Entanglement universality at criticality

Question

In $(d+1)$-dimensional quantum systems described by conformal field theories at criticality, I have been under the impression that the entanglement entropy is described by a non-universal area law plus universal logarithmic corrections.

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This area+log is true in $(1+1)$-dimensional systems, which has $$S(l) = \frac{c}{3}\ln \left(\frac{L}{\pi} \sin{\frac{\pi l}{L} }\right) +C$$ as seen in the transverse field ising model in another question, though please see another question of mine about this when $c$ is large compared to the local degrees of freedom. Note that the area law in $1d$ is just a constant law.

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Fradkin and Moore in "Hearing the Shape of a Quantum Drum" showed that in $(2+1)$-dimensional systems, the area+log statement is still true, though the log correction can sometimes vanish depending on the geometry of the region:

$$S(R) = 2f_s(l/a) + \alpha c \log(l/a) + O(1)$$

where $l$ is the length of the boundary of the region $R$, $a$ is the ultraviolet cutoff, $f_s$ is the non-universal leading area-law term, and there is the requisite universal $c \log(l/a)$ term. In said paper, they speculate that the $O(1)$ term is not universal. $\alpha$ here depends on the shape of the region $R$ and can be $0$.

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However, I was recently perusing a paper that noted that for a a (d-1 + 1)-dimensional theory, the entanglement entropy of a spherically shaped region with radius $r$ is

$$S(r) = \mu_{d-2}r^{d-2} + \mu_{d-4}r^{d-4} + ... + \begin{cases} (-1)^{d/2-1}4A\log(r/a), & \text{if $d$ is even} \\ (-1)^{(d-1)/2}F, & \text{if $d$ is odd} \end{cases} $$ where this last term is the universal term. Note that the above predicts a logarithmic term in the case $(1+1)$-dimensions with $d=2$ but not in $(2+1)$ dimensions with $d=3$.

This seems to partially contradict Fradkin and Moore's result above. While it's likely that the lack of a log term in $(2+1)$-dimensions for a sphere comes from that particular geometry suffering $\alpha=0$ in Fradkin and Moore's result, it seems that the $O(1)$ piece is indeed universal.

Is it that in $(2+1)$-dimensions both the logarithmic and constant terms are universal? Or is the constant term only universal in some geometries of the region? How about for other dimensions?

Answer 1

Indeed, it has to do with the geometry of the entangling region. In (2+1)d, when the boundary of the region is smooth (e.g. a disk region in (2+1)d), the subleading correction to the area law is a universal constant and is related to the free energy of the theory on $S^3$. If the region contains sharp corners (e.g. rectangle), the subleading term is logarithmic in $l$. In that case, the constant piece is not universal as it can be absorbed into $\ln (l/a)$, by redefining $a$. But the coefficient of the log is a universal function of the opening angles of the corners.

The higher-dimensional result you quote applies to sphere boundary, which is all smooth. When there are singularities on the boundary additional logarithmic corrections occur.