Jeudi 01 novembre 2012 à 14h15
Salle 234, Ecole de Physique

Topological phases and edge states in 1-dimensional quantum walks

Janos K. Asboth, Wigner Research Centre for Physics, Budapest

A Discrete Time Quantum Walk is a periodically driven lattice system. Taking the logarithm of the time evolution operator for one period (Floquet operator) associates a time-independent effective Hamiltonian with the quantum walk. Thus, the quantum walk simulates the effective Hamiltonian stroboscopically. Kitagawa and coworkers have shown [1] that quantum walks can simulate all classes of topological insulators and superconductors in 1 and 2 spatial dimensions. Quantum Walks have topological features that are not described by the effective Hamiltonian. This is most strikingly illustrated by an example [2]: Two bulks with the same effective Hamiltonian, but different driving cycles, at whose boundary a walker can be trapped in two different, topologically protected states. This trapping is a consequence of the bulk--boundary correspondence, showing that the bulks must have different topology. I show a simple way to define and evaluate the Z2xZ2 invariant characterizing Particle--Hole Symmetric quantum walks. This extends the approach suggested by Jiang et al. [3]. Apart from Particle--Hole Symmetry, there are two other symmetries used for the classification of topological phases of Hamiltonians: Time Reversal Symmetry and Chiral Symmetry. I discuss chiral symmetry for quantum walks, a concept that is not entirely trivial to define, and show that it leads to a ZxZ invariant. To show this, I introduce a generalization of the quantum walk with tunable symmetries that can host more than one topologically protected states at the same edge with the same energy.

[1]: T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler , Phys. Rev. A 82, 033429 (2010)
[2]: J. K. Asboth, http://arxiv.org/abs/1208.2143
[3]: L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Phys. Rev. Lett. 106, 220402 (2011)