_{Re}|_{r\rightarrow \infty}=0$, while $\Sigma_{Im}$ satisfies $t^{\prime}_{Im}|_{r\rightarrow r_0}=\infty$ and $t^{\prime}_{Im}|_{r\rightarrow 0}=\infty$.
• Figure 5: This surface corresponds to the case $\mathbf{A_0}$, (\ref{['caseah']}). The equation of motion for the spacelike hypersurfaces $\Sigma_{Re}$ follow the boundary conditions $t^{\prime}_{Re}|_{r\rightarrow 0}=0$ and $t^{\prime}_{Re}|_{r\rightarrow \infty}=\infty$, whereas for the timelike hypersurface $\Sigma_{Im}$, the boundary conditions are given by $t^{\prime}_{Im}|_{r\rightarrow r_0}=\infty$ and $t^{\prime}_{Im}|_{r\rightarrow \infty}=\infty$. The red curve in the plot describes the timelike surfaces and the green curves are for spacelike ones. We choose $d=4$ and the turning point of the timelike surface is fixed at $r_0=5$. These choices of $d$ and $r_0$ are same for all the neighboring figures of this section. The other parameters here are set to be $z=2$ and $\theta=1/2$.