TY - JOUR

T1 - Sensitivity of quantum walks to a boundary of two-dimensional lattices

T2 - Approaches based on the CGMV method and topological phases

AU - Endo, Takako

AU - Konno, Norio

AU - Obuse, Hideaki

AU - Segawa, Etsuo

N1 - Funding Information:
NK were supported by the Grant-in-Aid for Scientific Research Challenging Exploratory Research (JSPS KAKENHI No. JP15K13443). ES acknowledges financial support from the Grant-in-Aid for Young Scientists (B) and of Scientific Research (B) Japan Society for the Promotion of Science (Grant No. 16K17637, No. 16K03939). HO was supported by a Grant-in-Aid for Scientific Research on Innovative Areas ‘Topological Materials Science’ (JSPS KAKENHI No. JP16H00975) and also JSPS KAKENHI (No. JP16K17760 and No. JP16K05466). Finally, authors would like to thank reviewers for providing invaluable suggestions to this paper.

PY - 2017/10/10

Y1 - 2017/10/10

N2 - In this paper, we treat quantum walks in a two-dimensional lattice with cutting edges along a straight boundary introduced by Asboth and Edge (2015 Phys. Rev. A 91 022324) in order to study one-dimensional edge states originating from topological phases of matter and to obtain collateral evidence of how a quantum walker reacts to the boundary. Firstly, we connect this model to the CMV matrix, which provides a 5-term recursion relation of the Laurent polynomial associated with spectral measure on the unit circle. Secondly, we explicitly derive the spectra of bulk and edge states of the quantum walk with the boundary using spectral analysis of the CMV matrix. Thirdly, while topological numbers of the model studied so far are well-defined only when gaps in the bulk spectrum exist, we find a new topological number defined only when there are no gaps in the bulk spectrum. We confirm that the existence of the spectrum for edge states derived from the CMV matrix is consistent with the prediction from a bulk-edge correspondence using topological numbers calculated in the cases where gaps in the bulk spectrum do or do not exist. Finally, we show how the edge states contribute to the asymptotic behavior of the quantum walk through limit theorems of the finding probability. Conversely, we also propose a differential equation using this limit distribution whose solution is the underlying edge state.

AB - In this paper, we treat quantum walks in a two-dimensional lattice with cutting edges along a straight boundary introduced by Asboth and Edge (2015 Phys. Rev. A 91 022324) in order to study one-dimensional edge states originating from topological phases of matter and to obtain collateral evidence of how a quantum walker reacts to the boundary. Firstly, we connect this model to the CMV matrix, which provides a 5-term recursion relation of the Laurent polynomial associated with spectral measure on the unit circle. Secondly, we explicitly derive the spectra of bulk and edge states of the quantum walk with the boundary using spectral analysis of the CMV matrix. Thirdly, while topological numbers of the model studied so far are well-defined only when gaps in the bulk spectrum exist, we find a new topological number defined only when there are no gaps in the bulk spectrum. We confirm that the existence of the spectrum for edge states derived from the CMV matrix is consistent with the prediction from a bulk-edge correspondence using topological numbers calculated in the cases where gaps in the bulk spectrum do or do not exist. Finally, we show how the edge states contribute to the asymptotic behavior of the quantum walk through limit theorems of the finding probability. Conversely, we also propose a differential equation using this limit distribution whose solution is the underlying edge state.

KW - CMV matrix

KW - quantum walks

KW - topological phase

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U2 - 10.1088/1751-8121/aa8c5e

DO - 10.1088/1751-8121/aa8c5e

M3 - Article

AN - SCOPUS:85032225941

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 45

M1 - 455302

ER -