Abstract
Motivated by applications to the study of ultracold atomic gases near the unitarity limit, we investigate the structure of the operator product expansion (OPE) in non-relativistic conformal field theories (NRCFTs). The main tool used in our analysis is the representation theory of charged (i.e. non-zero particle number) operators in the NR-CFT, in particular the mapping between operators and states in a non-relativistic “radial quantization” Hilbert space. Our results include: a determination of the OPE coefficients of descendant operators in terms of those of the underlying primary state, a demonstration of convergence of the (imaginary time) OPE in certain kinematic limits, and an estimate of the decay rate of the OPE tail inside matrix elements which, as in relativistic CFTs, depends exponentially on operator dimensions. To illustrate our results we consider several examples, including a strongly interacting field theory of bosons tuned to the unitarity limit, as well as a class of holographic models. Given the similarity with known statements about the OPE in SO(2, d) invariant field theories, our results suggest the existence of a bootstrap approach to constraining NRCFTs, with applications to bound state spectra and interactions. We briefly comment on a possible implementation of this non-relativistic conformal bootstrap program.
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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].
C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation., Helv. Phys. Acta 45 (1972) 802 [INSPIRE].
R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D 42 (1990) 3500 [Erratum ibid. D 48 (1993) 3929] [INSPIRE].
R. Jackiw and S. Pi, Finite and infinite symmetries in (2 + 1)-dimensional field theory, Nucl. Phys. Proc. Suppl. 33C (1993) 104 [INSPIRE].
U. Niederer, The connections between the schroedinger group and the conformal group, Helv. Phys. Acta 47 (1974) 119 [INSPIRE].
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].
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].
J.L. Roberts et al., Resonant magnetic field control of elastic scattering in cold R-85b, Phys. Rev. Lett. 81 (1998) 5109 [INSPIRE].
C. Chin et al., High resolution feshbach spectroscopy of Cesium, Phys. Rev. Lett. 85 (2000) 2717.
P.J. Leo et al., Collision properties of ultracold 133 Cs atoms, Phys. Rev. Lett. 85 (2000) 2721.
T. Loftus et al., Resonant control of elastic collisions in an optically trapped Fermi gas of atoms, Phys. Rev. Lett. 88 (2002) 173201.
C.A. Regal, M. Greiner and D.S. Jin, Observation of resonance condensation of fermionic atom pairs, Phys. Rev. Lett. 92 (2004) 040403 [INSPIRE].
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].
D.M. Eagles, Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors, Phys. Rev. 186 (1969) 456 [INSPIRE].
A. Leggett, Diatomic molecules and cooper pairs, in Modern trends in the theory of condensed matter, A. Pekalski and J. Przystawa eds., Lecture Notes in Physics volume 115, Springer, Berlin Germany (1980).
A. J. Leggett, Cooper pairing in spin-polarized Fermi systems, J. Phys. (Paris) Colloq. 41 (1980) C7.
P. Nozieres and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: from weak to strong coupling superconduct ivity, J. Low. Temp. Phys. 59 (1985) 195 [INSPIRE].
W. Zwerger, The BCS-BEC crossover and the unitary Fermi gas, Springer, Berlin Germany (2012).
E. Braaten and L. Platter, Exact relations for a strongly interacting Fermi gas from the operator product expansion, Phys. Rev. Lett. 100 (2008) 205301 [arXiv:0803.1125] [INSPIRE].
E. Braaten, D. Kang and L. Platter, Exact relations for a strongly-interacting Fermi gas near a Feshbach resonance, Phys. Rev. A 78 (2008) 053606 [arXiv:0806.2277] [INSPIRE].
E. Braaten, D. Kang and L. Platter, Universal relations for identical bosons from 3-body physics, Phys. Rev. Lett. 106 (2011) 153005 [arXiv:1101.2854] [INSPIRE].
M. Barth and W. Zwerger, Tan relations in one dimension, Ann. Phys. 326 (2011) 2544 [arXiv:1101.5594].
J. Hofmann, M. Barth and W. Zwerger, Short-distance properties of Coulomb systems, Phys. Rev. B 87 (2013) 235125 [arXiv:1304.2891] [INSPIRE].
D.T. Son and E.G. Thompson, Short-distance and short-time structure of a unitary Fermi gas, Phys. Rev. A 81 (2010) 063634 [arXiv:1002.0922] [INSPIRE].
W.D. Goldberger and I.Z. Rothstein, Structure function sum rules for systems with large scattering lengths, Phys. Rev. A 85 (2012) 013613 [arXiv:1012.5975] [INSPIRE].
J. Hofmann, Current response, structure factor and hydrodynamic quantities of a two- and three-dimensional Fermi gas from the operator product expansion, Phys. Rev. A 84 (2011) 043603 [arXiv:1106.6035] [INSPIRE].
W.D. Goldberger and Z.U. Khandker, Viscosity sum rules at large scattering lengths, Phys. Rev. A 85 (2012) 013624 [arXiv:1107.1472] [INSPIRE].
E. Braaten and H.W. Hammer, Universal relation for the inelastic two-body loss rate, J. Phys. B 46 (2013) 215203 [arXiv:1302.5617] [INSPIRE].
E. Braaten, Universal relations for fermions with large scattering length, Lect. Notes Phys. 836 (2012) 193 [arXiv:1008.2922] [INSPIRE].
O. Bergman, Nonrelativistic field theoretic scale anomaly, Phys. Rev. D 46 (1992) 5474 [INSPIRE].
M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Statist. Phys. 75 (1994) 1023 [hep-th/9310081] [INSPIRE].
M. Henkel and J. Unterberger, Schrödinger invariance and space-time symmetries, Nucl. Phys. B 660 (2003) 407 [hep-th/0302187] [INSPIRE].
Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].
T. Mehen, On non-relativistic conformal field theory and trapped atoms: virial theorems and the state-operator correspondence in three dimensions, Phys. Rev. A 78 (2008) 013614 [arXiv:0712.0867] [INSPIRE].
A. Volovich and C. Wen, Correlation functions in non-relativistic holography, JHEP 05 (2009) 087 [arXiv:0903.2455] [INSPIRE].
C.A. Fuertes and S. Moroz, Correlation functions in the non-relativistic AdS/CFT correspondence, Phys. Rev. D 79 (2009) 106004 [arXiv:0903.1844] [INSPIRE].
R.G. Leigh and N.N. Hoang, Real-time correlators and non-relativistic holography, JHEP 11 (2009) 010 [arXiv:0904.4270] [INSPIRE].
E. Barnes, D. Vaman and C. Wu, Holographic real-time non-relativistic correlators at zero and finite temperature, Phys. Rev. D 82 (2010) 125042 [arXiv:1007.1644] [INSPIRE].
R. Jackiw and S.Y. Pi, Conformal blocks for the 4-point function in conformal quantum mechanics, Phys. Rev. D 86 (2012) 045017 [Erratum ibid. D 86 (2012) 089905] [arXiv:1205.0443] [INSPIRE].
C. Duval, P.A. Horvathy and L. Palla, Conformal symmetry of the coupled Chern-Simons and gauged nonlinear Schrödinger equations, Phys. Lett. B 325 (1994) 39 [hep-th/9401065] [INSPIRE].
C. Duval, P.A. Horvathy and L. Palla, Conformal properties of Chern-Simons vortices in external fields, Phys. Rev. D 50 (1994) 6658 [hep-th/9404047] [INSPIRE].
C. Duval, M. Hassaine and P.A. Horvathy, The geometry of Schrödinger symmetry in gravity background/non-relativistic CFT, Annals Phys. 324 (2009) 1158 [arXiv:0809.3128] [INSPIRE].
C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].
A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2D, JHEP 08 (2010) 004 [arXiv:0912.1090] [INSPIRE].
M. Luscher, Operator product expansions on the vacuum in conformal quantum field theory in two spacetime dimensions, Commun. Math. Phys. 50 (1976) 23 [INSPIRE].
G. Mack, Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory, Commun. Math. Phys. 53 (1977) 155 [INSPIRE].
D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
X. Bekaert, E. Meunier and S. Moroz, Symmetries and currents of the ideal and unitary Fermi gases, JHEP 02 (2012) 113 [arXiv:1111.3656] [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
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].
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].
M. Perroud, Projective representations of the Schrödinger group, Helv. Phys. Acta 50 (1977) 233 [INSPIRE].
S. Ferrara, R. Gatto and A.F. Grillo, Conformal invariance on the light cone and canonical dimensions, Nucl. Phys. B 34 (1971) 349 [INSPIRE].
S. Ferrara, A. Grillo and R. Gatto, Manifestly conformal covariant operator-product expansion, Lett. Nuovo Cim. 2S2 (1971) 1363 [INSPIRE]
S. Ferrara, A.F. Grillo and R. Gatto, Manifestly conformal-covariant expansion on the light cone, Phys. Rev. D 5 (1972) 3102 [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the harmonic oscillator, Helv. Phys. Acta 46 (1973) 191 [INSPIRE].
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].
M. Reed and B. Simon, Methods of modern mathematical physics 1: functional analysis, Academic Press, San Diego U.S.A. (1980).
J. Polchinski, String theory. Volume 1: an introduction to the bosonic string, Cambridge University Press, Cmabridge U.K. (1998).
J. Korevaar, Tauberian theory, a century of developments, Springer, Berlin Germany (2004).
V. Efimov, Energy levels arising form the resonant two-body forces in a three-body system, Phys. Lett. B 33 (1970) 563 [INSPIRE].
V.N. Efimov, Weakly-bound states of 3 resonantly-interacting particles, Sov. J. Nucl. Phys. 12 (1971) 589 [INSPIRE].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].
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].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3D Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
J. Golden and M.F. Paulos, No unitary bootstrap for the fractal Ising model, JHEP 03 (2015) 167 [arXiv:1411.7932] [INSPIRE].
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Goldberger, W.D., Khandker, Z.U. & Prabhu, S. OPE convergence in non-relativistic conformal field theories. J. High Energ. Phys. 2015, 1–31 (2015). https://doi.org/10.1007/JHEP12(2015)048
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DOI: https://doi.org/10.1007/JHEP12(2015)048