# A Guide to the Bott Index and Localizer Index

@article{Loring2019AGT, title={A Guide to the Bott Index and Localizer Index}, author={Terry A. Loring}, journal={arXiv: Mathematical Physics}, year={2019} }

The Bott index is inherently global. The pseudospectal index is inherently local, and so now the preferred name is the localizer index. We look at these on a rather standard model for a Chern insulator, with an emphasis how to program these effectively. We also discuss how to tune the localizer index so it behaves like a global index.

## 13 Citations

### The Bott index of two unitary operators and the integer quantum Hall effect

- Mathematics
- 2021

The Bott index of two unitary operators on an infinite dimensional Hilbert space is defined. Homotopic invariance with respect to multiplicative unitary perturbations of the type identity plus trace…

### On the Bott index of unitary matrices on a finite torus

- MathematicsLetters in Mathematical Physics
- 2022

This article reviews the foundations of the theory of the Bott index of a pair of unitary matrices in the context of condensed matter theory, as developed by Hastings and Loring (J. Math. Phys. 51,…

### Quadratic pseudospectrum for identifying localized states

- MathematicsJournal of Mathematical Physics
- 2023

It is proved that the quadratic pseudospectrum is local, and the bounds on the errors that are incurred by truncating the system in the vicinity of where the pseudospects are being calculated are derived.

### Higher-dimensional Hofstadter butterfly on the Penrose lattice

- PhysicsPhysical Review B
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Quasicrystal is now open to search for novel topological phenomena enhanced by its peculiar structure characterized by an irrational number and high-dimensional primitive vectors. Here we extend the…

### Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter.

- PhysicsPhysical review letters
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Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern…

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It is proved that the wLDOS is local in the sense that it can be computed up to arbitrarily small error using spatial truncations of the system Hamiltonian, and well-deﬁned and computable for in-time systems satisfying some natural assumptions.

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- Physics
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Metals and topological insulators have in common that they cannot be described by exponentially localized wave-functions. Here we establish a relationship between these two seemingly unrelated…

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We consider a model of electrons in a finite topological insulator. We numerically study the propagation of electronic wave-packets localized near edges of the structure in the presence of defects…

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- PhysicsNature communications
- 2023

Topological metals are conducting materials with gapless band structures and nontrivial edge-localized resonances. Their discovery has proven elusive because traditional topological classification…

### Classifying topology in photonic heterostructures with gapless environments

- Physics
- 2023

Photonic topological insulators exhibit bulk-boundary correspondence, which requires that boundary-localized states appear at the interface formed between topologically distinct insulating materials.…

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