Abstract
In this note we search for the ground state of the D = 3 Wilson-Fisher conformal O(4) model, at nonzero values of the two independent charge densities ρ1,2, on the torus spatial slice. Using an effective theory valid on scales longer than the scale defined by the charge density, we show that the ground-state configuration is inhomogeneous for generic ratios ρ1/ρ2. This result confirms, within the context of a well-defined effective theory, a recent no-go result of [1]. We also show that any spatially periodic ground state solutions have an energetic preference towards longer periods, within some range of ρ1/ρ2 containing a neighborhood of zero. This suggests that the scale of variation of the ground state solution in finite volume will be the infrared scale, and that the use of the effective theory at large charge in finite volume is self-consistent. Note added: the statements in this paper are true for arbitrary ratio of ρ1/ρ2, which we proved after we uploaded this paper. See [2].
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References
L. Álvarez-Gaumé, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP04 (2017) 059 [arXiv:1610.04495] [INSPIRE].
S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, Observables in inhomogeneous ground states at large global charge, arXiv:1804.06495 [INSPIRE].
S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT operator spectrum at large global charge, JHEP12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone bosons and CFT data, JHEP06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
O. Loukas, Abelian scalar theory at large global charge, Fortsch. Phys.65 (2017) 1700028 [arXiv:1612.08985] [INSPIRE].
L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett.119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
L.F. Alday, Solving CFTs with weakly broken higher spin symmetry, JHEP10 (2017) 161 [arXiv:1612.00696] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N) models, JHEP08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Bootstrapping the O(N) archipelago, JHEP11 (2015) 106 [arXiv:1504.07997] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
P. Dey, A. Kaviraj and K. Sen, More on analytic bootstrap for O(N) models, JHEP06 (2016) 136 [arXiv:1602.04928] [INSPIRE].
K. Diab, L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, On CJand CTin the Gross-Neveu and O(N) models, J. Phys.A 49 (2016) 405402 [arXiv:1601.07198] [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising model with the conformal bootstrap, Phys. Rev.D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3d Ising model with the conformal bootstrap II. c-minimization and precise critical exponents, J. Stat. Phys.157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett.28 (1972) 240 [INSPIRE].
D. Banerjee, S. Chandrasekharan, D. Orlando and S. Reffert, Conformal dimensions in the large charge sectors at the O(4) Wilson-Fisher fixed point, Phys. Rev. Lett.123 (2019) 051603 [arXiv:1902.09542] [INSPIRE].
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Hellerman, S., Kobayashi, N., Maeda, S. et al. A note on inhomogeneous ground states at large global charge. J. High Energ. Phys. 2019, 38 (2019). https://doi.org/10.1007/JHEP10(2019)038
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DOI: https://doi.org/10.1007/JHEP10(2019)038