Colloquium: Disclination loops, point defects, and all that in nematic liquid crystals

The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress was made in controlling and measuring colloidal inclusions in liquid crystalline phases. The topological structure of these systems is quite rich but, at the same time, subtle. Motivated by experiment and the power of topological reasoning, the classification of defects in uniaxial nematic liquid crystals was reviewed and expounded upon. Particular attention was paid to the ambiguities that arise in these systems, which have no counterpart in the much-storied XY model or the Heisenberg ferromagnet.

[1]  J J de Pablo,et al.  Defect structure around two colloids in a liquid crystal. , 2003, Physical review letters.

[2]  Miha Ravnik,et al.  Nematic colloids entangled by topological defects , 2009 .

[3]  P. E. Cladis,et al.  Non-singular disclinations of strength S = + 1 in nematics , 1972 .

[4]  S. Žumer,et al.  Entangled nematic colloidal dimers and wires. , 2007, Physical review letters.

[5]  S. Simon,et al.  Non-Abelian Anyons and Topological Quantum Computation , 2007, 0707.1889.

[6]  J M Yeomans,et al.  Numerical calculations of the phase diagram of cubic blue phases in cholesteric liquid crystals. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Alexei Abrikosov,et al.  Magnetic properties of superconductors of the second group , 1956 .

[8]  Randall D. Kamien,et al.  Molecular chirality and chiral parameters , 1999 .

[9]  E. Terentjev,et al.  Disclination loops, standing alone and around solid particles, in nematic liquid crystals. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Y. Bouligand,et al.  Fils et disinclinaisons dans un nématique en tube capillaire , 1974 .

[11]  Miha Ravnik,et al.  Two-Dimensional Nematic Colloidal Crystals Self-Assembled by Topological Defects , 2006, Science.

[12]  Holger Stark,et al.  Novel Colloidal Interactions in Anisotropic Fluids , 1997, Science.

[13]  A. Isihara,et al.  Theory of Liquid Crystals , 1972 .

[14]  G. Toulouse,et al.  The crossing of defects in ordered media and the topology of 3-manifolds , 1977 .

[15]  S. Žumer,et al.  Two-dimensional dipolar nematic colloidal crystals. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Hans-Rainer Trebin,et al.  The topology of non-uniform media in condensed matter physics , 1982 .

[17]  C. Pfleiderer,et al.  Spontaneous skyrmion ground states in magnetic metals , 2006, Nature.

[18]  R. Hornreich,et al.  Landau theory of cholesteric blue phases , 1983 .

[19]  Alexander M. Polyakov,et al.  Particle spectrum in quantum field theory , 1974 .

[20]  K. Jänich,et al.  Topological properties of ordinary nematics in 3-space , 1987 .

[21]  A. Kitaev.  Fault tolerant quantum computation by anyons , 1997, quant-ph/9707021.

[22]  Simon vCopar,et al.  Nematic braids: topological invariants and rewiring of disclinations , 2011 .

[23]  M. Hien,et al.  The Global Defect Index , 1999 .

[24]  Randall D. Kamien,et al.  The Geometry of Soft Materials: A Primer , 2002 .

[25]  N. D. Mermin,et al.  The topological theory of defects in ordered media , 1979 .

[26]  R. Kamien,et al.  Symmetry breaking in smectics and surface models of their singularities , 2009, Proceedings of the National Academy of Sciences.

[27]  D. Thouless,et al.  Ordering, metastability and phase transitions in two-dimensional systems , 1973 .

[28]  F. C. Frank,et al.  I. Liquid crystals. On the theory of liquid crystals , 1958 .

[29]  Noel A. Clark,et al.  Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids. , 2010, Nature materials.

[30]  P. Böni,et al.  Skyrmion Lattice in a Chiral Magnet , 2009, Science.

[31]  T. Lubensky,et al.  Principles of condensed matter physics , 1995 .

[32]  M. Cates,et al.  Colloids in liquid crystals: a lattice Boltzmann study , 2010, 1102.2193.

[33]  S. Coleman.  Aspects of Symmetry: Selected Erice Lectures , 1988 .

[34]  H. Nakanishi,et al.  Topological classification of unknotted ring defects , 1988 .

[35]  N. D. Mermin,et al.  Circulation and Angular Momentum in the A Phase of Superfluid Helium-3 , 1976 .

[36]  Robert B. Meyer,et al.  On the existence of even indexed disclinations in nematic liquid crystals , 1973 .

[37]  S. Coleman.  Quantum sine-Gordon equation as the massive Thirring model , 1975 .

[38]  O. Lavrentovich,et al.  Defects in liquid crystals: homotopy theory and experimental studies , 1988 .

[39]  Louis Michel,et al.  SYMMETRY DEFECTS AND BROKEN SYMMETRY. CONFIGURATIONS - HIDDEN SYMMETRY , 1980 .

[40]  R. Ho.  Algebraic Topology , 2022 .

[41]  Igor Muševič,et al.  Reconfigurable Knots and Links in Chiral Nematic Colloids , 2011, Science.

[42]  S. Žumer,et al.  Interactions of quadrupolar nematic colloids. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  N. David Mermin,et al.  Crystalline liquids: the blue phases , 1989 .

[44]  Nelson,et al.  Vortex entanglement in high-Tc superconductors. , 1988, Physical review letters.

[45]  R. Kamien.  Knot Your Simple Defect Lines? , 2011, Science.

[46]  T. Lubensky,et al.  Biaxial model of cholesteric liquid crystals , 1974 .