Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2411.18514 · Journal of High Energy Physics

Holographic timelike entanglement entropy in non-relativistic theories

Mir Afrasiar, Jaydeep Kumar Basak, Dimitrios Giataganas

Published May 23, 2025• 14 citations
arXivHTMLPDF

Abstract

A bstract 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.

DOI
10.1007/jhep05(2025)205

Related papers

Timelike entanglement entropy

Kazuki Doi, Jonathan Harper, Ali Mollabashi +2 more

Abstract We define a new complex-valued measure of information called the timelike entanglement entropy (EE) which in the boundary theory can be viewed as a Wick rotation that changes a spacelike boundary subregion to a timelike one. An explicit definition of the timelike EE in 2d field theories is provided followed by numerical computations which agree with the analytic continuation of the replica method for CFTs. We argue that timelike EE should be correctly interpreted as another measure previously considered, the pseudo entropy, which is the von Neumann entropy of a reduced transition matrix. Our results strongly imply that the imaginary part of the pseudo entropy describes an emergent time which generalizes the notion of an emergent space from quantum entanglement. For holographic systems we define the timelike EE as the total complex valued area of a particular stationary combination of both space and timelike extremal surfaces which are homologous to the boundary region. For the examples considered we find explicit matching of our optimization procedure and the careful implementation of the Wick rotation in the boundary CFT. We also make progress on higher dimensional generalizations and relations to holographic pseudo entropy in de Sitter space.

arXiv:2302.11695•Journal of High Energy Physics•May 8, 2023•104 citations

Aspects of holographic timelike entanglement entropy in black hole backgrounds

Mir Afrasiar, Jaydeep Kumar Basak, Keun-Young Kim

We study the holographic construction of timelike entanglement entropy (tEE) in black hole backgrounds in Lorentzian geometries. The holographic tEE is realized through extremal surfaces consisting of spacelike and timelike branches that encode its real and imaginary components, respectively. In the BTZ black hole, these surfaces extend into the interior of the black hole and reproduce the field-theoretic results. The analysis is further generalized to higher-dimensional AdS-Schwarzschild black holes, where the characteristics of tEE are obtained with increasing size of the boundary subsystem. Besides, we also show that the boundary subsystem length diverges at a dimension-dependent critical turning point. Notably, this critical point moves closer to the black hole horizon as the dimensionality of the bulk increases. For large subsystem lengths, the finite part of the tEE displays a characteristic volume-plus-area structure, with a real volume term and a complex coefficient of the area term approaching constant values at large dimensions. Besides, we also study the monotonicity of a new quantity, timelike entanglement density, which offers insights into a timelike area theorem in specific limits. Subsequently, we investigate the near-horizon dynamics in various black hole backgrounds, where the spacelike and timelike surfaces exhibit exponential growth of the form $e^{\frac{2π}β Δt}$ with inverse black hole temperature $β$.

arXiv:2512.21327•Dec 24, 2025•0 citations

Temporal entanglement entropy as a probe of renormalization group flow

Sebastian Grieninger, Kazuki Ikeda, Dmitri E. Kharzeev

A bstract The recently introduced concept of timelike entanglement entropy has sparked a lot of interest. Unlike the traditional spacelike entanglement entropy, timelike entanglement entropy involves tracing over a timelike subsystem. In this work, we propose an extension of timelike entanglement entropy to Euclidean space (“temporal entanglement entropy”), and relate it to the renormalization group (RG) flow. Specifically, we show that tracing over a period of Euclidean time corresponds to coarse-graining the system and can be connected to momentum space entanglement. We employ Holography, a framework naturally embedding RG flow, to illustrate our proposal. Within cutoff holography, we establish a direct link between the UV cutoff and the smallest resolvable time interval within the effective theory through the irrelevant $$T\overline{T }$$ deformation. Increasing the UV cutoff results in an enhanced capability to resolve finer time intervals, while reducing it has the opposite effect. Moreover, we show that tracing over a larger Euclidean time interval is formally equivalent to integrating out more UV degrees of freedom (or lowering the temperature). As an application, we point out that the temporal entanglement entropy can detect the critical Lifshitz exponent z in non-relativistic theories which is not accessible from spatial entanglement at zero temperature and density.

arXiv:2312.08534•Journal of High Energy Physics•May 3, 2024•27 citations

Holographic timelike c -function

Dimitrios Giataganas

The integration of high-energy degrees of freedom along the renormalization group (RG) flow in Poincaré-invariant theories is captured by a monotonic c -function. For such theories, holographic c -functions have been constructed using entanglement entropy. However, in theories with broken Lorentz invariance, these constructions generally fail, reflecting both the violation of entanglement RG monotonicity and its limitations in capturing essential features of nonrelativistic RG flows. Since many quantum many-body systems lack Lorentz invariance, it is important to identify a quantity that captures the decrease of effective degrees of freedom along such flows. We show that the recently introduced holographic timelike entanglement entropy naturally gives rise to a new c -function applicable to all such theories, which moreover obeys a geometric upper bound on its rate of change. We further demonstrate the existence of this c -function in theories with Lifshitz and hyperscaling-violating fixed points, showing that, provided the null energy condition and thermodynamic stability are satisfied, it obeys the expected RG monotonicity.

arXiv:2505.20459•Physical review. D/Physical review. D.•Jun 2, 2026•0 citations

Holographic entanglement entropy on generic time slices

Yuya Kusuki, Tadashi Takayanagi, Koji Umemoto

We study the holographic entanglement entropy and mutual information for Lorentz boosted subsystems. In holographic CFTs at zero and finite temperature, we find that the mutual information gets divergent in a universal way when the end points of two subsystems are light-like separated. In Lifshitz and hyperscaling violating geometries dual to non-relativistic theories, we show that the holographic entanglement entropy is not well-defined for Lorentz boosted subsystems in general. This strongly suggests that in non-relativistic theories, we cannot make a real space factorization of the Hilbert space on a generic time slice except the constant time slice, as opposed to relativistic field theories.

arXiv:1703.00915•Journal of High Energy Physics•Jun 1, 2017•17 citations

Timelike entanglement entropy with gravitational anomalies

Chong‐Sun Chu, Himanshu Parihar

A bstract We study the timelike entanglement entropy (TEE) in two dimensional conformal field theories (CFT) with gravitational anomalies. We employ analytical continuation to compute the timelike entanglement entropy for a pure timelike interval in such CFTs. We find that, unlike the real part, the imaginary part of the TEE displays an asymmetric dependence on the central charges of the left and right moving modes. We propose that the asymmetric dependence on central charges of the imaginary part of the TEE can be used to probe the presence of gravitational anomalies in chiral CFT. Furthermore, we propose a holographic construction to obtain the timelike entanglement entropy from the bulk dual geometries involving topologically massive gravity in AdS 3 . The holographic results obtained match exactly with the dual field theory results.

arXiv:2504.19694•Journal of High Energy Physics•Aug 5, 2025•5 citations
A bstract 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.
Abstract We define a new complex-valued measure of information called the timelike entanglement entropy (EE) which in the boundary theory can be viewed as a Wick rotation that changes a spacelike boundary subregion to a timelike one. An explicit definition of the timelike EE in 2d field theories is provided followed by numerical computations which agree with the analytic continuation of the replica method for CFTs. We argue that timelike EE should be correctly interpreted as another measure previously considered, the pseudo entropy, which is the von Neumann entropy of a reduced transition matrix. Our results strongly imply that the imaginary part of the pseudo entropy describes an emergent time which generalizes the notion of an emergent space from quantum entanglement. For holographic systems we define the timelike EE as the total complex valued area of a particular stationary combination of both space and timelike extremal surfaces which are homologous to the boundary region. For the examples considered we find explicit matching of our optimization procedure and the careful implementation of the Wick rotation in the boundary CFT. We also make progress on higher dimensional generalizations and relations to holographic pseudo entropy in de Sitter space.
We study the holographic construction of timelike entanglement entropy (tEE) in black hole backgrounds in Lorentzian geometries. The holographic tEE is realized through extremal surfaces consisting of spacelike and timelike branches that encode its real and imaginary components, respectively. In the BTZ black hole, these surfaces extend into the interior of the black hole and reproduce the field-theoretic results. The analysis is further generalized to higher-dimensional AdS-Schwarzschild black holes, where the characteristics of tEE are obtained with increasing size of the boundary subsystem. Besides, we also show that the boundary subsystem length diverges at a dimension-dependent critical turning point. Notably, this critical point moves closer to the black hole horizon as the dimensionality of the bulk increases. For large subsystem lengths, the finite part of the tEE displays a characteristic volume-plus-area structure, with a real volume term and a complex coefficient of the area term approaching constant values at large dimensions. Besides, we also study the monotonicity of a new quantity, timelike entanglement density, which offers insights into a timelike area theorem in specific limits. Subsequently, we investigate the near-horizon dynamics in various black hole backgrounds, where the spacelike and timelike surfaces exhibit exponential growth of the form e2πβΔte^{\frac{2π}β Δt}eβ2π​Δt with inverse black hole temperature βββ.
A bstract The recently introduced concept of timelike entanglement entropy has sparked a lot of interest. Unlike the traditional spacelike entanglement entropy, timelike entanglement entropy involves tracing over a timelike subsystem. In this work, we propose an extension of timelike entanglement entropy to Euclidean space (“temporal entanglement entropy”), and relate it to the renormalization group (RG) flow. Specifically, we show that tracing over a period of Euclidean time corresponds to coarse-graining the system and can be connected to momentum space entanglement. We employ Holography, a framework naturally embedding RG flow, to illustrate our proposal. Within cutoff holography, we establish a direct link between the UV cutoff and the smallest resolvable time interval within the effective theory through the irrelevant TT‾T\overline{T }TT deformation. Increasing the UV cutoff results in an enhanced capability to resolve finer time intervals, while reducing it has the opposite effect. Moreover, we show that tracing over a larger Euclidean time interval is formally equivalent to integrating out more UV degrees of freedom (or lowering the temperature). As an application, we point out that the temporal entanglement entropy can detect the critical Lifshitz exponent z in non-relativistic theories which is not accessible from spatial entanglement at zero temperature and density.
The integration of high-energy degrees of freedom along the renormalization group (RG) flow in Poincaré-invariant theories is captured by a monotonic c -function. For such theories, holographic c -functions have been constructed using entanglement entropy. However, in theories with broken Lorentz invariance, these constructions generally fail, reflecting both the violation of entanglement RG monotonicity and its limitations in capturing essential features of nonrelativistic RG flows. Since many quantum many-body systems lack Lorentz invariance, it is important to identify a quantity that captures the decrease of effective degrees of freedom along such flows. We show that the recently introduced holographic timelike entanglement entropy naturally gives rise to a new c -function applicable to all such theories, which moreover obeys a geometric upper bound on its rate of change. We further demonstrate the existence of this c -function in theories with Lifshitz and hyperscaling-violating fixed points, showing that, provided the null energy condition and thermodynamic stability are satisfied, it obeys the expected RG monotonicity.
We study the holographic entanglement entropy and mutual information for Lorentz boosted subsystems. In holographic CFTs at zero and finite temperature, we find that the mutual information gets divergent in a universal way when the end points of two subsystems are light-like separated. In Lifshitz and hyperscaling violating geometries dual to non-relativistic theories, we show that the holographic entanglement entropy is not well-defined for Lorentz boosted subsystems in general. This strongly suggests that in non-relativistic theories, we cannot make a real space factorization of the Hilbert space on a generic time slice except the constant time slice, as opposed to relativistic field theories.
A bstract We study the timelike entanglement entropy (TEE) in two dimensional conformal field theories (CFT) with gravitational anomalies. We employ analytical continuation to compute the timelike entanglement entropy for a pure timelike interval in such CFTs. We find that, unlike the real part, the imaginary part of the TEE displays an asymmetric dependence on the central charges of the left and right moving modes. We propose that the asymmetric dependence on central charges of the imaginary part of the TEE can be used to probe the presence of gravitational anomalies in chiral CFT. Furthermore, we propose a holographic construction to obtain the timelike entanglement entropy from the bulk dual geometries involving topologically massive gravity in AdS 3 . The holographic results obtained match exactly with the dual field theory results.