Abstract
We consider unitary CFTs with continuous global symmetries in d > 2. We consider a state created by the lightest operator of large charge Q ≫ 1 and analyze the correlator of two light charged operators in this state. We assume that the correlator admits a well-defined large Q expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists. We find that the crossing equations admit a consistent truncation, where only a finite number N of Regge trajectories contribute to the correlator at leading nontrivial order. We classify all such truncated solutions to the crossing. For one Regge trajectory N = 1, the solution is unique and given by the effective field theory of a Goldstone mode. For two or more Regge trajectories N ≥ 2, the solutions are encoded in roots of a certain degree N polynomial. Some of the solutions admit a simple weakly coupled EFT description, whereas others do not. In the weakly coupled case, each Regge trajectory corresponds to a field in the effective Lagrangian.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Pelissetto and E. Vicari, Critical phenomena and renormalization group theory, Phys. Rept. 368 (2002) 549 [cond-mat/0012164].
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].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, goldstone bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
L. Álvarez-Gaumé, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP 04 (2017) 059 [arXiv:1610.04495] [INSPIRE].
S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A note on inhomogeneous ground states at large global charge, arXiv:1705.05825 [INSPIRE].
O. Loukas, D. Orlando and S. Reffert, Matrix models at large charge, JHEP 10 (2017) 085 [arXiv:1707.00710] [INSPIRE].
M. Hasenbusch and E. Vicari, Anisotropic perturbations in three-dimensional O(N )-symmetric vector models, Phys. Rev. B 84 (2011) 125136 [arXiv:1108.0491].
D. Banerjee, S. Chandrasekharan and D. Orlando, Conformal dimensions via large charge expansion, Phys. Rev. Lett. 120 (2018) 061603 [arXiv:1707.00711] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [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].
N. Lashkari, A. Dymarsky and H. Liu, Eigenstate thermalization hypothesis in conformal field theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051].
M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].
L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 239 [arXiv:1509.06411] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro conformal blocks and thermality from classical background fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
A. Monin, Partition function on spheres: how to use zeta function regularization, Phys. Rev. D 94 (2016) 085013 [arXiv:1607.06493] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
S. Hellerman, S. Maeda and M. Watanabe, Operator dimensions from moduli, JHEP 10 (2017) 089 [arXiv:1706.05743] [INSPIRE].
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].
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The string landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
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
Corresponding author
Additional information
ArXiv ePrint: 1710.11161
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/.
About this article
Cite this article
Jafferis, D., Mukhametzhanov, B. & Zhiboedov, A. Conformal bootstrap at large charge. J. High Energ. Phys. 2018, 43 (2018). https://doi.org/10.1007/JHEP05(2018)043
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2018)043