English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In topological insulators and topological superconductors, the discrete jumpof the topological invariant upon tuning a certain system parameter defines atopological phase transition. A unified framework is employed to address thequantum criticality of the topological phase transitions in one to threespatial dimensions, which simultaneously incorporates the symmetryclassification, order of band crossing, $m$-fold rotational symmetry,correlation functions, critical exponents, scaling laws, and renormalizationgroup approach. We first classify higher-order Dirac models according to thetime-reversal, particle-hole, and chiral symmetries, and determine theeven-oddness of the order of band crossing in each symmetry class. Theeven-oddness further constrains the rotational symmetry $m$ permitted in asymmetry class. Expressing the topological invariant in terms of a momentumspace integration over a curvature function, the order of band crossingdetermines the critical exponent of the curvature function, as well as that ofthe Wannier state correlation function introduced through the Fourier transformof the curvature function. The conservation of topological invariant furtheryields a scaling law between critical exponents. In addition, a renormalizationgroup approach based on deforming the curvature function is demonstrated forall dimensions and symmetry classes. Through clarification of how the criticalquantities, including the jump of the topological invariant and criticalexponents, depend on the nonspatial and the rotational symmetry, our workintroduces the notion of universality class into the description of topologicalphase transitions.

Universality classes of topological phase transitions with higher-order band crossing