[Paper Review] Entanglement entropy and kinematic space in BCFT and RG flow

Samrat Bhowmick, Suchetan Das+1 more|arXiv (Cornell University)|Mar 6, 2017

Black Holes and Theoretical Physics37 references3 citations

TL;DR

This paper proposes a novel reverse approach to computing entanglement entropy (EE) in two-dimensional boundary conformal field theories (BCFT) by treating the kinematic space metric as a differential equation source. Using boundary conditions in the small-interval and large-distance limits, it derives EE expressions that match known results, demonstrating universality and consistency. The method successfully recovers the universal logarithmic term in EE for BCFT on the upper half-plane and extends to excited states via conformal mapping.

ABSTRACT

The relation between kinematic space metric and entanglement entropy provides us with a differential equation for entanglement entropy. For BCFT on upper half plane we solve this equation to obtain an expression for entanglement entropy consistent with known results in the literature. We also discuss how this relation can be used to recast the RG flow, under relevant deformations of a CFT, as a flow in the space of kinematic space metrics.

Motivation & Objective

• To reverse the standard AdS/CFT kinematic space formalism by deriving entanglement entropy from the kinematic space metric as a differential equation.
• To establish boundary conditions—small interval and large distance limits—that constrain the form of entanglement entropy in BCFT on the upper half-plane.
• To demonstrate that the derived EE expression matches known results from conformal invariance, validating the method’s consistency.
• To extend the method to deformed CFTs under relevant operators, recasting RG flow as a flow in kinematic space metric geometry.
• To apply the formalism to excited states, such as vacuum descendants, and reproduce known EE results via conformal mapping in kinematic space.

Proposed method

• Formulates the kinematic space metric as a second-order partial differential equation: $ \partial_u \partial_v S_{EE} = \text{source} $, where the source is derived from the metric of the k-space.
• Imposes two physical boundary conditions: (1) small interval limit ($ \Delta \to 0 $) matching CFT vacuum EE plus corrections, and (2) equivalence of $ \Delta \to 0 $ and $ T \to \infty $ limits.
• Solves the resulting PDE using separation of variables, assuming $ S_{EE} = S_1(\Delta) + S_2(T) $, to obtain a general form with universal and theory-dependent terms.
• Applies the method to the finite strip BCFT by transforming coordinates and solving the PDE in terms of $ \alpha = \sin(\pi T / 2L) $, $ \beta = \sin(\pi \Delta / 2L) $.
• Uses conformal mapping to generalize the EE for vacuum descendant states by transforming the metric under $ z \to h(z) $, preserving the differential structure.
• Derives the full EE expression via inverse Radon transform and boundary condition matching, fixing universal coefficients like $ c/6 $ and $ c/3 $.

Experimental results

Research questions

• RQ1 Can the kinematic space metric be used to derive entanglement entropy as a differential equation, rather than the other way around?
• RQ2 What boundary conditions are physically necessary and sufficient to uniquely determine the entanglement entropy in BCFT on the upper half-plane?
• RQ3 Does the derived EE expression recover the universal logarithmic term $ \frac{c}{6} \ln(\frac{2L}{\pi\epsilon} \sin(\frac{\pi x_1}{L})) $ known from conformal field theory?
• RQ4 How can RG flows under relevant deformations be reformulated as a flow in the space of kinematic space metrics?
• RQ5 Can the method reproduce known EE results for vacuum descendant states using conformal mapping in kinematic space?

Key findings

• The entanglement entropy for a single interval on the upper half-plane is derived as $ S(x_1) = \frac{c}{6} \ln\left(\frac{2L}{\pi\epsilon} \sin\frac{\pi x_1}{L}\right) + a $, matching the universal CFT result.
• The solution contains a universal logarithmic term fixed by boundary conditions, and a theory-dependent piece determined by the source function.
• For a constant-source metric $ ds^2 = \frac{a}{\sin^2(\pi\Delta/L)} + \frac{b}{\sin^2(\pi T/L)} dx_1 dx_2 $, the EE takes the form $ S = c' \ln(k' \sin(\pi\Delta/L)) + b'\Delta^2 + \cdots $, with coefficients fixed by limits.
• The method successfully reproduces the known EE for vacuum descendant states via conformal mapping: $ S^{(h)} = \frac{c}{3} \ln\left(\frac{h(x_1)-h(x_2)}{\epsilon}\right) - \frac{c}{6} \ln(h'(x_1)h'(x_2)) $.
• The RG flow of a CFT under relevant deformations is recast as a flow in the space of kinematic space metrics, with the metric evolving via the source function $ g_2(\eta), g_3(\eta) $.
• The analysis confirms that the small-interval limit is insensitive to the form of the source, as long as the metric is locally constant, ensuring robustness of the universal term.