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arXiv:2402.02310 · Physical Review Letters

Topological Effect on the Anderson Transition in Chiral Symmetry Classes

Pengwei Zhao, Zhenyu Xiao, Yeyang Zhang, Ryuichi Shindou

Published Nov 25, 2024• 11 citations
arXivHTMLPDF

Abstract

In this Letter, we propose a mechanism of an emergent quasilocalized phase in chiral symmetry classes, where the wave function along a spatial direction with weak topology is delocalized but exponentially localized along the other directions. The Anderson transition in two-dimensional chiral symmetry classes is induced by the proliferation of vortex-antivortex pairs of a U(1) phase degree of freedom, while the weak topology endows the pair with a Berry phase. We argue that the Berry phase induces spatial polarizations of the pairs along the topological direction through the quantum interference effect, and the proliferation of the polarized vortex pairs results in the quasilocalized phase.

DOI
10.1103/physrevlett.133.226601

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In this Letter, we propose a mechanism of an emergent quasilocalized phase in chiral symmetry classes, where the wave function along a spatial direction with weak topology is delocalized but exponentially localized along the other directions. The Anderson transition in two-dimensional chiral symmetry classes is induced by the proliferation of vortex-antivortex pairs of a U(1) phase degree of freedom, while the weak topology endows the pair with a Berry phase. We argue that the Berry phase induces spatial polarizations of the pairs along the topological direction through the quantum interference effect, and the proliferation of the polarized vortex pairs results in the quasilocalized phase.
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.
We investigate one-dimensional (1D) discrete-time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with the bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at nonzero energy in their dynamics. In contrast to results of previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust to spatial disorder but not to temporal disorder.
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.
Quantum walks are promising for information processing tasks because in regular graphs they spread quadratically more rapidly than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, with their wave function staying within a finite region even in the infinite time limit, with a probability exponentially close to 1. It is thus important to understand when a quantum walk will be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a one-dimensional quantum walk---the split-step walk---to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization transition occurs, and the walk spreads subdiffusively to infinity. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk and find that the disorder-induced Anderson localization and delocalization transitions are governed by the topological phases of the quantum walk.
Fathoming interplay between symmetry and topology of many-electron wave-functions has deepened understanding of quantum many body systems, especially after the discovery of topological insulators. Topology of electron wave-functions enforces and protects emergent gapless excitations, and symmetry is intrinsically tied to the topological protection in a certain class. Namely, unless the symmetry is broken, the topological nature is intact. We show novel interplay phenomena between symmetry and topology in topological phase transitions associated with line-nodal superconductors. The interplay may induce an exotic universality class in sharp contrast to that of the phenomenological Landau-Ginzburg theory. Hyper-scaling violation and emergent relativistic scaling are main characteristics, and the interplay even induces unusually large quantum critical region. We propose characteristic experimental signatures around the phase transitions in three spatial dimensions, for example, a linear phase boundary in a temperature-tuning parameter phase-diagram.
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.