Thermal entanglement transitions from strong $\textrm{SU}(2)$ symmetry

Abstract

We study mixed-state entanglement in Gibbs states with strong $\mathrm{SU}(2)$ symmetry, focusing on locally interacting spin systems with ferromagnetic interactions. Our main result is to show that finite-temperature ordering transitions are associated with entanglement transitions, and therefore the steady state of strongly symmetric Lindbladians can exhibit entanglement transitions. While in the paramagnetic phase it is known that the distillable entanglement and logarithmic entanglement negativity between two halves of a large system grow logarithmically with the square root of the number of spins, we show that in ferromagnetic phases these quantities grow parametrically faster with system size. To arrive at this result we first establish relations between these mixed-state entanglement measures and spin correlations in states that are singlets under global $\rm{SU}(2)$ symmetry transformations. We then introduce a semiclassical theory for $\rm{SU}(2)$ singlet thermal states. While the global singlet constraint generally enters this theory as a complicated function of the full semiclassical spin configuration, we show that in large $S$ limit it simplifies drastically to a Gaussian suppression of total magnetization in disordered phases as well as in ordered phases in the vicinity of continuous thermal phase transitions. This leads us to a field theory describing spin correlations in the singlet sector. Using this we determine the behavior of various probes of mixed-state entanglement at low and at high temperatures, supporting our analytical results using numerical Monte Carlo simulations of a three-dimensional lattice realization of our semiclassical theory. We also use exact numerics in one-dimensional spin-$1/2$ systems to confirm our predictions for the scaling of mixed-state entanglement with correlation length and system size in the paramagnetic phase.