Skip to main content
Download PDF

The large-charge expansion for Schrödinger systems

Journal of High Energy Physics volume 2018, Article number: 52 (2018) Cite this article

A preprint version of the article is available at arXiv.

Abstract

In this note, we perform the large-charge expansion for non-relativistic systems with a global U(1) symmetry in 3 + 1 and 2 + 1 space-time dimensions, motivated by applications to the unitary Fermi gas and anyons. These systems do not have full conformal invariance, but are invariant under the Schrödinger group. Also here, the low-energy physics is encoded by a Goldstone boson which is due to the breaking of the global symmetry when fixing the charge. We find that in 2 + 1 dimensions and higher, there is a large-charge expansion in which quantum corrections are suppressed with respect to the next-to-leading order terms in the Lagrangian. We give the next-to-leading-order expressions for the ground state energy and the speed of sound.

References

  1. S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT Operator Spectrum at Large Global Charge, JHEP 12 (2015) 071 [arXiv:1505.01537] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  2. L. Álvarez-Gaumé, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP 04 (2017) 059 [arXiv:1610.04495] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  3. O. Loukas, Abelian scalar theory at large global charge, Fortsch. Phys. 65 (2017) 1700028 [arXiv:1612.08985] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  4. A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A Note on Inhomogeneous Ground States at Large Global Charge, arXiv:1705.05825 [INSPIRE].

  6. S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, Observables in Inhomogeneous Ground States at Large Global Charge, arXiv:1804.06495 [INSPIRE].

  7. O. Loukas, D. Orlando and S. Reffert, Matrix models at large charge, JHEP 10 (2017) 085 [arXiv:1707.00710] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. O. Loukas, A matrix CFT at multiple large charges, JHEP 06 (2018) 164 [arXiv:1711.07990] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. S. Hellerman and S. Maeda, On the Large R-charge Expansion in \( \mathcal{N}=2 \) Superconformal Field Theories, JHEP 12 (2017) 135 [arXiv:1710.07336] [INSPIRE].

  10. S. Hellerman, S. Maeda and M. Watanabe, Operator Dimensions from Moduli, JHEP 10 (2017) 089 [arXiv:1706.05743] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. S. Hellerman, S. Maeda, D. Orlando, S. Reffert and M. Watanabe, Universal correlation functions in rank 1 SCFTs, arXiv:1804.01535 [INSPIRE].

  12. A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in \( \mathcal{N}=2 \) theories, JHEP 05 (2018) 074 [arXiv:1803.00580] [INSPIRE].

  13. G. Cuomo, A. de la Fuente, A. Monin, D. Pirtskhalava and R. Rattazzi, Rotating superfluids and spinning charged operators in conformal field theory, Phys. Rev. D 97 (2018) 045012 [arXiv:1711.02108] [INSPIRE].

  14. D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, Conformal Bootstrap At Large Charge, JHEP 05 (2018) 043 [arXiv:1710.11161] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. O. Loukas, D. Orlando, S. Reffert and D. Sarkar, An AdS/EFT correspondence at large charge, Nucl. Phys. B 934 (2018) 437 [arXiv:1804.04151] [INSPIRE].

  16. D. Banerjee, S. Chandrasekharan and D. Orlando, Conformal dimensions via large charge expansion, Phys. Rev. Lett. 120 (2018) 061603 [arXiv:1707.00711] [INSPIRE].

  17. A. De La Fuente, The large charge expansion at large N , JHEP 08 (2018) 041 [arXiv:1805.00501] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  18. M. Randeria, W. Zwerger and M. Zwierlein, The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, Lect. Notes Phys. 836 (2012) 1.

  19. W. Bakr et al., Strongly interacting Fermi gases, EPJ Web Conf. 57 (2013) 01002.

  20. D.T. Son and M. Wingate, General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas, Annals Phys. 321 (2006) 197 [cond-mat/0509786] [INSPIRE].

  21. Y.-H. Chen, F. Wilczek, E. Witten and B.I. Halperin, On Anyon Superconductivity, Int. J. Mod. Phys. B 3 (1989) 1001 [INSPIRE].

  22. R. Jackiw and S.-Y. Pi, Finite and infinite symmetries in (2 + 1)-dimensional field theory, hep-th/9206092 [INSPIRE].

  23. O. Bergman and G. Lozano, Aharonov-Bohm scattering, contact interactions and scale invariance, Annals Phys. 229 (1994) 416 [hep-th/9302116] [INSPIRE].

    ADS  Article  Google Scholar 

  24. Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].

  25. S.M. Kravec and S. Pal, Nonrelativistic Conformal Field Theories in the Large Charge Sector, arXiv:1809.08188 [INSPIRE].

  26. M. Greiter, F. Wilczek and E. Witten, Hydrodynamic Relations in Superconductivity, Mod. Phys. Lett. B 3 (1989) 903 [INSPIRE].

  27. H. Leutwyler, Nonrelativistic effective Lagrangians, Phys. Rev. D 49 (1994) 3033 [hep-ph/9311264] [INSPIRE].

  28. N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].

    ADS  Article  Google Scholar 

  29. S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, ETH Zürich, Zürich, CH-8093, Switzerland

    Samuel Favrod

  2. INFN sezione di Torino, Arnold-Regge Center, via Pietro Giuria 1, Turin, 10125, Italy

    Domenico Orlando

  3. Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, Bern, CH-3012, Switzerland

    Domenico Orlando & Susanne Reffert

Authors
  1. Samuel Favrod
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Domenico Orlando
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Susanne Reffert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Susanne Reffert.

Additional information

ArXiv ePrint: 1809.06371

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Favrod, S., Orlando, D. & Reffert, S. The large-charge expansion for Schrödinger systems. J. High Energ. Phys. 2018, 52 (2018). https://doi.org/10.1007/JHEP12(2018)052

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP12(2018)052

Keywords

  • Effective Field Theories
  • Global Symmetries
  • Conformal Field Theory