自旋空间中奇异相互作用各向同性的经典晶格自旋模型
Classical lattice spin models involving singular interactions isotropic in spin space
摘要：
我们在这里解决几个经典的晶格自旋模型，涉及$n-$组件单位矢量（$n=2,3$），与$D-$维晶格$\mathbb{Z}^D,\,D=1,2$，并通过限制到最近的邻居和各向同性的自旋空间，即由相互作用的自旋之间的标量积的函数定义的对势相互作用。当势函数为标量积的连续函数时，Mermin-Wagner定理及其推广在热力学极限下排除了所有有限温度下的取向序，并排除了当$D=1$ ;另一方面，我们在这里考虑了一些比较简单的标量积函数，它们从下面有界，对于某些相互方向发散到$+\infty$，且几乎处处连续且具有可积奇点。精确的解决方案，提出了$D=1$，显示没有相变和取向顺序的情况下，在所有有限的温度下的热力学极限;对于$D=2$，并在没有更严格的数学结果，广泛的模拟进行了对其中一些点的取向顺序的情况下，在所有有限的温度，并建议存在Berezinski \v-Kosterlitz无过渡。
英文摘要：
We address here a few classical lattice--spin models, involving $n-$component unit vectors ($n=2,3$), associated with a $D-$dimensional lattice $\mathbb{Z}^D,\,D=1,2$, and interacting via a pair potential restricted to nearest neighbours and being isotropic in spin space, i.e. defined by a function of the scalar product between the interacting spins. When the potential involves a continuous function of the scalar product, the Mermin--Wagner theorem and its generalizations exclude orientational order at all finite temperatures in the thermodynamic limit, and exclude phase transitions at finite temperatures when $D=1$; on the other hand, we have considered here some comparatively simple functions of the scalar product which are bounded from below, diverge to $+\infty$ for certain mutual orientations, and are continuous almost everywhere with integrable singularities. Exact solutions are presented for $D=1$, showing absence of phase transitions and absence of orientational order at all finite temperatures in the thermodynamic limit; for $D=2$, and in the absence of more stringent mathematical results, extensive simulations carried out on some of them point to the absence of orientational order at all finite temperatures, and suggest the existence of a Berezinski\vı-Kosterlitz-Thouless transition.
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