Rényi Entropies and Cosmic-Brane Backreaction

At finite replica index $n$, the replica defect is a physical source in the quotient geometry. Its tension $T_n=(n-1)/(4nG_N)$ backreacts on the metric, so the $n=1$ RT surface cannot simply be reused. The quantity directly equal to the backreacted brane area is the refined Rényi entropy, not $S_n$ itself; ordinary Rényi entropy follows by integration in $n$. Saddle transitions can make this integration piecewise. We use Euclidean replica saddles and return to Lorentzian statements only after specifying the continuation.

Required background. Replica cosmic-brane derivation supplies the quotient and tension, while Rényi analytic continuation supplies the boundary definition and branch problem.

Helpful background. FLM/QES separates loop corrections from finite-$n$ classical backreaction, and entropy verification tracks normalization and limits.

Refined Rényi entropy measures the brane area

Define

$$S_n(A)=\frac{1}{1-n}\log\operatorname{Tr}\rho_A^n.$$

The refined Rényi entropy is

$$\widetilde S_n \equiv n^2\partial_n\!\left(\frac{n-1}{n}S_n\right).$$

For a replica-symmetric semiclassical Einstein saddle, the quotient contains a brane $C_n$ with

$$T_n=\frac{n-1}{4nG_N}, \qquad 8\pi G_NT_n=2\pi\left(1-\frac1n\right),$$

and Dong’s area law is

$$\widetilde S_n(A)=\frac{\operatorname{Area}(C_n)}{4G_N}.$$

The backreacted metric solves

$$G_{ab}+\Lambda g_{ab} =8\pi G_NT_n\,h_{ab}\,\delta_{C_n}$$

in distributional notation. Thus $C_n$, its area, and the ambient saddle all depend on $n$. Integrating the refined entropy gives

$$\frac{n-1}{n}S_n =\int_1^n\frac{dn'}{n'^2}\,\widetilde S_{n'},$$

using regularity at $n=1$. The cosmic-brane area formula and this integral relation were derived explicitly by Dong (Dong 2016, §§2–3).

Vacuum interval in AdS3

For one interval of length $\ell$ in the vacuum of a holographic CFT$_2$, replica symmetry fixes a locally AdS$_3$ quotient with conical opening $2\pi/n$. The cosmic brane follows the interval geodesic, while its backreaction changes the transverse angular identification. The regulated brane length gives

$$\frac{\operatorname{Length}(C_n)}{4G_3} =\frac{c}{3n}\log\!\left(\frac{\ell}{\epsilon}\right),$$

where $c=3L/(2G_3)$ and the same Fefferman–Graham/CFT cutoff matching as on the RT page is used. Hence

$$\widetilde S_n =\frac{c}{3n}\log\!\left(\frac{\ell}{\epsilon}\right).$$

Integrating,

$$\begin{aligned} \frac{n-1}{n}S_n &=\frac{c}{3}\log\!\left(\frac{\ell}{\epsilon}\right) \int_1^n\frac{dn'}{n'^3}\\ &=\frac{c}{6}\left(1-\frac1{n^2}\right) \log\!\left(\frac{\ell}{\epsilon}\right), \end{aligned}$$

so

$$S_n =\frac{c}{6}\left(1+\frac1n\right) \log\!\left(\frac{\ell}{\epsilon}\right).$$

The $n\to1$ limit gives $c/3\log(\ell/\epsilon)$, the RT entropy, while finite $n$ probes the backreacted quotient. This is the first application and an exact leading-large-$c$ match to the universal CFT$_2$ interval result.

For a ball in a higher-dimensional CFT vacuum, a conformal map sends the reduced state to a thermal state on $\mathbb R\times H^{d-1}$ at $T_0=1/(2\pi R)$. The replica geometry is a hyperbolic black hole at $T_0/n$, and

$$S_n=\frac{n}{n-1}\frac1{T_0} \int_{T_0/n}^{T_0}dT\,S_{\rm thermal}(T).$$

Solving the regularity condition at the Euclidean horizon is the hyperbolic-black-hole version of including brane backreaction.

Branches and phase transitions

Suppose two replica-symmetric saddles have actions $I_a(n)$ and $I_b(n)$, crossing at $n_c>1$. Then the dominant refined entropy is derived from a piecewise action,

$$I_{\rm dom}(n)= \begin{cases} I_a(n),&1\leq n<n_c,\\ I_b(n),&n>n_c. \end{cases}$$

The integral for $S_n$ must be split at $n_c$. Continuing $C_n^{(a)}$ smoothly to large $n$ gives a mathematically valid subdominant branch but not the physical Rényi entropy.

An explicit mechanism occurs for hyperbolic black holes coupled to a sufficiently light scalar. As $n$ grows, the effective temperature $T_0/n$ falls; the bald hyperbolic black hole can become unstable and a scalar-hairy saddle dominates. The Rényi entropy then has a phase transition at finite $n$ (Belin, Maloney, and Matsuura 2013, §§3–5).

This supplies the adversarial test. The $n\to1$ limit follows the branch connected to the original state, where $T_n\to0$. The large-$n$ limit has $T_n\to1/(4G_N)$ and probes the lowest entanglement-spectrum levels; it may lie on the hairy branch. Agreement of the two limits after naive continuation is not required, and a mismatch at the known crossing is evidence for saddle competition, not a failure of the Rényi definition.

Limits and corrections

The area relation assumes a classical replica-symmetric Einstein saddle. Higher-curvature gravity replaces the brane area by an action-dependent functional. Bulk loops add quantum corrections on the $n$-dependent geometry. Replica-symmetry breaking removes the single-brane quotient. Noninteger continuation still needs a physical branch and contour.

Bit threads returns to the $n=1$, static Einstein optimization. Mixed-state pages later use different replicas and must not borrow the ordinary Rényi brane without deriving their gluing and continuation.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Belin, A., Maloney, A., and Matsuura, S. (2013). “Holographic phases of Rényi entropies.” Journal of High Energy Physics 2013(12), 050. DOI.
• Dong, X. (2016). “The gravity dual of Rényi entropy.” Nature Communications 7, 12472. DOI.