A Constructive Approach to Topological Invariants for One-dimensional Strictly Local Operators
arXiv:2010.14466 · doi:10.1016/j.jmaa.2021.125072
Abstract
In this paper we shall focus on one-dimensional strictly local operators, the notion of which naturally arises in the context of discrete-time quantum walks on the one-dimensional integer lattice. In particular, we give an elementary constructive approach to the following two topological invariants associated with such operators: Fredholm index and essential spectrum. As a direct application, we shall explicitly compute and fully classify these topological invariants for a well-known quantum walk model.
References in corpus (22)
- Observation of topologically protected bound states in a one dimensional photonic system
- Exploring Topological Phases With Quantum Walks
- From quantum cellular automata to quantum lattice gases
- Localization of Two-Dimensional Quantum Walks
- Relativistic quantum walks
- The topological classification of one-dimensional symmetric quantum walks
- Free Dirac evolution as a quantum random walk
- Quantum lattice gases and their invariants
- Quantum random walks in one dimension
- Quantum walks with an anisotropic coin I: spectral theory
- Spectral Stability of Unitary Network Models
- Quantum walks with an anisotropic coin II: scattering theory
- Relativistic effects in quantum walks: Klein's paradox and Zitterbewegung
- Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
- Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations
- Generalized eigenfunctions and scattering matrices for position-dependent quantum walks
- Continuous limits of linear and nonlinear quantum walks
- Weak limit theorem for a one-dimensional split-step quantum walk
- The Witten Index for 1D Supersymmetric Quantum Walks with Anisotropic Coins
- Absence of wave operators for one-dimensional quantum walks
- Quantum time delay for unitary operators: general theory
- Localization of quantum walks induced by recurrence properties of random walks
Cited by in corpus (5)
- The bulk-edge correspondence for the split-step quantum walk on the one-dimensional integer lattice
- Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
- The Witten index for one-dimensional split-step quantum walks under the non-Fredholm condition
- Spectral analysis of three-state quantum walks with general coin matrices
- The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution