[Paper Review] Note On The Dilaton Effective Action And Entanglement Entropy

Shamik Banerjee|arXiv (Cornell University)|Jun 11, 2014

Black Holes and Theoretical Physics33 references3 citations

TL;DR

This paper establishes a direct equivalence between the dilaton effective action approach and the Green's function method for computing entanglement entropy in non-conformal field theories. By coupling the theory to a constant background dilaton field, the authors show that the trace of the energy-momentum tensor—key to entanglement entropy—can be extracted from the dilaton effective action, reproducing known results for 2D and 4D theories flowing between UV and IR fixed points, including contributions from central charges and scale anomalies.

ABSTRACT

In this note we do the analysis of entanglement entropy more carefully when the non-conformal theory flows to a non-trivial IR fixed point. In particular we emphasize the role of the trace of the energy-momentum tensor in these calculations. We also compare the current technique for evaluating the entanglement entropy, particularly the Green's function method for gaussian theories, with the dilaton effective action approach and show that they compute identical quantities. As a result of this, the dilaton effective action approach can be thought of as an extension of Green's function technique to interacting theories.

Motivation & Objective

• To clarify the role of the trace of the energy-momentum tensor in entanglement entropy calculations for non-conformal field theories flowing to non-trivial IR fixed points.
• To demonstrate that the dilaton effective action method reproduces the same entanglement entropy as the Green's function technique in Gaussian theories.
• To extend the Green's function method to interacting theories by showing the dilaton effective action provides a natural generalization.
• To investigate the contribution of scale anomalies and universal terms in entanglement entropy when UV and IR fixed points are scale invariant but not conformally invariant.
• To explore the implications of additional R² terms in the dilaton effective action for theories with scale-invariant but non-conformal fixed points.

Proposed method

• Use the dilaton effective action on a conical geometry to compute the entanglement entropy, where the dilaton couples to the trace of the energy-momentum tensor.
• Apply trace anomaly matching to derive the universal part of the dilaton effective action, which depends on the difference in central charges (c_UV - c_IR) in 2D and (a_UV - a_IR) in 4D.
• Compute the response of the effective action to a constant dilaton background field τ, extracting the linear term in τ to obtain the integrated trace of T_μ^μ.
• Use the relation μ d/dμ F = -∫√h ⟨T_μ^μ⟩ to connect the effective action to the renormalization group flow of entanglement entropy.
• Derive the entanglement entropy by integrating the RG flow of S_EE, obtaining logarithmic divergences in the cutoffs (a, L_IR).
• Generalize the method to four dimensions by computing geometric invariants (R, R²) on the replica cone and including contributions from Weyl anomalies and scale anomalies.

Experimental results

Research questions

• RQ1 How does the dilaton effective action reproduce the entanglement entropy of a non-conformal 2D field theory with c_UV ≠ c_IR?
• RQ2 Can the dilaton effective action method be shown to be equivalent to the Green's function method in Gaussian theories, and does it extend to interacting theories?
• RQ3 What is the role of the trace of the energy-momentum tensor in the entanglement entropy of non-conformal field theories?
• RQ4 How do universal terms in entanglement entropy change when the UV and IR fixed points are scale-invariant but not conformally invariant?
• RQ5 What is the physical significance of the R² term in the dilaton effective action for scale-invariant but non-conformal fixed points, and how does it affect entanglement entropy?

Key findings

• The dilaton effective action on a cone reproduces the entanglement entropy of a 2D non-conformal field theory with c_UV ≠ c_IR, matching the result from the replica trick and trace anomaly matching.
• The entanglement entropy for an infinite half-line is given by S_EE = -c_UV/6 ln(μa) + c_IR/6 ln(μL_IR), with IR divergence requiring an IR cutoff L_IR.
• The RG flow of entanglement entropy is μ d/dμ S_EE = -(c_UV - c_IR)/6, confirming the result from Calabrese and Cardy via a different method.
• In four dimensions, the universal part of the entanglement entropy is μ d/dμ S_EE = 4((a_UV - a_IR) - (c_UV - c_IR)/3), generalizing the 2D result.
• For scale-invariant but non-conformal fixed points, an additional R² term in the dilaton effective action contributes a new universal term to entanglement entropy, absent in CFTs.
• The method provides a systematic way to compute renormalized entanglement entropy and may help test whether all unitary scale-invariant theories are conformally invariant, especially in higher dimensions like six.