Shannon and entanglement entropies of one- and two-dimensional critical wave functions

Abstract

We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant ${S}_{0}$. In $c=1$ critical systems (Tomonaga-Luttinger liquids), we find that ${S}_{0}$ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 $XXZ$ chain. In a massive (crystal) phase, ${S}_{0}$ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a $c=1/2$ critical point.