Advertisement

Journal of High Energy Physics

, 2018:52 | Cite as

The large-charge expansion for Schrödinger systems

  • Samuel Favrod
  • Domenico Orlando
  • Susanne ReffertEmail author
Open Access
Regular Article - Theoretical Physics

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.

Keywords

Effective Field Theories Global Symmetries Conformal Field Theory 

Notes

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.

References

  1. [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].ADSMathSciNetzbMATHGoogle Scholar
  2. [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].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    O. Loukas, Abelian scalar theory at large global charge, Fortsch. Phys. 65 (2017) 1700028 [arXiv:1612.08985] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  4. [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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [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. [6]
    S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, Observables in Inhomogeneous Ground States at Large Global Charge, arXiv:1804.06495 [INSPIRE].
  7. [7]
    O. Loukas, D. Orlando and S. Reffert, Matrix models at large charge, JHEP 10 (2017) 085 [arXiv:1707.00710] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    O. Loukas, A matrix CFT at multiple large charges, JHEP 06 (2018) 164 [arXiv:1711.07990] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [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. [10]
    S. Hellerman, S. Maeda and M. Watanabe, Operator Dimensions from Moduli, JHEP 10 (2017) 089 [arXiv:1706.05743] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. Hellerman, S. Maeda, D. Orlando, S. Reffert and M. Watanabe, Universal correlation functions in rank 1 SCFTs, arXiv:1804.01535 [INSPIRE].
  12. [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. [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. [14]
    D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, Conformal Bootstrap At Large Charge, JHEP 05 (2018) 043 [arXiv:1710.11161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [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. [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. [17]
    A. De La Fuente, The large charge expansion at large N , JHEP 08 (2018) 041 [arXiv:1805.00501] [INSPIRE].CrossRefzbMATHGoogle Scholar
  18. [18]
    M. Randeria, W. Zwerger and M. Zwierlein, The BCS-BEC Crossover and the Unitary Fermi Gas, Springer, Lect. Notes Phys. 836 (2012) 1.Google Scholar
  19. [19]
    W. Bakr et al., Strongly interacting Fermi gases, EPJ Web Conf. 57 (2013) 01002.Google Scholar
  20. [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. [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. [22]
    R. Jackiw and S.-Y. Pi, Finite and infinite symmetries in (2 + 1)-dimensional field theory, hep-th/9206092 [INSPIRE].
  23. [23]
    O. Bergman and G. Lozano, Aharonov-Bohm scattering, contact interactions and scale invariance, Annals Phys. 229 (1994) 416 [hep-th/9302116] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].
  25. [25]
    S.M. Kravec and S. Pal, Nonrelativistic Conformal Field Theories in the Large Charge Sector, arXiv:1809.08188 [INSPIRE].
  26. [26]
    M. Greiter, F. Wilczek and E. Witten, Hydrodynamic Relations in Superconductivity, Mod. Phys. Lett. B 3 (1989) 903 [INSPIRE].
  27. [27]
    H. Leutwyler, Nonrelativistic effective Lagrangians, Phys. Rev. D 49 (1994) 3033 [hep-ph/9311264] [INSPIRE].
  28. [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].ADSCrossRefGoogle Scholar
  29. [29]
    S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Samuel Favrod
    • 1
  • Domenico Orlando
    • 2
    • 3
  • Susanne Reffert
    • 3
    Email author
  1. 1.Institut für Theoretische Physik, ETH ZürichZürichSwitzerland
  2. 2.INFN sezione di Torino, Arnold-Regge CenterTurinItaly
  3. 3.Albert Einstein Center for Fundamental Physics, Institute for Theoretical PhysicsUniversity of BernBernSwitzerland

Personalised recommendations