具有高阶能带交叉的拓扑相变的普适类
Universality classes of topological phase transitions with higher-order band crossing
摘要：
在拓扑绝缘体和拓扑超导体中，拓扑不变量在调整某个系统参数时的离散跳跃定义了拓扑相变。本文采用统一的框架来研究一维到三维拓扑相变的量子临界性，同时结合了对称性分类、能带交叉阶数、$m$ -折叠旋转对称性、关联函数、临界指数、标度律和重整化群方法.我们首先根据时间反演、粒子-空穴和手征对称性对高阶狄拉克模型进行分类，并确定每类对称中能带交叉阶数的奇偶性。奇偶性进一步限制了对称类中允许的旋转对称性$m$。用曲率函数上的动量空间积分表示拓扑不变量，能带交叉的阶数决定了曲率函数的临界指数，以及通过曲率函数的傅里叶变换引入的Wannier态关联函数的临界指数.拓扑不变量的守恒进一步给出了临界指数之间的标度律。此外，一个重整化群方法的基础上变形的曲率函数证明了所有的尺寸和对称类。通过阐明临界量（包括拓扑不变量和临界指数的跳跃）如何依赖于非空间对称性和旋转对称性，我们的工作将普适类的概念引入到拓扑相变的描述中.
英文摘要：
In topological insulators and topological superconductors, the discrete jump of the topological invariant upon tuning a certain system parameter defines a topological phase transition. A unified framework is employed to address the quantum criticality of the topological phase transitions in one to three spatial dimensions, which simultaneously incorporates the symmetry classification, order of band crossing, $m$-fold rotational symmetry, correlation functions, critical exponents, scaling laws, and renormalization group approach. We first classify higher-order Dirac models according to the time-reversal, particle-hole, and chiral symmetries, and determine the even-oddness of the order of band crossing in each symmetry class. The even-oddness further constrains the rotational symmetry $m$ permitted in a symmetry class. Expressing the topological invariant in terms of a momentum space integration over a curvature function, the order of band crossing determines the critical exponent of the curvature function, as well as that of the Wannier state correlation function introduced through the Fourier transform of the curvature function. The conservation of topological invariant further yields a scaling law between critical exponents. In addition, a renormalization group approach based on deforming the curvature function is demonstrated for all dimensions and symmetry classes. Through clarification of how the critical quantities, including the jump of the topological invariant and critical exponents, depend on the nonspatial and the rotational symmetry, our work introduces the notion of universality class into the description of topological phase transitions.
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