The entanglement entropy of a region of finite length $l$ in the ground state, in 2d CFT diverges as $$S \approx \frac{c}{6} log(l)$$ (Is it c/3 or c/6 ?) Why does this diverge. In the derivation of this formula in the paper by Caleberse ( arXiv:hep-th/0405152) where is this divergence arising ?
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$\begingroup$ Is the divergence from the thermodynamic limit $L \rightarrow \infty$ $\endgroup$– M111Commented Nov 29, 2017 at 0:50
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The following discussion is within the scope of 1+1D quantum systems, which are more or less equivalent to 2D classical systems.
- It is not anomalous that entanglement entropy diverges at large $l$. Generally speaking, the ground state entanglement entropy of a gapped system (probed rigorously for 1D gapped systems) satisfies area law, while a generic state (e.g. a highly excited state) entanglement entropy satisfies volume law. Thus for a generic 1D quantum state, the entanglement entropy $S_E \propto l$ diverges linearly. The entanglement entropy is a constant only for the ground state of 1D gapped system. Entanglement entropy diverges even faster for higher-D systems.
- The reason why the entanglement entropy of ground state of 1+1D CFT does not satisfy area law (i.e. a constant) is due to the gaplessness and emergence of conformal symmetry. For example, the correlation function (i.e. two-point function) decays exponentially for gapped systems, while decaying algebraically for 1+1D CFT. This has already hinted the need for a correction for area law. For a 1+1D system with periodic boundary condition in thermodynamic limit, entanglement entropy has a logarithm correction in addition to a non-universal constant, i.e. $S_E \sim \frac{c}{3} \log (l)$ (for open boundary condition, $S_E \sim \frac{c}{6} \log (l)$). The derivation is shown in Calabrese and Cardy.
