zbMATH — the first resource for mathematics

Spin conductance and spin conductivity in topological insulators: analysis of kubo-like terms. (English) Zbl 07063411
Summary: We investigate spin transport in 2-dimensional insulators, with the long-term goal of establishing whether any of the transport coefficients corresponds to the Fu-Kane-Mele index which characterizes 2\(d\) time-reversal-symmetric topological insulators. Inspired by the Kubo theory of charge transport, and by using a proper definition of the spin current operator (Shi et al. in Phys Rev Lett 96:076604, 2006), we define the Kubo-like spin conductance \({G_K^{s_z}}\) and spin conductivity \({\sigma _K^{s_z}}\). We prove that for any gapped, periodic, near-sighted discrete Hamiltonian, the above quantities are mathematically well defined and the equality \({G_K^{s_z} = \sigma _K^{s_z}}\) holds true. Moreover, we argue that the physically relevant condition to obtain the equality above is the vanishing of the mesoscopic average of the spin-torque response, which holds true under our hypotheses on the Hamiltonian operator. A central role in the proof is played by the trace per unit volume and by two generalizations of the trace, the principal value trace and its directional version.
47A General theory of linear operators
81T Quantum field theory; related classical field theories
81Q General mathematical topics and methods in quantum theory
81U Quantum scattering theory
47L Linear spaces and algebras of operators
Full Text: DOI
[1] An, Z.; Liu, FQ; Lin, Y.; Liu, C., The universal definition of spin current, Sci. Rep., 2, 388, (2012)
[2] Ando, Y., Topological insulator materials, J. Phys. Soc. Jpn., 82, 102001, (2013)
[3] Aizenman, M.; Graf, GM, Localization bounds for an electron gas, J. Phys. A Math. Gen., 31, 6783, (1998) · Zbl 0953.82009
[4] Aizenman, M., Warzel, S.: Random Operators. Graduate Studies in Mathematics, vol. 168. American Mathematical Society, Providence (2015) · Zbl 1333.82001
[5] Avila, JC; 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
[6] Avron, JE; Seiler, R., Quantization of the Hall conductance for general, multiparticle Schrödinger Hamiltonians, Phys. Rev. Lett., 54, 259-262, (1985)
[7] Avron, J.; Seiler, R.; Simon, B., Charge deficiency, charge transport and comparison of dimensions, Commun. Math. Phys., 159, 399-422, (1994) · Zbl 0822.47056
[8] Bellissard, J.; Elst, A.; Schulz-Baldes, H., The non-commutative geometry of the quantum Hall effect, J. Math. Phys., 35, 5373, (1994) · Zbl 0824.46086
[9] Bouclet, JM; Germinet, F.; Klein, A.; Schenker, JH, Linear response theory for magnetic Schrödinger operators in disordered media, J. Funct. Anal., 226, 301-372, (2005) · Zbl 1088.82013
[10] Bray-Ali, N.; Nussinov, Z., Conservation and persistence of spin currents and their relation to the Lieb-Schulz-Mattis twist operators, Phys. Rev. B, 80, 012401, (2009)
[11] Carpentier, D.; Delplace, P.; Fruchart, M.; Gawedzki, K., Topological index for periodically driven time-reversal invariant 2D systems, Phys. Rev. Lett., 114, 106806, (2015) · Zbl 1331.82065
[12] Carpentier, D.; Delplace, P.; Fruchart, M.; Gawedzki, K.; Tauber, C., Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals, Nucl. Phys. B, 896, 779-834, (2015) · Zbl 1331.82065
[13] Cornean, HD; Monaco, D.; Teufel, S., Wannier functions and \(\mathbb{Z}_2\) invariants in time-reversal symmetric topological insulators, Rev. Math. Phys., 29, 1730001, (2017) · Zbl 1370.81081
[14] Elgart, A.; Graf, GM; Schenker, JH, Equality of the bulk and edge Hall conductances in a mobility gap, Commun. Math. Phys., 259, 185-221, (2005) · Zbl 1086.81081
[15] Elgart, A.; Schlein, B., Adiabatic charge transport and the Kubo formula for Landau-type Hamiltonians, Commun. Pure Appl. Math., 57, 590-615, (2004) · Zbl 1053.81099
[16] Fiorenza, D.; Monaco, D.; Panati, G., \(\mathbb{Z}_{2}\) invariants of topological insulators as geometric obstructions, Commun. Math. Phys., 343, 1115-1157, (2016) · Zbl 1346.81158
[17] Fröhlich, J.; Werner, P., Gauge theory of topological phases of matter, EPL, 101, 47007, (2013)
[18] Fu, L.; Kane, CL, Time reversal polarization and a \({\mathbb{Z}}_2\) adiabatic spin pump, Phys. Rev. B, 74, 195312, (2006)
[19] Fu, L.; Kane, CL; Mele, EJ, Topological insulators in three dimensions, Phys. Rev. Lett., 98, 106803, (2007)
[20] Gawedzki, K., Square root of gerbe holonomy and invariants of time-reversal-symmetric topological insulators, J. Geom. Phys., 120, 169-191, (2017) · Zbl 1430.53027
[21] Graf, GM, Aspects of the integer quantum Hall effect, Proc. Symp. Pure Math., 76, 429-442, (2007) · Zbl 1132.81356
[22] Graf, GM; Porta, M., Bulk-edge correspondence for two-dimensional topological insulators, Commun. Math. Phys., 324, 851-895, (2013) · Zbl 1291.82120
[23] Haldane, FDM, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett., 61, 2017, (1988)
[24] Hasan, MZ; Kane, CL, Colloquium: topological insulators, Rev. Mod. Phys., 82, 3045-3067, (2010)
[25] Kane, CL; Mele, EJ, \({\mathbb{Z}}_2\) topological order and the quantum spin Hall effect, Phys. Rev. Lett., 95, 146802, (2005)
[26] Kane, CL; Mele, EJ, Quantum spin Hall effect in graphene, Phys. Rev. Lett., 95, 226801, (2005)
[27] Katsura, H.; Koma, T., The \(\mathbb{Z}_2\) index of disordered topological insulators with time reversal symmetry, J. Math. Phys., 57, 021903, (2016) · Zbl 1341.82043
[28] Kirsch, W.: An Invitation to Random Schrödinger Operators, Preprint arXiv:0709.3707
[29] Kitaev, A., Periodic table for topological insulators and superconductors, AIP Conf. Proc., 1134, 22, (2009) · Zbl 1180.82221
[30] Kohn, W., Density functional and density matrix method scaling linearly with the number of atoms, Phys. Rev. Lett., 76, 3168, (1996)
[31] Marcelli, G.: A mathematical analysis of spin and charge transport in topological insulators, Ph.D. thesis in Mathematics, La Sapienza Universitá di Roma, Rome (2017)
[32] Marcelli, G., Monaco, D., Panati, G., Teufel, S.: Quantum (Spin) Hall Conductivity: Kubo Formula (and beyond), to appear (2019)
[33] Marcelli, G., Panati, G., Tauber, C.: Quantum Spin Hall conductance: a first principle analysis, in preparation (2019)
[34] Monaco, D.; Panati, G., Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry, Acta Appl. Math., 137, 185-203, (2015) · Zbl 1318.82045
[35] Monaco, D.; Tauber, C., Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess-Zumino, and Fu-Kane-Mele, Lett. Math. Phys., 107, 1315-1343, (2017) · Zbl 1370.35093
[36] Moore, JE; Balents, L., Topological invariants of time-reversal-invariant band structures, Phys. Rev. B, 75, 121306(r), (2007)
[37] Murakami, S., Quantum spin Hall effect and enhanced magnetic response by spin-orbit coupling, Phys. Rev. Lett., 97, 236805, (2006)
[38] Panati, G., Triviality of Bloch and Bloch-Dirac bundles, Ann. Henri Poincaré, 8, 995-1011, (2007) · Zbl 1375.81102
[39] Prodan, E.; Kohn, W., Nearsightedness of electronic matter, Proc. Natl. Acad. Sci. USA, 102, 11635-11638, (2005)
[40] Prodan, E., Robustness of the spin-Chern number, Phys. Rev. B, 80, 125327, (2009)
[41] Prodan, E., Manifestly gauge-independent formulations of the \(\mathbb{Z}_2\) invariants, Phys. Rev. B, 83, 235115, (2011)
[42] Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1979) · Zbl 0405.47007
[43] Ryu, S.; Schnyder, AP; Furusaki, A.; Ludwig, AWW, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys., 12, 065010, (2010)
[44] Schulz-Baldes, H., Persistence of spin edge currents in disordered quantum spin Hall systems, Commun. Math. Phys., 324, 589-600, (2013) · Zbl 1278.82065
[45] Schulz-Baldes, H., \(\mathbb{Z}_2\)-indices and factorization properties of odd symmetric Fredholm operators, Doc. Math., 20, 1481-1500, (2015) · Zbl 1341.47014
[46] Shi, J.; Zhang, P.; Xiao, D.; Niu, Q., Proper definition of spin current in spin-orbit coupled systems, Phys. Rev. Lett., 96, 076604, (2006)
[47] Simon, B.: Trace Ideals and Their Applications. American Mathematical Society, Providence (2005) · Zbl 1074.47001
[48] Sun, QF; Xie, XC; Wang, J., Persistent spin current in nano-devices and definition of the spin current, Phys. Rev. B, 77, 035327, (2008)
[49] Thouless, DJ; Kohmoto, M.; Nightingale, MP; Nijs, M., Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., 49, 405-408, (1982)
[50] Wu, S.; etal., Observation of the quantum spin Hall effect up to \(100\) Kelvin in a monolayer crystal, Science, 359, 76-79, (2018)
[51] Zhang, P.; Wang, Z.; Shi, J.; Xiao, D.; Niu, Q., Theory of conserved spin current and its application to a two-dimensional hole gas, Phys. Rev. B, 77, 075304, (2008)
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.