Skip to main content
SpringerLink
Log in

The Topology of the Quantum Vacuum

  • Chapter
Analogue Gravity Phenomenology

Part of the book series: Lecture Notes in Physics ((LNP,volume 870))

Abstract

Topology in momentum space is the main characteristic of the ground state of a system at zero temperature, the quantum vacuum. The gaplessness of fermions in bulk, on the surface or inside the vortex core is protected by topology, and is not sensitive to the details of the microscopic physics (atomic or trans-Planckian). Irrespective of the deformation of the parameters of the microscopic theory, the energy spectrum of these fermions remains strictly gapless. This solves the main hierarchy problem in particle physics: for fermionic vacua with Fermi points the masses of elementary particles are naturally small. The quantum vacuum of the Standard Model is one of the representatives of topological matter alongside with topological superfluids and superconductors, topological insulators and semi-metals, etc. There is a number of topological invariants in momentum space of different dimensions. They determine the universality classes of the topological matter and the type of the effective theory which emerges at low energy. In many cases they also give rise to emergent symmetries, including the effective Lorentz invariance, and emergent phenomena such as effective gauge and gravitational fields. The topological invariants in extended momentum and coordinate space determine the bulk-surface and bulk-vortex correspondence. They connect the momentum space topology in bulk with the real space. These invariants determine the gapless fermions living on the surface of a system or in the core of topological defects (vortices, strings, domain walls, solitons, monopoles, etc.). The momentum space topology gives some lessons for quantum gravity. In effective gravity emerging at low energy, the collective variables are the tetrad field and spin connections, while the metric is the composite object of tetrad field. This suggests that the Einstein-Cartan-Sciama-Kibble theory with torsion field is more relevant. There are also several scenarios of Lorentz invariance violation governed by topology, including splitting of Fermi point and development of the Dirac points with quadratic and cubic spectrum. The latter leads to the natural emergence of the Hořava-Lifshitz gravity.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Volovik, G.E.: The Universe in a Helium Droplet. Clarendon, Oxford (2003)

    MATH  Google Scholar 

  2. Volovik, G.E.: Quantum phase transitions from topology in momentum space. In: Unruh, W.G., Schützhold, R. (eds.) Quantum Analogues: From Phase Transitions to Black Holes and Cosmology. Springer Lecture Notes in Physics, vol. 718, pp. 31–73 (2007). arXiv:cond-mat/0601372

    Chapter  Google Scholar 

  3. Hasan, M.Z., Kane, C.L.: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)

    Article  ADS  Google Scholar 

  4. Qi, X.-L., Zhang, S.-C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011)

    Article  ADS  Google Scholar 

  5. Nielsen, H.B., Ninomiya, M.: Absence of neutrinos on a lattice, I: proof by homotopy theory. Nucl. Phys. B 185, 20 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  6. Nielsen, H.B., Ninomiya, M.: Absence of neutrinos on a lattice, II: intuitive homotopy proof. Nucl. Phys. B 193, 173 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  7. Volovik, G.E., Mineev, V.P.: Current in superfluid Fermi liquids and the vortex core structure. J. Exp. Theor. Phys. 56, 579–586 (1982)

    Google Scholar 

  8. Nielsen, H.B., Ninomiya, M.: The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B 130, 389–396 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  9. Avron, J.E., Seiler, R., Simon, B.: Homotopy and quantization in condensed matter physics. Phys. Rev. Lett. 51, 51–53 (1983)

    Article  ADS  Google Scholar 

  10. Semenoff, G.W.: Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53, 2449–2452 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  11. Niu, Q., Thouless, D.J., Wu, Y.-S.: Quantized Hall conductance as a topological invariant. Phys. Rev. B 31, 3372–3377 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  12. So, H.: Induced topological invariants by lattice fermions in odd dimensions. Prog. Theor. Phys. 74, 585–593 (1985)

    Article  ADS  Google Scholar 

  13. Ishikawa, K., Matsuyama, T.: Magnetic field induced multi component QED in three-dimensions and quantum Hall effect. Z. Phys. C 33, 41–45 (1986)

    Article  ADS  Google Scholar 

  14. Ishikawa, K., Matsuyama, T.: A microscopic theory of the quantum Hall effect. Nucl. Phys. B 280, 523–548 (1987)

    Article  ADS  Google Scholar 

  15. Salomaa, M.M., Volovik, G.E.: Cosmiclike domain walls in superfluid 3He-B: instantons and diabolical points in (k,r) space. Phys. Rev. B 37, 9298–9311 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  16. Haldane, F.D.M.: Model for a quantum Hall effect without landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  17. Volovik, G.E.: Analogue of quantum Hall effect in superfluid 3He film. J. Exp. Theor. Phys. 67, 1804–1811 (1988)

    Google Scholar 

  18. Yakovenko, V.M.: Spin, statistics and charge of solitons in (2+1)-dimensional theories. Fizika (Zagreb) 21(suppl. 3), 231 (1989). arXiv:cond-mat/9703195

    Google Scholar 

  19. Hořava, P.: Stability of Fermi surfaces and K-theory. Phys. Rev. Lett. 95, 016405 (2005)

    Article  ADS  Google Scholar 

  20. Schnyder, A.P., Ryu, S., Furusaki, A., Ludwig, A.W.W.: Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008)

    Article  ADS  Google Scholar 

  21. Schnyder, A.P., Ryu, S., Furusaki, A., Ludwig, A.W.W.: Classification of topological insulators and superconductors. AIP Conf. Proc. 1134, 10 (2009). arXiv:0905.2029

    Article  ADS  Google Scholar 

  22. Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009). arXiv:0901.2686

    Article  MathSciNet  ADS  Google Scholar 

  23. Volovik, G.E., Yakovenko, V.M.: Fractional charge, spin and statistics of solitons in superfluid 3He film. J. Phys. Condens. Matter 1, 5263–5274 (1989)

    Article  ADS  Google Scholar 

  24. Mackenzie, A.P., Maeno, Y.: The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing. Rev. Mod. Phys. 75, 657–712 (2003)

    Article  ADS  Google Scholar 

  25. Volovik, G.E.: Topological invariants for Standard Model: from semi-metal to topological insulator. JETP Lett. 91, 55–61 (2010). arXiv:0912.0502

    Article  ADS  Google Scholar 

  26. Abrikosov, A.A., Beneslavskii, S.D.: Possible existence of substances intermediate between metals and dielectrics. Sov. Phys. JETP 32, 699 (1971)

    ADS  Google Scholar 

  27. Abrikosov, A.A.: Quantum magnetoresistance. Phys. Rev. B 58, 2788 (1998)

    Article  ADS  Google Scholar 

  28. Burkov, A.A., Balents, L.: Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011)

    Article  ADS  Google Scholar 

  29. Burkov, A.A., Hook, M.D., Balents, L.: Topological nodal semimetals. Phys. Rev. B 84, 235126 (2011)

    Article  ADS  Google Scholar 

  30. Wan, X., Turner, A.M., Vishwanath, A., Savrasov, S.Y.: Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011)

    Article  ADS  Google Scholar 

  31. Ryu, S., Hatsugai, Y.: Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002)

    Article  ADS  Google Scholar 

  32. Manes, J.L., Guinea, F., Vozmediano, M.A.H.: Existence and topological stability of Fermi points in multilayered graphene. Phys. Rev. B 75, 155424 (2007)

    Article  ADS  Google Scholar 

  33. Vozmediano, M.A.H., Katsnelson, M.I., Guinea, F.: Gauge fields in graphene. Phys. Rep. 496, 109 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  34. Cortijo, A., Guinea, F., Vozmediano, M.A.H.: Geometrical and topological aspects of graphene and related materials. arXiv:1112.2054

  35. Schnyder, A.P., Ryu, S.: Topological phases and flat surface bands in superconductors without inversion symmetry. Phys. Rev. B 84, 060504(R) (2011). arXiv:1011.1438

    ADS  Google Scholar 

  36. Schnyder, A.P., Brydon, P.M.R., Timm, C.: Types of topological surface states in nodal noncentrosymmetric superconductors. arXiv:1111.1207

  37. Essin, A.M., Gurarie, V.: Bulk-boundary correspondence of topological insulators from their Green’s functions. Phys. Rev. B 84, 125132 (2011)

    Article  ADS  Google Scholar 

  38. Zubkov, M.A.: Generalized unparticles, zeros of the Green function, and momentum space topology of the lattice model with overlap fermions. arXiv:1202.2524

  39. Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)

    Article  ADS  MATH  Google Scholar 

  40. Volovik, G.E.: Zeros in the fermion spectrum in superfluid systems as diabolical points. Pis’ma Zh. Eksp. Teor. Fiz. 46, 81–84 (1987). JETP Lett. 46, 98–102 (1987)

    Google Scholar 

  41. Grinevich, P.G., Volovik, G.E.: Topology of gap nodes in superfluid 3He: π 4 homotopy group for 3He-B disclination. J. Low Temp. Phys. 72, 371–380 (1988)

    Article  ADS  Google Scholar 

  42. Volovik, G.E.: Gapless fermionic excitations on the quantized vortices in superfluids and superconductors. JETP Lett. 49, 391–395 (1989)

    ADS  Google Scholar 

  43. Silaev, M.A., Volovik, G.E.: Topological superfluid 3He-B: fermion zero modes on interfaces and in the vortex core. J. Low Temp. Phys. 161, 460–473 (2010). arXiv:1005.4672

    Article  ADS  Google Scholar 

  44. Khodel, V.A., Shaginyan, V.R.: Superfluidity in system with fermion condensate. JETP Lett. 51, 553 (1990)

    ADS  Google Scholar 

  45. Volovik, G.E.: A new class of normal Fermi liquids. JETP Lett. 53, 222 (1991)

    ADS  Google Scholar 

  46. Tsutsumi, Y., Ichioka, M., Machida, K.: Majorana surface states of superfluid 3He A and B phases in a slab. Phys. Rev. B 83, 094510 (2011)

    Article  ADS  Google Scholar 

  47. Volovik, G.E.: Flat band in the core of topological defects: bulk-vortex correspondence in topological superfluids with Fermi points. JETP Lett. 93, 66 (2011)

    Article  ADS  Google Scholar 

  48. Froggatt, C.D., Nielsen, H.B.: Origin of Symmetry. World Scientific, Singapore (1991)

    Book  Google Scholar 

  49. von Neumann, J., Wigner, E.P.: Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Phys. Z. 30, 467–470 (1929)

    MATH  Google Scholar 

  50. Novikov, S.P.: Magnetic Bloch functions and vector bundles. Typical dispersion laws and their quantum numbers. Sov. Math. Dokl. 23, 298–303 (1981)

    MATH  Google Scholar 

  51. Georgi, H.: Another odd thing about unparticle physics. Phys. Lett. B 650, 275–278 (2007). arXiv:0704.2457

    Article  ADS  Google Scholar 

  52. Luo, M., Zhu, G.: Some phenomenologies of unparticle physics. Phys. Lett. B 659, 341 (2008)

    Article  ADS  Google Scholar 

  53. Pepea, M., Wieseb, U.J.: Exceptional deconfinement in G(2) gauge theory. Nucl. Phys. B 768, 21–37 (2007)

    Article  ADS  Google Scholar 

  54. Kadastik, M., Kannike, K., Raidal, M.: Dark matter as the signal of grand unification. Phys. Rev. D 80, 085020 (2009)

    Article  ADS  Google Scholar 

  55. Klinkhamer, F.R., Volovik, G.E.: Emergent CPT violation from the splitting of Fermi points. Int. J. Mod. Phys. A 20, 2795–2812 (2005). arXiv:hep-th/0403037

    Article  ADS  MATH  Google Scholar 

  56. Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G., Esslinger, T.: Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. arXiv:1111.5020

  57. Klinkhamer, F.R.: Possible new source of T and CP violation in neutrino oscillations. Phys. Rev. D 73, 057301 (2006)

    Article  ADS  Google Scholar 

  58. Volovik, G.E., Gorkov, L.P.: Superconductivity classes in the heavy fermion systems. J. Exp. Theor. Phys. 61, 843–854 (1985)

    Google Scholar 

  59. Kaplan, D.B., Sun, S.: Spacetime as a topological insulator: mechanism for the origin of the fermion generations. Phys. Rev. Lett. 108, 181807 (2012)

    Article  ADS  Google Scholar 

  60. Volovik, G.E., Konyshev, V.A.: Properties of the superfluid systems with multiple zeros in fermion spectrum. JETP Lett. 47, 250–254 (1988)

    ADS  Google Scholar 

  61. Hořava, P.: Spectral dimension of the Universe in quantum gravity at a Lifshitz point. Phys. Rev. Lett. 102, 161301 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  62. Hořava: Quantum gravity at a Lifshitz point. Phys. Rev. D 79, 084008 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  63. Hořava, P.: Membranes at quantum criticality. J. High Energy Phys. 0903, 020 (2009). arXiv:0812.4287

    Article  ADS  Google Scholar 

  64. Xu, C., Hořava, P.: Emergent gravity at a Lifshitz point from a Bose liquid on the lattice. Phys. Rev. D 81, 104033 (2010)

    Article  ADS  Google Scholar 

  65. Lifshitz, E.M.: On the theory of second-order phase transitions I. Zh. Eksp. Teor. Fiz. 11, 255 (1941)

    Google Scholar 

  66. Lifshitz, E.M.: On the theory of second-order phase transitions II. Zh. Eksp. Teor. Fiz. 11, 269 (1941)

    Google Scholar 

  67. Volovik, G.E.: Reentrant violation of special relativity in the low-energy corner. JETP Lett. 73, 162–165 (2001). arXiv:hep-ph/0101286

    Article  ADS  Google Scholar 

  68. Dietl, P., Piechon, F., Montambaux, G.: New magnetic field dependence of Landau levels in a graphenelike structure. Phys. Rev. Lett. 100, 236405 (2008)

    Article  ADS  Google Scholar 

  69. Montambaux, G., Piechon, F., Fuchs, J.-N., Goerbig, M.O.: A universal Hamiltonian for motion and merging of Dirac points in a two-dimensional crystal. Eur. Phys. J. B 72, 509–520 (2009). arXiv:0907.0500

    Article  ADS  MATH  Google Scholar 

  70. deGail, R., Fuchs, J.-N., Goerbig, M.O., Piechon, F., Montambaux, G.: Manipulation of Dirac points in graphene-like crystals. Physica B 407, 1948–1952 (2012)

    Article  ADS  Google Scholar 

  71. Chong, Y.D., Wen, X.G., Soljacic, M.: Effective theory of quadratic degeneracies. Phys. Rev. B 77, 235125 (2008)

    Article  ADS  Google Scholar 

  72. Banerjee, S., Singh, R.R., Pardo, V., Pickett, W.E.: Tight-binding modeling and low-energy behavior of the semi-Dirac point. Phys. Rev. Lett. 103, 016402 (2009)

    Article  ADS  Google Scholar 

  73. Sun, K., Yao, H., Fradkin, E., Kivelson, S.A.: Topological insulators and nematic phases from spontaneous symmetry breaking in 2D Fermi systems with a quadratic band crossing. Phys. Rev. Lett. 103, 046811 (2009)

    Article  ADS  Google Scholar 

  74. Fu, L.: Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011)

    Article  ADS  Google Scholar 

  75. Wen, X.G., Zee, A.: Gapless fermions and quantum order. Phys. Rev. B 66, 235110 (2002)

    Article  ADS  Google Scholar 

  76. Beri, B.: Topologically stable gapless phases of time-reversal invariant superconductors. Phys. Rev. Lett. 103, 016402 (2009)

    Article  ADS  Google Scholar 

  77. McCann, E., Fal’ko, V.I.: Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006)

    Article  ADS  Google Scholar 

  78. Koshino, M., Ando, T.: Transport in bilayer graphene: calculations within a self-consistent Born approximation. Phys. Rev. B 73, 245403 (2006)

    Article  ADS  Google Scholar 

  79. Klinkhamer, F.R., Volovik, G.E.: Superluminal neutrino and spontaneous breaking of Lorentz invariance. Pis’ma Zh. Eksp. Teor. Fiz. 94, 731–733 (2011). arXiv:1109.6624

    Google Scholar 

  80. Klinkhamer, F.R.: OPREA’s superluminal muon-neutrino velocity and a Fermi-point-splitting model of Lorentz violation. arXiv:1109.5671

  81. Klinkhamer, F.R.: Superluminal neutrino, flavor, and relativity. arXiv:1110.2146

  82. Heikkilä, T.T., Volovik, G.E.: Fermions with cubic and quartic spectrum. Pis’ma Zh. Eksp. Teor. Fiz. 92, 751–756 (2010). JETP Lett. 92, 681–686 (2010). arXiv:1010.0393

    Google Scholar 

  83. Guinea, F., Castro Neto, A.H., Peres, N.M.R.: Electronic states and Landau levels in graphene stacks. Phys. Rev. B 73, 245426 (2006)

    Article  ADS  Google Scholar 

  84. Mak, K.F., Shan, J., Heinz, T.F.: Electronic structure of few-layer graphene: experimental demonstration of strong dependence on stacking sequence. Phys. Rev. Lett. 104, 176404 (2010)

    Article  ADS  Google Scholar 

  85. Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)

    Article  ADS  Google Scholar 

  86. Heikkilä, T.T., Volovik, G.E.: Dimensional crossover in topological matter: evolution of the multiple Dirac point in the layered system to the flat band on the surface. Pis’ma Zh. Eksp. Teor. Fiz. 93, 63–68 (2011). JETP Lett. 93, 59–65 (2011). arXiv:1011.4185

    Google Scholar 

  87. Altarelli, G., Feruglio, F.: Discrete flavor symmetries and models of neutrino mixing. Rev. Mod. Phys. 82, 2701–2729 (2010)

    Article  ADS  Google Scholar 

  88. Heikkilä, T.T., Kopnin, N.B., Volovik, G.E.: Flat bands in topological media. Pis’ma Zh. Eksp. Teor. Fiz. 94, 252–258 (2011). JETP Lett. 94, 233–239 (2011). arXiv:1012.0905

    Google Scholar 

  89. Andersen, J.O., Haugset, T.: Magnetization in (2+1)-dimensional QED at finite temperature and density. Phys. Rev. D 51, 3073–3080 (1995)

    Article  ADS  Google Scholar 

  90. Katsnelson, M.I., Volovik, G.E.: Quantum electrodynamics with anisotropic scaling: Heisenberg-Euler action and Schwinger pair production in the bilayer graphene. JETP Lett. 95, 411–415 (2012). arXiv:1203.1578

    Article  ADS  Google Scholar 

  91. Zubkov, M.A.: Schwinger pair creation in multilayer graphene. Pis’ma Zh. Eksp. Teor. Fiz. 95, 540–543 (2012). arXiv:1204.0138

    Google Scholar 

  92. Kane, C.L., Mele, E.: Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005)

    Article  ADS  Google Scholar 

  93. Volkov, B.A., Gorbatsevich, A.A., Kopaev, Yu.V., Tugushev, V.V.: Macroscopic current states in crystals. J. Exp. Theor. Phys. 54, 391–397 (1981)

    Google Scholar 

  94. Volkov, B.A., Pankratov, O.A.: Two-dimensional massless electrons in an inverted contact. JETP Lett. 42, 178–181 (1985)

    ADS  Google Scholar 

  95. Volovik, G.E.: Fermion zero modes at the boundary of superfluid 3He-B. Pis’ma Zh. Eksp. Teor. Fiz. 90, 440–442 (2009). JETP Lett. 90, 398–401 (2009). arXiv:0907.5389

    Google Scholar 

  96. Volovik, G.E.: Exotic Properties of Superfluid 3He. World Scientific, Singapore (1992)

    Book  Google Scholar 

  97. Volovik, G.E.: Fractional statistics and analogs of quantum Hall effect in superfluid 3He films. In: Ihas, G.G., Takano, Y. (eds.) Quantum Fluids and Solids. AIP Conference Proceedings, vol. 194, pp. 136–146 (1989)

    Google Scholar 

  98. Mineev, V.P., Volovik, G.E.: Planar and linear solitons in superfluid 3He. Phys. Rev. B 18, 3197–3203 (1978)

    Article  ADS  Google Scholar 

  99. Salomaa, M.M., Volovik, G.E.: Quantized vortices in superfluid 3He. Rev. Mod. Phys. 59, 533–613 (1987)

    Article  ADS  Google Scholar 

  100. Volovik, G.E.: Topological invariant for superfluid 3He-B and quantum phase transitions. Pis’ma Zh. Eksp. Teor. Fiz. 90, 639–643 (2009). JETP Lett. 90, 587–591 (2009). arXiv:0909.3084

    Google Scholar 

  101. Volovik, G.E.: Topological superfluid 3He-B in magnetic field and Ising variable. JETP Lett. 91, 201–205 (2010). arXiv:1001.1514

    Article  ADS  Google Scholar 

  102. Gurarie, V., Radzihovsky, L.: Resonantly-paired fermionic superfluids. Ann. Phys. 322, 2–119 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  103. Jackiw, R., Rebbi, C.: Solitons with fermion number 1/2. Phys. Rev. D 13, 3398–3409 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  104. Nishida, Y.: Is a color superconductor topological? Phys. Rev. D 81, 074004 (2010)

    Article  ADS  Google Scholar 

  105. Ohsaku, T.: BCS and generalized BCS superconductivity in relativistic quantum field theory: formulation. Phys. Rev. B 65, 024512 (2001)

    Article  ADS  Google Scholar 

  106. Nishida, Y., Santos, L., Chamon, C.: Topological superconductors as nonrelativistic limits of Jackiw-Rossi and Jackiw-Rebbi models. arXiv:1007.2201

  107. Rombouts, S.M.A., Dukelsky, J., Ortiz, G.: Quantum phase diagram of the integrable p x +ip y fermionic superfluid. arXiv:1008.3406

  108. Jackiw, R., Rossi, P.: Zero modes of the vortex-fermion system. Nucl. Phys. B 190, 681–691 (1981)

    Article  ADS  Google Scholar 

  109. Volovik, G.E.: Localized fermions on quantized vortices in superfluid 3He-B. J. Phys. Condens. Matter 3, 357–368 (1991)

    Article  ADS  Google Scholar 

  110. Teo, J.C.Y., Kane, C.L.: Majorana fermions and non-Abelian statistics in three dimensions. Phys. Rev. Lett. 104, 046401 (2010)

    Article  ADS  Google Scholar 

  111. Teo, J.C.Y., Kane, C.L.: Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B 82, 115120 (2010)

    Article  ADS  Google Scholar 

  112. Lu, C.-K., Herbut, I.F.: Pairing symmetry and vortex zero-mode for superconducting Dirac fermions. Phys. Rev. B 82, 144505 (2010)

    Article  ADS  Google Scholar 

  113. Caroli, C., de Gennes, P.G., Matricon, J.: Phys. Lett. 9, 307 (1964)

    Article  ADS  MATH  Google Scholar 

  114. Volovik, G.E.: Vortex motion in Fermi superfluids and Callan-Harvey effect. JETP Lett. 57, 244–248 (1993)

    ADS  Google Scholar 

  115. Misirpashaev, T.Sh., Volovik, G.E.: Fermion zero modes in symmetric vortices in superfluid 3He. Physica B 210, 338–346 (1995)

    Article  ADS  Google Scholar 

  116. Mizushima, T., Machida, K.: Vortex structures and zero-energy states in the BCS-to-BEC evolution of p-wave resonant Fermi gases. Phys. Rev. A 81, 053605 (2010)

    Article  ADS  Google Scholar 

  117. Lutchyn, R.M., Sau, J.D., Das Sarma, S.: Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010)

    Article  ADS  Google Scholar 

  118. Oreg, Y., Refael, G., von Oppen, F.: Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010)

    Article  ADS  Google Scholar 

  119. Beenakker, C.W.J.: Search for Majorana fermions in superconductors. arXiv:1112.1950

  120. Shiozaki, K., Fujimoto, S.: Green’s function method for line defects and gapless modes in topological insulators: beyond semiclassical approach. arXiv:1111.1685

  121. Wang, Z., Qi, X.-L., Zhang, S.-C.: General theory of interacting topological insulators. arXiv:1004.4229

  122. Kaplan, D.B.: Method for simulating chiral fermions on the lattice. Phys. Lett. B 288, 342–347 (1992). arXiv:hep-lat/9206013

    Article  MathSciNet  ADS  Google Scholar 

  123. Golterman, M.F.L., Jansen, K., Kaplan, D.B.: Chern-Simons currents and chiral fermions on the lattice. Phys. Lett. B 301, 219–223 (1993). arXiv:hep-lat/9209003

    Article  ADS  Google Scholar 

  124. Zubkov, M.A., Volovik, G.E.: Topological invariants for the 4D systems with mass gap. Nucl. Phys. B 860(2), 295–309 (2012). arXiv:1201.4185

    Article  MathSciNet  ADS  MATH  Google Scholar 

  125. Kopnin, N.B., Salomaa, M.M.: Mutual friction in superfluid 3He: effects of bound states in the vortex core. Phys. Rev. B 44, 9667–9677 (1991)

    Article  ADS  Google Scholar 

  126. Akama, K.: An attempt at pregeometry—gravity with composite metric. Prog. Theor. Phys. 60, 1900 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  127. Volovik, G.E.: Superfluid 3He-B and gravity. Physica B 162, 222 (1990)

    Article  ADS  Google Scholar 

  128. Wetterich, C.: Gravity from spinors. Phys. Rev. D 70, 105004 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  129. Diakonov, D.: Towards lattice-regularized quantum gravity. arXiv:1109.0091

Download references

Acknowledgements

This work is supported in part by the Academy of Finland and its COE program 2006–2011.

Author information

Authors and Affiliations

  1. Low Temperature Laboratory, Aalto University, P.O. Box 15100, 00076, Aalto, Finland

    Grigorii E. Volovik

  2. L.D. Landau Institute for Theoretical Physics, Kosygina 2, 119334, Moscow, Russia

    Grigorii E. Volovik

Authors
  1. Grigorii E. Volovik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Grigorii E. Volovik .

Editor information

Editors and Affiliations

  1. School of Engineering and Physical Science, Heriot-Watt University, Edinburgh, United Kingdom

    Daniele Faccio

  2. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Francesco Belgiorno

  3. Dipartimento di Fisica e Matematica, Universita’ dell’Insubria, Como, Italy

    Sergio Cacciatori

  4. Dipartimento di Fisica e Matematica, Universita’ dell’Insubria, Como, Italy

    Vittorio Gorini

  5. Int. School for Adv. Studies (SISSA), Trieste, Italy

    Stefano Liberati

  6. Dipartimento di Fisica e Matematica, Universita’ dell’Insubria, Como, Italy

    Ugo Moschella

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Volovik, G.E. (2013). The Topology of the Quantum Vacuum. In: Faccio, D., Belgiorno, F., Cacciatori, S., Gorini, V., Liberati, S., Moschella, U. (eds) Analogue Gravity Phenomenology. Lecture Notes in Physics, vol 870. Springer, Cham. https://doi.org/10.1007/978-3-319-00266-8_14

Download citation

Publish with us

Policies and ethics

Search

Navigation