Quantum vorticity at thermal equilibrium for spins systems with continuous symmetry

We propose a definition of vorticity at inverse temperature \beta for Gibbs states in quantum XY spin systems on the lattice by testing \exp[-\beta H] on a complete set of observables ("one-point functions"). We show in particular that it is independent of the choice of a particular basis. Imposing a compression of Pauli matrices at the boudary, which stands for the classical environment, we make some numerical simulations on finite lattices, and exhibit usual vortex patterns.