确定中心自旋模型累积量展开的有效性
Determining the validity of cumulant expansions for central spin models
AI总结：
本文研究中心自旋模型中累积量展开的有效性，发现平均场理论对大N极限的正确性依赖于参数缩放方式，且高阶展开的收敛性和误差特性与预期不同。
摘要：
对于具有多对一连接性的模型，人们普遍预期平均场理论能够精确捕获多粒子$N\to\infty$极限，并且海森堡方程的高阶累积量展开会收敛到同一极限，同时在有限$N$下提供改进的近似。这里我们表明，事实上情况并非总是如此。相反，平均场理论是否正确描述大$N$极限取决于模型参数如何随$N$缩放，且累积量展开的收敛性在偶数阶和奇数阶之间可能是不均匀的。此外，即使高阶累积量展开确实恢复了正确极限，误差也不随$N$单调变化，并且可能超过平均场理论的误差。
英文摘要：
For a model with many-to-one connectivity it is widely expected that mean-field theory captures the exact many-particle $N\to\infty$ limit, and that higher-order cumulant expansions of the Heisenberg equations converge to this same limit whilst providing improved approximations at finite $N$. Here we show that this is in fact not always the case. Instead, whether mean-field theory correctly describes the large-$N$ limit depends on how the model parameters scale with $N$, and the convergence of cumulant expansions may be non-uniform across even and odd orders. Further, even when a higher-order cumulant expansion does recover the correct limit, the error is not monotonic with $N$ and may exceed that of mean-field theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
