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.

  • Article
  • Published:

Interplay of quantum criticality and geometric frustration in columbite

Nature Physics volume 6, pages 702–706 (2010)Cite this article

Abstract

CoNb2O6 is a material with remarkable properties that are determined by an exciting interplay of quantum mechanics and geometric frustration. On the one hand, weakly coupled ferromagnetic Ising chains of Co2+ ions can be tuned by an applied magnetic field through a quantum critical point to be paramagnetic; on the other hand, the Ising chains must contend with residual interactions on a frustrated triangular lattice in their choice of how to order. Motivated by the material, we theoretically study the phase diagram of quantum ferromagnetic Ising chains coupled antiferromagnetically on a triangular lattice in the plane perpendicular to the chain direction. We combine exact solutions of the quantum criticality in the isolated chains with perturbative approximations for the frustrated interchain couplings. When the triangular lattice has an isosceles distortion, which occurs in the real material, the phase diagram at absolute zero temperature is rich with five different states of matter: ferrimagnetic, Néel, antiferromagnetic, paramagnetic and incommensurate phases, separated by quantum phase transitions. Implications of our results on experiments in CoNb2O6 are discussed.

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: Schematic structure of Co2+ ions in CoNb2O6.
Figure 2: Magnetic ordering patterns of FR, AF, N1 and N2 phases.
Figure 3: Calculated phase diagram of the model (equation (1)) for perfect triangles, α≡J2/J1=1, and with an isosceles distortion, α=0.99, and with a larger distortion, α=0.9.
Figure 4: Primary ordering vector in the IC phase.

Similar content being viewed by others

References

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

    Book  Google Scholar 

  2. Wannier, G. H. Antiferromagnetism. The triangular Ising net. Phys. Rev. 79, 357–364 (1950).

    Article  ADS  MathSciNet  Google Scholar 

  3. Coldea, R. et al. Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry. Science 327, 177–180 (2010).

    Article  ADS  Google Scholar 

  4. Lieb, E., Schultz, T. & Mattis, D. Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961).

    Article  ADS  MathSciNet  Google Scholar 

  5. Zamolodchikov, A. B. Integrals of motion and S-matrix of the (scaled) T=Tc Ising model with magnetic field. Int. J. Mod. Phys. A 4, 4235–4248 (1989).

    Article  ADS  Google Scholar 

  6. Kobayashi, S., Mitsuda, S., Ishikawa, M., Miyatani, K. & Kohn, K. Three-dimensional magnetic ordering in the quasi-one-dimensional Ising magnet CoNb2O6 with partially released geometrical frustration. Phys. Rev. B 60, 3331–3345 (1999).

    Article  ADS  Google Scholar 

  7. Heid, C. et al. Magnetic phase-diagram of CoNb2O6—a neutron-diffraction study. J. Magn. Magn. Mater. 151, 123–131 (1995).

    Article  ADS  Google Scholar 

  8. Cao, G., Durairaj, V., Chikara, S., Parkin, S. & Schlottmann, P. Partial antiferromagnetism in spin-chain Sr5Rh4O12, Ca5Ir3O12, and Ca4IrO6 single crystals. Phys. Rev. B 75, 134402 (2007).

    Article  ADS  Google Scholar 

  9. Moessner, R., Sondhi, S. L. & Chandra, P. Two-dimensional periodic frustrated Ising models in a transverse field. Phys. Rev. Lett. 84, 4457–4460 (2000).

    Article  ADS  Google Scholar 

  10. Pfeuty, P. The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970).

    Article  ADS  Google Scholar 

  11. Fonseca, P. & Zamolodchikov, A. Ising field theory in a magnetic field: Analytic properties of the free energy. J. Stat. Phys. 110, 527–590 (2003).

    Article  MathSciNet  Google Scholar 

  12. Scalapino, D. J., Imry, Y. & Pincus, P. Generalized Ginzburg–Landau theory of pseudo-one-dimensional systems. Phys. Rev. B 11, 2042–2048 (1975).

    Article  ADS  Google Scholar 

  13. Schulz, H. J. Dynamics of coupled quantum spin chains. Phys. Rev. Lett. 77, 2790–2793 (1996).

    Article  ADS  Google Scholar 

  14. Carr, S. T. & Tsvelik, A. M. Spectrum and correlation functions of a quasi-one-dimensional quantum Ising model. Phys. Rev. Lett. 90, 177206 (2003).

    Article  ADS  Google Scholar 

  15. Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 2000).

    Google Scholar 

Download references

Acknowledgements

We thank R. Coldea for stimulating discussions and acknowledge support from the Packard Foundation and National Science Foundation through grants DMR-0804564 and PHY05-51164.

Author information

Authors and Affiliations

  1. Department of Physics, University of California, Santa Barbara, California 93106-9530, USA

    SungBin Lee

  2. Department of Physics, University of Kentucky, Lexington, Kentucky 40506-0055, USA

    Ribhu K. Kaul

  3. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-9530, USA

    Leon Balents

  4. Present address: Microsoft Station Q, University of California, Santa Barbara, California 93106-6105, USA,

    Ribhu K. Kaul

Authors
  1. SungBin Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ribhu K. Kaul
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leon Balents
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The research reported in the manuscript was carried out by S.B.L. and R.K.K. under the supervision of L.B. The manuscript was written by R.K.K.

Corresponding author

Correspondence to Ribhu K. Kaul.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 210 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, S., Kaul, R. & Balents, L. Interplay of quantum criticality and geometric frustration in columbite. Nature Phys 6, 702–706 (2010). https://doi.org/10.1038/nphys1696

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys1696

This article is cited by

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