论文标题
相关长度的非分析性在具有指数衰减相互作用的系统中
Non-analyticity of the correlation length in systems with exponentially decaying interactions
论文作者
论文摘要
我们在$ \ mathbb {z}^d $上考虑各种晶格旋转系统（包括Ising，Potts和XY型号），具有$ j_x =ψ（x）e^{ - | x |} $的远距离交互，其中$ c | x |} $，其中$ψ（x）= e^{e^{\ mathsf {\ mathsf {o} {o}（o x | x |）我们明确表征了预先事实$ψ$，这些$ψ$引起相关长度，而相关的外部参数（s）（倒数温度$β$，磁场$ h $等）中的相关长度未分析。我们的结果适用于任何维度。作为一个有趣的特殊情况，我们证明，在一维系统中，相关长度在总结$ψ$的情况下是未分析的，与所有标准热力学量的众所周知的分析行为形成鲜明对比。我们还指出，这种非分析性在存在时也体现了2分函数行为的定性变化。特别是，我们将相关长度的分析性缺乏分析性与Ornstein中的质量间隙条件的失败 - 相关的相关理论。
We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $\mathbb{Z}^d$ with long-range interactions of the form $J_x = ψ(x) e^{-|x|}$, where $ψ(x) = e^{\mathsf{o}(|x|)}$ and $|\cdot|$ is an arbitrary norm. We characterize explicitly the prefactors $ψ$ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $β$, magnetic field $h$, etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $ψ$ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein--Zernike theory of correlations.
