Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2512.03022 · Physical review. B./Physical review. B

Topological marker in three dimensions based on kernel polynomial method

Ranadeep Roy, Wei Chen

Published Feb 2, 2026• 1 citation
arXivHTMLPDF

Abstract

The atomic-scale influence of disorder on the topological order can be quantified by a universal topological marker, although the practical calculation of the marker becomes numerically very costly in higher dimensions. We propose that for any symmetry class in higher dimensions, the topological marker can be calculated in a very efficient way by adopting the kernel polynomial method. Using class AII in three dimensions as an example, which is relevant to realistic topological insulators like Bi2Se3 and Bi2Te3, this method reveals the criteria for the invariance of topological order in the presence of disorder, as well as the possibility of a smooth cross over between two topological phases caused by disorder. In addition, the significantly enlarged system size in the numerical calculation implies that this method is capable of capturing the quantum criticality much closer to topological phase transitions, as demonstrated by a nonlocal topological marker.

DOI
10.1103/3jkb-jf29

Related papers

Regularized universal topological markers for Dirac systems

Yulin Qin, Chang-An Li, Jian Li

Topological markers provide an efficient and powerful characterization of topological features of many systems, especially when the translation symmetry is broken. Recently, a universal topological marker applicable in different symmetry classes of topological systems is proposed. However, it suffers from irregular behaviors at the boundary and its connection to other topological indexes remains elusive. In this work, we construct regularized universal topological markers that apply to Dirac systems by utilizing position operators that are compatible with periodic boundary conditions. The regularized markers eliminate the obstructive boundary irregularities successfully and give rise to the desired global topological invariants, such as the Chern number, consistently when integrated over all the lattice sites. Furthermore, the regularized form allows us to establish an explicit connection between the markers and some other known topological indices in two dimensions. For instance, it turns out to be equivalent to the Bott index in classes A, D, and C and equivalent to the spin Chern number in classes DIII and AII. We further examine the utility and stability of this marker in disordered scenarios. We find that its variance shows peaks at the phase boundaries, which promotes it as a useful indicator for detecting disorder-induced topological phase transitions.

arXiv:2512.24869•Physical review. B./Physical review. B•May 12, 2026•1 citation

Universal topological marker

Wei Chen

The topological order of Dirac models in any dimension and symmetry class can be mapped to lattice sites by a universal topological marker. From the topological operator that extracts this marker, one can further define a nonlocal marker that detects topological phase transitions. Several prototype models such as the Su-Schrieffer-Heeger model, Majorana chain, Chern insulator, Bernevig-Hughes-Zhang model, chiral and helical $p$-wave superconductors, three-dimensional time-reversal symmetric topological insulators, among many others, are employed to demonstrate the ubiquity of these markers.

arXiv:2209.10703•Physical review. B./Physical review. B•Jan 10, 2023•39 citations

Robustness of topological order against disorder

Lucas A. Oliveira, Wei Chen

A universal topological marker has been proposed recently to map the topological invariants of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the global topological order, represented by the spatially averaged topological marker, remains unchanged in the presence of disorder for 1D and 2D systems. We find that if an impurity corresponds to varying a nonzero-matrix element of the lattice Hamiltonian, regardless the element represents hopping, chemical potential, pairing, etc., then the average topological marker is conserved. However, if there are many strong impurities and the average distance between them is shorter than a correlation length, then the average marker is no longer conserved. In addition, strong and dense impurities can be used to continuously interpolate between one topological phase and another. A number of prototype lattice models including Su-Schrieffer-Heeger model, Kitaev chain, Chern insulators, Bernevig-Hughes-Zhang model, and chiral $p$-wave superconductors are used to elaborate the ubiquity of these statements.

arXiv:2401.02270•Physical review. B./Physical review. B•Mar 8, 2024•22 citations

Evaluation of the real-space second Chern number using the kernel polynomial method

Rui Chen, Bin Zhou

We evaluate the real-space second Chern number of four-dimensional Chern insulators using the kernel polynomial method. Our calculations are performed on a four-dimensional system with ${30}^{4}$ sites, and the numerical results agree well with theoretical expectations. Moreover, we show that the method is capable of capturing the disorder effects. This is evidenced by the phase diagram obtained for disordered systems, which agrees well with predictions from the self-consistent Born approximation. Furthermore, we extend the method to six dimensions and perform an exploratory real-space calculation of the third Chern number. Although finite-size effects prevent full quantization, the numerical results show qualitative agreement with theoretical expectations. The study represents a step forward in the real-space characterization of higher-dimensional topological phases.

arXiv:2507.18919•Physical review. B./Physical review. B•Sep 30, 2025•3 citations

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

Wei Chen, Andreas P. Schnyder

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.

arXiv:1901.11468•New Journal of Physics•Jun 17, 2019•62 citations

Disorder-induced quantum phase transitions in three-dimensional topological insulators and superconductors

Shinsei Ryu, Kentaro Nomura

We discuss the effects of disorder in time-reversal-invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in the symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field-theory analysis (the large-${N}_{f}$ expansion, where ${N}_{f}$ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, and several anomalous dimensions at a nontrivial critical point separating a metallic phase and a Dirac semimetal. A similar-field theory approach is developed for disorder-driven transitions in symmetry classes AIII, CI, and DIII. For these three symmetry classes, where topological superconductors are characterized by integer topological invariant, a complementary description is given in terms of the nonlinear sigma model supplemented with a topological term, which is a three-dimensional analog of the Pruisken term in the integer quantum Hall effect.

arXiv:1109.2942•Physical Review B•Apr 26, 2012•56 citations
The atomic-scale influence of disorder on the topological order can be quantified by a universal topological marker, although the practical calculation of the marker becomes numerically very costly in higher dimensions. We propose that for any symmetry class in higher dimensions, the topological marker can be calculated in a very efficient way by adopting the kernel polynomial method. Using class AII in three dimensions as an example, which is relevant to realistic topological insulators like Bi2Se3 and Bi2Te3, this method reveals the criteria for the invariance of topological order in the presence of disorder, as well as the possibility of a smooth cross over between two topological phases caused by disorder. In addition, the significantly enlarged system size in the numerical calculation implies that this method is capable of capturing the quantum criticality much closer to topological phase transitions, as demonstrated by a nonlocal topological marker.
Topological markers provide an efficient and powerful characterization of topological features of many systems, especially when the translation symmetry is broken. Recently, a universal topological marker applicable in different symmetry classes of topological systems is proposed. However, it suffers from irregular behaviors at the boundary and its connection to other topological indexes remains elusive. In this work, we construct regularized universal topological markers that apply to Dirac systems by utilizing position operators that are compatible with periodic boundary conditions. The regularized markers eliminate the obstructive boundary irregularities successfully and give rise to the desired global topological invariants, such as the Chern number, consistently when integrated over all the lattice sites. Furthermore, the regularized form allows us to establish an explicit connection between the markers and some other known topological indices in two dimensions. For instance, it turns out to be equivalent to the Bott index in classes A, D, and C and equivalent to the spin Chern number in classes DIII and AII. We further examine the utility and stability of this marker in disordered scenarios. We find that its variance shows peaks at the phase boundaries, which promotes it as a useful indicator for detecting disorder-induced topological phase transitions.
The topological order of Dirac models in any dimension and symmetry class can be mapped to lattice sites by a universal topological marker. From the topological operator that extracts this marker, one can further define a nonlocal marker that detects topological phase transitions. Several prototype models such as the Su-Schrieffer-Heeger model, Majorana chain, Chern insulator, Bernevig-Hughes-Zhang model, chiral and helical ppp-wave superconductors, three-dimensional time-reversal symmetric topological insulators, among many others, are employed to demonstrate the ubiquity of these markers.
A universal topological marker has been proposed recently to map the topological invariants of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the global topological order, represented by the spatially averaged topological marker, remains unchanged in the presence of disorder for 1D and 2D systems. We find that if an impurity corresponds to varying a nonzero-matrix element of the lattice Hamiltonian, regardless the element represents hopping, chemical potential, pairing, etc., then the average topological marker is conserved. However, if there are many strong impurities and the average distance between them is shorter than a correlation length, then the average marker is no longer conserved. In addition, strong and dense impurities can be used to continuously interpolate between one topological phase and another. A number of prototype lattice models including Su-Schrieffer-Heeger model, Kitaev chain, Chern insulators, Bernevig-Hughes-Zhang model, and chiral ppp-wave superconductors are used to elaborate the ubiquity of these statements.
We evaluate the real-space second Chern number of four-dimensional Chern insulators using the kernel polynomial method. Our calculations are performed on a four-dimensional system with 304{30}^{4}304 sites, and the numerical results agree well with theoretical expectations. Moreover, we show that the method is capable of capturing the disorder effects. This is evidenced by the phase diagram obtained for disordered systems, which agrees well with predictions from the self-consistent Born approximation. Furthermore, we extend the method to six dimensions and perform an exploratory real-space calculation of the third Chern number. Although finite-size effects prevent full quantization, the numerical results show qualitative agreement with theoretical expectations. The study represents a step forward in the real-space characterization of higher-dimensional topological phases.
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.
We discuss the effects of disorder in time-reversal-invariant topological insulators and superconductors in three spatial dimensions. For three-dimensional topological insulator in the symplectic (AII) symmetry class, the phase diagram in the presence of disorder and a mass term, which drives a transition between trivial and topological insulator phases, is computed numerically by the transfer matrix method. The numerics is supplemented by a field-theory analysis (the large-Nf{N}_{f}Nf​ expansion, where Nf{N}_{f}Nf​ is the number of valleys or Dirac cones), from which we obtain the correlation length exponent, and several anomalous dimensions at a nontrivial critical point separating a metallic phase and a Dirac semimetal. A similar-field theory approach is developed for disorder-driven transitions in symmetry classes AIII, CI, and DIII. For these three symmetry classes, where topological superconductors are characterized by integer topological invariant, a complementary description is given in terms of the nonlinear sigma model supplemented with a topological term, which is a three-dimensional analog of the Pruisken term in the integer quantum Hall effect.