AI 中文总结
研究平均场海森堡模型的戴维斯生成器谱隙，通过比较论证和分解可观测量空间为球张量算符的方法，给出了不同温度下谱隙的渐近估计，揭示了低温下与连续对称性破缺相关的慢化现象。
AI 中文摘要
平均场海森堡铁磁体是具有各向同性自旋 - 1/2 相互作用的完全图上的量子自旋模型。该非对易哈密顿量具有置换和\(\mathsf{SU}(2)\)不变性，其吉布斯态在逆温度\(\beta = 2\)时经历\(\mathsf{SU}(2)\)对称破缺相变。我们考虑相关的戴维斯生成器，证明了其在所有非临界温度下谱隙的严格渐近估计。固定\(\beta < 2\)时，谱隙与量子比特数\(n\)无关；\(\beta > 2\)时，谱隙为\(\Theta(n^{-1})\)。谱隙的匹配上界由总磁化序参量给出，表明低温（\(\beta > 2\)）下的减速与连续对称性破缺有关。我们方法的两个关键要素是比较论证和将可观测量空间分解为球张量算符。
英文摘要
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.
Comments63 pages, 1 figure
