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.

  • Review Article
  • Published:

Quantum criticality in heavy-fermion metals

Nature Physics volume 4, pages 186–197 (2008)Cite this article

Abstract

Quantum criticality describes the collective fluctuations of matter undergoing a second-order phase transition at zero temperature. Heavy-fermion metals have in recent years emerged as prototypical systems to study quantum critical points. There have been considerable efforts, both experimental and theoretical, that use these magnetic systems to address problems that are central to the broad understanding of strongly correlated quantum matter. Here, we summarize some of the basic issues, including the extent to which the quantum criticality in heavy-fermion metals goes beyond the standard theory of order-parameter fluctuations, the nature of the Kondo effect in the quantum-critical regime, the non-Fermi-liquid phenomena that accompany quantum criticality and the interplay between quantum criticality and unconventional superconductivity.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Quantum critical points in HF metals.
Figure 2: Schematic phase diagrams showing two classes of QCPs.
Figure 3: Divergence of the dimensionless critical Grüneisen ratio at QCPs.
Figure 4: Thermodynamic and transport properties close to the QCP in YbRh2Si2.
Figure 5: Evidence for an additional low-energy scale in the Hall-effect and thermodynamic data of YbRh2Si2.
Figure 6: The temperature-versus-magnetic-field phase diagram for YbRh2Si2.
Figure 7: Pressure dependence of magnetic order and superconductivity in CeCu2(Si1−xGex)2.
Figure 8: Changes of Fermi surface properties across a likely QCP in CeRhIn5.

Similar content being viewed by others

References

  1. Rosenberg, R. Why is ice slippery? Phys. Today 58, 50–55 (2005).

    Article  ADS  Google Scholar 

  2. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, New York, 1999).

    MATH  Google Scholar 

  3. Coleman, P. & Schofield, A. J. Quantum criticality. Nature 433, 226–229 (2005).

    Article  ADS  Google Scholar 

  4. Stewart, G. R. Non-Fermi-liquid behavior in d- and f-electron metals. Rev. Mod. Phys. 73, 797–855 (2001) 78, 743–753 (2006).

    Article  ADS  Google Scholar 

  5. Löhneysen, H. v., Rosch, A., Vojta, M. & Wölfle, P. Fermi-liquid instabilities at magnetic quantum phase transitions. Rev. Mod. Phys. 79, 1015–1075 (2007).

    Article  ADS  Google Scholar 

  6. Löhneysen, H. v. et al. Non-Fermi-liquid behavior in a heavy-fermion alloy at a magnetic instability. Phys. Rev. Lett. 72, 3262–3265 (1994).

    Article  ADS  Google Scholar 

  7. Schröder, A. et al. Onset of antiferromagnetism in heavy-fermion metals. Nature 407, 351–355 (2000).

    Article  ADS  Google Scholar 

  8. Trovarelli, O. et al. YbRh2Si2: Pronounced non-Fermi-liquid effects above a low-lying magnetic phase transition. Phys. Rev. Lett. 85, 626–629 (2000).

    Article  ADS  Google Scholar 

  9. Hertz, J. A. Quantum critical phenomena. Phys. Rev. B 14, 1165–1184 (1976).

    Article  ADS  Google Scholar 

  10. Millis, A. J. Effect of a nonzero temperature on quantum critical points in itinerant fermion systems. Phys. Rev. B 48, 7183–7196 (1993).

    Article  ADS  Google Scholar 

  11. Moriya, T. & Takimoto, T. Anomalous properties around magnetic instability in heavy electron systems. J. Phys. Soc. Jpn 64, 960–969 (1995).

    Article  ADS  Google Scholar 

  12. Si, Q., Rabello, S., Ingersent, K. & Smith, J. L. Locally critical quantum phase transitions in strongly correlated metals. Nature 413, 804–808 (2001).

    Article  ADS  Google Scholar 

  13. Coleman, P., Pépin, C., Si, Q. & Ramazashvili, R. How do Fermi liquids get heavy and die? J. Phys. Condens. Matter 13, R723–R738 (2001).

    Article  ADS  Google Scholar 

  14. Senthil, T., Vojta, M. & Sachdev, S. Weak magnetism and non-Fermi liquids near heavy-fermion critical points. Phys. Rev. B 69, 035111 (2004).

    Article  ADS  Google Scholar 

  15. Andres, K., Graebner, J. E. & Ott, H. R. 4f-virtual-bound-state formation in CeAl3 at low temperatures. Phys. Rev. Lett. 35, 1779–1782 (1975).

    Article  ADS  Google Scholar 

  16. Steglich, F. et al. Superconductivity in the presence of strong Pauli paramagnetism: CeCu2Si2 . Phys. Rev. Lett. 43, 1892–1896 (1979).

    Article  ADS  Google Scholar 

  17. Assmus, W. et al. Superconductivity in CeCu2Si2 single crystals. Phys. Rev. Lett. 52, 469–472 (1984).

    Article  ADS  Google Scholar 

  18. Ott, H. R., Rudigier, H., Fisk, Z. & Smith, J. L. UBe13: An unconventional actinide superconductor. Phys. Rev. Lett. 50, 1595–1598 (1983).

    Article  ADS  Google Scholar 

  19. Stewart, G. R., Fisk, Z., Willis, J. O. & Smith, J. L. Possibility of coexistence of bulk superconductivity and spin fluctuations in UPt3 . Phys. Rev. Lett. 52, 679–682 (1984).

    Article  ADS  Google Scholar 

  20. Schlabitz, W. et al. Superconductivity and magnetic order in a strongly interacting Fermi system: URu2Si2 . Z. Phys. B 62, 171–177 (1986).

    Article  ADS  Google Scholar 

  21. Grewe, N. & Steglich, F. in Handbook on the Physics and Chemistry of Rare Earths Vol. 14 (eds Gschneidner, K. A. Jr & Eyring, L.) 343–474 (Elsevier, Amsterdam, 1991).

    Google Scholar 

  22. Kondo, J. Resistance minimum in dilute magnetic alloys. Prog. Theor. Phys. 32, 37–49 (1964).

    Article  ADS  Google Scholar 

  23. Hewson, A. C. The Kondo Problem to Heavy Fermions (Cambridge Univ. Press, Cambridge, 1993).

    Book  Google Scholar 

  24. Kotliar, G. & Vollhardt, D. Strongly correlated materials: Insights from dynamical mean-field theory. Phys. Today 57, 53–59 (2004).

    Article  Google Scholar 

  25. Doniach, S. The Kondo lattice and weak antiferromagnetism. Physica B+C 91, 231–234 (1977).

    Article  ADS  Google Scholar 

  26. Varma, C. M. Mixed-valence compounds. Rev. Mod. Phys. 48, 219–238 (1976).

    Article  ADS  Google Scholar 

  27. Seaman, C. L. et al. Evidence for non-Fermi liquid behavior in the Kondo alloy Y1−xUxPd3 . Phys. Rev. Lett. 67, 2882–2885 (1991).

    Article  ADS  Google Scholar 

  28. Andraka, B. & Tsvelik, A. M. Observation of non-Fermi-liquid behavior in U0.2Y0.8Pd3 . Phys. Rev. Lett. 67, 2886–2889 (1991).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  30. Continentino, M. A., Japiassu, G. M. & Troper, A. Critical approach to the coherence transition in Kondo lattices. Phys. Rev. B 39, 9734–9737 (1993).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  32. Si, Q., Rabello, S., Ingersent, K. & Smith, J. L. Local fluctuations in quantum critical metals. Phys. Rev. B 68, 115103 (2003).

    Article  ADS  Google Scholar 

  33. Burdin, S., Grempel, D. R. & Georges, A. Heavy-fermion and spin-liquid behavior in a Kondo lattice with magnetic frustration. Phys. Rev. B 66, 045111 (2002).

    Article  ADS  Google Scholar 

  34. Steglich, F. in Festkörperprobleme (Adv. Solid State Physics) Vol. XVII (ed. Treusch, J.) 319–350 (Vieweg, Braunschweig, 1977).

    Google Scholar 

  35. Melnikov, V. I. Thermodynamics of the Kondo problem. Sov. Phys. JETP Lett. 35, 511–515 (1982).

    ADS  Google Scholar 

  36. Zhu, J.-X., Grempel, D. R. & Si, Q. Continuous quantum phase transition in a Kondo lattice model. Phys. Rev. Lett. 91, 156404 (2003).

    Article  ADS  Google Scholar 

  37. Glossop, M. T. & Ingersent, K. Magnetic quantum phase transition in an anisotropic Kondo lattice. Phys. Rev. Lett. 99, 227203 (2007).

    Article  ADS  Google Scholar 

  38. Zhu, J.-X., Kirchner, S., Bulla, R. & Si, Q. Zero-temperature magnetic transition in an easy-axis Kondo lattice model. Phys. Rev. Lett. 99, 227204 (2007).

    Article  ADS  Google Scholar 

  39. Zhu, L., Garst, M., Rosch, A. & Si, Q. Universally diverging Grüneisen parameter and the magnetocaloric effect close to quantum critical points. Phys. Rev. Lett. 91, 066404 (2003).

    Article  ADS  Google Scholar 

  40. Garst, M. & Rosch, A. Sign change of the Grüneisen parameter and magnetocaloric effect near quantum critical points. Phys. Rev. B 72, 205129 (2005).

    Article  ADS  Google Scholar 

  41. Küchler, R. et al. Divergence of the Grüneisen ratio at quantum critical points in heavy fermion metals. Phys. Rev. Lett. 91, 066405 (2003).

    Article  ADS  Google Scholar 

  42. Küchler, R. et al. Quantum criticality in the cubic heavy-fermion system CeIn3−xSnx . Phys. Rev. Lett. 96, 256403 (2006).

    Article  ADS  Google Scholar 

  43. Gegenwart, P., Weickert, F., Garst, M., Perry, R. S. & Maeno, Y. Metamagnetic quantum criticality in Sr3Ru2O7 studied by thermal expansion. Phys. Rev. Lett. 96, 136402 (2006).

    Article  ADS  Google Scholar 

  44. Gegenwart, P. et al. Multiple energy scales at a quantum critical point. Science 315, 969–971 (2007).

    Article  ADS  Google Scholar 

  45. Zhu, L. Quantum Phase Transitions in Strongly Correlated Metals. Thesis, Rice Univ. (2005).

  46. Pépin, C. Fractionalization and Fermi-surface volume in heavy-fermion compounds: The case of YbRh2Si2 . Phys. Rev. Lett. 94, 066402 (2005).

    Article  ADS  Google Scholar 

  47. Küchler, R. et al. Grüneisen ratio divergence at the quantum critical point in CeCu6−xAgx . Phys. Rev. Lett. 93, 096402 (2004).

    Article  ADS  Google Scholar 

  48. Varma, C. M., Littlewood, P. B., Schmitt-Rink, S., Abrahams, E. & Ruckenstein, A. E. Phenomenology of the normal state of Cu–O high temperature superconductors. Phys. Rev. Lett. 63, 1996–1999 (1989).

    Article  ADS  Google Scholar 

  49. Aronson, M. C. et al. Non-Fermi-liquid scaling of the magnetic response in UCu5−xPdx(x=1,1.5). Phys. Rev. Lett. 75, 725–728 (1995).

    Article  ADS  Google Scholar 

  50. Stockert, O., Löhneysen, H. v., Rosch, A., Pyka, N. & Loewenhaupt, M. Two-dimensional fluctuations at the quantum critical point of CeCu6−xAux . Phys. Rev. Lett. 80, 5627–5630 (1998).

    Article  ADS  Google Scholar 

  51. Gegenwart, P., Custers, J., Tokiwa, Y., Geibel, C. & Steglich, F. Ferromagnetic quantum critical fluctuations in YbRh2(Si0.95Ge0.05)2 . Phys. Rev. Lett. 94, 076402 (2005).

    Article  ADS  Google Scholar 

  52. Misawa, T., Yamaji, Y. & Imada, M. YbRh2Si2: Quantum tricritical behavior in itinerant electron systems. Preprint at <http://arxiv.org/abs/cond-mat/0710.3260> (2007).

  53. Sichelschmidt, J., Ivanshin, V. A., Ferstl, J., Geibel, C. & Steglich, F. Low temperature electron spin resonance of the Kondo ion in a heavy fermion metal: YbRh2Si2 . Phys. Rev. Lett. 91, 156401 (2003).

    Article  ADS  Google Scholar 

  54. Kadowaki, H. et al. Quantum critical point of an itinerant antiferromagnet in a heavy fermion. Phys. Rev. Lett. 96, 016401 (2006).

    Article  ADS  Google Scholar 

  55. Custers, J. et al. The break-up of heavy electrons at a quantum critical point. Nature 424, 524–527 (2003).

    Article  ADS  Google Scholar 

  56. Gegenwart, P. et al. Breakup of heavy fermions on the brink of phase A in CeCu2Si2 . Phys. Rev. Lett. 81, 1501–1504 (1998).

    Article  ADS  Google Scholar 

  57. Hlubina, R. & Rice, T. M. Resistivity as a function of temperature for models with hot spots on the Fermi surface. Phys. Rev. B 51, 9253–9260 (1995).

    Article  ADS  Google Scholar 

  58. Rosch, A. Interplay of disorder and spin fluctuations in the resistivity near a quantum critical point. Phys. Rev. Lett. 82, 4280–4283 (1999).

    Article  ADS  Google Scholar 

  59. Sereni, J. et al. Scaling the C∝TlnT dependence in Ce-systems. Physica B 230–232, 580–582 (1997).

    Article  ADS  Google Scholar 

  60. Paschen, S. et al. Hall-effect evolution across a heavy-fermion quantum critical point. Nature 432, 881–885 (2004).

    Article  ADS  Google Scholar 

  61. Norman, M. R., Si, Q., Bazaliy, Ya. B. & Ramazashvili, R. Hall effect in nested antiferromagnets near the quantum critical point. Phys. Rev. Lett. 90, 116601 (2003).

    Article  ADS  Google Scholar 

  62. Lee, M., Husmann, A., Rosenbaum, T. F. & Aeppli, G. High resolution study of magnetic ordering at absolute zero. Phys. Rev. Lett. 92, 187201 (2004).

    Article  ADS  Google Scholar 

  63. Shishido, H., Settai, R., Harima, H. & Ônuki, Y. A drastic change of the Fermi surface at a critical pressure in CeRhIn5: dHvA study under pressure. J. Phys. Soc. Jpn 74, 1103–1106 (2005).

    Article  ADS  Google Scholar 

  64. Park, T. et al. Hidden magnetism and quantum criticality in the heavy fermion superconductor CeRhIn5 . Nature 440, 65–68 (2006).

    Article  ADS  Google Scholar 

  65. Paul, I., Pépin, C. & Norman, M. R. Kondo breakdown and hybridization fluctuations in the Kondo–Heisenberg lattice. Phys. Rev. Lett. 98, 026402 (2007).

    Article  ADS  Google Scholar 

  66. Yeh, A. et al. Quantum phase transition in a common metal. Nature 419, 459–462 (2002).

    Article  ADS  Google Scholar 

  67. Bud’ko, S. L., Morosan, E. & Canfield, P. C. Magnetic field induced non-Fermi liquid behavior in YbAgGe single crystals. Phys. Rev. B 69, 014415 (2004).

    Article  ADS  Google Scholar 

  68. Bud’ko, S. L., Zapf, V., Morosan, E. & Canfield, P. C. Field-dependent Hall effect in single-crystal heavy fermion YbAgGe below 1 K. Phys. Rev. B 72, 172413 (2005).

    Article  ADS  Google Scholar 

  69. Tokiwa, Y. et al. Low-temperature thermodynamic properties of the heavy-fermion compound YbAgGe close to the field-induced quantum critical point. Phys. Rev. B 73, 094435 (2006).

    Article  ADS  Google Scholar 

  70. Onishi, Y. & Miyake, K. Enhanced valence fluctuations caused by f–c Coulomb interaction in Ce-based heavy electrons: Possible origin of pressure-induced enhancement of superconducting transition temperature in CeCu2Ge2 and related compounds. J. Phys. Soc. Jpn 69, 3955–3964 (2000).

    Article  ADS  Google Scholar 

  71. Jaccard, D., Behnia, K. & Sierro, J. Pressure-induced heavy fermion superconductivity in CeCu2Ge2 . Phys. Lett. A 163, 475–480 (1992).

    Article  ADS  Google Scholar 

  72. Yuan, H. Q. et al. Observation of two distinct superconducting phases in CeCu2Si2 . Science 302, 2104–2107 (2003).

    Article  ADS  Google Scholar 

  73. Sato, N. K. et al. Strong coupling between local moments and superconducting ‘heavy’ electrons in UPd2Al3 . Nature 410, 340–343 (2001).

    Article  ADS  Google Scholar 

  74. Saxena, S. S. et al. Superconductivity on the border of itinerant-electron ferromagnetism in UGe2 . Nature 406, 587–592 (2000).

    Article  ADS  Google Scholar 

  75. Lévy, F., Sheikin, I., Grenier, B. & Huxley, A. D. Magnetic field-induced superconductivity in the ferromagnet URhGe. Science 309, 1343–1346 (2005).

    Article  ADS  Google Scholar 

  76. Huy, N. T. et al. Superconductivity on the border of weak itinerant ferromagnetism in UCoGe. Phys. Rev. Lett. 99, 067006 (2007).

    Article  ADS  Google Scholar 

  77. Stockert, O. et al. Nature of the A phase in CeCu2Si2 . Phys. Rev. Lett. 92, 136401 (2004).

    Article  ADS  Google Scholar 

  78. Stockert, O. et al. Physica B (2007, in the press).

  79. Mathur, N. D. et al. Magnetically mediated superconductivity in heavy fermion compounds. Nature 394, 39–43 (1998).

    Article  ADS  Google Scholar 

  80. Scalapino, D. J., Loh, E. & Hirsch, J. E. d-wave pairing near a spin-density-wave instability. Phys. Rev. B 34, 8190–8192 (1986).

    Article  ADS  Google Scholar 

  81. Petrovic, C. et al. Heavy-fermion superconductivity in CeCoIn5 at 2.3 K. J. Phys. Condens. Matter 13, L337–L342 (2001).

    Article  Google Scholar 

  82. Gegenwart, P. et al. Non-Fermi liquid normal state of the heavy-fermion superconductor UBe13 . Physica C 408–410, 157–160 (2004).

    Article  ADS  Google Scholar 

  83. Anderson, P. W. Is there glue in cuprate superconductors? Science 316, 1705–1707 (2007).

    Article  Google Scholar 

  84. Yamamoto, S. J. & Si, Q. Fermi surface and antiferromagnetism in the Kondo lattice: An asymptotically exact solution in d>1 dimensions. Phys. Rev. Lett. 99, 016401 (2007).

    Article  ADS  Google Scholar 

  85. Si, Q. Global magnetic phase diagram and local quantum criticality in heavy fermion metals. Physica B 378, 23–27 (2006).

    Article  ADS  Google Scholar 

  86. Settai, R., Takeuchi, T. & Ônuki, Y. Recent advances in Ce-based heavy-fermion superconductivity and Fermi surface properties. J. Phys. Soc. Jpn 76, 051003 (2007).

    Article  ADS  Google Scholar 

  87. Gegenwart, P. et al. Magnetic-field induced quantum critical point in YbRh2Si2 . Phys. Rev. Lett. 89, 056402 (2002).

    Article  ADS  Google Scholar 

  88. Zhu, L. & Si, Q. Critical local moment fluctuations in the Bose–Fermi Kondo model. Phys. Rev. B 66, 024426 (2002).

    Article  ADS  Google Scholar 

  89. Zaránd, G. & Demler, E. Quantum phase transitions in the Bose–Fermi Kondo model. Phys. Rev. B 66, 024427 (2002).

    Article  ADS  Google Scholar 

  90. Vojta, M. & Kirćan, M. Pseudogapped Fermi–Bose Kondo model. Phys. Rev. Lett. 90, 157203 (2003).

    Article  ADS  Google Scholar 

  91. Glossop, M. T. & Ingersent, K. Numerical renormalization-group study of the Bose–Fermi Kondo model. Phys. Rev. Lett. 95, 067202 (2005).

    Article  ADS  Google Scholar 

  92. Kirchner, S. & Si, Q. Scaling and enhanced symmetry at the quantum critical point of the sub-Ohmic Bose–Fermi Kondo model. Phys. Rev. Lett. 100, 026403 (2008).

    Article  ADS  Google Scholar 

  93. Maebashi, H., Miyake, K. & Varma, C. M. Undressing the Kondo effect near the antiferromagnetic quantum critical point. Phys. Rev. Lett. 95, 207207 (2005).

    Article  ADS  Google Scholar 

  94. Rech, J., Coleman, P., Zaránd, G. & Parcollet, O. Schwinger Boson approach to the fully screened Kondo model. Phys. Rev. Lett. 96, 016601 (2006).

    Article  ADS  Google Scholar 

  95. Knebel, G., Aoki, D., Braithwaite, D., Salce, B. & Flouquet, J. Coexistence of antiferromagnetism and superconductivity in CeRhIn5 under high pressure and magnetic field. Phys. Rev. B 74, 020501 (2006).

    Article  ADS  Google Scholar 

  96. Paglione, J. et al. Field-induced quantum critical point in CeCoIn5 . Phys. Rev. Lett. 91, 246405 (2003).

    Article  ADS  Google Scholar 

  97. Bianchi, A., Movshovich, R., Vekhter, I., Pagliuso, P. G. & Sarrao, J. L. Avoided antiferromagnetic order and quantum critical point in CeCoIn5 . Phys. Rev. Lett. 91, 257001 (2003).

    Article  ADS  Google Scholar 

  98. Singh, S. et al. Probing the quantum critical behavior in CeCoIn5 via Hall effect measurements. Phys. Rev. Lett. 98, 057001 (2007).

    Article  ADS  Google Scholar 

  99. Donath, J. G., Gegenwart, P., Steglich, F., Bauer, E. D. & Sarrao, J. L. Dimensional crossover of quantum critical behavior in CeCoIn5. Preprint at <http://arxiv.org/abs/cond-mat/0704.0506> (2007).

  100. Pham, L. D., Park, T., Maquilon, S., Thompson, J. D. & Fisk, Z. Reversible tuning of the heavy-fermion ground state in CeCoIn5 . Phys. Rev. Lett. 97, 056404 (2006).

    Article  ADS  Google Scholar 

  101. Gegenwart, P. et al. Non-Fermi liquid effects at ambient pressure in a stoichiometric heavy-fermion compound with very low disorder: CeNi2Ge2 . Phys. Rev. Lett. 82, 1293 (1999).

    Article  ADS  Google Scholar 

  102. Loidl, A. et al. Local-moment and itinerant antiferromagnetism in the heavy-fermion system Ce(Cu1−xNix)2Ge2 . Ann. Phys. 1, 78–91 (1992).

    Article  Google Scholar 

  103. Knafo, W. et al. Anomalous scaling behavior of the dynamical spin susceptibility of Ce0.925La0.075Ru2Si2 . Phys. Rev. B 70, 174401 (2004).

    Article  ADS  Google Scholar 

  104. Montfrooij, W. et al. Non-local quantum criticality in Ce(Ru1−xFex)2Ge2 (x=xc=0.77). Phys. Rev. Lett. 91, 087202 (2003).

    Article  ADS  Google Scholar 

  105. Heuser, K., Scheidt, E.-W., Schreiner, T. & Stewart, G. R. Disappearance of hyperscaling at low temperatures in non-Fermi liquid CeCu5.2Ag0.8 . Phys. Rev. B 58, 15959–15961 (1998).

    Article  ADS  Google Scholar 

  106. Stockert, O., Enderle, M. & Löhneysen, v. H. Magnetic fluctuations at a field-induced quantum phase transition. Phys. Rev. Lett. 99, 237203 (2007).

    Article  ADS  Google Scholar 

  107. Westerkamp, T. et al. Physica B (2007, in the press).

  108. Brando, M. et al. (in the press).

  109. Belitz, D., Kirkpatrick, T. R. & Vojta, T. How generic scale invariance influences quantum and classical phase transitions. Rev. Mod. Phys. 77, 579–632 (2005).

    Article  ADS  Google Scholar 

  110. Abanov, Ar. & Chubukov, A. V. Spin-fermion model near the quantum critical point: One-loop renormalization group results. Phys. Rev. Lett. 84, 5608–5611 (2000).

    Article  ADS  Google Scholar 

  111. Süllow, S., Aronson, M. C., Rainford, B. D. & Haen, P. Doniach phase diagram, revisited: From ferromagnet to Fermi liquid in pressurized CeRu2Ge2 . Phys. Rev. Lett. 82, 2963–2966 (1999).

    Article  ADS  Google Scholar 

  112. Sidorov, V. A. et al. Magnetic phase diagram of the ferromagnetic Kondo-lattice compound CeAgSb2 up to 80 kbar. Phys. Rev. B 67, 224419 (2003).

    Article  ADS  Google Scholar 

  113. Lonzarich, G. G. in Electron (ed. Springford, M.) Ch. 6, 109–147 (Cambridge Univ. Press, Cambridge, 1997).

    Google Scholar 

  114. Doiron-Leyraud, N. et al. Fermi-liquid breakdown in the paramagnetic phase of a pure metal. Nature 425, 595–599 (2003).

    Article  ADS  Google Scholar 

  115. Uhlarz, M., Pfleiderer, C. & Hayden, S. M. Quantum phase transitions in the itinerant ferromagnet ZrZn2 . Phys. Rev. Lett. 93, 256404 (2004).

    Article  ADS  Google Scholar 

  116. Pedrazzini, P. et al. Metallic state in cubic FeGe beyond its quantum phase transition. Phys. Rev. Lett. 98, 047204 (2007).

    Article  ADS  Google Scholar 

  117. Millis, A. J., Schofield, A. J., Lonzarich, G. G. & Grigera, S. A. Metamagnetic quantum criticality in metals. Phys. Rev. Lett. 88, 217204 (2002).

    Article  ADS  Google Scholar 

  118. Grigera, S. A. et al. Magnetic field-tuned quantum criticality in the metallic ruthenate Sr3Ru2O7 . Science 294, 329–332 (2001).

    Article  ADS  Google Scholar 

  119. Grigera, S. A. et al. Disorder-sensitive phase formation linked to metamagnetic quantum criticality. Science 306, 1154–1157 (2004).

    Article  ADS  Google Scholar 

  120. Flouquet, J., Haen, P., Raymond, S., Aoki, D. & Knebel, G. Itinerant metamagnetism of CeRu2Si2: Bringing out the dead. Comparison with the new Sr3Ru2O7 case. Physica B 319, 251–261 (2002).

    Article  ADS  Google Scholar 

  121. Weickert, F. et al. Search for a quantum critical end-point in CeRu2(Si1−xGex)2 . Physica B 359, 68–70 (2005).

    Article  ADS  Google Scholar 

  122. Daou, R., Bergemann, C. & Julian, S. R. Continuous evolution of the Fermi surface of CeRu2Si2 across the metamagnetic transition. Phys. Rev. Lett. 96, 026401 (2006).

    Article  ADS  Google Scholar 

  123. Kim, K. H., Harrison, N., Jaime, M., Boebinger, G. S. & Mydosh, J. A. Magnetic-field-induced quantum critical point and competing order parameters in URu2Si2 . Phys. Rev. Lett. 91, 256401 (2003).

    Article  ADS  Google Scholar 

  124. Oeschler, N. et al. Physica B (2007, in the press).

  125. Gegenwart, P. et al. Physica B (2007, in the press).

  126. Friedemann, S. et al. Physica B (2007, in the press).

  127. Tokiwa, Y. et al. Field-induced suppression of the heavy-fermion state in YbRh2Si2 . Phys. Rev. Lett. 94, 226402 (2005).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We would like to thank E. Abrahams, C. J. Bolech, M. Brando, M. T. Glossop, S. Kirchner, M. Nicklas, P. Nikolic, S. Paschen, T. Westerkamp, S. Wirth and S. J. Yamamoto for their input on the manuscript. The work at Göttingen and Dresden has been supported in part by the DFG Research Unit 960 (‘Quantum Phase Transitions’) and the work at Rice by NSF Grant No. DMR-0706625 and the Robert A. Welch Foundation.

Author information

Authors and Affiliations

  1. I. Physik. Institut, Georg-August Universität Göttingen, D-37077 Göttingen, Germany

    Philipp Gegenwart

  2. Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

    Qimiao Si

  3. Max-Planck-Institut für Chemische Physik fester Stoffe, D-01187 Dresden, Germany

    Frank Steglich

Authors
  1. Philipp Gegenwart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qimiao Si
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Frank Steglich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qimiao Si.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gegenwart, P., Si, Q. & Steglich, F. Quantum criticality in heavy-fermion metals. Nature Phys 4, 186–197 (2008). https://doi.org/10.1038/nphys892

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys892

Search

Advanced 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