Nonrelativistic conformal field theories in the large charge sector

  • S. M. KravecEmail author
  • Sridip Pal
Open Access
Regular Article - Theoretical Physics


We study Schrödinger invariant field theories (nonrelativistic conformal field theories) in the large charge (particle number) sector. We do so by constructing the effective field theory (EFT) for a Goldstone boson of the associated U(1) symmetry in a harmonic potential. This EFT can be studied semi-classically in a large charge expansion. We calculate the dimensions of the lowest lying operators, as well as correlation functions of charged operators. We find universal behavior of three point function in large charge sector. We comment on potential applications to fermions at unitarity and critical anyon systems.


Conformal Field Theory Effective Field Theories Global Symmetries Space-Time Symmetries 


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.


  1. [1]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    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
  3. [3]
    S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A Note on Inhomogeneous Ground States at Large Global Charge, arXiv:1705.05825 [INSPIRE].
  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]
    D. Banerjee, S. Chandrasekharan and D. Orlando, Conformal dimensions via large charge expansion, Phys. Rev. Lett. 120 (2018) 061603 [arXiv:1707.00711] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    A. De La Fuente, The large charge expansion at large N, JHEP 08 (2018) 041 [arXiv:1805.00501] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean Bootstrap, arXiv:1808.03212 [INSPIRE].
  8. [8]
    T. Mehen, I.W. Stewart and M.B. Wise, Conformal invariance for nonrelativistic field theory, Phys. Lett. B 474 (2000) 145 [hep-th/9910025] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    Y. Nishida and D.T. Son, Unitary Fermi gas, ϵ-expansion and nonrelativistic conformal field theories, Lect. Notes Phys. 836 (2012) 233 [arXiv:1004.3597] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    W.D. Goldberger, Z.U. Khandker and S. Prabhu, OPE convergence in non-relativistic conformal field theories, JHEP 12 (2015) 048 [arXiv:1412.8507] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  12. [12]
    S. Golkar and D.T. Son, Operator Product Expansion and Conservation Laws in Non-Relativistic Conformal Field Theories, JHEP 12 (2014) 063 [arXiv:1408.3629] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S. Pal, Unitarity and universality in nonrelativistic conformal field theory, Phys. Rev. D 97 (2018) 105031 [arXiv:1802.02262] [INSPIRE].ADSGoogle Scholar
  14. [14]
    C.A. Regal, M. Greiner and D.S. Jin, Observation of Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett. 92 (2004) 040403 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, A.J. Kerman and W. Ketterle, Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance, Phys. Rev. Lett. 92 (2004) 120403 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    X. Chen, E. Fradkin and W. Witczak-Krempa, Gapless quantum spin chains: multiple dynamics and conformal wavefunctions, J. Phys. A 50 (2017) 464002 [arXiv:1707.02317] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  17. [17]
    D.B. Kaplan, M.J. Savage and M.B. Wise, A New expansion for nucleon-nucleon interactions, Phys. Lett. B 424 (1998) 390 [nucl-th/9801034] [INSPIRE].
  18. [18]
    D.B. Kaplan, M.J. Savage and M.B. Wise, Two nucleon systems from effective field theory, Nucl. Phys. B 534 (1998) 329 [nucl-th/9802075] [INSPIRE].
  19. [19]
    C. Chin, V. Vuletić, A.J. Kerman and S. Chu, High precision Feshbach spectroscopy of ultracold cesium collisions, Nucl. Phys. A 684 (2001) 641 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    J.L. Roberts, N.R. Claussen, J.P. Burke, C.H. Greene, E.A. Cornell and C.E. Wieman, Resonant Magnetic Field Control of Elastic Scattering in Cold R-85b, Phys. Rev. Lett. 81 (1998) 5109 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    T. Loftus, C. Regal, C. Ticknor, J. Bohn and D.S. Jin, Resonant control of elastic collisions in an optically trapped fermi gas of atoms, Phys. Rev. Lett. 88 (2002) 173201.ADSCrossRefGoogle Scholar
  22. [22]
    I.Z. Rothstein and P. Shrivastava, Symmetry Realization via a Dynamical Inverse Higgs Mechanism, JHEP 05 (2018) 014 [arXiv:1712.07795] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    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].
  24. [24]
    S. Favrod, D. Orlando and S. Reffert, The large-charge expansion for Schrödinger systems, JHEP 12 (2018) 052 [arXiv:1809.06371] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
  26. [26]
    C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
  27. [27]
    V. Ogievetsky, Nonlinear realizations of internal and space-time symmetries, in proceedings of x-th winter school of theoretical physics in karpacz, Universitas Wratislaviensis, Wroclaw, Poland (1974).Google Scholar
  28. [28]
    L.V. Delacrétaz, S. Endlich, A. Monin, R. Penco and F. Riva, (Re-)Inventing the Relativistic Wheel: Gravity, Cosets and Spinning Objects, JHEP 11 (2014) 008 [arXiv:1405.7384] [INSPIRE].
  29. [29]
    I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstone’s theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    A. Nicolis, R. Penco, F. Piazza and R.A. Rosen, More on gapped Goldstones at finite density: More gapped Goldstones, JHEP 11 (2013) 055 [arXiv:1306.1240] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    T. Brauner, S. Endlich, A. Monin and R. Penco, General coordinate invariance in quantum many-body systems, Phys. Rev. D 90 (2014) 105016 [arXiv:1407.7730] [INSPIRE].ADSGoogle Scholar
  32. [32]
    N. Doroud, D. Tong and C. Turner, The Conformal Spectrum of Non-Abelian Anyons, SciPost Phys. 4 (2018) 022 [arXiv:1611.05848] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    N. Doroud, D. Tong and C. Turner, On Superconformal Anyons, JHEP 01 (2016) 138 [arXiv:1511.01491] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    R. Jackiw and S.-Y. Pi, Selfdual Chern-Simons solitons, Prog. Theor. Phys. Suppl. 107 (1992) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    D.T. Son, Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].ADSGoogle Scholar
  37. [37]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    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].ADSMathSciNetGoogle Scholar
  39. [39]
    C. Hoyos, S. Moroz and D.T. Son, Effective theory of chiral two-dimensional superfluids, Phys. Rev. B 89 (2014) 174507 [arXiv:1305.3925] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    S. Moroz, C. Hoyos, C. Benzoni and D.T. Son, Effective field theory of a vortex lattice in a bosonic superfluid, SciPost Phys. 5 (2018) 039 [arXiv:1803.10934] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of California at San DiegoLa JollaU.S.A.

Personalised recommendations