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Physical review. B./Physical review. B

Four-dimensional topological Anderson insulator with an emergent second Chern number

Rui Chen, Xiao-Xia Yi, Bin Zhou

Published Aug 29, 2023• 15 citations
Publisher

Abstract

Four-dimensional (4D) topological insulators, which are impossible in real materials, have attracted much attention by virtue of the recent progress achieved in quantum simulations of higher-dimensional systems. In this paper, we employ the supercell approximation to investigate the disorder effects on a system that supports the 4D topological insulator phases characterized by quantized second Chern numbers and the normal insulator phase. We demonstrate that the 4D topological insulator phases are robust against weak disorders. Moreover, we reveal that disorder can transform a normal insulator to a 4D topological insulator with an emergent second Chern number, referred to as a 4D topological Anderson insulator. An effective-medium theory based on the Born approximation further confirms the numerical conclusions.

DOI
10.1103/physrevb.108.085306

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Four-dimensional (4D) topological insulators, which are impossible in real materials, have attracted much attention by virtue of the recent progress achieved in quantum simulations of higher-dimensional systems. In this paper, we employ the supercell approximation to investigate the disorder effects on a system that supports the 4D topological insulator phases characterized by quantized second Chern numbers and the normal insulator phase. We demonstrate that the 4D topological insulator phases are robust against weak disorders. Moreover, we reveal that disorder can transform a normal insulator to a 4D topological insulator with an emergent second Chern number, referred to as a 4D topological Anderson insulator. An effective-medium theory based on the Born approximation further confirms the numerical conclusions.
While the effects of off-diagonal disorder have been extensively studied in low-dimensional topological systems, such research remains insufficient in four-dimensional (4D) topological systems. In this work, we investigate the effect of off-diagonal disorders on a 4D topological insulator. We find that off-diagonal disorder can induce a 4D topological Anderson insulator (TAI) phase. This differs from previous finding in two-dimensional topological insulators, where off-diagonal disorder fails to induce the TAI phase. Specifically, we consider 16 different configurations of disorder and identify two distinct types based on their effect on the system. Furthermore, using the self-consistent Born approximation, we elucidate the unique role of off-diagonal disorder in these phase transitions compared to the diagonal Anderson disorder.
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 Topological insulators are exotic material that possess conducting surface states protected by the topology of the system. They can be classified in terms of their properties under discrete symmetries and are characterized by topological invariants. The latter has been measured experimentally for several models in one, two and three dimensions in both condensed matter and quantum simulation platforms. The recent progress in quantum simulation opens the road to the simulation of higher dimensional Hamiltonians and in particular of the 4D quantum Hall effect. These systems are characterized by the second Chern number, a topological invariant that appears in the quantization of the transverse conductivity for the non-linear response to both external magnetic and electric fields. This quantity cannot always be computed analytically and there is therefore a need of an algorithm to compute it numerically. In this work, we propose an efficient algorithm to compute the second Chern number in 4D systems. We construct the algorithm with the help of lattice gauge theory and discuss the convergence to the continuous gauge theory. We benchmark the algorithm on several relevant models, including the 4D Dirac Hamiltonian and the 4D quantum Hall effect and verify numerically its rapid convergence.
We study the two-dimensional disordered topological superconductor with Hubbard interactions. When the magnitude of the pairing potential is tuned to special values, this interacting model is exactly solvable even when disorders are imposed on the potential term or coupling constants. The topology of this model is investigated in detail by the real space Chern number formula, which computes the topological index of disordered systems to high precisions. It is found that the disorders can drive the system from topological trivial phase to a non-trivial phase, which generalizes the topological Anderson phenomena to interacting models. The self-consistent Born approximation is also employed to understand the influence of the disorders on the parameters of the interacting topological superconductor. It provide an alternative way to understand the topological transitions at weak disordered region.
We investigate disorder-driven transitions between trivial and topological insulator (TI) phases in two-dimensional (2D) systems. Our study primarily focuses on the Bernevig-Hughes-Zhang (BHZ) model with Anderson disorder, while other standard 2D TI models exhibit equivalent features. The analysis is based on the local Chern marker (LCM), a local quantity that allows for the characterization of topological transitions in finite and disordered systems. Our simulations indicate that disorder-driven trivial to topological insulator transitions are nicely characterized by C0{\mathcal{C}}_{0}C0​, the disorder-averaged LCM near the central cell of the system. We show that C0{\mathcal{C}}_{0}C0​ is characterized by a single-parameter scaling, namely, ${\mathcal{C}}_{0}(M,W,L)\ensuremath{\equiv}{\mathcal{C}}_{0}(z)$, with $z=[{W}^{\ensuremath{\mu}}\ensuremath{-}{W}_{c}^{\ensuremath{\mu}}(M)]L$, where MMM is the Dirac mass, WWW is the disorder strength, and LLL is the system size, while ${W}_{c}(M)\ensuremath{\propto}\sqrt{M}$ and $\ensuremath{\mu}\ensuremath{\approx}2$ stand for the critical disorder strength and the critical exponent, respectively. Our numerical results are in agreement with a theoretical prediction based on a first-order Born approximation analysis. These observations lead us to speculate that the universal scaling function we have found is rather general for amorphous and disorder-driven topological phase transitions.
Topological states of matter in disordered systems without translation symmetry have attracted great interest in recent years. These states with topological characters are not only robust against certain disorders, but also can be counterintuitively induced by disorders from a topologically trivial phase in the clean limit. In this review, we summarize the current theoretical and experimental progress on disorder-induced topological phases in both condensed-matter and artificial systems. We first introduce the topological Anderson insulators (TAIs) induced by random disorders and their topological characterizations and experimental realizations. We then discuss various extensions of TAIs with unique localization phenomena in quasiperiodic and non-Hermitian systems. We also review the theoretical and experimental studies on the disorder-induced topology in dynamical and many-body systems, including topological Anderson-Thouless pumps, disordered correlated topological insulators and average-symmetry protected topological orders acting as interacting TAI phases. Finally, we conclude the review by highlighting potential directions for future explorations.