Abstract
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.
Highlights
A recently emerged issue in the research of topological insulators (TIs) and topological superconductors (TSCs) is the understanding of quantum criticality near topological phase transitions[1, 2, 3, 4, 5, 6]
Our approach generalizes the symmetry classification[7, 8, 9, 10] to higherorder Dirac models in all physically relevant cases, and reveals the following features due to the interplay between the topological invariant C, the {T, C, S} symmetries, order of band crossing n, and the m-fold rotational symmetry in 2D and 3D: (1) The {T, C, S} symmetries constrain the order of band crossing to be even, odd, or integer in each dimension × symmetry class
It is possible that only certain rotational symmetries m are compatible with a specific symmetry class, which may depend on the system being fermionic or bosonic
Summary
A recently emerged issue in the research of topological insulators (TIs) and topological superconductors (TSCs) is the understanding of quantum criticality near topological phase transitions[1, 2, 3, 4, 5, 6]. We investigate the quantum criticality of topological phase transitions within a framework that simultaneously incorporates all the aforementioned ingredients, namely (1) symmetry classification, (2) order of band crossing, (3) m-fold rotational symmetry (for 2D and 3D), (4) correlation function, (4) critical exponents, and (5) the discrete jump of topological invariant. We will demonstrate that a previously proposed scaling law[6], as well as a renormalization group approach based on the curvature function[11, 12], remain valid for all the 15 cases All these features together give rise to a coherent picture for the quantum criticality of topological phase transitions, especially how the critical quantities {∆C, n, ν} depend on the dimension and symmetry, which we refer to as the universality classes of TIs and TSCs. The article is organized in the following manner.
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