Holographic Timelike Entanglement Entropy

• HTEE is a nonlocal information-theoretic functional that extends entanglement entropy from spacelike to timelike separations in quantum field theory using complex extremal surfaces.
• Its real part captures ultraviolet divergences while the imaginary part encodes causal ordering and phase coherence, resulting in universal phase factors across different holographic models.
• HTEE methodologies, including geodesic-combination and coordinate complexification, effectively probe IR dynamics, black hole interiors, and emergent time phenomena in holography.

Holographic Timelike Entanglement Entropy (HTEE) is a complex, nonlocal information-theoretic functional defined for subsystems separated along timelike (rather than spacelike) directions in quantum field theory. As an extension of the Ryu–Takayanagi and Hubeny–Rangamani–Takayanagi prescriptions for holographic entanglement entropy to Lorentzian settings, HTEE is realized via extremal (generally complex-valued) bulk surfaces in gravitational duals, encoding entanglement and coherence across temporal regions. Both real and imaginary parts of HTEE are crucial: the real part exhibits ultraviolet divergences and is akin to ordinary entanglement entropy, while the imaginary part reflects causal ordering, pseudo-entropy, and, in some cases, the presence of anomalies or confinement. HTEE is sensitive both to global geometric features (e.g., black hole interiors, Lifshitz scaling) and IR dynamics (e.g., phase transitions, scalar hair), offering a probe of Lorentz symmetry breaking, quantum chaos, and emergent time.

1. Definition and Holographic Prescriptions

HTEE generalizes the concept of entanglement entropy from spacelike to timelike boundary subregions. For a boundary interval $A=\{(t,0):t\in[-T/2,+T/2]\}$, the HTEE is computed as the fractional (typically $1/4G_N$) bulk "area" of a codimension-2 extremal surface, now anchored at endpoints separated in time. This extremal surface generically contains both spacelike and timelike segments, resulting in a complex-valued area: $$S_{\rm HTEE} = \frac{\textrm{Area(spacelike)}}{4G_N} + i\,\frac{\textrm{Area(timelike)}}{4G_N}$$ and is sometimes termed a pseudo-entropy.

Several equivalent holographic schemes have been established, especially in three-dimensional AdS or Lifshitz backgrounds (Jena et al., 2024):

• Geodesic-combination method: Construct two spacelike bulk geodesics from the endpoints to null infinity (yielding the real part), and a timelike geodesic connecting their asymptotic ends (yielding the imaginary part). Combine their lengths.
• Complexified Weak Extremal Surface (CWES): Form a piecewise extremal surface (spacelike and timelike patches), matching normal derivatives at joining points, and minimize a natural complex ordering (imaginary part first, then real).
• Smooth Merging: Continuously glue spacelike and timelike segments at a merging bulk point, ensuring differentiability and matching integration constants.
• Coordinate Complexification: Allow all coordinates to be complex, and identify a single codimension-2 extremal surface whose complexified area reproduces HTEE.

In three dimensions, these proposals yield coincident results due to the tractability of geodesic equations (Jena et al., 2024), but in higher dimensions and with additional matter or curvature terms, more general complex extremal surfaces are required (Heller et al., 2024, Zhao et al., 4 Sep 2025).

2. Analytic Results and Universal Formulas

For three-dimensional holographic spacetimes, especially Lifshitz backgrounds (dynamical exponent $z\geq1$), the HTEE for a timelike interval of duration $T$ is (Jena et al., 2024): $$S_{\rm TEE} = \frac{L}{4G_3} \left( \frac{2}{z} \ln\frac{\alpha T}{\epsilon^z} + i\frac{\pi}{z} \right)$$ where $\alpha$ is the bulk anisotropic scaling parameter, $L$ is the curvature radius, and $\epsilon$ is a UV cutoff. The real part captures $\sim (2/z)\ln T$ ultraviolet divergence, while the imaginary part is a universal constant $\sim \pi/z$.

In relativistic AdS$_3$/CFT$_2$, one recovers (Grieninger et al., 2023, He et al., 2023, Doi et al., 2023, Li et al., 2022): $$S_{\rm HTEE} = \frac{c}{3} \ln \frac{T}{\epsilon} + i\,\frac{\pi c}{6}$$ with $c$ the central charge. Finite-temperature (BTZ) and boundary CFT cases generalize these formulas, introducing hyperbolic/trigonometric functions and multiple phase regimes depending on the interval and system size (Chu et al., 2023).

In higher dimensions, the general pattern holds: vacuum contributions to timelike and spacelike entanglement entropies differ by universal phase factors $(-i)^d$ (Zhao et al., 4 Sep 2025). Excited-state or deformation contributions (e.g. relevant scalar operators, black brane parameters, higher-curvature corrections) enter the real part and alter the asymptotic scaling but do not affect the universal imaginary phase.

3. Physical Interpretation of the Imaginary Component

The imaginary part of HTEE is not present in standard (spacelike) entanglement entropy and arises due to the signature of timelike separation. In Lorentzian signature, the proper length of a timelike geodesic is imaginary; holographically, this encodes the passage from a Hermitian reduced density matrix to a non-Hermitian "transition matrix," giving rise to pseudo-entropy (Li et al., 2022, Doi et al., 2023). The imaginary term can be interpreted as a measure of time-ordering or phase coherence across the interval, with implications for the emergence of the time dimension in holography and non-unitary dynamics.

In theories with Lifshitz scaling, the magnitude of the imaginary part is rescaled by the dynamical exponent ($\propto 1/z$), encoding the anisotropic redshift between time and space (Jena et al., 2024). In Euclidean CFTs with gravitational anomaly, only one chiral central charge contributes to the imaginary part, making $\operatorname{Im} S_{\rm HTEE}$ a probe of chirality (Chu et al., 28 Apr 2025). In nonrelativistic and confining models, the presence of a constant or vanishing imaginary part can mark Fermi surfaces or deconfined/confining phases (Afrasiar et al., 2024, Afrasiar et al., 2024).

4. Effects of Deformations, Anisotropy, and Higher Curvature

HTEE is sensitive to deformations and the underlying IR structure of the field theory or its gravitational dual:

• Scalar Hair and IR Backreaction: Introducing relevant deformations (e.g., scalar operators sourcing "hairy" black holes) breaks the invariance of the imaginary component seen in pure AdS/BZT spacetime. The imaginary part develops nontrivial dependence on the interval length and deformation parameters; for example, in $d=3$, the imaginary component can decrease and then rise with interval size and becomes suppressed at late times (Prihadi et al., 26 Jan 2026).
• Anisotropy: In anisotropic holographic backgrounds (e.g., black branes with different $f_3$, $f_4$), the behavior of the imaginary part is governed non-monotonically by the anisotropy parameter, revealing crossovers in quantum correlations and transitions between regimes (Goki et al., 25 Jan 2026).
• Higher-Curvature Corrections: Lovelock and Gauss–Bonnet gravity introduce dimension-dependent corrections to the imaginary component. For vacuum, the real and imaginary pieces maintain proportionality up to universal phase factors $(-i)^{d}$, while excitation terms (e.g., mass deformations) contribute only to the real part (Zhao et al., 4 Sep 2025).
• Confinement and phase transitions: In confining geometries, HTEE displays a critical interval size above which the imaginary part vanishes, signaling a "first-order" transition akin to spatial entanglement (Afrasiar et al., 2024, Nunez et al., 26 May 2025, Nunez et al., 18 Aug 2025).

5. Generalizations and Applications

HTEE has been adapted and studied across a wide set of contexts:

• Nonrelativistic and Lifshitz Theories: The scaling of both real and imaginary parts of HTEE encodes data about the Lifshitz exponent and hyperscaling violation. In particular, the presence of a logarithm in $\operatorname{Re} S$ or a nonvanishing constant $\operatorname{Im} S$ signals Fermi surface physics (Afrasiar et al., 2024).
• Black Hole Interiors and Singularity Probes: In black hole spacetimes, the timelike extremal surface extends behind the horizon and often probes the deep interior, including the singularity (Afrasiar et al., 24 Dec 2025, Anegawa et al., 2024). For certain singularities, the dominant HTEE saddle is complex, requiring careful contour selection in function space—a phenomenon outside standard replica derivations.
• Cosmological Applications: In FLRW universes, the timelike analog of HEE probes temporal "entanglement" and is only strongly subadditive if the cosmological background is accelerating, directly linking energy conditions, causal horizons, and information-theoretic inequalities (Noumi et al., 14 Apr 2025).
• Subregion Complexity and RG Flow: HTEE complements CV- and CA-proposals for subregion complexity, with certain regimes where the holographic complexity is entirely supported behind horizons or singularities. The temporal extension, i.e. "temporal entanglement entropy," directly maps onto coarse-graining and the renormalization group (Grieninger et al., 2023).
• Central Charges: Both the "slab" and "Liu–Mezei" central charge functions extracted from HTEE reproduce known central charge data in CFT and track RG flows and backreaction effects in nonconformal setups (Nunez et al., 18 Aug 2025, Roychowdhury, 26 Jul 2025, Nunez et al., 26 May 2025).

6. Algorithmic Principles and Physical Implications

In all instances, the prescription for HTEE involves selecting, among possibly many complex extremal surfaces, the saddle minimizing the imaginary part and, in case of degeneracy, the real part. For strip geometries or higher dimensions where the splitting into purely spacelike and timelike extremal surfaces may be ambiguous or lead to nonuniqueness, a global minimization principle on the complexified area functional is essential for consistency (Li et al., 2022, Heller et al., 2024).

The distinctive features of HTEE—its universal imaginary part, sensitivity to symmetry breaking, and dynamical response to geometry—make it a potent probe of deep IR phenomena, causal structure, and the quantum origin of time in gravitational and field-theoretic systems.

---

Select Analytic Results for 3D Lifshitz Theory

Term	Formula	Scaling with $z$
Real part	$$\frac{L}{4G_3} \frac{2}{z} \ln\left( \frac{\alpha T}{\epsilon^z} \right)$$	Inverse with $z$
Imaginary part	$\frac{L}{4G_3} \frac{\pi}{z}$	Inverse with $z$
Universal HTEE formula	$$S_{\rm TEE} = \frac{L}{4G_3} \left( \frac{2}{z} \ln\frac{\alpha T}{\epsilon^z} + i\frac{\pi}{z} \right)$$	Both contributions scaled by $1/z$

This illustrates the robust structure: real part dominated by the UV-divergent logarithm, imaginary part UV-finite and determined by the redshift parameter $z$ (Jena et al., 2024). The same pattern holds generically, with model-dependent coefficients and possible additional interval-dependent corrections under deformations (Prihadi et al., 26 Jan 2026).

---

HTEE thus unifies the Lorentzian generalization of spatial entanglement entropy, pseudo-entropy, and emergent time in holography, providing a rigorous and sensitive probe of both geometric and quantum features beyond reach of standard entanglement diagnostics.