Letter | Published:

Dimensional reduction at a quantum critical point

Naturevolume 441pages617620 (2006) | Download Citation



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 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, New York, 1999)

  2. 2

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

  3. 3

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

  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)

  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)

  6. 6

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

  7. 7

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

  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)

  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)

  10. 10

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

  11. 11

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

  12. 12

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

  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)

  14. 14

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

  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)

  16. 16

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

  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)

  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)

  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)

  20. 20

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

  21. 21

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

  22. 22

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

  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)

  24. 24

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

  25. 25

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

  26. 26

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

  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)

  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)

  29. 29

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

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


  1. Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University, Stanford, California, 94305, USA

    • S. E. Sebastian
    •  & I. R. Fisher
  2. NHMFL, MS-E536, Los Alamos National Laboratory

    • N. Harrison
    • , M. Jaime
    •  & P. A. Sharma
  3. Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA

    • C. D. Batista
  4. National High Magnetic Field Laboratory, Tallahassee, Florida, 32310, USA

    • L. Balicas
  5. Institute for Solid State Physics, University of Tokyo, Kashiwa, 277-8581, Chiba, Japan

    • N. Kawashima


  1. Search for S. E. Sebastian in:

  2. Search for N. Harrison in:

  3. Search for C. D. Batista in:

  4. Search for L. Balicas in:

  5. Search for M. Jaime in:

  6. Search for P. A. Sharma in:

  7. Search for N. Kawashima in:

  8. Search for I. R. Fisher in:

Competing interests

Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.

Corresponding author

Correspondence to S. E. Sebastian.

About this article

Publication history



Issue Date



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.