Complete homotopy invariants for translation invariant symmetric quantum walks on a chain

C. Cedzich; T. Geib; C. Stahl; L. Velázquez; A. H. Werner; R. F. Werner

doi:10.22331/q-2018-09-24-95

Complete homotopy invariants for translation invariant symmetric quantum walks on a chain

C. Cedzich^1,2, T. Geib¹, C. Stahl¹, L. Velázquez³, A. H. Werner^4,5, and R. F. Werner¹

¹Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany
²Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany
³Departamento de Matemática Aplicada & IUMA, Universidad de Zaragoza, María de Luna 3, 50018 Zaragoza, Spain
⁴QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark,
⁵NBIA, Niels Bohr Institute, University of Copenhagen, Denmark

Published:	2018-09-24, volume 2, page 95
Eprint:	arXiv:1804.04520v2
Doi:	https://doi.org/10.22331/q-2018-09-24-95
Citation:	Quantum 2, 95 (2018).

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Abstract

We 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 Schrödinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.

► BibTeX data

@article{Cedzich2018completehomotopy,
  doi = {10.22331/q-2018-09-24-95},
  url = {https://doi.org/10.22331/q-2018-09-24-95},
  title = {Complete homotopy invariants for translation invariant symmetric quantum walks on a chain},
  author = {Cedzich, C. and Geib, T. and Stahl, C. and Vel{\'{a}}zquez, L. and Werner, A. H. and Werner, R. F.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {95},
  month = sep,
  year = {2018}
}

► References

[1] A. Ahlbrecht, A. Alberti, D. Meschede, V. B. Scholz, A. H. Werner, and R. F. Werner. Molecular binding in interacting quantum walks. New J. Phys., 14:073050, 2012. 10.15488/1301, arXiv:1105.1051.
https://doi.org/10.15488/1301
arXiv:1105.1051

[2] A. Ahlbrecht, H. Vogts, A. H. Werner, and R. F. Werner. Asymptotic evolution of quantum walks with random coin. J. Math. Phys., 52:042201, 2011. 10.1063/1.3575568, arXiv:1009.2019.
https://doi.org/10.1063/1.3575568
arXiv:1009.2019

[3] A. Alberti, W. Alt, R. Werner, and D. Meschede. Decoherence models for discrete-time quantum walks and their application to neutral atom experiments. New J. Phys., 16(12):123052, 2014. 10.1088/1367-2630/16/12/123052, arXiv:1409.6145.
https://doi.org/10.1088/1367-2630/16/12/123052
arXiv:1409.6145

[4] A. Altland and M. R. Zirnbauer. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B, 55:1142–1161, 1997. 10.1103/PhysRevB.55.1142, arXiv:cond-mat/9602137.
https://doi.org/10.1103/PhysRevB.55.1142
arXiv:cond-mat/9602137

[5] A. Arvanitogeorgos. An introduction to Lie groups and the geometry of homogeneous spaces, volume 22. Amer. Math. Soc., 2003. 10.1090/stml/022.
https://doi.org/10.1090/stml/022

[6] J. K. Asbóth. Symmetries, topological phases, and bound states in the one-dimensional quantum walk. Phys. Rev. B, 86:195414, 2012. 10.1103/PhysRevB.86.195414, arXiv:1208.2143.
https://doi.org/10.1103/PhysRevB.86.195414
arXiv:1208.2143

[7] J. K. Asbóth, C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner. Nongentle perturbations and the topological classification of one-dimensional quantum walks. In preparation.

[8] J. K. Asbóth and H. Obuse. Bulk-boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B, 88:121406, 2013. 10.1103/PhysRevB.88.121406, arXiv:1303.1199.
https://doi.org/10.1103/PhysRevB.88.121406
arXiv:1303.1199

[9] J. K. Asbóth, B. Tarasinski, and P. Delplace. Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems. Phys. Rev. B, 90:125143, 2014. 10.1103/PhysRevA.89.042327, arXiv:1401.2673.
https://doi.org/10.1103/PhysRevA.89.042327
arXiv:1401.2673

[10] J. E. Avron and L. Sadun. Fredholm indices and the phase diagram of quantum hall systems. J. Math. Phys., 42(1):1, 2001. 10.1063/1.1331317, arXiv:math-ph/0008040.
https://doi.org/10.1063/1.1331317
arXiv:math-ph/0008040

[11] C. Cedzich, T. Geib, F. A. Grünbaum, C. Stahl, L. Velázquez, A. H. Werner, and R. F. Werner. The topological classification of one-dimensional symmetric quantum walks. Ann. Inst, Poincaré A, 19(2):325–383, 2016. 10.1007/s00023-017-0630-x, arXiv:1611.04439.
https://doi.org/10.1007/s00023-017-0630-x
arXiv:1611.04439

[12] C. Cedzich, T. Geib, C. Stahl, and R. F. Werner. Involutive symmetries for unitary operators: The 43-fold way. In preparation.

[13] C. Cedzich, F. A. Grünbaum, L. Velázquez, A. H. Werner, and R. F. Werner. A quantum dynamical approach to matrix Khrushchev's formulas. Commun. Pure Appl. Math., 69(5):909–957. 10.1002/cpa.21579, arXiv:1405.0985.
https://doi.org/10.1002/cpa.21579
arXiv:1405.0985

[14] C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner. Propagation of quantum walks in electric fields. Phys. Rev. Lett., 111:160601, 2013. 10.1103/PhysRevLett.111.160601, arXiv:1302.2081.
https://doi.org/10.1103/PhysRevLett.111.160601
arXiv:1302.2081

[15] X. Chen, Z.-C. Gu, and X.-G. Wen. Classification of gapped symmetric phases in one-dimensional spin systems. Phys. Rev. B, 83:035107, 2011. 10.1103/PhysRevB.83.035107, arXiv:1008.3745.
https://doi.org/10.1103/PhysRevB.83.035107
arXiv:1008.3745

[16] R. G. Douglas. Toeplitz and Wiener-Hopf operators in ${H}^\infty +{C}$. Bull. Amer. Math. Soc., 74(5):895–899, 09 1968. 10.1090/S0002-9904-1968-12071-3.
https://doi.org/10.1090/S0002-9904-1968-12071-3

[17] H. Dym and M. P. McKean. Fourier Series and Integrals. Academic Press, San Diego, 1972.

[18] E. E. Ewert and R. Meyer. Coarse geometry and topological phases. 2018. arXiv:1802.05579.
arXiv:1802.05579

[19] J. Garnett. Bounded analytic functions, volume 236. Springer Science & Business Media, 2007. 10.1007/0-387-49763-3.
https://doi.org/10.1007/0-387-49763-3

[20] I. Gohberg, S. Goldberg, and M. A. Kaashoeck. Classes of Linear Operators, vol. I and II, volume 49/64. Operator Theory: Advances and Applications, 1990/1993. 10.1007/978-3-0348-7509-7.
https://doi.org/10.1007/978-3-0348-7509-7

[21] G. M. Graf and C. Tauber. Bulk–edge correspondence for two-dimensional Floquet topological insulators. 19(3):709–741, 2018. 10.1007/s00023-018-0657-7, arXiv:1707.09212.
https://doi.org/10.1007/s00023-018-0657-7
arXiv:1707.09212

[22] G. Grimmett, S. Janson, and P. F. Scudo. Weak limits for quantum random walks. Phys. Rev. E, 69:026119, 2004. 10.1103/PhysRevE.69.026119, arXiv:quant-ph/0309135.
https://doi.org/10.1103/PhysRevE.69.026119
arXiv:quant-ph/0309135

[23] D. Gross, V. Nesme, H. Vogts, and R. Werner. Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys., 310:419–454, 2012. 10.1007/s00220-012-1423-1, arXiv:0910.3675.
https://doi.org/10.1007/s00220-012-1423-1
arXiv:0910.3675

[24] P. Hartman. On completely continuous Hankel matrices. Proc. Am. Math. Soc., 9:862–866, 1958. 10.1090/S0002-9939-1958-0108684-8.
https://doi.org/10.1090/S0002-9939-1958-0108684-8

[25] M. Hasan and C. L. Kane. Colloquium : Topological insulators. Rev. Mod. Phys., 82:3045–3067, 2010. 10.1103/RevModPhys.82.3045, arXiv:1002.3895.
https://doi.org/10.1103/RevModPhys.82.3045
arXiv:1002.3895

[26] M. Hastings. Classifying quantum phases with the Kirby torus trick. Phys. Rev. B, 88(16):165114, 2013. 10.1103/PhysRevB.88.165114, arXiv:1305.6625.
https://doi.org/10.1103/PhysRevB.88.165114
arXiv:1305.6625

[27] P. Heinzner, A. Huckleberry, and M. R. Zirnbauer. Symmetry classes of disordered fermions. Commun. Math. Phys., 257(3):725–771, 2005. 10.1007/s00220-005-1330-9, arXiv:math-ph/0411040.
https://doi.org/10.1007/s00220-005-1330-9
arXiv:math-ph/0411040

[28] A. Joye. Dynamical localization for d-dimensional random quantum walks. Quant. Inf. Process., 11:1251–1269, 2012. 10.1007/s11128-012-0406-7, arXiv:1201.4759.
https://doi.org/10.1007/s11128-012-0406-7
arXiv:1201.4759

[29] C. L. Kane and E. J. Mele. Quantum spin Hall effect in graphene. Phys. Rev. Lett., 95:226801, 2005. 10.1103/PhysRevLett.95.226801, arXiv:cond-mat/0411737.
https://doi.org/10.1103/PhysRevLett.95.226801
arXiv:cond-mat/0411737

[30] C. L. Kane and E. J. Mele. ${Z}_{2}$ topological order and the quantum spin Hall effect. Phys. Rev. Lett., 95:146802, 2005. 10.1103/PhysRevLett.95.146802, arXiv:cond-mat/0506581.
https://doi.org/10.1103/PhysRevLett.95.146802
arXiv:cond-mat/0506581

[31] A. Kitaev. Periodic table for topological insulators and superconductors. AIP Conference Proceedings, 1134:22–30, 2009. 10.1063/1.3149495, arXiv:0901.2686.
https://doi.org/10.1063/1.3149495
arXiv:0901.2686

[32] A. Kitaev and C. Laumann. Topological phases and quantum computation. Les Houches Summer School: Exact methods in low-dimensional physics and quantum computing, 89:101, 2009. arXiv:0904.2771.
arXiv:0904.2771

[33] A. Y. Kitaev. Unpaired majorana fermions in quantum wires. Phys. Usp., 44(10S):131, 2001. 10.1070/1063-7869/44/10S/S29, arXiv:cond-mat/0010440.
https://doi.org/10.1070/1063-7869/44/10S/S29
arXiv:cond-mat/0010440

[34] T. Kitagawa. Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quant. Inf. Process., 11:1107–1148, 2012. 10.1007/s11128-012-0425-4, arXiv:1112.1882.
https://doi.org/10.1007/s11128-012-0425-4
arXiv:1112.1882

[35] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White. Observation of topologically protected bound states in photonic quantum walks. Nature Comm., 3:882, 2012. 10.1038/ncomms1872, arXiv:1105.5334.
https://doi.org/10.1038/ncomms1872
arXiv:1105.5334

[36] T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler. Exploring topological phases with quantum walks. Phys. Rev. A, 82:033429, 2010. 10.1103/PhysRevA.82.033429, arXiv:1003.1729.
https://doi.org/10.1103/PhysRevA.82.033429
arXiv:1003.1729

[37] Y. Kubota. Controlled topological phases and bulk-edge correspondence. Commun. Math. Phys., 349(2):493–525, Jan 2017. 10.1007/s00220-016-2699-3, arXiv:1511.05314.
https://doi.org/10.1007/s00220-016-2699-3
arXiv:1511.05314

[38] X. Liu, F. Harper, and R. Roy. Chiral flow in one-dimensional Floquet topological insulators. 2018. arXiv:1806.00026.
arXiv:1806.00026

[39] H. Obuse, J. K. Asbóth, Y. Nishimura, and N. Kawakami. Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk. Phys. Rev. B, 92(4):045424, 2015. 10.1103/PhysRevB.92.045424, arXiv:1505.03264.
https://doi.org/10.1103/PhysRevB.92.045424
arXiv:1505.03264

[40] V. Peller. Hankel Operators and Their Applications. Springer Monographs in Mathematics. Springer New York, 2003. 10.1007/978-0-387-21681-2.
https://doi.org/10.1007/978-0-387-21681-2

[41] E. Prodan. Disordered topological insulators: a non-commutative geometry perspective. J. Phys. A-Math. Theor., 44(11):113001, 2011. 10.1088/1751-8113/44/11/113001, arXiv:1010.0595.
https://doi.org/10.1088/1751-8113/44/11/113001
arXiv:1010.0595

[42] E. Prodan and H. Schulz-Baldes. Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics. Springer, 2016. 10.1007/978-3-319-29351-6, arXiv:1510.08744.
https://doi.org/10.1007/978-3-319-29351-6
arXiv:1510.08744

[43] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang. Time-reversal-invariant topological superconductors and superfluids in two and three dimensions. Phys. Rev. Lett., 102:187001, 2009. 10.1103/PhysRevLett.102.187001, arXiv:0803.3614.
https://doi.org/10.1103/PhysRevLett.102.187001
arXiv:0803.3614

[44] X.-L. Qi, T. L. Hughes, and S.-C. Zhang. Topological invariants for the Fermi surface of a time-reversal-invariant superconductor. Phys. Rev. B, 81(13):134508, 2010. 10.1103/PhysRevB.81.134508, arXiv:0908.3550.
https://doi.org/10.1103/PhysRevB.81.134508
arXiv:0908.3550

[45] X.-L. Qi and S.-C. Zhang. Topological insulators and superconductors. Rev. Mod. Phys., 83(4):1057, 2011. 10.1103/RevModPhys.83.1057, arXiv:1008.2026.
https://doi.org/10.1103/RevModPhys.83.1057
arXiv:1008.2026

[46] T. Rakovszky and J. K. Asboth. Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk. Phys. Rev. A, 92:052311, 2015. 10.1103/PhysRevA.92.052311, arXiv:1505.04513.
https://doi.org/10.1103/PhysRevA.92.052311
arXiv:1505.04513

[47] J. Roe. Index theory, coarse geometry and topology of manifolds. Regional conference series in Mathematics #90. Am. Math. Soc, 1996. 10.1090/cbms/090.
https://doi.org/10.1090/cbms/090

[48] J. Roe. Lectures on coarse geometry. University lecture series #31. Am. Math. Soc, 2003. 10.1090/ulect/031.
https://doi.org/10.1090/ulect/031

[49] M. Rosenblum. On the Hilbert matrix, II. Proc. Am. Math. Soc., 9:581–585, 1958. 10.1090/S0002-9939-1958-0099599-2.
https://doi.org/10.1090/S0002-9939-1958-0099599-2

[50] J. Rotman. An Introduction to Algebraic Topology. Graduate Texts in Mathematics. Springer New York, 1998. 10.1007/978-1-4612-4576-6.
https://doi.org/10.1007/978-1-4612-4576-6

[51] R. Roy. Topological superfluids with time reversal symmetry. 2008. arXiv:0803.2868.
arXiv:0803.2868

[52] R. Roy and F. Harper. Periodic Table for Floquet Topological Insulators. Phys. Rev. B, 96:155118, Oct 2017. 10.1103/PhysRevB.96.155118, arXiv:1603.06944.
https://doi.org/10.1103/PhysRevB.96.155118
arXiv:1603.06944

[53] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X, 3:031005, 2013. 10.1103/PhysRevX.3.031005, arXiv:1212.3324.
https://doi.org/10.1103/PhysRevX.3.031005
arXiv:1212.3324

[54] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. of Phys., 12(6):065010, 2010. 10.1088/1367-2630/12/6/065010, arXiv:0912.2157.
https://doi.org/10.1088/1367-2630/12/6/065010
arXiv:0912.2157

[55] C. Sadel and H. Schulz-Baldes. Topological boundary invariants for Floquet systems and quantum walks. Math. Phys. Ana. Geom., 20(4):22, 2017. 10.1007/s11040-017-9253-1, arXiv:1708.01173.
https://doi.org/10.1007/s11040-017-9253-1
arXiv:1708.01173

[56] D. Sarason. Algebras of functions on the unit circle. Bull. Amer. Math. Soc., 79(2):286–299, 1973. 10.1090/S0002-9904-1973-13144-1.
https://doi.org/10.1090/S0002-9904-1973-13144-1

[57] A. Schnyder, S. Ryu, A. Furusaki, and A. Ludwig. Classification of topological insulators and superconductors. AIP Conference Proceedings, 1134:10–21, 2009. 10.1063/1.3149481, arXiv:0905.2029.
https://doi.org/10.1063/1.3149481
arXiv:0905.2029

[58] H. Schulz-Baldes. $\mathbb{Z}_2$-indices and factorization properties of odd symmetric Fredholm operators. Doc. Math., 20:1481–1500, 2015. arXiv:1311.0379.
arXiv:1311.0379

[59] C. Stahl. Interactive tool at https://qig.itp.uni-hannover.de/bulkedge/sse.
https://qig.itp.uni-hannover.de/bulkedge/sse

[60] N. Steenrod. The Topology of Fibre Bundles. Princeton Mathematical series. University Press, 1951.

[61] M. H. Stone. The generalized Weierstrass approximation theorem. Mathematics Magazine, 21(4):167–184, 1948. 10.2307/3029750.
https://doi.org/10.2307/3029750

[62] B. Tarasinski, J. K. Asbóth, and J. P. Dahlhaus. Scattering theory of topological phases in discrete-time quantum walks. Phys. Rev. A, 89:042327, 2014. 10.1103/PhysRevA.89.042327, arXiv:1401.2673.
https://doi.org/10.1103/PhysRevA.89.042327
arXiv:1401.2673

[63] K. Weierstrass. Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 2:633–639, 1885.

[64] E. P. Wigner. Normal form of antiunitary operators. J. Math. Phys., 1:409–413, 1960. 10.1007/978-3-662-02781-3_38.
https://doi.org/10.1007/978-3-662-02781-3_38

[65] M. R. Zirnbauer. Symmetry classes. 2010. arXiv:1001.0722.
arXiv:1001.0722

[66] B. Zumino. Normal forms of complex matrices. J. Math. Phys., 3(5):1055–1057, 1962. 10.1063/1.1724294.
https://doi.org/10.1063/1.1724294

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