Holographic entanglement entropy in Chern-Simons gravity with torsion

1. Problem Statement

The holographic entanglement entropy (HEE) is a cornerstone of the AdS/CFT correspondence, typically calculated using the Ryu-Takayanagi (RT) formula or its higher-curvature generalizations (e.g., Dong's formula). However, standard holographic calculations implicitly assume the bulk spacetime geometry is Riemannian (torsion-free), utilizing the Levi-Civita connection.

While Riemann-Cartan (RC) geometry, where torsion is an independent dynamical field, is well-motivated in extensions of General Relativity and condensed matter physics (e.g., spin transport), there is a lack of understanding regarding how torsion affects holographic entanglement entropy. Specifically, it is unknown how the presence of torsion in the bulk modifies the universal divergent terms of the entanglement entropy in the dual boundary Conformal Field Theory (CFT).

2. Methodology

The authors investigate this problem within the framework of 5-dimensional Chern-Simons (CS) gravity in an Anti-de Sitter (AdS$_5$) bulk with axial torsion. Their approach involves two complementary conjectures and explicit calculations:

A. Theoretical Framework

• Bulk Theory: They utilize 5D CS gravity, which is dual to a 4D CFT. Unlike standard Einstein-Hilbert gravity, CS gravity in 5D allows for a non-trivial boundary torsion induced via the Fefferman-Graham (FG) expansion.
• Boundary Conditions: They consider a specific solution representing AdS$_5$ with axial torsion, where the boundary metric is flat, but the boundary torsion is non-vanishing ($T^i = -C \epsilon_{ijk} dx^j \wedge dx^k$).
• Prescription 1 (Direct Substitution):
  • They start with the known formula for HEE in a 4D CFT with vanishing $c$-type central charge (characteristic of CS gravity duals). In the torsion-free case, the universal logarithmic divergence depends on the induced Ricci scalar ($R_\Sigma$) on the entangling surface $\Sigma$.
  • Hypothesis: They propose that in the presence of torsion, one should simply replace the Riemannian Ricci scalar with the Riemann-Cartan (RC) Ricci scalar ($R^{(RC)}_\Sigma$) in the entropy formula.
• Prescription 2 (Wald Entropy Analogy):
  • They draw an analogy between HEE and Wald's black hole entropy formula. In torsion-free gravity, the extremal surface for HEE coincides with the black hole horizon.
  • They note that Wald's entropy in CS gravity with torsion is obtained by replacing the Riemann curvature with the RC curvature in the functional.
  • Hypothesis: They propose upgrading the HEE functional similarly ($R \to R^{(RC)}$).
  • Technical Challenge: The RC functional does not yield a simple analytic extremal surface with proper asymptotics. To overcome this, they evaluate the torsion-full entropy functional on the torsion-free extremal surface (the standard RT surface), arguing that for 5D CS gravity, the logarithmic divergence depends only on the boundary condition ($r(0)=R$) and not the higher-order coefficients of the surface profile.

B. Explicit Calculation

• Setup: They calculate the HEE for a solid cylinder (and a ball) in the boundary theory.
• Geometry: The bulk metric is AdS$_5$ with axial torsion. The entangling surface is a 2-sphere (for the ball) or a cylinder boundary.
• Computation:
  1. They derive the induced RC Ricci scalar ($R^{(RC)}_\Sigma$) on the entangling surface using the generalized Gauss-Codazzi equations, which include contorsion terms.
  2. They substitute this into the entropy functional.
  3. They integrate the functional over the radial coordinate $\rho$ (introducing a UV cutoff $\epsilon$) to isolate the divergent terms.

3. Key Contributions and Results

A. Discovery of a Torsion-Induced Logarithmic Term

The primary result is the derivation of a universal, torsion-induced logarithmic divergence in the entanglement entropy.

• In the torsion-free case, the entropy for a 4D CFT dual to 5D CS gravity (where the trace anomaly $c=0$) contains a term proportional to the area of the entangling surface times the induced Ricci scalar.
• With torsion, the induced RC Ricci scalar on a flat boundary surface (like a sphere) acquires a constant term proportional to the square of the torsion strength ($C^2$).
• Result: The entanglement entropy acquires an additional divergent term:
  $$S_{\text{torsion}} \sim A_\Sigma C^2 \ln(\epsilon)$$
  where $A_\Sigma$ is the area of the entangling surface, $C$ is the torsion strength, and $\epsilon$ is the UV cutoff.

B. Consistency Between Two Approaches

The authors demonstrate that both proposed methodologies yield the exact same result:

1. Direct Substitution: Replacing $R_\Sigma$ with $R^{(RC)}_\Sigma$ in the standard HEE formula yields the $C^2 \ln \epsilon$ term.
2. Wald Analogy: Evaluating the Wald-like entropy functional (with RC curvature) on the torsion-free extremal surface also yields the $C^2 \ln \epsilon$ term.
   This agreement serves as a strong consistency check for their prescription, suggesting that the RC Ricci scalar correctly captures the torsional contribution to entanglement entropy in this holographic model.

C. Physical Interpretation

The authors provide a heuristic physical interpretation of the result. They compare the torsion-induced term to the logarithmic divergence found in massive QFTs ($\sim m^2 \ln \epsilon$).

• They show that a massless scalar field non-minimally coupled to the background with torsion effectively acquires a mass term $m^2 \propto -C^2$.
• This suggests that axial torsion acts as an emergent infrared scale in the boundary effective description, mimicking the effect of a mass term in the entanglement entropy.

4. Significance

• First Explicit Holographic Computation: This paper provides the first explicit calculation of torsion-induced contributions to holographic entanglement entropy.
• Extension of Holography: It successfully extends the holographic dictionary to include Riemann-Cartan geometries, moving beyond the standard assumption of torsion-free bulk spacetimes.
• Validation of RC Geometry in Holography: The consistency between the direct substitution method and the Wald entropy analogy validates the use of RC curvature in holographic entropy functionals.
• Implications for Condensed Matter: Since torsion is used to model spin transport and other phenomena in condensed matter, this work offers a new tool for understanding how geometric defects (torsion) affect quantum information measures like entanglement entropy in dual field theories.
• Future Directions: The authors suggest that while their derivation relies on the specific simplicity of 5D CS gravity, the mechanism (RC Ricci scalar replacement) likely generalizes to other higher-curvature gravity theories, opening a path for studying entanglement in more general torsion-full holographic models.

In summary, the paper establishes that torsion generates a universal logarithmic divergence in holographic entanglement entropy, proportional to the square of the torsion strength, and provides a robust prescription for calculating this effect in 5D Chern-Simons gravity.