Magnetic topological quantum chemistry

For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids – Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states.

[1]  C. Kane,et al.  Dirac Semimetals in Two Dimensions. , 2015, Physical review letters.

[2]  B. Bradlyn,et al.  Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice , 2018, Physical Review B.

[3]  Peter Fulde,et al.  Superconductivity in a Strong Spin-Exchange Field , 1964 .

[4]  N. Regnault,et al.  Fragile Phases as Affine Monoids: Classification and Material Examples , 2019 .

[5]  Liang Fu,et al.  Topological insulators in three dimensions. , 2006, Physical review letters.

[6]  M. Vergniory,et al.  Higher-order topological insulators , 2017, Science Advances.

[7]  B. Bradlyn,et al.  Strong and fragile topological Dirac semimetals with higher-order Fermi arcs , 2019, Nature Communications.

[8]  F. Crick,et al.  Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid , 1974, Nature.

[9]  L. Landau,et al.  The Theory of Phase Transitions , 1936, Nature.

[10]  Claudia Felser,et al.  A complete catalogue of high-quality topological materials , 2019, Nature.

[11]  C. Felser,et al.  Topology of Disconnected Elementary Band Representations. , 2017, Physical review letters.

[12]  Joel E. Moore,et al.  Antiferromagnetic topological insulators , 2010, 1004.1403.

[13]  R. Fraser.  The structure of deoxyribose nucleic acid. , 2004, Journal of structural biology.

[14]  A. Cracknell.  The application of Landau's theory of continuous phase transitions to magnetic phase transitions , 1971 .

[15]  E. J. Mele,et al.  Quantum spin Hall effect in graphene. , 2004, Physical review letters.

[16]  Leon Balents,et al.  Mott physics and band topology in materials with strong spin-orbit interaction , 2009, 0907.2962.

[17]  M. Vergniory,et al.  Higher-Order Topology in Bismuth , 2018, Nature Physics.

[18]  O. Halpern,et al.  Magnetic Scattering of Slow Neutrons , 1937 .

[19]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

[20]  D. Son.  Is the Composite Fermion a Dirac Particle , 2015, 1502.03446.

[21]  A. Vishwanath,et al.  Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates , 2010, 1007.0016.

[22]  D. Hsieh,et al.  A topological Dirac insulator in a quantum spin Hall phase , 2008, Nature.

[23]  M. Metlitski.  A 1d lattice model for the boundary of the quantum spin-Hall insulator. , 2019, 1908.08958.

[24]  A. Gromov.  Towards Classification of Fracton Phases: The Multipole Algebra , 2018, Physical Review X.

[25]  Daniel B. Litvin,et al.  Magnetic Group Tables: 1-, 2- and 3-dimensional magnetic subperiodic groups and magnetic space groups , 2013 .

[26]  N. Read,et al.  Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect , 1999, cond-mat/9906453.

[27]  Benedikt Ernst,et al.  Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet , 2019, Science.

[28]  A. Vishwanath,et al.  Fragile Topology and Wannier Obstructions. , 2017, Physical review letters.

[29]  Yuqing He,et al.  Catalogue of topological electronic materials , 2018, Nature.

[30]  A. Larkin,et al.  Nonuniform state of superconductors , 1964 .

[31]  B. Bernevig,et al.  Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators , 2017, 1708.04230.

[32]  E. Wigner.  Ueber die Operation der Zeitumkehr in der Quantenmechanik , 1993 .

[33]  Z. Fang,et al.  Topological classification and diagnosis in magnetically ordered electronic materials , 2021, Physical Review B.

[34]  C. Kane,et al.  Topological Classification of Crystalline Insulators through Band Structure Combinatorics , 2016, 1612.02007.

[35]  Feng Tang,et al.  Comprehensive search for topological materials using symmetry indicators , 2018, Nature.

[36]  C. Felser,et al.  Building blocks of topological quantum chemistry: Elementary band representations , 2017, 1709.01935.

[37]  Ashvin Vishwanath,et al.  Subject Areas : Strongly Correlated Materials A Viewpoint on : Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates , 2011 .

[38]  T. Nomoto,et al.  Multipole Superconductivity in Nonsymmorphic Sr_{2}IrO_{4}. , 2017, Physical review letters.

[39]  A. Ney,et al.  Large magnetic gap at the Dirac point in Bi2Te3/MnBi2Te4 heterostructures , 2018, Nature.

[40]  Feng Tang,et al.  Comprehensive search for topological materials using symmetry indicators , 2019, Nature.

[41]  D. Vanderbilt,et al.  Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. , 2008, Physical review letters.

[42]  X. Dai,et al.  Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides , 2014, 1501.00060.

[43]  R. Arita,et al.  Cluster multipole theory for anomalous Hall effect in antiferromagnets , 2016, 1611.06042.

[44]  Hans Wondratschek,et al.  Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups. , 2006, Acta crystallographica. Section A, Foundations of crystallography.

[45]  S. Young,et al.  Filling-Enforced Magnetic Dirac Semimetals in Two Dimensions. , 2016, Physical review letters.

[46]  C. Felser,et al.  Fermi-arc diversity on surface terminations of the magnetic Weyl semimetal Co3Sn2S2 , 2019, Science.

[47]  Alexei Kitaev,et al.  Periodic table for topological insulators and superconductors , 2009, 0901.2686.

[48]  Wei Zhang,et al.  Quantized Anomalous Hall Effect in Magnetic Topological Insulators , 2010, Science.

[49]  Liang Fu,et al.  Topological crystalline insulators. , 2010, Physical review letters.

[50]  R. Evarestov,et al.  Band Corepresentations of Magnetic Space Groups , 1989 .

[51]  M. I. Aroyo,et al.  Topological quantum chemistry , 2017, Nature.

[52]  A. Georges,et al.  Slave-rotor mean-field theories of strongly correlated systems and the Mott transition in finite dimensions , 2004, cond-mat/0404334.

[53]  Leon Balents,et al.  Deconfined Quantum Critical Points , 2003, Science.

[55]  G. Fitzgerald,et al.  'I. , 2019, Australian journal of primary health.

[56]  Haldane,et al.  Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the "parity anomaly" , 1988, Physical review letters.

[57]  M. Norman,et al.  Odd parity and line nodes in nonsymmorphic superconductors , 2009, 0907.0753.

[58]  Liang Fu,et al.  Topological insulators with inversion symmetry , 2006, cond-mat/0611341.

[59]  R. Arita,et al.  Multipole expansion for magnetic structures: A generation scheme for a symmetry-adapted orthonormal basis set in the crystallographic point group , 2019, Physical Review B.

[60]  F. Crick,et al.  Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid , 1953, Nature.

[61]  Stanley C. Miller,et al.  Tables of irreducible representations of space groups and co-representations of magnetic space groups , 1967 .

[62]  Xiao-Liang Qi,et al.  Topological field theory of time-reversal invariant insulators , 2008, 0802.3537.

[63]  Q. Xue,et al.  Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator , 2013, Science.

[64]  A. Vishwanath,et al.  Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators , 2017, Physical Review X.

[65]  C. Kane,et al.  Wallpaper fermions and the nonsymmorphic Dirac insulator , 2017, Science.

[66]  E. J. Mele,et al.  Weyl and Dirac semimetals in three-dimensional solids , 2017, 1705.01111.

[67]  Z. K. Liu,et al.  Magnetic Weyl semimetal phase in a Kagomé crystal , 2019, Science.

[68]  Shuang Jia,et al.  Discovery of a Weyl fermion semimetal and topological Fermi arcs , 2015, Science.

[69]  A. Cracknell,et al.  The mathematical theory of symmetry in solids;: Representation theory for point groups and space groups, , 1972 .

[70]  A. Vishwanath,et al.  Symmetry-based indicators of band topology in the 230 space groups , 2017, Nature Communications.

[71]  C. Fang,et al.  New classes of topological crystalline insulators having surface rotation anomaly , 2017, Science Advances.

[72]  Wladimir A. Benalcazar,et al.  Quantized electric multipole insulators , 2016, Science.

[73]  Shou-Cheng Zhang,et al.  Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells , 2006, Science.

[74]  Z. Fang,et al.  Quantitative mappings between symmetry and topology in solids , 2017, Nature Communications.

[75]  J. Zak.  Band representations and symmetry types of bands in solids , 1981 .

[76]  A. Vishwanath,et al.  Realizing topological surface states in a lower-dimensional flat band , 2016, 1609.08618.

[77]  B. Bernevig,et al.  The Axion Insulator as a Pump of Fragile Topology , 2018, 1810.02373.

[78]  V. N. Zverev,et al.  Prediction and observation of an antiferromagnetic topological insulator , 2018, Nature.

[79]  Zhijun Wang,et al.  Hourglass fermions , 2016, Nature.

[80]  Hsin Lin,et al.  Topological crystalline insulators in the SnTe material class , 2012, Nature Communications.

[81]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[82]  A. Vishwanath,et al.  Structure and topology of band structures in the 1651 magnetic space groups , 2017, Science Advances.

[83]  Hans Wondratschek,et al.  Bilbao Crystallographic Server: I. Databases and crystallographic computing programs , 2006 .

[84]  Binghai Yan,et al.  Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe_{2} (X=Mo,W). , 2018, Physical review letters.