On the Bott index of unitary matrices on a finite torus

Toniolo, Daniele

doi:10.48550/arxiv.1708.05912

Cited by 8 publications

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“…We have numerically demonstrated very good agreement between the total transmission and the Bott index, which indicates that the bulk-boundary correspondence holds in the disordered topological magnon system. Our numerical studies pave the way for a rigorous mathematical proof of the equivalence between the bosonic Bott index and Chern number [46], as well as the bulk-boundary correspondence in topological magnonic systems, which are open issues for future research. All the discussions above also apply for other deformed honeycomb lattices, providing that the clean system is gapped [23].…”

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confidence: 96%

“…In electronic systems, the Bott index was introduced to study nonperiodic systems such as disordered topological insulators [38] and quasicrystals [39,40]. The Bott index quantifies the obstruction to construct an orthogonal basis of localized Wannier functions that span the occupied states [37], and it has been proven to be equivalent to the Chern number in the large-system limit [46].…”

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confidence: 99%

“…Γ is a diagonal matrix taking a value of 1 for the j th diagonal elements when j ∈ {ε n }, and 0 otherwise. Note that for fermionic systems, η = 1, and the above definition returns to the electronic Bott index [40,46]. B is always an integer as long as V U V † U † is nonsingular [38,46], and specifically, B = 0 when the matrices U and V commute.…”

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confidence: 99%

“…Note that for fermionic systems, η = 1, and the above definition returns to the electronic Bott index [40,46]. B is always an integer as long as V U V † U † is nonsingular [38,46], and specifically, B = 0 when the matrices U and V commute. For a clean system with well-defined gaps, the Bott index of each band separated by gaps is well defined.…”

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confidence: 99%

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Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

2020

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We investigate the role of disorder on the various topological magnonic phases present in deformed honeycomb ferromagnets. To this end, we introduce a bosonic Bott index to characterize the topology of magnon spectra in finite, disordered systems. The consistency between the Bott index and Chern number is numerically established in the clean limit. We demonstrate that topologically protected magnon edge states are robust to moderate disorder and, as anticipated, localized in the strong regime. We predict a disorder-driven topological phase transition, a magnonic analog of the "topological Anderson insulator" in electronic systems, where the disorder is responsible for the emergence of the nontrivial topology. Combining the results for the Bott index and transport properties, we show that bulk-boundary correspondence holds for disordered topological magnons. Our results open the door for research on topological magnonics as well as other bosonic excitations in finite and disordered systems.

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confidence: 96%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

2020

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“…Again, gapped Chern bands exist in the 2D generalization. The Chern band is characterized by Bott index [42], which is calculated in the real space and equivalent to the Chern number [43,44]. The Bott indices for the gapped bands are shown by the blue numbers in the figure.…”

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confidence: 99%

Topological phases and Anderson localization in off-diagonal mosaic lattices

Zeng,

Lü

2021

Preprint

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We introduce a one-dimensional lattice model whose hopping amplitudes are modulated for equally spaced sites. Such mosaic lattice exhibits many interesting topological and localization phenomena that do not exist in the regular off-diagonal lattices. When the mosaic modulation is commensurate with the underlying lattice, topologically nontrivial phases with zero-and nonzero-energy edge modes are observed as we tune the modulation, where the nontrivial regimes are characterized by quantized Berry phases. If the modulation becomes incommensurate, Anderson localization will emerge. The eigenstate is found to be localized on two neighboring sites connected by the quasiperiodic hopping terms. This is in sharp contrast with the conventional localization process where the state is localized only on a single site, indicating a new mechanism for Anderson localization. Furthermore, both the commensurate and incommensurate off-diagonal mosaic lattices can host Chern insulators in their two-dimensional generalizations. Our work provides a new platform for exploring topological phases and Anderson localization in low-dimensional systems.

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Driven quantum many-body systems and out-of-equilibrium topology

Bandyopadhyay

Bhattacharjee

Sen

2021

J. Phys.: Condens. Matter

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In this review we present some of the work done in India in the area of driven and out-of-equilibrium systems with topological phases. After presenting some well-known examples of topological systems in one and two dimensions, we discuss the effects of periodic driving in some of them. We discuss the unitary as well as the non-unitary dynamical preparation of topologically non-trivial states in one and two dimensional systems. We then discuss the effects of Majorana end modes on transport through a Kitaev chain and a junction of three Kitaev chains. Following this, transport through the surface states of a three-dimensional topological insulator has also been reviewed. The effects of hybridization between the top and bottom surfaces in such systems and the application of electromagnetic radiation on a strip-like region on the top surface are described. Two unusual topological systems are mentioned briefly, namely, a spin system on a kagome lattice and a Josephson junction of three superconducting wires. We have also included a pedagogical discussion on topology and topological invariants in the appendices, where the connection between topological properties and the intrinsic geometry of many-body quantum states is also elucidated.

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On the Bott index of unitary matrices on a finite torus

Cited by 8 publications

References 0 publications

Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

Topological phases and Anderson localization in off-diagonal mosaic lattices

Driven quantum many-body systems and out-of-equilibrium topology

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