Abstract
We study operators in Schrödinger invariant field theories (non-relativistic conformal field theories or NRCFTs) with large charge (particle number) and spin. Via the state-operator correspondence for NRCFTs, such operators correspond to states of a superfluid in a harmonic trap with phonons or vortices. Using the effective field theory of the Goldstone mode, we compute the dimensions of operators to leading order in the angular momentum L and charge Q. We find a diverse set of scaling behaviors for NRCFTs in both d = 2 and d = 3 spatial dimensions. These results apply to theories with a superfluid phase, such as unitary fermions or critical anyon systems.
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References
R.J. Donnelly, Quantized Vortices in Helium II, Cambridge University Press, (1991).
S. Vitiello, L. Reatto, G. Chester and M. Kalos, Vortex line in superfluid he 4: A variational monte carlo calculation, Phys. Rev. B 54 (1996) 1205.
G. Ortiz and D.M. Ceperley, Core structure of a vortex in superfluid 4 He, Phys. Rev. Lett. 75 (1995) 4642.
S. Giorgini, J. Boronat and J. Casulleras, Vortex excitation in superfluid 4 He: A diffusion monte carlo study, Phys. Rev. Lett. 77 (1996) 2754.
G. Baym, C. Pethick and D. Pines, Superfluidity in neutron stars, Nature 224 (1969) 673.
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
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].
S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A Note on Inhomogeneous Ground States at Large Global Charge, arXiv:1705.05825 [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
D. Banerjee, S. Chandrasekharan and D. Orlando, Conformal dimensions via large charge expansion, Phys. Rev. Lett. 120 (2018) 061603 [arXiv:1707.00711] [INSPIRE].
A. De La Fuente, The large charge expansion at large N, JHEP 08 (2018) 041 [arXiv:1805.00501] [INSPIRE].
D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, Conformal Bootstrap At Large Charge, JHEP 05 (2018) 043 [arXiv:1710.11161] [INSPIRE].
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].
M.W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. Schunck and W. Ketterle, Vortices and superfluidity in a strongly interacting fermi gas, Nature 435 (2005) 1047.
A. Kobach and S. Pal, Conformal Structure of the Heavy Particle EFT Operator Basis, Phys. Lett. B 783 (2018) 311 [arXiv:1804.01534] [INSPIRE].
M. Henkel and J. Unterberger, Schrödinger invariance and space-time symmetries, Nucl. Phys. B 660 (2003) 407 [hep-th/0302187] [INSPIRE].
C.A. Regal, M. Greiner and D.S. Jin, Observation of Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett. 92 (2004) 040403 [cond-mat/0401554] [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 [cond-mat/0403049] [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].
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].
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].
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.
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].
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].
Y. Nishida and D.T. Son, Unitary Fermi gas, epsilon expansion, and nonrelativistic conformal field theories, Lect. Notes Phys. 836 (2012) 233 [arXiv:1004.3597].
Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].
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].
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].
S. Pal, Unitarity and universality in nonrelativistic conformal field theory, Phys. Rev. D 97 (2018) 105031 [arXiv:1802.02262] [INSPIRE].
K. Ohashi, T. Fujimori and M. Nitta, Conformal symmetry of trapped Bose-Einstein condensates and massive Nambu-Goldstone modes, Phys. Rev. A 96 (2017) 051601 [arXiv:1705.09118] [INSPIRE].
S.M. Kravec and S. Pal, Nonrelativistic Conformal Field Theories in the Large Charge Sector, JHEP 02 (2019) 008 [arXiv:1809.08188] [INSPIRE].
S. Favrod, D. Orlando and S. Reffert, The large-charge expansion for Schrödinger systems, JHEP 12 (2018) 052 [arXiv:1809.06371] [INSPIRE].
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].
C.J. Pethick and H. Smith, Bose-einstein condensation in dilute gases, Cambridge University Press, (2008).
G. Bruun and L. Viverit, Vortex state in superfluid trapped fermi gases at zero temperature, Phys. Rev. A 64 (2001) 063606.
D.E. Sheehy and L. Radzihovsky, Vortices in spatially inhomogeneous superfluids, Phys. Rev. A 70 (2004) 063620.
A.J. Groszek, D.M. Paganin, K. Helmerson and T.P. Simula, Motion of vortices in inhomogeneous Bose-Einstein condensates, Phys. Rev. A 97 (2018) 023617 [arXiv:1708.09202] [INSPIRE].
A.L. Fetter, Kelvin mode of a vortex in a nonuniform bose-einstein condensate, Phys. Rev. A 69 (2004) 043617.
V. Tkachenko, On vortex lattices, Sov. Phys. JETP 22 (1966) 1282.
L. Campbell and R.M. Ziff, Vortex patterns and energies in a rotating superfluid, Phys. Rev. B 20 (1979) 1886.
N.R. Cooper, Rapidly rotating atomic gases, Adv. Phys. 57 (2008) 539.
N. Doroud, D. Tong and C. Turner, On Superconformal Anyons, JHEP 01 (2016) 138 [arXiv:1511.01491] [INSPIRE].
N. Doroud, D. Tong and C. Turner, The Conformal Spectrum of Non-Abelian Anyons, SciPost Phys. 4 (2018) 022 [arXiv:1611.05848] [INSPIRE].
R. Jackiw and S.-Y. Pi, Selfdual Chern-Simons solitons, Prog. Theor. Phys. Suppl. 107 (1992) 1 [INSPIRE].
R. Chitra and D. Sen, Ground state of many anyons in a harmonic potential, Phys. Rev. B 46 (1992) 10923 [INSPIRE].
S. Hellerman and I. Swanson, Boundary Operators in Effective String Theory, JHEP 04 (2017) 085 [arXiv:1609.01736] [INSPIRE].
S. Hellerman and I. Swanson, String Theory of the Regge Intercept, Phys. Rev. Lett. 114 (2015) 111601 [arXiv:1312.0999] [INSPIRE].
B. Horn, A. Nicolis and R. Penco, Effective string theory for vortex lines in fluids and superfluids, JHEP 10 (2015) 153 [arXiv:1507.05635] [INSPIRE].
S. Hellerman, private communication, (2019).
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].
V. Tkachenko, Elasticity of vortex lattices, Soviet J. Exp. Theor. Phys. 29 (1969) 945.
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].
I.Z. Rothstein and P. Shrivastava, Symmetry Realization via a Dynamical Inverse Higgs Mechanism, JHEP 05 (2018) 014 [arXiv:1712.07795] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
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Kravec, S.M., Pal, S. The spinful large charge sector of non-relativistic CFTs: from phonons to vortex crystals. J. High Energ. Phys. 2019, 194 (2019). https://doi.org/10.1007/JHEP05(2019)194
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DOI: https://doi.org/10.1007/JHEP05(2019)194