000078217 001__ 78217
000078217 005__ 20190314124411.0
000078217 0247_ $$2doi$$a10.22331/q-2018-09-24-95
000078217 0248_ $$2sideral$$a110744
000078217 037__ $$aART-2018-110744
000078217 041__ $$aeng
000078217 100__ $$aCedzich, C.
000078217 245__ $$aComplete homotopy invariants for translation invariant symmetric quantum walks on a chain
000078217 260__ $$c2018
000078217 5060_ $$aAccess copy available to the general public$$fUnrestricted
000078217 5203_ $$aWe provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv: 1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if walks are considered equivalent when they differ only by adding a trivial walk, i.e., one that allows no jumps between cells, then the classification collapses also to the general one. The indices of the general classification can be computed in practice only for walks closely related to some translation invariant ones. We prove a completed collection of simple formulas in terms of winding numbers of band structures covering all symmetry types. Furthermore, we determine the strength of the locality conditions, and show that the continuity of the band structure, which is a minimal requirement for topological classifications in terms of winding numbers to make sense, implies the compactness of the commutator of the walk with a half-space projection, a condition which was also the basis of the general theory. In order to apply the theory to the joining of large but finite bulk pieces, one needs to determine the asymptotic behaviour of a stationary Schrodinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.
000078217 536__ $$9info:eu-repo/grantAgreement/ES/DGA/E64$$9info:eu-repo/grantAgreement/ES/MINECO-FEDER/MTM2014-53963-P
000078217 540__ $$9info:eu-repo/semantics/openAccess$$aAll rights reserved$$uhttp://www.europeana.eu/rights/rr-f/
000078217 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/acceptedVersion
000078217 700__ $$aGeib, T.
000078217 700__ $$aStahl, C.
000078217 700__ $$0(orcid)0000-0002-3050-9540$$aVelazquez, L.$$uUniversidad de Zaragoza
000078217 700__ $$aWerner, A.H.
000078217 700__ $$aWerner, R.F.
000078217 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada
000078217 773__ $$g2 (2018), [33 pp]$$pQuantum$$tQuantum$$x2521-327X
000078217 8564_ $$s795079$$uhttps://zaguan.unizar.es/record/78217/files/texto_completo.pdf$$yPostprint
000078217 8564_ $$s87310$$uhttps://zaguan.unizar.es/record/78217/files/texto_completo.jpg?subformat=icon$$xicon$$yPostprint
000078217 909CO $$ooai:zaguan.unizar.es:78217$$particulos$$pdriver
000078217 951__ $$a2019-03-14-12:01:06
000078217 980__ $$aARTICLE