TY - JOUR

T1 - Bulk-edge correspondence of one-dimensional quantum walks

AU - Cedzich, Christopher

AU - Grünbaum, F. A.

AU - Stahl, C.

AU - Velázquez, L.

AU - Werner, A. H.

AU - Werner, R. F.

N1 - Funding Information: We thank Tobias J Osborne, Andrea Alberti, and János Asbóth for a critical reading of the manuscript. C Cedzich, C Stahl and R F Werner acknowledge support from the ERC grant DQSIM and the European project SIQS. F A Grünbaum was partially supported by the Applied Math. Sciences subprogram of the Office of Energy Research, USDOE, under Contract DE-AC03-76SF00098, and by AFOSR grant FA95501210087 through a subcontract to Carnegie Mellon University. The work of L Velázquez is partially supported by the research project MTM2011-28952-C02-01 and MTM2014-53963-P from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF), and by Project E-64 of Diputación General de Aragón (Spain). A H Werner acknowledges support from the ERC grant TAQ.

PY - 2016/4/20

Y1 - 2016/4/20

N2 - We outline a theory of symmetry protected topological phases of one-dimensional quantum walks. We assume spectral gaps around the symmetry-distinguished points +1 and -1, in which only discrete eigenvalues are allowed. The phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure. However, our theory requires no translation invariance whatsoever, and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations, even those that cannot be continuously contracted to the identity. More precisely we define two indices for every walk, characterizing the behavior far to the right and far to the left, respectively. Their sum is a lower bound on the number of eigenstates at +1 and -1. For a translation invariant system the indices add up to zero, so one of them already characterizes the phase. By joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel, so the theory predicts bound states at +1 or -1. This is a rigorous statement of bulk-edge correspondence. The results also apply to the Hamiltonian case with a single gap at zero.

AB - We outline a theory of symmetry protected topological phases of one-dimensional quantum walks. We assume spectral gaps around the symmetry-distinguished points +1 and -1, in which only discrete eigenvalues are allowed. The phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure. However, our theory requires no translation invariance whatsoever, and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations, even those that cannot be continuously contracted to the identity. More precisely we define two indices for every walk, characterizing the behavior far to the right and far to the left, respectively. Their sum is a lower bound on the number of eigenstates at +1 and -1. For a translation invariant system the indices add up to zero, so one of them already characterizes the phase. By joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel, so the theory predicts bound states at +1 or -1. This is a rigorous statement of bulk-edge correspondence. The results also apply to the Hamiltonian case with a single gap at zero.

UR - http://www.scopus.com/inward/record.url?scp=84964952427&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/49/21/21LT01

DO - 10.1088/1751-8113/49/21/21LT01

M3 - Article

VL - 49

JO - J. Phys. A

JF - J. Phys. A

IS - 21

M1 - 21LT01

ER -