Spectral Gap of the Davies Generator for the Mean-Field Heisenberg Model

Abstract

The mean-field Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin-1/2 interactions. This non-commuting Hamiltonian is permutation and $\mathsf{SU}(2)$ invariant, and its Gibbs states undergo an $\mathsf{SU}(2)$ symmetry breaking phase transition at inverse temperature $β=2$. We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight asymptotic estimates for its spectral gap at all noncritical temperatures. For fixed $β<2$, the gap as a function of number of qubits $n$ is $Θ(1)$, while for fixed $β>2$ the gap is $Θ(n^{-1})$. The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature ($β>2$) slowdown is associated with broken continuous symmetry. Two key ingredients in our approach are a comparison argument, which introduces auxiliary generators to bound dissipation on nontrivial representations of the symmetry groups $\mathsf{SU}(2)$ and $\mathsf{S}_n$, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.