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 $\beta =2$. 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 $\beta <2$, the gap as a function of number of qubits $n$ is $\Theta (1)$, while for fixed $\beta >2$ the gap is $\Theta (n^{-1})$. The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature ($\beta >2$) 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.

63 pages, 1 figure