Article | Published:

Locally critical quantum phase transitions in strongly correlated metals


When a metal undergoes a continuous quantum phase transition, non-Fermi-liquid behaviour arises near the critical point. All the low-energy degrees of freedom induced by quantum criticality are usually assumed to be spatially extended, corresponding to long-wavelength fluctuations of the order parameter. But this picture has been contradicted by the results of recent experiments on a prototype system: heavy fermion metals at a zero-temperature magnetic transition. In particular, neutron scattering from CeCu6-x Aux has revealed anomalous dynamics at atomic length scales, leading to much debate as to the fate of the local moments in the quantum-critical regime. Here we report our theoretical finding of a locally critical quantum phase transition in a model of heavy fermions. The dynamics at the critical point are in agreement with experiment. We propose local criticality to be a phenomenon of general relevance to strongly correlated metals.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.


  1. 1

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

  2. 2

    Maple, M. B. et al. Non-Fermi-liquid behavior in strongly correlated f-electron materials. J. Low Temp. Phys. 95, 225–243 (1994).

  3. 3

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

  4. 4

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

  5. 5

    Trovarelli, O. et al. YbRb2Si2: Pronounced non-Fermi-liquid effects above a low-lying magnetic phase transition. Phys. Rev. Lett. 85, 626–629 (2000).

  6. 6

    Heuser, K. et al. Inducement of non-Fermi-liquid behavior with a magnetic field. Phys. Rev. B 57, R4198–R4201 (1998).

  7. 7

    Estrela, P., de Visser, A., Naka, T., de Boer, F. R. & Pereira, L. C. J. High-pressure study of the non-Fermi liquid material U2Pt2In. Preprint cond-mat/0009324 at 〈〉 (2000).

  8. 8

    Steward, G. Non-Fermi liquid behavior in d- and f- electron metals. Rev. Mod. Phys. 73, in the press (2001).

  9. 9

    Schröder, A., Aeppli, G., Bucher, E., Ramazashvili, R. & Coleman, P. Scaling of magnetic fluctuations near a quantum phase transition. Phys. Rev. Lett. 80, 5623–5626 (1998).

  10. 10

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

  11. 11

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

  12. 12

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

  13. 13

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

  14. 14

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

  15. 15

    Coleman, P. Theories of non-Fermi liquid behavior in heavy fermions. Physica B 259–261, 353–358 (1999).

  16. 16

    Si, Q., Smith, J. L. & Ingersent, K. Quantum critical behavior in Kondo systems. Int. J. Mod. Phys. B 13, 2331–2342 (1999).

  17. 17

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

  18. 18

    Coleman, P., Pépin, C. & Tsvelik, A. M. Supersymmetric spin operators. Phys. Rev. B 62, 3852–3868 (2000).

  19. 19

    Leggett, A. J. et al. Dynamics of the dissipative two-state system. Rev. Mod. Phys. 59, 1–86 (1987).

  20. 20

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

  21. 21

    Anderson, P. W. Localized magnetic states in metals. Phys. Rev. 124, 41–53 (1961).

  22. 22

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

  23. 23

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

  24. 24

    Smith, J. L. & Si, Q. Spatial correlations in dynamical mean-field theory. Phys. Rev. B 61, 5184–5192 (2000).

  25. 25

    Si, Q. & Smith, J. L. Kosterlitz-Thouless transition and short range spatial correlations in an extended Hubbard model. Phys. Rev. Lett. 77, 3391–3394 (1996).

  26. 26

    Chitra, R. & Kotliar, G. Effect of Coulomb long-range interactions on the Mott transition. Phys. Rev. Lett. 84, 3678–3681 (2000).

  27. 27

    Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996).

  28. 28

    Smith, J. L. & Si, Q. Non-Fermi liquids in the two-band extended Hubbard model. Europhys. Lett. 45, 228–234 (1999).

  29. 29

    Sengupta, A. M. Spin in a fluctuating field: the Bose (+Fermi) Kondo models. Phys. Rev. B 61, 4041–4043 (2000).

  30. 30

    Sachdev, S. & Ye, J. Gapless spin-fluid ground state in a random quantum Heisenberg magnet. Phys. Rev. Lett. 70, 3339–3342 (1993).

  31. 31

    Continentino, M. A. Universal behavior in heavy fermions. Phys. Rev. B 47, 11587–11590 (1993).

  32. 32

    Rosch, A., Schröder, A., Stockert, O. & von Löhneysen, H. Mechanism for the non-Fermi-liquid behavior in CeCu6-xAux. Phys. Rev. Lett. 79, 159–162 (1997).

  33. 33

    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).

  34. 34

    Vojta, M., Buragohain, C. & Sachdev, S. Quantum impurity dynamics in two-dimensional antiferromagnets and superconductors. Phys. Rev. B 61, 15152–15184 (2000).

  35. 35

    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).

  36. 36

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

  37. 37

    Raymond, S., Regnault, L. P., Flouquet, J., Wildes, A. & Lejay, P. Pressure dependence of the spin dynamics around a quantum critical point: an inelastic neutron scattering study of Ce0.87La0.13Ru2Si2. Preprint cond-mat/0102427 at 〈〉 (2001).

  38. 38

    Sachdev, S. Theory of finite-temperature crossovers near quantum critical points close to, or above, their upper-critical dimensions. Phys. Rev. B 55, 142–163 (1997).

  39. 39

    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).

  40. 40

    Wilson, K. G. & Kogut, J. The renormalization group and the ε expansion. Phys. Rep. C 12, 75–200 (1974).

Download references


We thank G. Aeppli, A. Chubukov, P. Coleman, A. J. Millis, A. Schröder, A. M. Sengupta, C. M. Varma and P. Wölfle for discussions. This work was supported by the NSF, TCSUH and the A. P. Sloan Foundation.

Author information

Correspondence to Qimiao Si.

Rights and permissions

Reprints and Permissions

About this article

Further reading

Figure 1: A theoretical model of heavy fermions.
Figure 2: Diagram of a conventional quantum phase transition in Kondo lattices.
Figure 3: Diagram of a locally critical quantum phase transition in Kondo lattices.


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.