We study and present the results of curvature for different symmetry classes (BDI, AIII and A) of model Hamiltonians and also present the transformation of model Hamiltonian from one distinct symmetry class to the other based on the curvature property. We observe the mirror symmetric curvature for the Hamiltonian with BDI symmetry class but there is no evidence of such behaviour for Hamiltonians of AIII symmetry class. We show the origin of torsion and its consequences on the parameter space of topological phase of the system. We find the evidence of torsion for the Hamiltonian of A symmetry class. We present Serret–Frenet equations for all model Hamiltonians in \(\mathbf {R}^3\) space. To the best of our knowledge, this is the first application of curvature theory to the model Hamiltonian of different symmetry classes which belong to the topological state of matter.

我们研究并展示了模型哈密顿量不同对称类（BDI、AIII 和 A）的曲率结果，还展示了模型哈密顿量从一个不同的对称类到另一个基于曲率特性的转换。我们观察到具有 BDI 对称类的哈密顿量的镜像对称曲率，但没有证据表明 AIII 对称类的哈密顿量有这种行为。我们展示了扭转的起源及其对系统拓扑相参数空间的影响。我们找到了 A 对称类的哈密顿量的扭转证据。我们在\(\mathbf {R}^3\) 中给出了所有模型哈密顿量的 Serret-Frenet 方程空间。据我们所知，这是曲率理论首次应用于属于物质拓扑状态的不同对称类的模型哈密顿量。