Thermodynamic extrinsic curvature is a new mathematical tool in thermodynamic geometry. By using the thermodynamic extrinsic curvature, one may obtain a more complete geometric representation of the critical phenomena and thermodynamics. We introduce nonperturbative thermodynamic extrinsic curvature of an ideal two-dimensional gas of anyons. Using extrinsic curvature, we find new fixed points in nonperturbative thermodynamics of the anyon gas that particles behave as semions. Here, we investigate the critical behavior of thermodynamic extrinsic curvature of two-dimensional Kagome Ising model near the critical point ${\beta }_c={(k_B{T}_c)}^{-1}$ in a constant magnetic field and show that it behaves as $|\beta -{\beta }_c{|}^{\alpha }$ with $\alpha =0$, where $\alpha$ denotes the critical exponent of the specific heat. Then, we consider the three-dimensional spherical model and show that the scaling behavior is $|\beta -{\beta }_c{|}^{\alpha }$, where $\alpha =-1$. Finally, using a general argument, we show that extrinsic curvature $K$ has two different scaling behaviors for positive and negative $\alpha$. For $\alpha >0$, our results indicate that $K\sim |\beta -{\beta }_c{|}^{\frac{1}{2}(\alpha -2)}$. However, for $\alpha <0$, we found a different scaling behavior, where $K\sim |\beta -{\beta }_c{|}^{\alpha }$.

热力学外曲率是热力学几何中的一种新的数学工具。通过使用热力学外曲率，可以获得临界现象和热力学的更完整的几何表示。我们引入任意子理想二维气体的非微扰热力学外在曲率。利用外在曲率，我们在任意子气体的非微扰热力学中找到了新的不动点，粒子表现为半子。在这里，我们研究了二维 Kagome Ising 模型在临界点附近的热力学外曲率的临界行为${\beta }_C={（k_{乙}{\mathrm{时间}}_C）}^{-1}$在恒定磁场中，并表明它的行为为$|\beta -{\beta }_C{|}^{\alpha }$和$\alpha =0$， 在哪里$\alpha$表示比热的临界指数。然后，我们考虑三维球形模型并表明缩放行为是$|\beta -{\beta }_C{|}^{\alpha }$， 在哪里$\alpha =-1$。最后，使用一般论证，我们证明外在曲率$K$对于正向和负向有两种不同的缩放行为$\alpha$。为了$\alpha >0$，我们的结果表明$K～|\beta -{\beta }_C{|}^{\frac{1}{2}（\alpha -2）}$。然而，对于$\alpha <0$，我们发现了不同的缩放行为，其中$K～|\beta -{\beta }_C{|}^{\alpha }$。