Holographic Entanglement Entropy

• Holographic entanglement entropy is the geometric encoding of quantum entanglement in gravitational duals via extremal surfaces, as defined by the RT and HRT prescriptions.
• It examines the competition between smooth connected and piecewise-smooth minimal surfaces, with connected surfaces generally dominating in standard thermal backgrounds.
• The analysis shows that except for special cases like nonextremal D6-branes, horizon effects do not trigger entanglement phase transitions, highlighting its role as a distinct thermodynamic potential.

Holographic entanglement entropy is a geometric realization of quantum entanglement in quantum field theories with gravitational duals. Central to the AdS/CFT correspondence, the entanglement entropy of a subregion in the boundary theory is computed by the area of an extremal surface in the dual AdS bulk, as prescribed by the Ryu–Takayanagi (RT) or the covariant Hubeny–Rangamani–Takayanagi (HRT) formula. At finite temperature, the presence of a black hole horizon in the bulk modifies the entanglement structure and raises the question of whether horizons induce new entanglement phase transitions, similar to those observed in holographic models of confinement. A systematic analysis in a variety of supergravity backgrounds, including BTZ black holes, non-extremal Dp-branes, dyonic black holes in AdS$_4$, and Schwarzschild-AdS black holes in global coordinates, reveals that for all standard cases the smooth connected minimal surface always dominates, and no order-parameter–like transition in entanglement entropy as a function of subsystem size occurs, except for a special case involving D6-branes (0710.5483). This behavior distinguishes holographic entanglement entropy as a unique thermodynamic potential, distinct from the free energy.

1. Foundational Framework: Holographic Entanglement Entropy

The entanglement entropy $S_A$ of a boundary subregion $A$ is related to the area of a codimension-2 minimal (or extremal) surface $\gamma_A$ in the bulk—anchored to $\partial A$—as

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

where $G_N$ is the Newton constant in the bulk. For static spacetimes, this is the RT prescription; for time-dependent cases, the extremal surface replaces the minimal one (HRT).

In the AdS$_3$/CFT$_2$ context, this reproduces the universal expression

$S_A = \frac{c}{3} \log\left(\frac{\ell}{a}\right)$

for an interval of length $\ell$, with $a$ the UV cutoff and $c$ the central charge related to the AdS scale by $c=\frac{3R}{2G_N^{(3)}}$. In the presence of horizons (finite temperature), the minimal surface may interact with the black hole, affecting $S_A$ both quantitatively and qualitatively.

2. Minimal Surface Competition and the Absence of Transitions

In gravitational backgrounds dual to finite-temperature field theories, minimal surfaces can be of two types:

• (i) Smooth (connected): a single surface stretching into the bulk, possibly approaching but not touching the horizon,
• (ii) Piecewise-smooth (disconnected or “cap + horizon”): segments extend from each boundary point down to the horizon and are joined by a horizontal segment along the horizon itself.

Generic formulas for their areas take the form

$$A_c = 2 \int_{r_*}^{r_\infty} \frac{\beta(r) H(r)}{\sqrt{H(r) - H(r_*)}} dr$$

for the connected surface, with $r_*$ the minimal radial depth, and

$$A_d = 2 \int_{r_0}^{r_\infty} \beta(r) \sqrt{H(r)} dr + \ell\sqrt{H(r_0)}$$

for the piecewise-smooth surface, with $r_0$ the horizon and $\ell$ the size of $A$.

A key question is whether, as $\ell$ is varied, the globally minimal configuration switches—signaling a phase transition in entanglement entropy analogous to the confining/deconfining transitions in gauge theory.

Case Study Table: Dominance of Smooth Surfaces

Geometry (Bulk)	Smooth Connected EE Dominates?	Transition Observed?
BTZ (AdS$_3$ BH)	Yes	No
Nonextremal D$p$-branes ($p<6$)	Yes	No
Nonextremal D6-brane	No (small $\ell$), Yes (large $\ell$)	Yes (only for $p=6$)
Dyonic BH (AdS$_4$)	Yes	No
Schwarzschild, global AdS$_p$	Yes	No

Smooth connected surfaces contribute the minimal entanglement entropy at all $\ell$, except for nonextremal D6-brane backgrounds, which are of limited gauge-theoretic interest. This absence of an entanglement phase transition distinguishes thermal backgrounds (with horizons) from confining geometries, where such transitions in minimal surfaces encode an order parameter for confinement (0710.5483).

3. Explicit Analysis in Key Backgrounds

3.1 BTZ Black Hole (AdS$_3$/CFT$_2$)

For the BTZ black hole, the minimal surface is a geodesic. Explicit formulas relate the geodesic’s turning point $r_*$ and subsystem size $\ell$: $$\frac{2\pi \ell}{L} = \frac{R}{r_+} \ln\left(\frac{r_* + r_+}{r_* - r_+}\right)$$ and the geodesic length yields

$$S_c = \frac{1}{4 G_N^{(3)}} \, 2R \ln\left(\frac{r_\infty}{r_+} \sinh \frac{\pi \ell \, r_+}{R L}\right)$$

which exactly matches the thermal CFT$_2$ entanglement entropy

$$S_A = \frac{c}{3} \log\left(\frac{\beta}{\pi a} \sinh \frac{\pi\ell}{\beta}\right) , \quad \beta = \frac{RL}{r_+}$$

The piecewise-smooth surface always gives a greater area, so the smooth configuration dominates for all $\ell$.

3.2 Nonextremal D$p$-Branes

For D$p$-branes with $p < 6$, no phase transition as $\ell$ varies is found; the smooth minimal surface is always favored. For $p=6$, a transition appears: for small $\ell$ the piece-wise surface is minimal, while for large $\ell$ the smooth surface dominates. However, $p=6$ is relevant primarily to higher-dimensional generalized theories, not to standard gauge theory models.

3.3 AdS$_4$ Dyonic Black Holes

With electric/magnetic charges parameterized by $\rho^2$, the minimal area analysis confirms that the smooth connected surface dominates and no transition is induced by electric/magnetic charges or by varying subsystem size.

3.4 Schwarzschild–AdS in Global Coordinates

In global AdS$_5$ and AdS$_4$, where the Hawking–Page transition (thermal AdS $\leftrightarrow$ black hole) is associated in AdS/CFT with confinement/deconfinement, the entanglement entropy does not exhibit any analogous "phase transition"—the smooth surface configuration remains globally minimal for all subsystem sizes.

4. Significance: Entanglement Entropy as a Distinct Thermodynamic Potential

Unlike the thermal free energy, which exhibits sharp transitions at the Hawking–Page point, the holographic entanglement entropy in these backgrounds is a smooth function as the subsystem size is varied. This implies that:

• The entanglement entropy, computed holographically, does not serve as a conventional order parameter for the deconfinement transition at finite temperature.
• Instead, it behaves as a thermodynamic potential distinct from the free energy, reflecting the underlying microstructure of entanglement in the dual field theory.
• The conjugate variables $(\ell, r_*)$ suggest an underlying rich thermodynamic structure, potentially hinting at new thermodynamic quantities beyond those encoded in the free energy.

In all cases, horizons set the overall scale and act as an IR cutoff, but do not trigger surface transitions in the sense familiar from confining backgrounds. The transition found for D6-branes is exceptional and of limited relevance for physical gauge theories.

5. Universal Formulas and Reproduction of Field Theory Results

The Ryu–Takayanagi relation and its generalizations yield precise matches with field theory predictions in notable cases. For instance, in AdS$_3$/CFT$_2$ the correspondence is explicit, and in higher dimensions the leading order area law divergence is automatically reproduced via the holographic prescription.

Key formulas:

• Ryu–Takayanagi: $S_A = \mathrm{Area}(\gamma_A)/(4G_N)$
• BTZ case: $$S_A = \frac{c}{3} \log\left(\frac{\beta}{\pi a} \sinh\left(\frac{\pi\ell}{\beta}\right)\right)$$
• Generic background:
  • $$A_c = 2 \int_{r_*}^{r_\infty} \frac{\beta(r) H(r)}{\sqrt{H(r) - H(r_*)}} dr$$
  • $$A_d = 2 \int_{r_0}^{r_\infty} \beta(r)\sqrt{H(r)} dr + \ell \sqrt{H(r_0)}$$

The lack of a phase transition in $S_A$ as $\ell$ is varied—for all geometries except D6—confirms the generality of these results.

6. Broader Perspective: Role of Horizons and Comparison to Confinement

Horizons, encoding finite temperature in the dual theory, modify the minimal surface configurations but do not generically induce transitions between different entanglement-phases. In contrast, in confining geometries realized by (e.g.) AdS solitons, a genuine transition between connected and disconnected minimal surfaces as a function of subsystem size is an unambiguous haLLMark of the confinement/deconfinement phase structure (0905.0932).

The analysis reveals that, for thermal states, the presence of a horizon primarily sets the IR scale (via parameters like $r_+$, $u_0$, $\beta$), imparting an overall scaling to the entanglement entropy but not inducing dynamical transitions in minimal surface dominance. This is in striking contrast to the phase structure of free energy and other order-parameter-like quantities.

7. Implications and Outlook

The results establish that:

• Entanglement entropy in standard finite-temperature holographic backgrounds is not a sharp probe of horizon-induced phase transitions.
• It nonetheless encodes key features of the thermal state, such as scaling with temperature and smooth crossover from quantum to thermal entanglement as one tunes subsystem size.
• The unique role of $(\ell, r_*)$ as conjugate variables may reflect deeper connections between geometric and thermodynamic data.
• These findings are crucial for interpreting entanglement entropy as a diagnostic of geometry and for distinguishing entanglement-driven versus thermal phenomena in holographic and field-theoretic contexts.

In summary, holographic entanglement entropy provides a robust, geometric probe that reproduces field theory results, respects thermodynamic scaling, and, except for exceptional backgrounds, eschews phase transitions as a function of subsystem size in finite-temperature settings—a fact that evidences its role as a thermodynamic potential distinct from conventional free energy (0710.5483).