Wannier functions and \(\mathbb{Z}_2\) invariants in time-reversal symmetric topological insulators.

*(English)*Zbl 1370.81081The paper gives constructive proofs of the existence of exponentionally localized Wannier functions in gapped periodic quantum systems with a fermionic time-reversal symmetric (TRS) topological insulators in dimension \(d\leq 2\). The paper also discusses the possibility of imposing a natural compatibility condition with the time-reversal operator and the relation of the latter with the \(\mathbb{Z}_2\) invariants. There are proved that analytic and periodic frames can indeed be constructed in dimension \(d\leq 2\) and encountered a topological obstruction in d = 2, when it is required also a TRS properties for the frame to hold. The 2d construction procedure can go through provided the Graf-Porta (GP) \(\mathbb{Z}_2\) index vanishes: the latter is introduced as a “bulk invariant” for 2d TRS topological insulators. The topological obstruction is encoded in the \(\mathbb{Z}_2\) Fiorenza-Monaco-Panati (FMP) invariant \(\delta\). The GP index becomes manifestly a topological invariant of the quantum system, but the topological obstruction \(\delta\) is brought into contact with the actual process of detection of topological phases in real experimental setups in view of the bulk-edge correspondence. These two quantities agree numerically. From its equality with the GP index, it is deduced an expression for invariant \(\delta\), which depends explicitly on the family of projections, to which it is associated via its Berry connection and Berry curvature. As a result, after formulation of main results of the study, it is reformulated the problem of the construction of analytic, periodic and possibly TRS Bloch frames for a \(d\)-dimensional family of projectors. Their properties are modeled after the once of the eigen-projectors of the Hamiltonian of a gapped periodic quantum system with fermionic TRS in terms of an equivalent problem for particular families of unitary matrices \(\alpha\). The latter emerge from the construction of a \(d\)-dimensional frame. The possibility to construct a periodic and TRS frame is then equivalent to the possibility of “rotating” this family \(\alpha\) to the identity matrix by preserving its properties. Then the problem is specialized and solved in dimensions \(d=1\) and 2. The problem is solved by constructing explicitly “good logarithms”. The topological obstruction to the existence of a “good logarithm” in \(d=2\) is encoded in GP index, for which the authors prove several useful properties, including that it characterizes completely the homotopy class of a continuous, periodic and TRS family of unitary matrices. Due to the GP index is a topological obstruction, it allows discussion of several approaches to the formulation of \(\mathbb{Z}_2\) invariants distinguishing the different topological phases in 2d TRS topological insulators. Equivalence between the GP and FMP invariants allows one to show that the \(\mathbb{Z}_2\) invariant can be expressed in geometric terms as a function of the family of projectors.

Reviewer: Ivan A. Parinov (Rostov-na-Donu)

##### MSC:

81Q70 | Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory |

35J10 | Schrödinger operator, Schrödinger equation |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

82D25 | Statistical mechanical studies of crystals |

##### Keywords:

Wannier functions; Bloch frames; fermionic time-reversal symmetry; topological insulators; \(\mathbb{Z}_2\) invariants##### References:

[1] | Hasan, M. Z.; Kane, C. L., Colloquium: topological insulators, Rev. Mod. Phys., 82, 3045-3067, (2010) |

[2] | Fu, L., Topological crystalline insulators, Phys. Rev. Lett., 106, 106802, (2011) |

[3] | Ando, Y., Topological insulator materials, J. Phys. Soc. Jpn., 82, 102001, (2013) |

[4] | Fruchart, M.; Carpentier, D., An introduction to topological insulators, Comptes Rendus Phy., 14, 779-815, (2013) |

[5] | Altland, A.; Zirnbauer, M. R., Non-standard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B, 55, 1142-1161, (1997) |

[6] | Kitaev, A., Periodic table for topological insulators and superconductors, AIP Conf. Proc., 1134, 22-30, (2009) · Zbl 1180.82221 |

[7] | Ryu, S.; Schnyder, A. P.; Furusaki, A.; Ludwig, A. W. W., Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys., 12, 065010, (2010) |

[8] | Thiang, G. C., On the \(K\)-theoretic classification of topological phases of matter, Ann. Henri Poincaré, 17, 757-794, (2016) · Zbl 1344.81144 |

[9] | Prodan, E.; Schulz-Baldes, H., Bulk and Boundary Invariants for Complex Topological Insulators: From \(K\)-Theory to Physics, (2016), Springer, Basel · Zbl 1342.82002 |

[10] | Kennedy, R.; Guggenheim, C., Homotopy theory of strong and weak topological insulators, Phys. Rev. B, 91, 245148, (2015) |

[11] | Kennedy, R.; Zirnbauer, M. R., Bott periodicity for \(\Bbb Z_2\) symmetric ground states of gapped free-fermion systems, Comm. Math. Phys., 342, 3, 909-963, (2016) · Zbl 1346.81159 |

[12] | Großmann, J.; Schulz-Baldes, H., Index pairings in presence of symmetries with applications to topological insulators, Comm. Math. Phys., 343, 477-513, (2016) · Zbl 1348.82083 |

[13] | Prodan, E., Disordered topological insulators: A non-commutative geometry perspective, J. Phys. A, 44, 113001, (2011) · Zbl 1213.82082 |

[14] | Bourne, C.; Carey, A. L.; Rennie, A., A noncommutative framework for topological insulators, Rev. Math. Phys., 28, 1650004, (2016) · Zbl 1364.81269 |

[15] | Kane, C. L.; Mele, E. J., \(\Bbb Z_2\) topological order and the quantum spin Hall effect, Phys. Rev. Lett., 95, 146802, (2005) |

[16] | Fu, L.; Kane, C. L., Time reversal polarization and a \(\Bbb Z_2\) adiabatic spin pump, Phys. Rev. B, 74, 195312, (2006) |

[17] | Bernevig, B. A.; Hughes, T. L.; Zhang, Sh.-Ch., Quantum spin Hall effect and topological phase transition in hgte quantum wells, Science, 15, 1757-1761, (2006) |

[18] | Monaco, D.; Panati, G., Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry, Acta App. Math., 137, 185-203, (2015) · Zbl 1318.82045 |

[19] | Cornean, H. D.; Herbst, I.; Nenciu, G., On the construction of composite Wannier functions, Ann. Henri Poincaré, 17, 3361-3398, (2016) · Zbl 1357.82069 |

[20] | Nenciu, G., Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians, Rev. Mod. Phys., 63, 91-127, (1991) |

[21] | Panati, G., Triviality of Bloch and Bloch-Dirac bundles, Ann. Henri Poincaré, 8, 995-1011, (2007) · Zbl 1375.81102 |

[22] | Fu, L.; Kane, C. L.; Mele, E. J., Topological insulators in three dimensions, Phys. Rev. Lett., 98, 106803, (2007) |

[23] | Avila, J. C.; Schulz-Baldes, H.; Villegas-Blas, C., Topological invariants of edge states for periodic two-dimensional models, Math. Phys. Anal. Geom., 16, 136-170, (2013) · Zbl 1271.81210 |

[24] | Graf, G. M.; Porta, M., Bulk-edge correspondence for two-dimensional topological insulators, Comm. Math. Phys., 324, 851-895, (2013) · Zbl 1291.82120 |

[25] | Schulz-Baldes, H., Persistence of spin edge currents in disordered quantum spin Hall systems, Comm. Math. Phys., 324, 589-600, (2013) · Zbl 1278.82065 |

[26] | De Nittis, G.; Gomi, K., Classification of “quaternionic” Bloch bundles, Comm. Math. Phys., 339, 1-55, (2015) · Zbl 1326.57047 |

[27] | Fiorenza, D.; Monaco, D.; Panati, G., \(\Bbb Z_2\) invariants of topological insulators as geometric obstructions, Comm. Math. Phys., 343, 1115-1157, (2016) · Zbl 1346.81158 |

[28] | Nenciu, A.; Nenciu, G., The existence of generalised Wannier functions for one-dimensional systems, Comm. Math. Phys., 190, 541-548, (1998) · Zbl 0907.34075 |

[29] | Fiorenza, D.; Monaco, D.; Panati, G., Construction of real-valued localized composite Wannier functions for insulators, Ann. Henri Poincaré, 17, 63-97, (2016) · Zbl 1338.82057 |

[30] | È. Cancés, A. Levitt, G. Panati and G. Stoltz, Robust determination of maximally-localized Wannier functions (2016); arXiv:1605.07201. |

[31] | Soluyanov, A. A.; Vanderbilt, D., Smooth gauge for topological insulators, Phys. Rev. B, 85, 115415, (2012) |

[32] | Winkler, G. W.; Soluyanov, A. A.; Troyer, M., Smooth gauge and Wannier functions for topological band structures in arbitrary dimensions, Phys. Rev. B, 93, 035453, (2016) |

[33] | Prodan, E., Manifestly gauge-independent formulations of the \(\Bbb Z_2\) invariants, Phys. Rev. B, 83, 235115, (2011) |

[34] | Hua, L.-K., On the theory of automorphic functions of a matrix variable I — geometrical basis, Amer. J. Math., 66, 470-488, (1944) · Zbl 0063.02919 |

[35] | Dubrovin, B. A.; Novikov, S. P.; Fomenko, A. T., Modern Geometry — Methods and Applications. Part II: The Geometry and Topology of Manifolds, (1985), Springer, New York · Zbl 0565.57001 |

[36] | Husemoller, D., Fibre Bundles, (1994), Springer, New York |

[37] | Freund, S.; Teufel, S., Peierls substitution for magnetic Bloch bands, Anal. PDE, 9, 773-811, (2016) · Zbl 1343.81088 |

[38] | H. D. Cornean and D. Monaco, in preparation. |

[39] | Schulz-Baldes, H., \(\Bbb Z_2\) indices and factorization properties of odd symmetric Fredholm operators, Doc. Math., 20, 1481-1500, (2015) · Zbl 1341.47014 |

[40] | Karp, R. L.; Mansouri, F.; Rno, J. S., Product integral representations of Wilson lines and Wilson loops and non-abelian Stokes theorem, Turk. J. Phy., 24, 365-384, (2000) |

[41] | Fiorenza, D.; Sati, H.; Schreiber, U., Mathematical Aspects of Quantum Field Theories, A higher stacky perspective on Chern-Simons theory, 153-211, (2015), Springer, Basel · Zbl 1315.81091 |

[42] | Berry, M. V., Quantal phase factors accompanying adiabatic changes, Proc. R. Lond. A, 392, 45-57, (1984) · Zbl 1113.81306 |

[43] | Kato, T., Perturbation Theory for Linear Operators, (1966), Springer, Berlin · Zbl 0148.12601 |

[44] | Guillemin, V.; Pollack, A., Differential Topology, (1974), Prentice Hall, Englewood Cliffs, New Jersey · Zbl 0361.57001 |

[45] | J. von Neumann and E. Wigner, On the behavior of eigenvalues in adiabatic processes, Phys. Z.30 (1929) 467-470; Republished in Quantum Chemistry: Classic Scientific Papers, ed. H. Hettema (World Scientific, Singapore, 2000), pp. 25-31. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.