The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution

Abstract

Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a pair of a unitary time-evolution $U$ and a ${\mathbb{Z}}_2$-grading operator $\Gamma$ satisfying the chiral symmetry condition $U^{\ast }=\Gamma U\Gamma .$ In this paper, this index theory will be extended to encompass non-unitary $U$. The existing literature for unitary $U$ makes use of the indispensable assumption that $U$ is essentially gapped; that is, we require that the essential spectrum of $U$ contains neither $-1$ nor $+1$ to define the associated index. It turns out that this assumption is no longer necessary, if the given time-evolution $U$ is non-unitary. As a concrete example, we shall consider a well-known non-unitary quantum walk model on the one-dimensional integer lattice, introduced by Mochizuki-Kim-Obuse.

References in corpus (26)

• Environment-Assisted Quantum Walks in Photosynthetic Energy Transfer
• Quantum walks of correlated particles
• Observation of topologically protected bound states in a one dimensional photonic system
• Exploring Topological Phases With Quantum Walks
• Bulk--Boundary Correspondence for Chiral Symmetric Quantum Walks
• Weak limits for quantum random walks
• Localization of Two-Dimensional Quantum Walks
• Topological phases and delocalization of quantum walks in random environments
• Explicit definition of $\mathcal{P}\mathcal{T}$ symmetry for non-unitary quantum walks with gain and loss
• The topological classification of one-dimensional symmetric quantum walks
• Bulk-edge correspondence of one-dimensional quantum walks
• Edge-state enhanced transport in a 2-dimensional quantum walk
• Staggered Quantum Walks on Graphs
• Quantum walks with an anisotropic coin I: spectral theory
• Quantum walks with an anisotropic coin II: scattering theory
• 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
• Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry
• 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
• A Constructive Approach to Topological Invariants for One-dimensional Strictly Local Operators
• Chiral Floquet systems and quantum walks at half period
• Quantum time delay for unitary operators: general theory
• Parameterization of quantum walks on cycles