Zero modes and divergence of entanglement entropy

We investigate the cause of the divergence of the entanglement entropy for the free scalar fields in $(1+1)$ and $(D+1)$ dimensional space-times. In a canonically equivalent set of variables, we show explicitly that the divergence in the entanglement entropy of the continuum field in $(1+1)$ dimensions is due to the accumulation of a large number of near-zero frequency modes as opposed to the commonly held view of divergence having UV origin. The feature revealing the divergence in zero modes is related to the observation that the entropy is invariant under a hidden scaling transformation even when the Hamiltonian is not. We discuss the role of dispersion relations and the dimensionality of the space-time on the behavior of entanglement entropy.

Figure 1

Entropy $v/s$ the size of reduced state, with fixed boundaries. For $\beta =1$, the area law is violated.