Skip to main content

Thank you for visiting 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.

Dimensional reduction at a quantum critical point


Competition between electronic ground states near a quantum critical point1,2 (QCP)—the location of a zero-temperature phase transition driven solely by quantum-mechanical fluctuations—is expected to lead to unconventional behaviour in low-dimensional systems3. New electronic phases of matter have been predicted to occur in the vicinity of a QCP by two-dimensional theories3,4,5,6,7,8, and explanations based on these ideas have been proposed for significant unsolved problems in condensed-matter physics, such as non-Fermi-liquid behaviour and high-temperature superconductivity. But the real materials to which these ideas have been applied are usually rendered three-dimensional by a finite electronic coupling between their component layers; a two-dimensional QCP has not been experimentally observed in any bulk three-dimensional system, and mechanisms for dimensional reduction have remained the subject of theoretical conjecture9,10,11. Here we show evidence that the Bose–Einstein condensate of spin triplets in the three-dimensional Mott insulator BaCuSi2O6 (refs 12–16) provides an experimentally verifiable example of dimensional reduction at a QCP. The interplay of correlations on a geometrically frustrated lattice causes the individual two-dimensional layers of spin-½ Cu2+ pairs (spin dimers) to become decoupled at the QCP, giving rise to a two-dimensional QCP characterized by linear power law scaling distinctly different from that of its three-dimensional counterpart. Thus the very notion of dimensionality can be said to acquire an ‘emergent’ nature: although the individual particles move on a three-dimensional lattice, their collective behaviour occurs in lower-dimensional space.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Experimentally obtained phase boundary of BaCuSi2O6.
Figure 2: Crossover from 3D to 2D BEC critical exponent.
Figure 3: 2D BEC power law behaviour.
Figure 4: Inter-layer decoupling at the QCP.


  1. 1

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

    MATH  Google Scholar 

  2. 2

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

    ADS  CAS  Article  Google Scholar 

  3. 3

    Sachdev, S. Quantum criticality: Competing ground states in low dimensions. Science 288, 475–480 (2000)

    ADS  CAS  Article  Google Scholar 

  4. 4

    Stockert, O., von Lohneysen, 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)

    ADS  CAS  Article  Google Scholar 

  5. 5

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

    ADS  CAS  Article  Google Scholar 

  6. 6

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

    ADS  CAS  Article  Google Scholar 

  7. 7

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

    ADS  CAS  Article  Google Scholar 

  8. 8

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

    ADS  CAS  Article  Google Scholar 

  9. 9

    Xu, C. & Moore, J. E. Geometric criticality for transitions between plaquette phases in integer-spin Kagome XXZ antiferromagnets. Phys. Rev. B 72, 064455 (2005)

    ADS  Article  Google Scholar 

  10. 10

    Batista, C. D. & Nussinov, Z. Generalized Elitzur's theorem and dimensional reductions. Phys. Rev. B 72, 045137 (2005)

    ADS  Article  Google Scholar 

  11. 11

    Tewari, S., Toner, J. & Chakravarty, S. Floating phase in a dissipative Josephson junction array. Phys. Rev. B 72, 060505(R) (2005)

    ADS  Article  Google Scholar 

  12. 12

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

    ADS  CAS  Article  Google Scholar 

  13. 13

    Sebastian, S. E. et al. Characteristic Bose-Einstein condensation scaling close to a quantum critical point in BaCuSi2O6 . Phys. Rev. B 72, 100404(R) (2005)

    ADS  Article  Google Scholar 

  14. 14

    FitzHugh, E. W. & Zycherman, L. A. A purple barium copper silicate pigment from early China. Studies Conserv. 37, 145–154 (1992)

    CAS  Article  Google Scholar 

  15. 15

    Nikuni, T., Oshikawa, M., Oosawa, A. & Tanaka, H. Bose-Einstein condensation of diluted magnons in TlCuCl3 . Phys. Rev. Lett. 84, 5868–5871 (2000)

    ADS  CAS  Article  Google Scholar 

  16. 16

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

    ADS  Article  Google Scholar 

  17. 17

    Sparta, K. M. & Roth, G. Reinvestigation of the structure of BaCuSi2O6–evidence for a phase transition at high temperature. Acta Crystallogr. B 60, 491–495 (2004)

    Article  Google Scholar 

  18. 18

    Samulon, E. C. et al. Low temperature structural phase transition and incommensurate lattice modulation in the spin gap compound BaCuSi2O6 . Phys. Rev. B 73, 100407(R) (2006)

    ADS  Article  Google Scholar 

  19. 19

    Sasago, Y., Uchinokura, K., Zheludev, A. & Shirane, A. Temperature-dependent spin gap and singlet ground state in BaCuSi2O6 . Phys. Rev. B 55, 8357–8360 (1991)

    ADS  Article  Google Scholar 

  20. 20

    Mila, F. Ladders in a magnetic field: a strong coupling approach. Eur. Phys. J. B 6, 201–205 (1998)

    ADS  CAS  Article  Google Scholar 

  21. 21

    Giamarchi, T. & Tsvelik, A. M. Coupled ladders in a magnetic field. Phys. Rev. B 59, 11398–11407 (1999)

    ADS  CAS  Article  Google Scholar 

  22. 22

    Rice, T. M. To condense or not to condense. Science 298, 760–761 (2002)

    CAS  Article  Google Scholar 

  23. 23

    Kawashima, N. Quantum critical point of the XY model and condensation of field-induced quasiparticles in dimer compounds. J. Phys. Soc. Jpn 73, 3219–3222 (2004)

    ADS  CAS  Article  Google Scholar 

  24. 24

    Fisher, M. P. A. et al. Boson localization and the superfluid-insulator transition. Phys. Rev. B 40, 546–570 (1989)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  25. 25

    Nohadani, O. et al. Universal scaling at field-induced magnetic phase transitions. Phys. Rev. B 69, 220402(R) (2004)

    ADS  Article  Google Scholar 

  26. 26

    Harris, A. B. Effect of random defects on the critical behaviour of Ising models. J. Phy. C 7, 1671–1692 (1974)

    ADS  Article  Google Scholar 

  27. 27

    Vojta, T. & Sknepnek, R. Critical points and quenched disorder: from Harris criterion to rare regions and smearing. Phys. Status Solidi B 241, 2118–2127 (2004)

    ADS  CAS  Article  Google Scholar 

  28. 28

    Maltseva, M. & Coleman, P. Failure of geometric frustration to preserve a quasi-two-dimensional spin fluid. Phys. Rev. B 72, 174415(R) (2005)

    ADS  Article  Google Scholar 

  29. 29

    Henley, C. L. Ordering due to disorder in a frustrated vector antiferromagnet. Phys. Rev. Lett. 62, 2056–2059 (1989)

    ADS  CAS  Article  Google Scholar 

Download references


N.H., C.D.B., M.J. and P.A.S. acknowledge Laboratory Directed Research and Development (LDRD) support at LANL. S.E.S and I.R.F. acknowledge National Science Foundation (NSF) support. Experiments performed at the NHMFL, Tallahassee, were supported by the NSF, the State of Florida, and the Department of Energy. We thank T. P. Murphy, E. C. Palm, P. Tanedo and P. B. Brooks for experimental assistance, and acknowledge discussions with A. G. Green, E.-A. Kim, S. A. Kivelson, D. I. Santiago and J. Schmalian. I.R.F. acknowledges support from the Alfred P. Sloan Foundation and S.E.S. from the Mustard Seed Foundation.

Author information



Corresponding author

Correspondence to S. E. Sebastian.

Ethics declarations

Competing interests

Reprints and permissions information is available at The authors declare no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sebastian, S., Harrison, N., Batista, C. et al. Dimensional reduction at a quantum critical point. Nature 441, 617–620 (2006).

Download citation

Further reading


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.


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