Zero modes and entanglement entropy

Abstract

Ultraviolet divergences are widely discussed in studies of entanglement entropy. Also present, but much less understood, are infrared divergences due to zero modes in the field theory. In this note, we discuss the importance of carefully handling zero modes in entanglement entropy. We give an explicit example for a chain of harmonic oscillators in 1D, where a mass regulator is necessary to avoid an infrared divergence due to a zero mode. We also comment on a surprising contribution of the zero mode to the UV-scaling of the entanglement entropy.

Entanglement entropy in causal set theory

“…Our choices of parameters (ρ for the causal set, and m and k for the oscillators) in this section were arbitrary. As we change the values of these parameters (as long as the UV cutoffs ρ and k remain large such that the asymptotic form of the entropy holds, and as long as the mass, m, remains finite in order to avoid the infrared divergence discussed in [17]), the qualitative results of this section do not change (as unpublished investigations have shown). The magnitude of the maximum of the quadratic relation (Figures 9-12) will, however, depend on these parameters.…”

When Is an Area Law Not an Area Law?

“…For this reason, we believe that most computations claiming to evaluate the vacuum entanglement entropy in 1 + 1 dimensions are incomplete and need to be corrected. Indeed, numerical simulations [17] strongly suggest that in these two cases the entanglement entropy will diverge in the absence of some sort of infrared regulator (the divergence being logarithmic in the regulator). In the analysis above, we have implicitly imposed such a regulator, for example Dirichlet conditions at one end of the spatial interval [0, L], or as in [18] the choice of "Sorkin-Johnston-vacuum" in some large "causal diamond" containing [0, L].…”

Zero mode suppression of superluminal signals in light-matter interactions

“…For this reason, it is perhaps desirable to be able to ignore or remove the zero mode from any calculation by hand. In some contexts, such as UDW model coupled via derivative coupling, its effect can indeed be made negligible at the level of detector responses in appropriate limits [16], but in some other contexts it has significant impact on detector dynamics and entanglement [16,17,23,24]. There are also cases when the zero mode has been excluded by assumption from a setup with periodic boundary conditions (e.g., in [25][26][27][28][29]), thus it is of interest to further study the impact that the removal of the zero mode may have on the relativistic nature of the interaction, and in particular in the causality of the whole particle detector model.…”