Anisotropic scaling for 3D topological models

S Rufo,M A R Griffith,Nei Lopes,Mucio A Continentino

doi:10.1038/s41598-021-01888-x

Abstract

A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent nu obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent theta =nu _{x}/nu _{z}. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for nu and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation nu _{bar{alpha }}z_{bar{alpha }}=2q and theta =nu _{x}/nu _{z}=z_{z}/z_{x} for nu and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction bar{alpha }. An anisotropic quantum hyperscaling relation is also obtained.

Highlights

Topological models, computing their anisotropic correlation length critical exponent ν obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point
Beyond the standard topological materials, we point out the Dirac semimetals (DSMs) that present band touching at some points in their band structures
The main purpose of the present work is to extend and characterize the critical behavior of topological materials that do not belong to the topological table of classification

Summary

Introduction

Topological models, computing their anisotropic correlation length critical exponent ν obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. Beyond the standard topological materials, we point out the Dirac semimetals (DSMs) that present band touching at some points in their band structures These are protected by time-reversal symmetry (TRS) and/ or inversions symmetry (IS)[32]. In a Weyl semimetal crystal, the chiralities associated with the Weyl nodes (Fermi points) can be understood as topological charges These leads to monopoles and anti-monopoles of Berry curvature in momentum space, which serve as the topological invariant of this phase[34]. Another property, induced by the projection of the bulk Weyl nodes is the appearance of Fermi arcs in the surface of the Brillouin z one[34,35]. The Weyl nodes with zero energy observed in Weyl semimetal materials (WSMs) can be described as low-energy fermion excitations using the Dirac equation[37,38,39]

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Conclusion