A c-theorem for the entanglement entropy

TL;DR

This work derives an entropic c-theorem in 1+1 dimensions from strong subadditivity and Lorentz invariance, introducing a monotone entropic function c(r)=r dS/dr that decreases along scale and yields a finite central-charge value at fixed points. The authors compare the resulting entropic c-function for free fields with Zamolodchikov’s c-function, finding notable differences, and provide analytic and numerical treatments via conical geometries and Painlevé-formulated correlators. They also critically assess the challenges of generalizing the entropic c-theorem to higher dimensions, outlining shape, commutativity, divergence, and logarithmic obstacles that currently preclude a straightforward extension. The work offers a novel, purely entropic perspective on RG irreversibility in 1+1D QFTs and clarifies the limitations of extending this approach to more complex spacetime dimensions.

Abstract

The combination of the Lorentz symmetry and the strong subadditive property of the entropy leads to a c-theorem for the entanglement entropy in 1+1 dimensions. We present a simple derivation of this theorem and compare the associated c-functions with the Zamolodchikov's ones for the case of free fields. We discuss the various difficulties which obstacle the naive generalizations of the entropic c-theorem to higher dimensions.

Figures (2)

• Figure 1: Time is the vertical axis, the horizontal axis is the spatial coordinate $x$ and the null lines are drawn at $45^\circ$. The causal domain of dependence (diamond shaped set drawn with dashed lines) corresponding to the spatial intervals $b$, $c$ have intersection given by the domain of dependence of $a$ and union (followed by causal completion) given by the domain of dependence of $d$.
• Figure 2: From top to bottom: one third of the Zamolodchikov c-functions for a real scalar and a Dirac field, and entropic c-functions for a Dirac (dashed curve) and a real scalar field (dotted curve).