Mutual information and the F-theorem

TL;DR

The paper introduces mutual information between concentric circles as a universal geometric regulator to extract a regulator-independent central charge ${\tilde c}_0$ in three dimensions, enabling a rigorous $F$-theorem-like monotonicity along RG flows. By establishing an expansion for $I(A^+,A^-)$ and choosing a symmetric regulator ($\alpha=0$), the authors show ${\tilde c}_0$ is UV-insensitive and aligns with the entanglement-entropy central charge $c_0$ at fixed points, as supported by holographic, free-scalar, and lattice analyses. They formulate a renormalized entropy $I_0(A)$ and a corresponding c-function $C(R)$ that decrease monotonically under RG evolution, thereby proving the 3d $F$-theorem within this framework. The approach also clarifies ambiguities in odd-dimensional entanglement entropy calculations, proves compatibility with gauge theories, and provides practical pathways for lattice computations, with potential generalizations to other dimensions and topological phases.

Abstract

Mutual information is used as a purely geometrical regularization of entanglement entropy applicable to any QFT. A coefficient in the mutual information between concentric circular entangling surfaces gives a precise universal prescription for the monotonous quantity in the c-theorem for d=3. This is in principle computable using any regularization for the entropy, and in particular is a definition suitable for lattice models. We rederive the proof of the c-theorem for d=3 in terms of mutual information, and check our arguments with holographic entanglement entropy, a free scalar field, and an extensive mutual information model.

Figures (12)

• Figure 1: The entropy of the set of lattice points inside circles of radius $R$ for a free scalar in a square lattice. The radius is taken from $3.5$ to $35$, with equal spacing of $0.2$. The values of $-2 \pi c_0$ shown are obtained by computing $S(R)$ and then subtracting the term proportional to $R$ in a linear fit to the data.
• Figure 2: (Colour online) We compute the mutual information between the interior of the circle of radius $R_-$ and the exterior of a concentric circle of radius $R_+$ with $R_+> R_-$.
• Figure 3: (Colour online) In both figures, the shaded areas represent regions on both sides of the boundaries where the correlations between $A^+$ and $A^-$ contributing to the mutual information are sensitive to higher UV scales. As indicated in panel (b), this band has a width of roughly $2L_m$ in the regime where $\varepsilon\ll L_m$. These (nonconformal) correlations give local contributions calculable from a small region, such as that enclosed by the dashed (red) box. In panel (a), these local regions tend to become flat in the limit of large $R$ and so the local factor can be written as a series in inverse powers of $R$, as in eq. (\ref{['stinkpot']}). Panel (b) illustrates the more general case where the local factor can be expressed in terms of local geometric quantities in terms of the unit normal and tangent vectors, ${\hat{\bf n}}$ and ${\hat{\bf t}}$, as in eq. (\ref{['inte']}).
• Figure 4: (Colour online) Panel (a) shows the mutual information $I(A^+,A^-)$ calculated on a square lattice for a free scalar. The mutual information is plotted as a function of $R/\delta$ while holding the ratio $\varepsilon/R$ fixed with $\varepsilon/R=2/11$. Panel (b) shows the same data as in panel (a) but now on the vertical axis, we have plotted the quantity $I_0(A)=I(A^+,A^-)-2\pi R\,{\tilde{a}}/\varepsilon$. For the limit of small $\varepsilon/R$, this quantity should converge to the universal value of the constant term $-4\pi {\tilde{c}}_0$. The green dashed line shows the approximate value of $-4\pi {\tilde{c}}_0$ obtained by averaging over the values for $R\in (25,33)$. The exact value shown with the red dashed line is slightly smaller.
• Figure 5: (Colour online) The averaged constant term in the entropy of circles $\bar{S}(R)$ in a square lattice as a function of the radius in lattice units, $R/\delta$. The blue points are for a radius step $\Delta R=0.2$ (as in figure \ref{['entro-circ']}). The red points are for $\Delta R=(3\pi)^{-1}\sim 0.106$. The green points are for the same $\Delta R$ but the circles are centered at the point $(1/3,1/2)$ in lattice units, rather than at the origin. The correct value of $2\pi c_0$ is shown with the dashed line.