Farey分数自旋链自由能在临界点的渐近性
Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point
摘要：
我们考虑外场$h$中的Farey分数自旋链。使用的想法从动力系统和功能分析，我们表明，自由能$f$在附近的二阶相变，确切地说，由$$ f \sim \frac t{\log t}-\frac1{2} \frac{h^2}t \quad \text{for} \quad h^2\ll t \ll 1 . $$这里$t=λ_{G}\log(2)(1-\fracβ{β_c})$是一个降低的温度，使偏离临界点的比例由高斯映射，$λ_G$的李雅普诺夫指数。因此，$λ_G$决定了比热和磁化率奇异性的幅度。据我们所知，只有另一个微观定义的相互作用模型，其相变附近的自由能是已知的两个变量的函数。我们的研究结果证实了以前发现的集群近似，并表明，集群机制实际上是负责的过渡。然而，结果不同意部分与再正常化组治疗。
英文摘要：
We consider the Farey fraction spin chain in an external field $h$. Using ideas from dynamical systems and functional analysis, we show that the free energy $f$ in the vicinity of the second-order phase transition is given, exactly, by $$ f \sim \frac t{\log t}-\frac1{2} \frac{h^2}t \quad \text{for} \quad h^2\ll t \ll 1 . $$ Here $t=λ_{G}\log(2)(1-\fracβ{β_c})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, $λ_G$. It follows that $λ_G$ determines the amplitude of both the specific heat and susceptibility singularities. To our knowledge, there is only one other microscopically defined interacting model for which the free energy near a phase transition is known as a function of two variables. Our results confirm what was found previously with a cluster approximation, and show that a clustering mechanism is in fact responsible for the transition. However, the results disagree in part with a renormalisation group treatment.
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