Topology in Condensed Matter

  • Laura Ortiz MartínEmail author
Part of the Springer Theses book series (Springer Theses)


Fundamental Physics focuses its efforts on finding out the smallest building blocks of matter. Following this purpose along the centuries, the elements were discovered in the 19th century and Chemistry achieved unprecedented significance. Throughout the 20th century, Physics has endorsed a feverishly search for elementary particles. On the contrary, Condensed Matter Physics plays around with the same atoms and electrons which have been studied for many centuries. Discoveries in this field are not new elements but the emergence of new phases using the same elements. Emergence describes the properties of a material by how the electrons and atoms are organised [1]. This implies that the wide range of materials does not come from the variety of the components. As a natural consequence, the main interest in this area is to find out how they combine together forming new states of matter. These very well-known components form a whole plethora of new states of matter: crystalline solids, magnets and superconductors are some representative examples. The effective field theory known as Landau-Ginzburg theory [2] gave a universal description for quantum phases of matter based on the dimensionality and the symmetries of local order parameters. The great success of the last century was to classify phases of matter following the principle of spontaneous symmetry breaking [3]. What Landau’s theory is lacking is precisely the quantum effects, since it is developed for systems at finite temperatures.


  1. 1.
    Wen X-G (2017) Colloquium: Zoo of quantum-topological phases of matter. Rev Mod Phys 89:041004ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Landau LD, Lifshitz EM (1980) Statistical physics. Pergamon Press, OxfordzbMATHGoogle Scholar
  3. 3.
    Anderson PW (1997) Basic notions of condensed matter physics. Pergamon Press, OxfordGoogle Scholar
  4. 4.
    Klitzing KV, Dorda G, Pepper M (1980) New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys Rev Lett 45:494–497ADSCrossRefGoogle Scholar
  5. 5.
    Tsui DC, Stormer HL, Gossard AC (1982) Two-dimensional magnetotransport in the extreme quantum limit. Phys Rev Lett 48:1559–1562ADSCrossRefGoogle Scholar
  6. 6.
    Bednorz JG, Müller KA (1986) Possible high-\(t_c\) superconductivity in the \(balacuo\) system. Zeitschrift für Physik B Condens Matter 64(2):189–193Google Scholar
  7. 7.
    Qi X-L, Hughes TL, Zhang S-C (2008) Topological field theory of time-reversal invariant insulators. Phys Rev B 78:195424ADSCrossRefGoogle Scholar
  8. 8.
    Wen X-G (2002) Quantum orders and symmetric spin liquids. Phys Rev B 65:165113ADSCrossRefGoogle Scholar
  9. 9.
    Bombin H (2010) Topological order with a twist: ising anyons from an abelian model. Phys Rev Lett 105:030403Google Scholar
  10. 10.
    Barkeshli M, Qi X-L (2014) Synthetic topological qubits in conventional bilayer quantum hall systems. Phys Rev X 4:041035Google Scholar
  11. 11.
    You Y-Z, Wen X-G (2012) Projective non-Abelian statistics of dislocation defects in a \({\mathbb{z}}_{N}\) rotor model. Phys Rev B 86:161107ADSCrossRefGoogle Scholar
  12. 12.
    Lindner NH, Berg E, Refael G, Stern A (2012) Fractionalizing majorana fermions: non-abelian statistics on the edges of abelian quantum hall states. Phys Rev X 2:041002Google Scholar
  13. 13.
    Brown BJ, Bartlett SD, Doherty AC, Barrett SD (2013) Topological entanglement entropy with a twist. Phys Rev Lett 111:220402ADSCrossRefGoogle Scholar
  14. 14.
    Petrova O, Tchernyshyov O (2011) Spin waves in a skyrmion crystal. Phys Rev B 84:214433ADSCrossRefGoogle Scholar
  15. 15.
    Hamma A, Cincio L, Santra S, Zanardi P, Amico L (2013) Local response of topological order to an external perturbation. Phys Rev Lett 110:210602ADSCrossRefGoogle Scholar
  16. 16.
    Kitaev AYu (2003) Fault-tolerant quantum computation by anyons. Ann Phys 303(1):2–30Google Scholar
  17. 17.
    Wen XG, Niu Q (1990) Ground-state degeneracy of the fractional quantum hall states in the presence of a random potential and on high-genus riemann surfaces. Phys Rev B 41:9377–9396ADSCrossRefGoogle Scholar
  18. 18.
    Wen X-G (1991) Topological orders and chern-simons theory in strongly correlated quantum liquid. Int J Mod Phys B 05(10):1641–1648ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Hasan MZ, Kane CL (2010) Colloquium: topological insulators. Rev Mod Phys 82:3045–3067ADSCrossRefGoogle Scholar
  20. 20.
    Qi X-L, Zhang S-C (2011) Topological insulators and superconductors. Rev Mod Phys 83:1057–1110ADSCrossRefGoogle Scholar
  21. 21.
    Fidkowski L, Kitaev A (2010) Effects of interactions on the topological classification of free fermion systems. Phys Rev B 81:134509ADSCrossRefGoogle Scholar
  22. 22.
    Fidkowski L, Kitaev A (2011) Topological phases of fermions in one dimension. Phys Rev B 83:075103ADSCrossRefGoogle Scholar
  23. 23.
    Essin AM, Hermele M (2013) Classifying fractionalization: symmetry classification of gapped \({\mathbb{z}}_{2}\) spin liquids in two dimensions. Phys Rev B 87:104406Google Scholar
  24. 24.
    Mesaros A, Ran Y (2013) Classification of symmetry enriched topological phases with exactly solvable models. Phys Rev B 87:155115Google Scholar
  25. 25.
    Turner AM, Zhang Y, Mong RSK, Vishwanath A (2012) Quantized response and topology of magnetic insulators with inversion symmetry. Phys Rev B 85:165120ADSCrossRefGoogle Scholar
  26. 26.
    Mong RSK, Essin AM, Moore JE (2010) Antiferromagnetic topological insulators. Phys Rev B 81:245209ADSCrossRefGoogle Scholar
  27. 27.
    Liang F (2011) Topological crystalline insulators. Phys Rev Lett 106:106802CrossRefGoogle Scholar
  28. 28.
    Hughes TL, Prodan E, Bernevig BA (2011) Inversion-symmetric topological insulators. Phys Rev B 83:245132ADSCrossRefGoogle Scholar
  29. 29.
    Hasan MZ, Moore JE (2011) Three-dimensional topological insulators. Ann Rev Condens Matter Phys 2(1):55–78ADSCrossRefGoogle Scholar
  30. 30.
    Lu Y-M, Vishwanath A (2012) Theory and classification of interacting integer topological phases in two dimensions: a Chern-Simons approach. Phys Rev B 86:125119ADSCrossRefGoogle Scholar
  31. 31.
    Chen X, Gu Z-C, Liu Z-X, Wen X-G (2013) Symmetry protected topological orders and the group cohomology of their symmetry group. Phys Rev B 87:155114Google Scholar
  32. 32.
    Turner AM, Pollmann F, Berg E (2011) Topological phases of one-dimensional fermions: an entanglement point of view. Phys Rev B 83:075102ADSCrossRefGoogle Scholar
  33. 33.
    Tang E, Wen X-G (2012) Interacting one-dimensional fermionic symmetry-protected topological phases. Phys Rev Lett 109:096403ADSCrossRefGoogle Scholar
  34. 34.
    Affleck I, Kennedy T, Lieb EH, Tasaki H (1987) Rigorous results on valence-bond ground states in antiferromagnets. Phys Rev Lett 59:799–802ADSCrossRefGoogle Scholar
  35. 35.
    Pollmann F, Berg E, Turner AM, Oshikawa M (2012) Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys Rev B 85:075125ADSCrossRefGoogle Scholar
  36. 36.
    Levin M, Gu Z-C (2012) Braiding statistics approach to symmetry-protected topological phases. Phys Rev B 86:115109Google Scholar
  37. 37.
    Wilczek F (1982) Quantum mechanics of fractional-spin particles. Phys Rev Lett 49:957–959ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Arovas D, Schrieffer JR, Wilczek F (1984) Fractional statistics and the quantum hall effect. Phys Rev Lett 53:722–723ADSCrossRefGoogle Scholar
  39. 39.
    Nayak C, Simon SH, Stern A, Freedman M, Sarma SD (2008) Non-Abelian anyons and topological quantum computation. Rev Mod Phys 80:1083–1159ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Chen X (2017) Symmetry fractionalization in two dimensional topological phases. Rev Phys 2:3–18CrossRefGoogle Scholar
  41. 41.
    Martin-Delgado MA, Sierra G (1997) Strongly correlated magnetic and superconducting systems. Springer, BerlinzbMATHGoogle Scholar
  42. 42.
    Laughlin RB (1983) Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys Rev Lett 50:1395–1398ADSCrossRefGoogle Scholar
  43. 43.
    Haldane FDM (1983) Fractional quantization of the hall effect: a hierarchy of incompressible quantum fluid states. Phys Rev Lett 51:605–608ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Halperin BI (1984) Statistics of quasiparticles and the hierarchy of fractional quantized hall states. Phys Rev Lett 52:1583–1586ADSCrossRefGoogle Scholar
  45. 45.
    Jain JK (1989) Composite-fermion approach for the fractional quantum hall effect. Phys Rev Lett 63:199–202ADSCrossRefGoogle Scholar
  46. 46.
    Kim C, Matsuura AY, Shen Z-X, Motoyama N, Eisaki H, Uchida S, Tohyama T, Maekawa S (1996) Observation of spin-charge separation in one-dimensional SrCuO\(_{2}\). Phys Rev Lett 77:4054–4057ADSCrossRefGoogle Scholar
  47. 47.
    Senthil T, Fisher MPA (2000) \({Z}_{2}\) gauge theory of electron fractionalization in strongly correlated systems. Phys Rev B 62:7850–7881ADSCrossRefGoogle Scholar
  48. 48.
    Recati A, Fedichev PO, Zwerger W, Zoller P (2003) Spin-charge separation in ultracold quantum gases. Phys Rev Lett 90:020401ADSCrossRefGoogle Scholar
  49. 49.
    Wang F, Vishwanath A (2006) Spin-liquid states on the triangular and Kagomé lattices: a projective-symmetry-group analysis of Schwinger boson states. Phys Rev B 74:174423ADSCrossRefGoogle Scholar
  50. 50.
    Levin M, Stern A (2009) Fractional topological insulators. Phys Rev Lett 103:196803ADSCrossRefGoogle Scholar
  51. 51.
    Barkeshli M, Bonderson P, Cheng M, Wang Z (2014) Symmetry, defects, and gauging of topological phases. arXiv:1410.4540
  52. 52.
    Song H, Huang S-J, Fu L, Hermele M (2017) Topological phases protected by point group symmetry. Phys Rev X 7:011020Google Scholar
  53. 53.
    Vishwanath A, Senthil T (2013) Physics of three-dimensional bosonic topological insulators: surface-deconfined criticality and quantized magnetoelectric effect. Phys Rev X 3:011016Google Scholar
  54. 54.
    Chen X, Burnell FJ, Vishwanath A, Fidkowski L (2015) Anomalous symmetry fractionalization and surface topological order. Phys Rev X 5:041013Google Scholar
  55. 55.
    Hsieh D, Qian D, Wray L, Xia Y, Hor YS, Cava RJ, Hasan MZ (2008) A topological Dirac insulator in a quantum spin hall phase. Nature 452:970ADSCrossRefGoogle Scholar
  56. 56.
    Hsieh D, Xia Y, Qian D, Wray L, Dil JH, Meier F, Osterwalder J, Patthey L, Checkelsky JG, Ong NP, Fedorov AV, Lin H, Bansil A, Grauer D, Hor YS, Cava RJ, Hasan MZ (2008) A tunable topological insulator in the spin helical Dirac transport regime. Nature 460:1101ADSCrossRefGoogle Scholar
  57. 57.
    Chiu C-K, Teo JCY, Schnyder AP, Ryu S (2016) Classification of topological quantum matter with symmetries. Rev Mod Phys 88:035005ADSCrossRefGoogle Scholar
  58. 58.
    Kane CL, Mele EJ (2005) Quantum spin hall effect in graphene. Phys Rev Lett 95:226801Google Scholar
  59. 59.
    Fu L, Kane CL (2006) Time reversal polarization and a \({Z}_{2}\) adiabatic spin pump. Phys Rev B 74:195312ADSCrossRefGoogle Scholar
  60. 60.
    Fu L, Kane CL (2007) Topological insulators with inversion symmetry. Phys Rev B 76:045302ADSCrossRefGoogle Scholar
  61. 61.
    Wunsch B, Guinea F, Sols F (2008) Dirac-point engineering and topological phase transitions in honeycomb optical lattices. New J Phys 10(10):103027CrossRefGoogle Scholar
  62. 62.
    Viyuela O, Rivas A, Gasparinetti S, Wallraff A, Filipp S, Martin-Delgado MA (2018) Observation of topological Uhlmann phases with superconducting qubits. npj Quantum Inf 4:10ADSCrossRefGoogle Scholar
  63. 63.
    Fu L, Kane CL (2008) Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys Rev Lett 100:096407Google Scholar
  64. 64.
    Zhang P, Yaji K, Hashimoto T, Ota Y, Kondo T, Okazaki K, Wang Z, Wen J, Gu GD, Ding H, Shin S (2018) Observation of topological superconductivity on the surface of an iron-based superconductor. Science 360(6385):182–186ADSCrossRefGoogle Scholar
  65. 65.
    Ryu S, Hatsugai Y (2002) Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys Rev Lett 89:077002ADSCrossRefGoogle Scholar
  66. 66.
    Schnyder AP, Ryu S (2011) Topological phases and surface flat bands in superconductors without inversion symmetry. Phys Rev B 84:060504ADSCrossRefGoogle Scholar
  67. 67.
    Brydon PMR, Schnyder AP, Timm C (2011) Topologically protected flat zero-energy surface bands in noncentrosymmetric superconductors. Phys Rev B 84:020501ADSCrossRefGoogle Scholar
  68. 68.
    Shitade A, Katsura H, Kuneš J, Qi X-L, Zhang S-C, Nagaosa N (2009) Quantum spin hall effect in a transition metal oxide Na\(_{2}\)IrO\(_{3}\). Phys Rev Lett 102:256403Google Scholar
  69. 69.
    Xiao D, Zhu W, Ran Y, Nagaosa N, Okamoto S (2009) Interface engineering of quantum hall effects in digital transition metal oxide heterostructures. Nat Commun 2:596ADSCrossRefGoogle Scholar
  70. 70.
    Dzero M, Sun K, Coleman P, Galitski V (2012) Theory of topological Kondo insulators. Phys Rev B 85:045130ADSCrossRefGoogle Scholar
  71. 71.
    Wolgast S, yan Kurdak Ç, Sun K, Allen JW, Kim D-J, Fisk Z (2013) Low-temperature surface conduction in the Kondo insulator SmB\({}_{6}\). Phys Rev B 88:180405 (2013)Google Scholar
  72. 72.
    Darriet J, Regnault LP (1993) The compound Y\(_{2}\)BaNiO\(_{5}\): a new example of a haldane gap in a \(S = 1\) magnetic chain. Solid State Commun 86(7):409–412ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Racah Institute of PhysicsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations