Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quantum magnetism and criticality

Abstract

Magnetic insulators have proved to be fertile ground for studying new types of quantum many-body state, and I survey recent experimental and theoretical examples. The insights and methods also transfer to novel superconducting and metallic states. Of particular interest are critical quantum states, sometimes found at quantum phase transitions, which have gapless excitations with no particle- or wave-like interpretation, and control a significant portion of the finite-temperature phase diagram. Remarkably, their theory is connected to holographic descriptions of Hawking radiation from black holes.

This is a preview of subscription content

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Néel states.
Figure 2: The coupled-dimer antiferromagnet.
Figure 3: Valence-bond solid states.
Figure 4: Caricature of a spin-liquid state.
Figure 5: Histogram of the VBS order parameter.

© 2007 APS

Figure 6: Bosons with repulsive interactions on a square lattice at filling f=1.
Figure 7: Bosons with repulsive interactions on a square lattice at filling f=1/2.
Figure 8: Structures of possible insulating states on the square lattice for paired electrons at a hole density of x=1/8 (f=1/16).

© 2005 APS

Figure 9: Phase diagram of the superfluid–insulator transition in two dimensions.
Figure 10: Superfluid–insulator transition applied to the cuprate superconductors.

© 2007 AAAS

References

  1. Einstein, A., Podolsky, B. & Rosen, N. Can quantum mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935).

    ADS  Article  Google Scholar 

  2. Matsumoto, M., Yasuda, C., Todo, S. & Takayama, H. Ground-state phase diagram of quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice. Phys. Rev. B 65, 014407 (2002).

    ADS  Article  Google Scholar 

  3. Chakravarty, A., Halperin, B. I. & Nelson, D. R. Two-dimensional quantum Heisenberg antiferromagnet at low temperatures. Phys. Rev. B 39, 2344–2371 (1989).

    ADS  Article  Google Scholar 

  4. Chubukov, A. V., Sachdev, S. & Ye, J. Theory of two-dimensional quantum antiferromagnets with a nearly-critical ground state. Phys. Rev. B 49, 11919–11961 (1994).

    ADS  Article  Google Scholar 

  5. Coldea, R., Tennant, D. A. & Tylczynski, Z. Extended scattering continua characteristic of spin fractionalization in the two-dimensional frustrated quantum magnet Cs2CuCl4 observed by neutron scattering. Phys. Rev. B 68, 134424 (2003).

    ADS  Article  Google Scholar 

  6. Bernu, B., Lecheminant, P., Lhuillier, C. & Pierre, L. Exact spectra, spin susceptibilities, and order parameter of the quantum Heisenberg antiferromagnet on the triangular lattice. Phys. Rev. B 50, 10048–10062 (1994).

    ADS  Article  Google Scholar 

  7. Chubukov, A. V., Senthil, T. & Sachdev, S. Universal magnetic properties of frustrated quantum antiferromagnets in two dimensions. Phys. Rev. Lett. 72, 2089–2092 (1994).

    ADS  Article  Google Scholar 

  8. Oosawa, A., Fujisawa, M., Osakabe, T., Kakurai, K. & Tanaka, H. Neutron diffraction study of the pressure-induced magnetic ordering in the spin gap system TlCuCl3 . J. Phys. Soc. Japan 72, 1026–1029 (2003).

    ADS  Article  Google Scholar 

  9. Rüegg, Ch. et al. Bose–Einstein condensation of the triplet states in the magnetic insulator TlCuCl3 . Nature 423, 62–65 (2003).

    ADS  Article  Google Scholar 

  10. Jaime, M. et al. Magnetic-field-induced condensation of triplons in Han purple pigment BaCuSi2O6 . Phys. Rev. Lett. 93, 087203 (2004).

    ADS  Article  Google Scholar 

  11. Cavadini, N. et al. Magnetic excitations in the quantum spin system TlCuCl3 . Phys. Rev. B 63, 172414 (2001).

    ADS  Article  Google Scholar 

  12. Xu, G. et al. Mesoscopic phase coherence in a quantum spin fluid. Science 317, 1049–1052 (2007).

    ADS  Article  Google Scholar 

  13. Tamura, M., Nakao, A. & Kato, R. Frustration-induced valence-bond ordering in a new quantum triangular antiferromagnet based on [Pd(dmit)2]. J. Phys. Soc. Japan 75, 093701 (2006).

    ADS  Article  Google Scholar 

  14. Lee, S.-H. et al. Quantum-spin-liquid states in the two-dimensional kagome antiferromagnets ZnxCu4−x(OD)6Cl2 . Nature Mater. 6, 853–857 (2007).

    ADS  Article  Google Scholar 

  15. Kohsaka, Y. et al. An intrinsic bond-centered electronic glass with unidirectional domains in underdoped cuprates. Science 315, 1380–1385 (2007).

    ADS  Article  Google Scholar 

  16. Raczkowski, M., Capello, M., Poilblanc, D., Frésard, R. & Olés, A. M. Unidirectional d-wave superconducting domains in the two-dimensional t-J model. Phys. Rev. B 76, 140505(R) (2007).

    ADS  Article  Google Scholar 

  17. Vojta, M. & Rösch, O. Superconducting d-wave stripes in cuprates: Valence bond order coexisting with nodal quasiparticles. Preprint at <http://arxiv.org/abs/cond-mat/0709.4244> (2007).

  18. Sachdev, S. & Read, N. Large N expansion for frustrated and doped quantum antiferromagnets. Int. J. Mod. Phys. B 5, 219–249 (1991).

    ADS  Article  Google Scholar 

  19. Sandvik, A. W. Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions. Phys. Rev. Lett. 98, 227202 (2007).

    ADS  Article  Google Scholar 

  20. Melko, R. G. & Kaul, R. K. Universal scaling in the fan of an unconventional quantum critical point. Phys. Rev. Lett. 100, 017203 (2008).

    ADS  Article  Google Scholar 

  21. Jiang, F.-J., Nyfeler, M., Chandrasekharan, S. & Wiese, U.-J. From an antiferromagnet to a valence bond solid: Evidence for a first order phase transition. Preprint at <http://arxiv.org/abs/cond-mat/0710.3926> (2007).

  22. Melko, R. G., Sandvik, A. W. & Scalapino, D. J. Two-dimensional quantum XY model with ring exchange and external field. Phys. Rev. B 69, 100408(R) (2004).

    ADS  Article  Google Scholar 

  23. Read, N. & Sachdev, S. Valence bond and spin-Peierls ground states of low-dimensional quantum antiferromagnets. Phys. Rev. Lett. 62, 1694–1697 (1989) ibid Spin-Peierls, valence-bond solid, and Néel ground-states of low-dimensional quantum antiferromagnets. Phys. Rev. B 42, 4568–4569 (1990).

    ADS  Article  Google Scholar 

  24. Motrunich, O. I. & Vishwanath, A. Emergent photons and transitions in the O(3) sigma model with hedgehog suppression. Phys. Rev. B 70, 075104 (2004).

    ADS  Article  Google Scholar 

  25. Fradkin, E. & Kivelson, S. Short range resonating valence bond theories and superconductivity. Mod. Phys. Lett. B 4, 225–232 (1990).

    ADS  Article  Google Scholar 

  26. Polyakov, A. M. Gauge Fields and Strings (Harwood Academic, New York, 1987).

    Google Scholar 

  27. Haldane, F. D. M. O(3) Nonlinear σ model and the topological distinction between integer- and half-integer-spin antiferromagnets in two dimensions. Phys. Rev. Lett. 61, 1029–1032 (1988).

    ADS  MathSciNet  Article  Google Scholar 

  28. Senthil, T., Balents, L., Sachdev, S, Vishwanath, A. & Fisher, M. P. A. Quantum criticality beyond the Landau–Ginzburg–Wilson paradigm. Phys. Rev. B 70, 144407 (2004).

    ADS  Article  Google Scholar 

  29. Read, N. & Sachdev, S. Large N expansion for frustrated quantum antiferromagnets. Phys. Rev. Lett. 66, 1773–1776 (1991).

    ADS  Article  Google Scholar 

  30. Sachdev, S. Kagome and triangular lattice Heisenberg antiferromagnets: ordering from quantum fluctuations and quantum-disordered ground states with deconfined bosonic spinons. Phys. Rev. B 45, 12377–12396 (1992).

    ADS  Article  Google Scholar 

  31. Wen, X.-G. Mean-field theory of spin-liquid states with finite energy gap and topological orders. Phys. Rev. B 44, 2664–2672 (1991).

    ADS  Article  Google Scholar 

  32. Jalabert, R. & Sachdev, S. Spontaneous alignment of frustrated bonds in an anisotropic, three dimensional Ising model. Phys. Rev. B 44, 686–690 (1991).

    ADS  Article  Google Scholar 

  33. Sachdev, S. & Vojta, M. Translational symmetry breaking in two-dimensional antiferromagnets and superconductors. J. Phys. Soc. Japan 69 (Suppl. B), 1–9 (2000).

    Google Scholar 

  34. Bais, F. A. Flux metamorphosis. Nucl. Phys. B 170, 32–43 (1980).

    ADS  MathSciNet  Article  Google Scholar 

  35. Bais, F. A., van Driel, P. & de Wild Propitius, M. Quantum symmetries in discrete gauge theories. Phys. Lett. B 280, 63–70 (1992).

    ADS  MathSciNet  Article  Google Scholar 

  36. Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    ADS  MathSciNet  Article  Google Scholar 

  37. Wang, C., Harrington, J. & Preskill, J. Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory. Ann. Phys. 303, 31–58 (2003).

    ADS  Article  Google Scholar 

  38. Wen, X.-G. Quantum orders in an exact soluble model. Phys. Rev. Lett. 90, 016803 (2003).

    ADS  Article  Google Scholar 

  39. Senthil, T. & Fisher, M. P. A. Z2 gauge theory of electron fractionalization in strongly correlated systems. Phys. Rev. B 62, 7850–7881 (2000).

    ADS  Article  Google Scholar 

  40. Moessner, R. & Sondhi, S. L. Resonating valence bond phase in the triangular lattice quantum dimer model. Phys. Rev. Lett. 86, 1881–1884 (2001).

    ADS  Article  Google Scholar 

  41. Freedman, M., Nayak, C., Shtengel, K., Walker, K. & Wang, Z. A class of P,T-invariant topological phases of interacting electrons. Ann. Phys. 310, 428–492 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  42. Rüegg, Ch. et al. Pressure-controlled quantum fluctuations and elementary excitations in quantum magnets (preprint).

  43. Liu, K.-S. & Fisher, M. E. Quantum lattice gas and the existence of a supersolid. J. Low. Temp. Phys. 10, 655–683 (1973).

    ADS  Article  Google Scholar 

  44. Senthil, T., Vishwanath, A., Balents, L., Sachdev, S. & Fisher, M. P. A. Deconfined quantum critical points. Science 303, 1490–1494 (2004).

    ADS  Article  Google Scholar 

  45. Kuklov, A. B., Prokofev, N. V., Svistunov, B. V. & Troyer, M. Deconfined criticality, runaway flow in the two-component scalar electrodynamics and weak first-order superfluid-solid transitions. Ann. Phys. 321, 1602–1621 (2006).

    ADS  Article  Google Scholar 

  46. Nogueira, F. S., Kragset, S. & Sudbo, A. Quantum critical scaling behaviour of deconfined spinons. Phys. Rev. B 76, 220403(R) (2007).

    ADS  Article  Google Scholar 

  47. Rantner, W. & Wen, X.-G. Electron spectral function and algebraic spin liquid for the normal state of underdoped high T c superconductors. Phys. Rev. Lett. 86, 3871–3874 (2001).

    ADS  Article  Google Scholar 

  48. Affleck, I. & Marston, J. B. Large-n limit of the Heisenberg-Hubbard model: Implications for high-T c superconductors. Phys. Rev. B 37, 3774–3777 (1988).

    ADS  Article  Google Scholar 

  49. Hermele, M. et al. Stability of U(1) spin liquids in two dimensions. Phys. Rev. B 70, 214437 (2004).

    ADS  Article  Google Scholar 

  50. Hermele, M., Senthil, T. & Fisher, M. P. A. Algebraic spin liquid as the mother of many competing orders. Phys. Rev. B 72, 104404 (2005).

    ADS  Article  Google Scholar 

  51. Ran, Y., Hermele, M., Lee, P. A. & Wen, X.-G. Projected wavefunction study of spin-1/2 Heisenberg model on the Kagome lattice. Phys. Rev. Lett. 98, 117205 (2007).

    ADS  Article  Google Scholar 

  52. Bloch, I. Ultracold quantum gases in optical lattices. Nature Phys. 1, 23–30 (2005).

    ADS  Article  Google Scholar 

  53. Fisher, M. P. A., Weichmann, P. B., Grinstein, G. & Fisher, D. S. Boson localization and the superfluid–insulator transition. Phys. Rev. B 40, 546–570 (1989).

    ADS  Article  Google Scholar 

  54. Balents, L., Bartosch, L., Burkov, A., Sachdev, S. & Sengupta, K. Putting competing orders in their place near the Mott transition. Phys. Rev. B 71, 144508 (2005).

    ADS  Article  Google Scholar 

  55. Lannert, L., Fisher, M. P. A. & Senthil, T. Quantum confinement transition in a d-wave superconductor. Phys. Rev. B 63, 134510 (2001).

    ADS  Article  Google Scholar 

  56. Balents, L. & Sachdev, S. Dual vortex theory of doped antiferromagnets. Ann. Phys. 322, 2635–2664 (2007).

    ADS  Article  Google Scholar 

  57. Tranquada, J. M. et al. Quantum magnetic excitations from stripes in copper oxide superconductors. Nature 429, 534–538 (2004).

    ADS  Article  Google Scholar 

  58. Senthil, T., Sachdev, S. & Vojta, M. Fractionalized Fermi liquids. Phys. Rev. Lett. 90, 216403 (2003).

    ADS  Article  Google Scholar 

  59. Senthil, T., Sachdev, S. & Vojta, M. Quantum phase transitions out of the heavy Fermi liquid. Physica B 359–361, 9–16 (2005).

    ADS  Article  Google Scholar 

  60. Kaul, R. K., Kim, Y. B., Sachdev, S. & Senthil, T. Algebraic charge liquids. Nature Phys. 4, 28–31 (2008).

    ADS  Article  Google Scholar 

  61. Damle, K. & Sachdev, S. Non-zero temperature transport near quantum critical points. Phys. Rev. B 56, 8714–8733 (1997).

    ADS  Article  Google Scholar 

  62. Giamarchi, T. Umklapp process and resistivity in one-dimensional fermion systems. Phys. Rev. B 44, 2905–2913 (1991).

    ADS  Article  Google Scholar 

  63. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 1999).

    MATH  Google Scholar 

  64. Zaanen, J. A black hole full of answers. Nature 448, 1000–1001 (2007).

    ADS  Article  Google Scholar 

  65. Strominger, A. & Vafa, C. Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B 379, 99–104 (1996).

    ADS  MathSciNet  Article  Google Scholar 

  66. Maldacena, J. M. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  67. Gubser, S. S., Klebanov, I. R. & Polyakov, A. M. Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105–114 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  68. Witten, E. Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–290 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  69. Policastro, G., Son, D. T. & Starinets, A. O. From AdS/CFT correspondence to hydrodynamics. JHEP 0209, 043 (2002).

    ADS  MathSciNet  Article  Google Scholar 

  70. Herzog, C. P., Kovtun, P. K., Sachdev, S. & Son, D. T. Quantum critical transport, duality, and M-theory. Phys. Rev. D 75, 085020 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  71. Herzog, C. P. The hydrodynamics of M-theory. JHEP 0212, 026 (2002).

    ADS  MathSciNet  Article  Google Scholar 

  72. Damle, K. & Sachdev, S. Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures. Phys. Rev. B 57, 8307–8339 (1998).

    ADS  Article  Google Scholar 

  73. Wang, Y. & Li, L. Ong, N. P. Nernst effect in high-T c superconductors. Phys. Rev. B 73, 024510 (2006).

    ADS  Article  Google Scholar 

  74. Müller, M. & Sachdev, S. Collective cyclotron motion of the relativistic plasma in graphene. Preprint at <http://arxiv.org/abs/cond-mat/0801.2970> (2008).

  75. Son, D. T. Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction. Phys. Rev. B 75, 235423 (2007).

    ADS  Article  Google Scholar 

  76. Hartnoll, S. A., Kovtun, P. K., Müller, M. & Sachdev, S. Theory of the Nernst effect near quantum phase transitions in condensed matter, and in dyonic black holes. Phys. Rev. B 76, 144502 (2007).

    ADS  Article  Google Scholar 

  77. Kadanoff, L. P. & Martin, P. C. Hydrodynamic equations and correlation functions. Ann. Phys. 24, 419–469 (1963).

    ADS  MathSciNet  Article  Google Scholar 

  78. Landau, L. D. & Lifshitz, E. M. Fluid Mechanics, Section 127 (Butterworth-Heinemann, Oxford, 1987).

    Google Scholar 

  79. Hartnoll, S. A. & Herzog, C. P. Impure AdS/CFT. Preprint at <http://arxiv.org/abs/cond-mat/0801.1693> (2008).

  80. Hartnoll, S. A. & Kovtun, P. K. Hall conductivity from dyonic black holes. Phys. Rev. D 76, 066001 (2007).

    ADS  Article  Google Scholar 

  81. Doiron-Leyraud, N. et al. Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor. Nature 447, 565–568 (2007).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

I thank S. Bais, S. Hartnoll, R. Kaul, R. Melko, M. Metlitski, M. Müller and A. Sandvik for valuable comments. This work was supported by NSF Grant No. DMR-0537077.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Subir Sachdev.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sachdev, S. Quantum magnetism and criticality. Nature Phys 4, 173–185 (2008). https://doi.org/10.1038/nphys894

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys894

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing