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:

Topological defects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics

Nature Physics volume 10, pages 970–977 (2014)Cite this article

Subjects

Abstract

Lars Onsager and Richard Feynman envisaged that the three-dimensional (3D) superfluid-to-normal λ transition in 4He occurs through the proliferation of vortices. This process should hold for every phase transition in the same universality class. The role of topological defects in symmetry-breaking phase transitions has become a prime topic in cosmology and high-temperature superconductivity, even though direct imaging of these defects is challenging. Here we show that the U(1) continuous symmetry that emerges at the ferroelectric critical point of multiferroic hexagonal manganites leads to a similar proliferation of vortices. Moreover, the disorder field (vortices) is coupled to an emergent U(1) gauge field, which becomes massive by means of the Higgs mechanism when vortices condense (span the whole system) on heating above the ferroelectric transition temperature. Direct imaging of the vortex network in hexagonal manganites offers unique experimental access to this dual description of the ferroelectric transition, while enabling tests of the Kibble–Zurek mechanism.

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: Dual description of a phase transition with Z2 × Z3 symmetry.
Figure 2: 3D picture of vortex cores: depth profiling of vortex domain patterns.
Figure 3: Vortex–antivortex pair correlation function.
Figure 4: Domain patterns for different initial annealing temperatures Ti (above, near and below Tc).
Figure 5: Dependence of vortex density nv on cooling rate.
Figure 6: Winding numbers for KZM defects.

Similar content being viewed by others

References

  1. Anderson, P. W. Basic Notions of Condensed Matter Physics (Benjamin/Cummings Pub. Co., Advanced Book Program, 1984).

    Google Scholar 

  2. Wilson, K. G. The renormalization group: Critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773–840 (1975).

    Article  ADS  MathSciNet  Google Scholar 

  3. José, J. V., Kadanoff, L. P., Kirkpatrick, S. & Nelson, D. R. Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model. Phys. Rev. B 16, 1217–1241 (1977).

    Article  ADS  Google Scholar 

  4. Onsager, L. The two fluid model for helium II [remarks by L. Onsager on pp. 249–250]. Nuovo Cimento Suppl. 6, 249–250 (1949).

    Article  MathSciNet  Google Scholar 

  5. Feynman, R. in Progress in Low Temperature Physics (ed Gorter, C.) Vol. I. (North-Holland, 1955).

    Google Scholar 

  6. Vilenkin, A. & Shellard, E. P. S. Cosmic Strings and Other Topological Defects (Cambridge Univ. Press, 1994).

    MATH  Google Scholar 

  7. Kibble, T. Phase-transition dynamics in the lab and the universe. Phys. Today 60, 47–52 (September, 2007).

    Article  Google Scholar 

  8. Shenoy, S. R. & Chattopadhyay, B. Anisotropic three-dimensional XY model and vortex-loop scaling. Phys. Rev. B 51, 9129–9147 (1995).

    Article  ADS  Google Scholar 

  9. Domínguez, D., Grønbech-Jensen, N., Bishop, A. R. & Shenoy, S. R. Transformer configuration in three dimensional Josephson lattices at zero magnetic field. Phys. Rev. Lett. 75, 717–720 (1995).

    Article  ADS  Google Scholar 

  10. Kiometzis, M., Kleinert, H. & Schakel, A. M. J. Critical exponents of the superconducting phase transition. Phys. Rev. Lett. 73, 1975–1977 (1994).

    Article  ADS  Google Scholar 

  11. Nguyen, A. K. & Sudbø, A. Onsager loop transition and first-order flux-line lattice melting in high-Tc superconductors. Phys. Rev. B 57, 3123–3143 (1998).

    Article  ADS  Google Scholar 

  12. Ryu, S. & Stroud, D. Nature of the low-field transition in the mixed state of high-temperature superconductors. Phys. Rev. B 57, 14476–14497 (1998).

    Article  ADS  Google Scholar 

  13. Golubchik, D., Polturak, E. & Koren, G. Evidence for long-range correlations within arrays of spontaneously created magnetic vortices in a Nb thin-film superconductor. Phys. Rev. Lett. 104, 247002 (2010).

    Article  ADS  Google Scholar 

  14. Kleinert, H. Gauge Fields in Condensed Matter (World Scientific Publishing, 1989).

    Book  Google Scholar 

  15. Kleinert, H. From Landau’s order parameter to modern disorder fields. Am. Inst. Phys. Conf. Ser. 1205, 103–107 (2010).

    ADS  Google Scholar 

  16. Kiometzis, M., Kleinert, H. & Schakel, A. M. J. Dual description of the superconducting phase transition. Fortschr. Phys.-Prog. Phys. 43, 697–732 (1995).

    Article  ADS  Google Scholar 

  17. Kibble, T. W. B. Topology of cosmic domains and strings. J. Phys. A: Math. Gen. 9, 1387–1398 (1976).

    Article  ADS  Google Scholar 

  18. Zurek, W. H. Cosmological experiments in superfluid helium? Nature 317, 505–508 (1985).

    Article  ADS  Google Scholar 

  19. del Campo, A. & Zurek, W. H. Universality of phase transition dynamics: Topological defects from symmetry breaking. Int. J. Mod. Phys. A 29, 1430018 (2014).

    Article  ADS  Google Scholar 

  20. Chuang, I., Durrer, R., Turok, N. & Yurke, B. Cosmology in the laboratory—Defect dynamics in liquid-crystals. Science 251, 1336–1342 (1991).

    Article  ADS  Google Scholar 

  21. Ruutu, V. M. H. et al. Vortex formation in neutron-irradiated superfluid He-3 as an analogue of cosmological defect formation. Nature 382, 334–336 (1996).

    Article  ADS  Google Scholar 

  22. Bauerle, C., Bunkov, Y. M., Fisher, S. N., Godfrin, H. & Pickett, G. R. Laboratory simulation of cosmic string formation in the early Universe using superfluid He-3. Nature 382, 332–334 (1996).

    Article  ADS  Google Scholar 

  23. Sadler, L. E., Higbie, J. M., Leslie, S. R., Vengalattore, M. & Stamper-Kurn, D. M. Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate. Nature 443, 312–315 (2006).

    Article  ADS  Google Scholar 

  24. Weiler, C. N. et al. Spontaneous vortices in the formation of Bose–Einstein condensates. Nature 455, 948–951 (2008).

    Article  ADS  Google Scholar 

  25. Ulm, S. et al. Observation of the Kibble–Zurek scaling law for defect formation in ion crystals. Nature Commun. 4, 2290 (2013).

    Article  ADS  Google Scholar 

  26. Pyka, K. et al. Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals. Nature Commun. 4, 2291 (2013).

    Article  ADS  Google Scholar 

  27. Lamporesi, G., Donadello, S., Serafini, S., Dalfovo, F. & Ferrari, G. Spontaneous creation of Kibble–Zurek solitons in a Bose–Einstein condensate. Nature Phys. 9, 655–659 (2013).

    Article  ADS  Google Scholar 

  28. Choi, T. et al. Insulating interlocked ferroelectric and structural antiphase domain walls in multiferroic YMnO3 . Nature Mater. 9, 253–258 (2010).

    Article  ADS  Google Scholar 

  29. Mostovoy, M. Multiferroics: A whirlwind of opportunities. Nature Mater. 9, 188–190 (2010).

    Article  ADS  Google Scholar 

  30. Jungk, T., Hoffmann, A., Fiebig, M. & Soergel, E. Electrostatic topology of ferroelectric domains in YMnO3 . Appl. Phys. Lett. 97, 012904 (2010).

    Article  ADS  Google Scholar 

  31. Lochocki, E. B., Park, S., Lee, N., Cheong, S. W. & Wu, W. D. Piezoresponse force microscopy of domains and walls in multiferroic HoMnO3 . Appl. Phys. Lett. 99, 232901 (2011).

    Article  ADS  Google Scholar 

  32. Chae, S. C. et al. Evolution of the domain topology in a ferroelectric. Phys. Rev. Lett. 110, 167601 (2013).

    Article  ADS  Google Scholar 

  33. Chae, S. C. et al. Self-organization, condensation, and annihilation of topological vortices and antivortices in a multiferroic. Proc. Natl Acad. Sci. USA 107, 21366–21370 (2010).

    Article  ADS  Google Scholar 

  34. Blankschtein, D., Ma, M., Berker, A. N., Grest, G. S. & Soukoulis, C. M. Ordering of a stacked frustrated triangular system in 3 dimensions. Phys. Rev. B 29, 5250–5252 (1984).

    Article  ADS  Google Scholar 

  35. Oshikawa, M. Ordered phase and scaling in Z(n) models and the three-state antiferromagnetic Potts model in three dimensions. Phys. Rev. B 61, 3430–3434 (2000).

    Article  ADS  Google Scholar 

  36. Vachaspati, T. & Vilenkin, A. Formation and evolution of cosmic strings. Phys. Rev. D 30, 2036–2045 (1984).

    Article  ADS  Google Scholar 

  37. Chae, S. C. et al. Direct observation of the proliferation of ferroelectric loop domains and vortex–antivortex pairs. Phys. Rev. Lett. 108, 167603 (2012).

    Article  ADS  Google Scholar 

  38. Griffin, S. M. et al. Scaling behavior and beyond equilibrium in the hexagonal manganites. Phys. Rev. X 2, 041022 (2012).

    Google Scholar 

  39. Campostrini, M., Hasenbusch, M., Pelissetto, A., Rossi, P. & Vicari, E. Critical behavior of the three-dimensional XY universality class. Phys. Rev. B 63, 214503 (2001).

    Article  ADS  Google Scholar 

  40. Hohenberg, P. C. & Halperin, B. I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977).

    Article  ADS  Google Scholar 

  41. Zurek, W. H. Topological relics of symmetry breaking: Winding numbers and scaling tilts from random vortex–antivortex pairs. J. Phys. Condens. Matter 25, 404209 (2013).

    Article  Google Scholar 

  42. Liu, F. & Mazenko, G. F. Defect–defect correlation in the dynamics of 1st-order phase-transition. Phys. Rev. B 46, 5963–5971 (1992).

    Article  ADS  Google Scholar 

  43. Hove, J. & Sudbo, A. Anomalous scaling dimensions and stable charged fixed point of type-II superconductors. Phys. Rev. Lett. 84, 3426–3429 (2000).

    Article  ADS  Google Scholar 

  44. Monaco, R., Aaroe, M., Mygind, J., Rivers, R. J. & Koshelets, V. P. Experiments on spontaneous vortex formation in Josephson tunnel junctions. Phys. Rev. B 74, 144513 (2006).

    Article  ADS  Google Scholar 

  45. Monaco, R., Mygind, J., Rivers, R. J. & Koshelets, V. P. Spontaneous fluxoid formation in superconducting loops. Phys. Rev. B 80, 180501 (2009).

    Article  ADS  Google Scholar 

  46. Kavoussanaki, E., Monaco, R. & Rivers, R. J. Testing the Kibble–Zurek scenario with annular Josephson tunnel junctions. Phys. Rev. Lett. 85, 3452–3455 (2000).

    Article  ADS  Google Scholar 

  47. Dodd, M. E., Hendry, P. C., Lawson, N. S., McClintock, P. V. E. & Williams, C. D. H. Nonappearance of vortices in fast mechanical expansions of liquid He-4 through the lambda transition. Phys. Rev. Lett. 81, 3703–3706 (1998).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank S. C. Chae, A. del campo and V. Zapf for stimulating discussion. This project was in part supported by the DOE under the LDRD program at the Los Alamos National Laboratory. The work at Rutgers University was supported by the DOE under Grant No. DE-FG02-07ER46382. Y.K. acknowledges the financial support by the RIKEN iTHES Project. The work was also supported by China Scholarship Council.

Author information

Author notes

  1. Shi-Zeng Lin and Xueyun Wang: These authors contributed equally to this work.

Authors and Affiliations

  1. Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

    Shi-Zeng Lin, Yoshitomo Kamiya, Gia-Wei Chern, Wojciech H. Zurek & Cristian D. Batista

  2. Rutgers Centre for Emergent Materials and Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road Piscataway, New Jersey 08854, USA,

    Xueyun Wang, Fei Fan, David Fan, Brian Casas, Yue Liu, Valery Kiryukhin & Sang-Wook Cheong

  3. iTHES Research Group and Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan

    Yoshitomo Kamiya

  4. Shaanxi Key Laboratory of Condensed Matter Structures and Properties, School of Science, Northwestern Polytechnical University, Xi’an 710129, China

    Fei Fan

  5. Montgomery High School, 1016 Route 601, Skillman New Jersey 08558, USA,

    David Fan

  6. Department of Physics, Functional Materials Laboratory, University of South Florida, Tampa, Florida 33613, USA

    Brian Casas

  7. The MOE key Laboratory of Weak-Light Nonlinear Photonics and TEDA Applied Physics School, Nankai University, Tianjin 300457, China

    Yue Liu

Authors
  1. Shi-Zeng Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xueyun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoshitomo Kamiya
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gia-Wei Chern
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Fei Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. David Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Brian Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Yue Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Valery Kiryukhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Wojciech H. Zurek
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. Cristian D. Batista
    View author publications

    You can also search for this author in PubMed Google Scholar

  12. Sang-Wook Cheong
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

S-W.C. designed and supervised the experiment. W.H.Z. and C.D.B. discussed the simulations and experiments, and wrote the section on the Kibble–Zurek mechanism. X.W. carried out annealing experiments, AFM and PFM work. F.F. performed PFM work. D.F., B.C. and Y.L. analysed vortex–antivortex optical images and V.K. calculated the experimental correlation functions. S-Z.L., Y.K. and G-W.C. simulated the theoretical results. S-Z.L., X.W., S-W.C., W.H.Z., C.D.B. and V.K. co-wrote the paper. All authors discussed the results.

Corresponding author

Correspondence to Sang-Wook Cheong.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 876 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, SZ., Wang, X., Kamiya, Y. et al. Topological defects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics. Nature Phys 10, 970–977 (2014). https://doi.org/10.1038/nphys3142

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys3142

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