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.

Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons


Quantum many-body systems can have phase transitions1 even at zero temperature; fluctuations arising from Heisenberg’s uncertainty principle, as opposed to thermal effects, drive the system from one phase to another. Typically, during the transition the relative strength of two competing terms in the system’s Hamiltonian changes across a finite critical value. A well-known example is the Mott–Hubbard quantum phase transition from a superfluid to an insulating phase2,3, which has been observed for weakly interacting bosonic atomic gases. However, for strongly interacting quantum systems confined to lower-dimensional geometry, a novel type4,5 of quantum phase transition may be induced and driven by an arbitrarily weak perturbation to the Hamiltonian. Here we observe such an effect—the sine–Gordon quantum phase transition from a superfluid Luttinger liquid to a Mott insulator6,7—in a one-dimensional quantum gas of bosonic caesium atoms with tunable interactions. For sufficiently strong interactions, the transition is induced by adding an arbitrarily weak optical lattice commensurate with the atomic granularity, which leads to immediate pinning of the atoms. We map out the phase diagram and find that our measurements in the strongly interacting regime agree well with a quantum field description based on the exactly solvable sine–Gordon model8. We trace the phase boundary all the way to the weakly interacting regime, where we find good agreement with the predictions of the one-dimensional Bose–Hubbard model. Our results open up the experimental study of quantum phase transitions, criticality and transport phenomena beyond Hubbard-type models in the context of ultracold gases.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout

Figure 1: Comparing two types of superfluid-to-Mott-insulator phase transition in one dimension.
Figure 2: Modulation spectroscopy on bosons in one dimension.
Figure 3: Transport measurements on the 1D Bose gas.
Figure 4: Phase diagram for the strongly interacting 1D Bose gas.


  1. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 2000)

    Book  MATH  Google Scholar 

  2. Jaksch, D., Bruder, C., Cirac, J., Gardiner, C. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998)

    Article  ADS  CAS  Google Scholar 

  3. Greiner, M. et al. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002)

    Article  ADS  CAS  Google Scholar 

  4. Giamarchi, T. Quantum Physics in One Dimension (Oxford Univ. Press, 2003)

    Book  MATH  Google Scholar 

  5. Gogolin, A. O., Tsvelik, A. M. & Nersesyan, A. A. Bosonization and Strongly Correlated Systems (Cambridge Univ. Press, 1998)

    Google Scholar 

  6. Büchler, H. P., Blatter, G. & Zwerger, W. Commensurate-incommensurate transition of cold atoms in an optical lattice. Phys. Rev. Lett. 90, 130401 (2003)

    Article  ADS  PubMed  Google Scholar 

  7. Pokrovsky, V. L. & Talapov, A. L. Ground state, spectrum, and phase diagram of two-dimensional incommensurate crystals. Phys. Rev. Lett. 42, 65–67 (1979)

    Article  ADS  CAS  Google Scholar 

  8. Coleman, S. Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D 11, 2088–2097 (1975)

    Article  ADS  Google Scholar 

  9. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008)

    Article  ADS  CAS  Google Scholar 

  10. Jördens, R., Strohmaier, N., Günter, K., Moritz, H. & Esslinger, T. A Mott insulator of fermionic atoms in an optical lattice. Nature 455, 204–207 (2008)

    Article  ADS  PubMed  Google Scholar 

  11. Schneider, U. et al. Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice. Science 322, 1520–1525 (2008)

    Article  ADS  CAS  PubMed  Google Scholar 

  12. Girardeau, M. Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. Phys. 1, 516–523 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Paredes, B. et al. Tonks–Girardeau gas of ultracold atoms in an optical lattice. Nature 429, 277–281 (2004)

    Article  ADS  CAS  PubMed  Google Scholar 

  14. Kinoshita, T., Wenger, T. & Weiss, D. S. Observation of a one-dimensional Tonks-Girardeau gas. Science 305, 1125–1128 (2004)

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Haller, E. et al. Realization of an excited, strongly correlated quantum gas phase. Science 325, 1224–1227 (2009)

    Article  ADS  CAS  PubMed  Google Scholar 

  16. Lieb, E. H. & Liniger, W. Exact analysis of an interacting Bose gas. Phys. Rev. 130, 1605–1616 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010)

    Article  ADS  CAS  Google Scholar 

  18. Kraemer, T. et al. Optimized production of a cesium Bose-Einstein condensate. Appl. Phys. B 79, 1013–1019 (2004)

    Article  ADS  CAS  Google Scholar 

  19. Stöferle, T., Moritz, H., Schori, C., Köhl, M. & Esslinger, T. Transition from a strongly interacting 1D superfluid to a Mott-insulator. Phys. Rev. Lett. 92, 130403 (2004)

    Article  ADS  PubMed  Google Scholar 

  20. Iucci, A., Cazalilla, M. A., Ho, A. F. & Giamarchi, T. Energy absorption of a Bose gas in a periodically modulated optical lattice. Phys. Rev. A 73, 041608(R) (2006)

    Article  ADS  Google Scholar 

  21. Clark, S. R. & Jaksch, D. Signatures of the superfluid to Mott-insulator transition in the excitation spectrum of ultracold atoms. N. J. Phys. 8, 160 (2005)

    Article  Google Scholar 

  22. Mun, J. et al. Phase diagram for a Bose-Einstein condensate moving in an optical lattice. Phys. Rev. Lett. 99, 150604 (2007)

    Article  ADS  PubMed  Google Scholar 

  23. Fertig, C. D. et al. Strongly inhibited transport of a degenerate 1D Bose gas in a lattice. Phys. Rev. Lett. 94, 120403 (2005)

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Altman, E., Polkovnikov, A., Demler, E., Halperin, B. I. & Lukin, M. D. Superfluid-insulator transition in a moving system of interacting bosons. Phys. Rev. Lett. 95, 020402 (2005)

    Article  ADS  CAS  PubMed  Google Scholar 

  25. Schachenmayer, J., Pupillo, G. & Daley, A. J. Time-dependent currents of one-dimensional bosons in an optical lattice. N. J. Phys. 12, 025014 (2010)

    Article  Google Scholar 

  26. Rapsch, S., Schollwck, U. & Zwerger, W. Density matrix renormalization group for disordered bosons in one dimension. Europhys. Lett. 46, 559–564 (1999)

    Article  ADS  CAS  Google Scholar 

  27. Cazalilla, M. A. Bosonizing one-dimensional cold atomic gases. J. Phys. B 37, S1–S47 (2004)

    Article  ADS  CAS  Google Scholar 

  28. Zamolodchikov, A. B. & Zamolodchikov, A. B. Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120, 253–291 (1979)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  29. Zamolodchikov, A. Mass scale in the sine-Gordon model and its reductions. Int. J. Mod. Phys. A 10, 1125–1150 (1995)

    Article  ADS  Google Scholar 

  30. Gould, P. L., Ruff, G. A. & Pritchard, D. E. Diffraction of atoms by light: the near-resonant Kapitza-Dirac effect. Phys. Rev. Lett. 56, 827–830 (1986)

    Article  ADS  CAS  PubMed  Google Scholar 

Download references


We thank W. Zwerger for discussions. We are indebted to R. Grimm for generous support. We gratefully acknowledge funding by the Austrian Ministry of Science and Research (Bundesministerium für Wissenschaft und Forschung) and the Austrian Science Fund (Fonds zur Förderung der wissenschaftlichen Forschung) in the form of a START prize grant, and by the European Union through the STREP FP7-ICT-2007-C project NAME-QUAM (Nanodesigning of Atomic and Molecular Quantum Matter) and within the framework of the EuroQUASAR collective research project QuDeGPM. R.H. is supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme.

Author information

Authors and Affiliations



The experimental work was done by E.H., R.H., M.J.M., J.G.D, L.R., M.G. and H.-C.N. Theoretical analysis and support was provided by M.D. and G.P. The manuscript was written with substantial contributions from all authors.

Corresponding author

Correspondence to Hanns-Christoph Nägerl.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Haller, E., Hart, R., Mark, M. et al. Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons. Nature 466, 597–600 (2010).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

This article is cited by


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