Localised Wannier functions in metallic systems. (English) Zbl 1412.35280
Summary: The existence and construction of exponentially localised Wannier functions for insulators are a well-studied problem. In comparison, the case of metallic systems has been much less explored, even though localised Wannier functions constitute an important and widely used tool for the numerical band interpolation of metallic condensed matter systems. In this paper, we prove that, under generic conditions, \(N\) energy bands of a metal can be exactly represented by \(N+1\) Wannier functions decaying faster than any polynomial. We also show that, in general, the lack of a spectral gap does not allow for exponential decay.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 81Q70 | Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory |
| 82D35 | Statistical mechanics of metals |
Software:
Wannier90References:
| [1] | Auckly, D., Kuchment, P.: On Parseval frames of exponentially decaying composite Wannier functions. In: Bonetto, F., Borthwick, D., Harrell, E., Loss M. (eds.) Mathematical Problems in Quantum Physics, pp. 227-240. Vol. 717 in Contemporary Mathematics Volume. American Mathematical Society (2018) · Zbl 1428.35417 |
| [2] | Brouder, Ch., Panati, G., Calandra, M., Mourougane, Ch., Marzari, N.: Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98(4), 046402 (2007) · doi:10.1103/PhysRevLett.98.046402 |
| [3] | Cancès, É., Levitt, A., Panati, G., Stoltz, G.: Robust determination of maximally localized Wannier functions. Phys. Rev. B 95(7), 075114 (2017) · doi:10.1103/PhysRevB.95.075114 |
| [4] | Cohen, M.L., Bergstresser, T.K.: Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures. Phys. Rev. 141(2), 789 (1966) · doi:10.1103/PhysRev.141.789 |
| [5] | Cornean, H.D., Herbst, I., Nenciu, Gh: On the construction of composite Wannier functions. Ann. Henri Poincarè 17(12), 3361-3398 (2016) · Zbl 1357.82069 · doi:10.1007/s00023-016-0489-2 |
| [6] | Cornean, H.D., Monaco, D.: On the construction of Wannier functions in topological insulators: the 3D case. Ann. Henri Poincarè 18(12), 3863-3902 (2017) · Zbl 1386.82068 · doi:10.1007/s00023-017-0621-y |
| [7] | Cornean, H.D., Moscolari, M., Monaco, D.: Parseval frames of exponentially localized magnetic Wannier functions. To appear in Commun. Math. Phys. (2019) · Zbl 1507.82073 |
| [8] | Cornean, H.D., Monaco, D., Teufel, S.: Wannier functions and \[\mathbb{Z}_2\] Z2 invariants in time-reversal symmetric topological insulators. Rev. Math. Phys. 29(2), 1730001 (2017) · Zbl 1370.81081 · doi:10.1142/S0129055X17300011 |
| [9] | Damle, A., Levitt, A., Lin, L.: Variational formulation for Wannier functions with entangled band structure. Preprint. arXiv:1801.08572 (2018) · Zbl 1412.82041 |
| [10] | Damle, A., Lin, L., Ying, L.: SCDM-k: localized orbitals for solids via selected columns of the density matrix. J. Comput. Phys. 334, 1-15 (2017) · Zbl 1375.81255 · doi:10.1016/j.jcp.2016.12.053 |
| [11] | Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. Henri Poincaré 17(1), 63-97 (2016) · Zbl 1338.82057 · doi:10.1007/s00023-015-0400-6 |
| [12] | Friedan, D.: A proof of the Nielsen-Ninomiya theorem. Commun. Math. Phys. 85(4), 481-490 (1982) · doi:10.1007/BF01403500 |
| [13] | Fefferman, C., Weinstein, M.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25(4), 1169-1220 (2012) · Zbl 1316.35214 · doi:10.1090/S0894-0347-2012-00745-0 |
| [14] | Gontier, D., Levitt, A., Siraj-Dine, S.: Numerical construction of Wannier functions through homotopy. Preprint. arXiv:1812.06746 (2018) · Zbl 1417.82030 |
| [15] | Guillemin, V., Pollack, A.: Differential Topology. Prentice Hall (1974) · Zbl 0361.57001 |
| [16] | Hasan, M.Z., Kane, C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045-3067 (2010) · doi:10.1103/RevModPhys.82.3045 |
| [17] | Jost, J.: Compact Riemann Surfaces: An Introduction to Contemporary Mathematics. Springer, Berlin (2013) · Zbl 0869.30001 |
| [18] | Kuchment, P.: An overview of periodic elliptic operators. Bull. AMS 53, 343-414 (2016) · Zbl 1346.35170 · doi:10.1090/bull/1528 |
| [19] | Marzari, N., Mostofi, A.A., Yates, J.R., Souza, I., Vanderbilt, D.: Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84(4), 1419 (2012) · doi:10.1103/RevModPhys.84.1419 |
| [20] | Marzari, N., Vanderbilt, D.: Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56(20), 12847 (1997) · doi:10.1103/PhysRevB.56.12847 |
| [21] | Monaco, D.: Chern and Fu-Kane-Mele invariants as topological obstructions. Chap. 12 In: Dell’Antonio, G., Michelangeli, A. (eds.) Advances in Quantum Mechanics: Contemporary Trends and Open Problems. Vol. 18 in Springer INdAM Series. Springer (2017) · Zbl 1374.81101 |
| [22] | Monaco, D., Panati, G.: Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene. J. Stat. Phys. 155(6), 1027-1071 (2014) · Zbl 1401.82064 · doi:10.1007/s10955-014-0918-x |
| [23] | Monaco, D., Panati, G.: Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry. Acta Appl. Math. 137(1), 185-203 (2015) · Zbl 1318.82045 · doi:10.1007/s10440-014-9995-8 |
| [24] | Monaco, D., Panati, G., Pisante, A., Teufel, S.: Optimal decay of Wannier functions in Chern and quantum Hall insulators. Commun. Math. Phys. 359(1), 61-100 (2018) · Zbl 1400.82249 · doi:10.1007/s00220-017-3067-7 |
| [25] | Mostofi, A.A., Yates, J.R., Lee, Y.-S., Souza, I., Vanderbilt, D., Marzari, N.: wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 178(9), 685-699 (2008) · Zbl 1196.81033 · doi:10.1016/j.cpc.2007.11.016 |
| [26] | Mathai, V., Thiang, GCh.: Global topology of Weyl semimetals and Fermi arcs. J. Phys. A 50(11), 11LT01 (2017) · Zbl 1364.82063 · doi:10.1088/1751-8121/aa59b2 |
| [27] | Mustafa, J.I., Coh, S., Cohen, M.L., Louie, S.G.: Automated construction of maximally localized Wannier functions: optimized projection functions method. Phys. Rev. B 92(16), 165134 (2015) · doi:10.1103/PhysRevB.92.165134 |
| [28] | Nielsen, H.B., Ninomiya, M.: A no-go theorem for regularizing chiral fermions. Phys. Lett. B 105(2-3), 219-223 (1981) · doi:10.1016/0370-2693(81)91026-1 |
| [29] | Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. Henri Poincaré 8(5), 995-1011 (2007) · Zbl 1375.81102 · doi:10.1007/s00023-007-0326-8 |
| [30] | Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators. Academic Press (1978) · Zbl 0401.47001 |
| [31] | Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51(24), 2167 (1983) · doi:10.1103/PhysRevLett.51.2167 |
| [32] | Souza, I., Marzari, N., Vanderbilt, D.: Maximally localized Wannier functions for entangled energy bands. Phys. Rev. B 65(3), 035109 (2001) · doi:10.1103/PhysRevB.65.035109 |
| [33] | Thiang, GCh., Sato, K., Gomi, K.: FuKaneMele monopoles in semimetals. Nucl. Phys. B 923, 107-125 (2017) · Zbl 1373.82099 · doi:10.1016/j.nuclphysb.2017.07.018 |
| [34] | von Neumann, J., Wigner, E.: On the behaviour of eigenvalues in adiabatic processes. Phys. Z. 30, 467 (1929). Republished in: Hettema, H. (ed.) Quantum Chemistry: Classic Scientific Papers, pp. 25-31. World Scientific (2000) |
| [35] | Yates, J.R., Wang, X., Vanderbilt, D., Souza, I.: Spectral and Fermi surface properties from Wannier interpolation. Phys. Rev. B 75(19), 195121 (2007) · doi:10.1103/PhysRevB.75.195121 |
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