Holographic Entanglement Entropy

• HEE is a nonlocal observable that quantifies quantum entanglement by relating boundary CFT regions to the areas of extremal surfaces in higher-dimensional AdS spacetimes.
• The geometric prescription, based on the Ryu–Takayanagi and HRT formulas, links entanglement entropy to minimal surface areas and highlights additional corrections for non-coherent and superposed states.
• HEE is applied to diagnose phase transitions in holographic superconductors, multi-region entanglement, and quantum critical phenomena, providing a bridge between geometry and quantum information.

Holographic Entanglement Entropy (HEE) is a key nonlocal observable in the paper of strongly coupled quantum systems via gauge/gravity duality. It encodes the quantum entanglement of spatial subregions in a boundary conformal field theory (CFT) by associating it with geometric quantities—specifically, areas of extremal surfaces—in higher-dimensional asymptotically Anti-de Sitter (AdS) spacetimes. The comprehensive paper of HEE, from foundational prescriptions to recent generalizations, has transformed both conceptual and practical approaches to quantum gravity and quantum information in high energy theory.

1. Geometric Prescription and the Ryu–Takayanagi Formula

The fundamental computation of HEE is governed by the Ryu–Takayanagi (RT) formula for static states. For a boundary region $A$ of a strongly coupled large-N CFT with a semiclassical AdS gravity dual, the HEE is given by

$S_A = \frac{\mathrm{Area}(\gamma_A)}{4 G_N}$

where $\gamma_A$ is the codimension‑2 minimal surface in the bulk ending on $\partial A$, and $G_N$ is Newton's constant. In the large-N limit, the leading contribution scales as $O(N^2)$ and corresponds to the classical area law. The prescription was extended to covariant situations (time-dependent settings) via the Hubeny–Rangamani–Takayanagi (HRT) proposal, with $\gamma_A$ replaced by an extremal surface.

The RT formula applies rigorously only to boundary states whose expectation values (e.g., of $T_{\mu\nu}$) define a unique, classical bulk metric; in other words, for “coherent” states dual to a single classical geometry (Asplund, 2012).

2. Beyond the Area Law: Non-Coherent States and Superpositions

Critically, states in the boundary CFT that are superpositions of coherent states—such as EPR/Bell-like entangled superpositions with well-separated support—do not possess a unique dual classical geometry. The entanglement entropy for these quantum superposed states cannot be fully captured by the classical RT area term. Instead, there are additional $O(N^2)$ contributions arising from quantum correlations beyond what is seen by the minimal surface in any single geometry (Asplund, 2012).

Toy models developed for such superposed states derive an entropy of the form

$$S_A \approx \frac{\mathrm{Area}(\gamma_A)}{4 G_N} + N^2 C \log 2$$

where $C$ parametrizes the fraction of the degrees of freedom entangled across the superposition. In the large-N limit, this reflects the presence of $O(N^2)$ fields, each contributing a non-geometric log-term to the entropy when maximally entangled.

These supplementary terms are especially important in dynamical processes like local quenches, which generate highly entangled superpositions that cannot be described by a single classical geometry. The implication is that refined prescriptions—perhaps involving explicit sums over quantum bulk geometries—are required for these cases.

3. HEE in Holographic Superconductors and Imbalanced Phases

In gravitational duals of superconducting phases, HEE provides a diagnostic for phase transitions and encodes the physical impact of symmetry breaking and condensate formation. In imbalanced holographic superconductors (chemical potential mismatch), the HEE in the superconducting (hairy black hole) phase is consistently lower than in the normal AdS–Reissner–Nordström (AdS–RN) black hole phase at the same thermodynamic parameters (Dutta et al., 2013).

A striking contrast is observed:

• In the AdS–RN (normal) phase, HEE increases with increasing chemical imbalance parameter $\beta$.
• In the superconducting phase, HEE decreases as $\beta$ increases.

This dichotomy is explained by the “locking in” of degrees of freedom via condensate formation, which reduces entanglement in the superconducting phase. While HEE functions as a sensitive indicator of the phase transition, it should be interpreted in conjunction with thermodynamic quantities such as free energy, as for large imbalance the state may cease to be superconducting despite lower HEE.

4. Multi-Region Entanglement and Geometric Phase Transitions

The paper of HEE for multiple disjoint regions (e.g., multiple strips) reveals rich phase structure due to the combinatorics of possible bulk minimal surfaces (Ben-Ami et al., 2014). When analyzing $m$ strips (equal length and separation), the possible minimal area surfaces reduce, by virtue of topology and strong subadditivity, to at most four types: disconnected, connected, and two “disconnected” types present in confining geometries.

Critical points in the entanglement phase diagram are marked by geometric transitions: as the relative size and separation of the strips change, the minimal surface topology (and thus HEE) undergoes first order transitions. In confining backgrounds or finite-temperature BTZ black hole geometries, additional “entanglement plateaux” and “Hagedorn” transitions further enrich the phase structure.

5. HEE in Higher Derivative Gravity and the Wald-Like Prescriptions

For gravitational actions with higher derivative corrections, the standard RT formula must be generalized. When the Lagrangian depends on the Riemann tensor and its derivatives, new terms appear in the entropy functional:

• A “generalized Wald entropy” accounting for extrinsic curvature couplings on generic entangling surfaces.
• An “anomaly of entropy” associated with logarithmic divergences, crucial for matching CFT trace anomaly calculations in 4d and 6d (solving the Hung–Myers–Smolkin puzzle).

The formal HEE expression is

$S_{HEE} = S_{G\text{-}Wald} + S_{Anomaly}$

with explicit expressions involving functional derivatives of the action with respect to curvature and its derivatives (Miao et al., 2014). For Einstein–like $f(R, R_{\mu\nu})$ theories, field redefinition invariance ensures that the HEE may still be computed with the Wald entropy functional; the minimizing surfaces necessarily have vanishing trace of extrinsic curvature, leading to significant simplification (Mozaffar et al., 2016).

6. Dynamical and Thermal Corrections to HEE

Time-dependent gravitational backgrounds (e.g., perturbed black branes, infalling shells) provide an arena for exploring how nonequilibrium physics imprints on HEE. Small deformations (mass, angular momentum, charge) allow a perturbative expansion of HEE in conserved charges. Second order corrections are typically negative (except in anomalous settings) and ensure that the entanglement entropy is bounded above, provided the cosmic censorship conjecture holds; this establishes a direct link between bulk geometric regularity and entropic bounds in the boundary CFT (He et al., 2014).

In hydrodynamical flows dual to relativistic black brane backgrounds, increasing the fluid velocity leads to a UV-divergent growth of the regulated HEE as the velocity approaches the speed of light. The mutual information between strips disappears beyond a critical velocity—an entanglement “phase transition”—while sound mode excitations induce nontrivial time evolution and, in some cases, novel dynamical UV divergences (Bhattacharya et al., 2022).

7. HEE and Quantum Criticality

HEE is a robust nonlocal probe of quantum phase transitions (QPT) in holographic models. In systems exhibiting metal-insulator transitions or lattice deformations (Q-lattice, Einstein–Maxwell–Dilaton), HEE shows pronounced maxima or extremal behavior of its derivatives at quantum critical points (Ling et al., 2015, Ling et al., 2016). This feature is conjectured to be universal for holographic QPTs.

Detailed modeling reveals that in holographic superconductors, the critical entanglement entropy is suppressed below the transition. In more complex settings (e.g., with dark matter sector coupling or vector condensates), the scaling of HEE near criticality picks out physical parameters—linear with respect to coupling, nonlinearly enhanced by chemical potential imbalance—providing a diagnostic for critical phenomena (Yao et al., 27 May 2025). In some exotic vectorized phases, the minimal surface is geometrically screened by an effective boundary (where a certain Christoffel symbol vanishes), leading to a “screened volume law” where the entropy density is strictly less than the thermal value (Chen et al., 11 Apr 2025). This mechanism highlights limitations in the ability of entanglement entropy to probe the complete Hilbert space in certain thermal phases.

8. Implications and Outlook

The paper of HEE has established profound connections between geometric aspects of gravity, quantum information theory, and strongly coupled field theory dynamics:

• It provides a geometric mechanism for understanding area laws and their violations in quantum systems with large central charge.
• It forces refinement of the bulk–boundary dictionary in regimes where the classical gravitational description fails, especially for non-coherent states and superpositions connected with local quenches or thermalization.
• It motivates the development of new entropy functionals in higher derivative gravity compatible with field redefinition invariance and anomaly matching.
• It uncovers entanglement-driven phase transitions and “screened” entanglement regimes with potentially novel critical properties and information-theoretic signatures.

The breadth of recent developments, from dynamical criticality in scalarized black holes (Li et al., 11 Jan 2025) to emergent entanglement screening in vectorized backgrounds, confirms HEE as a central observable in the modern landscape of holographic duality, with continuing implications for the encoding of quantum information in spacetime geometry and the quantum structure of black holes and strongly correlated phases.