A refinement of entanglement entropy and the number of degrees of freedom

TL;DR

This work addresses the UV-divergent nature of entanglement entropy by introducing a UV-finite renormalized entanglement entropy ${\mathcal{S}}_d^{(\Sigma)}(R)$ that isolates universal, scale-specific entanglement at size $R$ and can be interpreted as an RG flow of entanglement. The authors derive its general properties, connect it to RG fixed points, and explore both free-field and holographic (gravity-dual) realizations, showing how ${\mathcal{S}}_d^{(\Sigma)}(R)$ encodes the number of effective degrees of freedom at scale $R$ and reduces to central charges at fixed points. They demonstrate in $d=3$ that ${\mathcal{S}}_3(R)$ is non-negative and monotone in Lorentz-invariant unitary QFTs, while in $d=4$ monotonicity can fail, with holographic flows revealing phase transitions in the entanglement surface topology. The work also extends the analysis to Fermi liquids, finite temperature/chemical potential, and Rényi entropies, illustrating a broad framework for understanding scale-dependent entanglement and its relation to the counting of degrees of freedom across RG flows.

Abstract

We introduce a "renormalized entanglement entropy" which is intrinsically UV finite and is most sensitive to the degrees of freedom at the scale of the size R of the entangled region. We illustrated the power of this construction by showing that the qualitative behavior of the entanglement entropy for a non-Fermi liquid can be obtained by simple dimensional analysis. We argue that the functional dependence of the "renormalized entanglement entropy" on R can be interpreted as describing the renormalization group flow of the entanglement entropy with distance scale. The corresponding quantity for a spherical region in the vacuum, has some particularly interesting properties. For a conformal field theory, it reduces to the previously proposed central charge in all dimensions, and for a general quantum field theory, it interpolates between the central charges of the UV and IR fixed points as R is varied from zero to infinity. We conjecture that in three (spacetime) dimensions, it is always non-negative and monotonic, and provides a measure of the number of degrees of freedom of a system at scale R. In four dimensions, however, we find examples in which it is neither monotonic nor non-negative.

Figures (11)

• Figure 1: ${\mathcal{S}}_3(R)$ for a free massive scalar field: The red point is the value for $m=0$. The black dashed line is the result of the asymptotic expansion \ref{['scle']}. The numerical results are computed following the method of Srednicki:1993im with a radial lattice discretization. We choose the system size to be $L_{IR}=200 a$, where $a$ is the lattice spacing. To avoid boundary effects the restriction to $10a \leq R \leq 45 a$ was made. To extend the range of $mR$ we obtained the results for $1/m=20 a, \ 40a, \ 120a$. In the plots, the orange dots are data points for $1/m= 120a$, the blue ones are for $1/m=40a$, and the green ones are for $1/m=20 a$. As expected all our data points collapse into one curve as ${\mathcal{S}}_3(R)$ can only depend on $mR$ in the continuum limit. For more details see Appendix \ref{['app:num']}.
• Figure 2: Cartoon of a minimal surface of disk topology (black) v.s. a minimal surface of cylinder topology (red). The cylinder type surface is possible only for \ref{['singf']} with $n > 2$.
• Figure 3: Left:$f(z)$ for the domain wall solution describing the flow of M2-brane theory to an IR fixed point preserving ${{\mathcal{N}}}=2$ supersymmetry. Middle: plot of $S(R) - S_{\rm UV} (R)$ where $S_{\rm UV}$ denotes that at the UV fixed point. The UV divergences cancel when taking the difference, but the resulting expression does not have a well-defined large $R$ limit, with a linear $R$-dependence. As in the case of a free massive scalar and Dirac field of Sec. \ref{['sec:free']}, the presence of such a linear term can be understood as a finite renormalization between the short distance cutoffs of the UV and IR fixed points, as discussed in Sec. \ref{['app:UV']}. Right:${\mathcal{S}}_3 (R)$ for this flow. We normalize the value at UV to be $1$. The horizontal dashed line denotes the expected value \ref{['cftir']} for the IR fixed point. The black line (lower line) is obtained from numerical calculation by using \ref{['rr1']}. For this flow, $\epsilon$ in \ref{['neke']} is $\epsilon \approx 0.3$, and thus equation \ref{['master1']} should be a reasonable approximation, whose results are plotted using the red line (upper line). Note that the part linear in $R$ in $-S_{\rm finite}$ as seen in the second plot is automatically eliminated when considering ${\mathcal{S}}_3 (R)$.
• Figure 4: Left: a steep domain wall (toy example) with $f (z) = 1 + {14 z^{100} \over 5^{100} + z^{100}}$. Middle: plot of $S (R)- S_{\rm conf} (R)$ where $S_{\rm conf} (R)$ denotes the entanglement entropy for the UV fixed point. The short-distance divergences cancel when taking the differences. For the indicated range of $R$, the action \ref{['minar']} has three extrema, all of disk type. The entanglement entropy of the system is given by the smallest of them. There is a first-order "phase transition" at $R_c=4.4$. Right:${\mathcal{S}}_3 (R)$ has a discontinuous jump, which is indicated by the vertical green line. The dashed horizontal line is the expected asymptotic value for the IR fixed point.
• Figure 5: Left:${\mathcal{S}}_3 (R)$ for $f(z) = 1 + z^2$. Middle:${\mathcal{S}}_3 (R)$ for $f (z) = 1 + z^3$ which exhibits a "second-order phase transition" from minimal surface of disk topology (black curve) to cylindrical topology (red curve). Right:${\mathcal{S}}_3 (R)$ for $f (z) = (1 + z^2)^2$, which exhibits a "first-order phase transition" between the surfaces of two topologies. The dashed curve corresponds to other extrema of the minimal surface action. There is a discontinuous jump in ${\mathcal{S}}_3 (R)$ which is indicated by the green vertical line.