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:1402.1738 · Physical Review Letters

Quantum Criticality of Quasi-One-Dimensional Topological Anderson Insulators

Alexander Altland, Dmitry Bagrets, Lars Fritz, Alex Kamenev, Hanno Schmiedt

Published May 23, 2014• 104 citations
arXivHTMLPDF

Abstract

We present an analytic theory of quantum criticality in the quasi-one-dimensional topological Anderson insulators of class AIII and BDI. We describe the systems in terms of two parameters ($g$, $\ensuremath{\chi}$) representing localization and topological properties, respectively. Surfaces of half-integer valued $\ensuremath{\chi}$ define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two-parameter flow describing the class $A$ quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given. We check the quantitative validity of our theory by comparison to numerical transfer matrix computations.

DOI
10.1103/physrevlett.112.206602

Related papers

Topology versus Anderson localization: Nonperturbative solutions in one dimension

Alexander Altland, Dmitry Bagrets, Alex Kamenev

We present an analytic theory of quantum criticality in quasi one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters $(g,χ)$ representing localization and topological properties, respectively. Certain critical values of $χ$ (half-integer for $\Bbb{Z}$ classes, or zero for $\Bbb{Z}_2$ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric non-linear sigma-models. In $\Bbb{Z}_2$ classes we uncover a hidden supersymmetry, present at the quantum critical point.

arXiv:1411.5992•Physical Review B•Feb 27, 2015•66 citations

Quantum Hall criticality in Floquet topological insulators

Kun Woo Kim, Dmitry Bagrets, Tobias Micklitz +1 more

Topological Floquet Anderson Insulators (TFAIs) are systems in which an interplay of topology, driving, and static disorder stabilizes a unique form of topological quantum matter. Here, the authors present a first-principles theory, describing the topological phases and quantum criticality of the two-dimensional TFAIs in symmetry class A. Within this framework, they demonstrate how disorder is key to the formation of topological quantization, and that its critical behavior is in the quantum Hall universality class.

arXiv:1910.06892•Physical review. B./Physical review. B•Apr 1, 2020•14 citations

Anisotropic Topological Anderson Transitions in Chiral Symmetry Classes

Zhenyu Xiao, Kohei Kawabata, Xunlong Luo +2 more

We study quantum phase transitions of three-dimensional disordered systems in the chiral classes (AIII and BDI) with and without weak topological indices. We show that the systems with a nontrivial weak topological index universally exhibit an emergent thermodynamic phase where wave functions are delocalized along one spatial direction but exponentially localized in the other two spatial directions, which we call the quasilocalized phase. Our extensive numerical study clarifies that the critical exponent of the Anderson transition between the metallic and quasilocalized phases, as well as that between the quasilocalized and localized phases, are different from that with no weak topological index, signaling the new universality classes induced by topology. The quasilocalized phase and concomitant topological Anderson transition manifest themselves in the anisotropic transport phenomena of disordered weak topological insulators and nodal-line semimetals, which exhibit the metallic behavior in one direction but the insulating behavior in the other directions.

arXiv:2211.09999•Physical Review Letters•Aug 1, 2023•18 citations

Spectrum-Wide Quantum Criticality at the Surface of Class AIII Topological Phases: An “Energy Stack” of Integer Quantum Hall Plateau Transitions

Björn Sbierski, Jonas F. Karcher, Matthew S. Foster

In the absence of spin-orbit coupling, the conventional dogma of Anderson localization asserts that all states localize in two dimensions, with a glaring exception: the quantum Hall plateau transition (QHPT).In that case, the localization length diverges and interference-induced quantum-critical spatial fluctuations appear at all length scales.Normally, QHPT states occur only at isolated energies; accessing them therefore requires fine-tuning of the electron density or magnetic field.In this paper we show that QHPT states can be realized throughout an energy continuum, i.e., as an "energy stack" of critical states wherein each state in the stack exhibits QHPT phenomenology.The stacking occurs without fine-tuning at the surface of a class AIII topological phase, where it is protected by U(1) and (anomalous) chiral or time-reversal symmetries.Spectrum-wide criticality is diagnosed by comparing numerics to universal results for the longitudinal Landauer conductance and wave function multifractality at the QHPT.Results are obtained from an effective 2D surface field theory and from a bulk 3D lattice model.We demonstrate that the stacking of quantum-critical QHPT states is a robust phenomenon that occurs for AIII topological phases with both odd and even winding numbers.The latter conclusion may have important implications for the still poorly understood logarithmic conformal field theory believed to describe the QHPT.

arXiv:1912.06748•Physical Review X•May 1, 2020•23 citations

Scaling of critical wave functions at topological Anderson transitions in one dimension

Eoin Quinn, Thomas Cope, Jens Hjorleifur Bardarson +1 more

Topological Anderson transitions, which are direct phase transitions between topologically distinct Anderson localized phases, allow for criticality in one-dimensional disordered systems. We analyze the statistical properties of an ensemble of critical wave functions at such transitions. We find that the local moments are strongly inhomogeneous, with significant amplification towards the edges of the system. In particular, we obtain an analytic expression for the spatial profile of the local moments, which is valid at all topological Anderson transitions in one dimension, as we verify by direct comparison with numerical simulations of various lattice models.

arXiv:1505.05374•Physical Review B•Sep 29, 2015•4 citations

Topology of critical chiral phases: Multiband insulators and superconductors

Oleksandr Balabanov, Daniel Erkensten, Henrik Johannesson

Recent works have proved the existence of symmetry-protected edge states in certain one-dimensional topological band insulators and superconductors at the gap-closing points which define quantum phase transitions between two topologically nontrivial phases. We show how this picture generalizes to multiband critical models belonging to any of the chiral symmetry classes AIII, BDI, or CII of noninteracting fermions in one dimension.

arXiv:2104.08853•Physical Review Research•Oct 18, 2021•19 citations
We present an analytic theory of quantum criticality in the quasi-one-dimensional topological Anderson insulators of class AIII and BDI. We describe the systems in terms of two parameters (ggg, \ensuremathχ\ensuremath{\chi}\ensuremathχ) representing localization and topological properties, respectively. Surfaces of half-integer valued \ensuremathχ\ensuremath{\chi}\ensuremathχ define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two-parameter flow describing the class AAA quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given. We check the quantitative validity of our theory by comparison to numerical transfer matrix computations.
We present an analytic theory of quantum criticality in quasi one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters (g,χ)(g,χ)(g,χ) representing localization and topological properties, respectively. Certain critical values of χχχ (half-integer for Z\Bbb{Z}Z classes, or zero for Z2\Bbb{Z}_2Z2​ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric non-linear sigma-models. In Z2\Bbb{Z}_2Z2​ classes we uncover a hidden supersymmetry, present at the quantum critical point.
Topological Floquet Anderson Insulators (TFAIs) are systems in which an interplay of topology, driving, and static disorder stabilizes a unique form of topological quantum matter. Here, the authors present a first-principles theory, describing the topological phases and quantum criticality of the two-dimensional TFAIs in symmetry class A. Within this framework, they demonstrate how disorder is key to the formation of topological quantization, and that its critical behavior is in the quantum Hall universality class.
We study quantum phase transitions of three-dimensional disordered systems in the chiral classes (AIII and BDI) with and without weak topological indices. We show that the systems with a nontrivial weak topological index universally exhibit an emergent thermodynamic phase where wave functions are delocalized along one spatial direction but exponentially localized in the other two spatial directions, which we call the quasilocalized phase. Our extensive numerical study clarifies that the critical exponent of the Anderson transition between the metallic and quasilocalized phases, as well as that between the quasilocalized and localized phases, are different from that with no weak topological index, signaling the new universality classes induced by topology. The quasilocalized phase and concomitant topological Anderson transition manifest themselves in the anisotropic transport phenomena of disordered weak topological insulators and nodal-line semimetals, which exhibit the metallic behavior in one direction but the insulating behavior in the other directions.
In the absence of spin-orbit coupling, the conventional dogma of Anderson localization asserts that all states localize in two dimensions, with a glaring exception: the quantum Hall plateau transition (QHPT).In that case, the localization length diverges and interference-induced quantum-critical spatial fluctuations appear at all length scales.Normally, QHPT states occur only at isolated energies; accessing them therefore requires fine-tuning of the electron density or magnetic field.In this paper we show that QHPT states can be realized throughout an energy continuum, i.e., as an "energy stack" of critical states wherein each state in the stack exhibits QHPT phenomenology.The stacking occurs without fine-tuning at the surface of a class AIII topological phase, where it is protected by U(1) and (anomalous) chiral or time-reversal symmetries.Spectrum-wide criticality is diagnosed by comparing numerics to universal results for the longitudinal Landauer conductance and wave function multifractality at the QHPT.Results are obtained from an effective 2D surface field theory and from a bulk 3D lattice model.We demonstrate that the stacking of quantum-critical QHPT states is a robust phenomenon that occurs for AIII topological phases with both odd and even winding numbers.The latter conclusion may have important implications for the still poorly understood logarithmic conformal field theory believed to describe the QHPT.
Topological Anderson transitions, which are direct phase transitions between topologically distinct Anderson localized phases, allow for criticality in one-dimensional disordered systems. We analyze the statistical properties of an ensemble of critical wave functions at such transitions. We find that the local moments are strongly inhomogeneous, with significant amplification towards the edges of the system. In particular, we obtain an analytic expression for the spatial profile of the local moments, which is valid at all topological Anderson transitions in one dimension, as we verify by direct comparison with numerical simulations of various lattice models.