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

• URL: http://arxiv.org/abs/2607.21798v1
• Date: Thu, 23 Jul 2026 20:30:28 GMT
• Title: Spectral Gap of the Davies Generator for the Mean-Field Heisenberg Model
• Abstract summary: Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin interactions.<n>We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight estimates for its spectral gap at all noncritical temperatures.<n>The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature slowdown is associated with broken continuous symmetry.
• Score: 3.940099144535651
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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