Entanglement and classical fluctuations at finite-temperature critical points

We investigate several entanglement-related quantities at finite-temperature criticality in the three-dimensional quantum spherical model, both as a function of temperature $T$ and of the quantum parameter $g$, which measures the strength of quantum fluctuations. While the von Neumann and the R\'enyi entropies exhibit the volume-law for any $g$ and $T$, the mutual information obeys an area law. The prefactors of the volume-law and of the area-law are regular across the transition, reflecting that universal singular terms vanish at the transition. This implies that the mutual information is dominated by nonuniversal contributions. This hampers its use as a witness of criticality, at least in the spherical model. We also study the logarithmic negativity. For any value of $g,T$, the negativity exhibits an area-law. The negativity vanishes deep in the paramagnetic phase, it is larger at small temperature, and it decreases upon increasing the temperature. For any $g$, it exhibits the so-called sudden death, i.e., it is exactly zero for large enough $T$. The vanishing of the negativity defines a "death line", which we characterise analytically at small $g$. Importantly, for any finite $T$ the area-law prefactor is regular across the transition, whereas it develops a cusp-like singularity in the limit $T\to 0$. Finally, we consider the single-particle entanglement and negativity spectra. The vast majority of the levels are regular across the transition. Only the larger ones exhibit singularities. These are related to the presence of a zero mode, which reflects the symmetry breaking. This implies the presence of sub-leading singular terms in the entanglement entropies. Interestingly, since the larger levels do not contribute to the negativity, sub-leading singular corrections are expected to be suppressed for the negativity.