Abstract
We combine perturbation theory with analytic and numerical bootstrap techniques to study the critical point of the long-range Ising (LRI) model in two and three dimensions. This model interpolates between short-range Ising (SRI) and mean-field behaviour. We use the Lorentzian inversion formula to compute infinitely many three-loop corrections in the two-dimensional LRI near the mean-field end. We further exploit the exact OPE relations that follow from bulk locality of the LRI to compute infinitely many two-loop corrections near the mean-field end, as well as some one-loop corrections near SRI. By including such exact OPE relations in the crossing equations for LRI we set up a very constrained bootstrap problem, which we solve numerically using SDPB. We find a family of sharp kinks for two- and three-dimensional theories which compare favourably to perturbative predictions, as well as some Monte Carlo simulations for the two-dimensional LRI.
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Acknowledgments
We would like to thank A. Gimenez-Grau, A. Tilloy, B. van Rees, D. Mazáč, G. Gori, J. Henriksson, K. Tiwana, L. F. Alday, M. Meineri, M. Paulos, N. Defenu, P. Liendo and S. Rychkov for discussions. Numerical calculations in this paper were done on the University of Oxford Advanced Research Computing (ARC) facility [101]. CB is supported by the São Paulo Research Foundation (FAPESP) grants 2023/03825-2 and 2019/24277-8, and also received funding from the European Union (ERC, Analytic Conformal Bootstrap project, Grant Agreement no. 787185, held by L.F. Alday). EL is funded by the European Union (ERC, QFT.zip project, Grant Agreement no. 101040260, held by A. Tilloy). The work of MN is supported by funding from the Mathematical Institute, University of Oxford. PvV is funded by the European Union (ERC, FUNBOOTS project, project number 101043588, held by M. Paulos) and acknowledges support from the DFG through the Emmy Noether research group ‘The Conformal Bootstrap Program’ project number 400570283, and through the German-Israeli Project Cooperation (DIP) grant ‘Holography and the Swampland’. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
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Behan, C., Lauria, E., Nocchi, M. et al. Analytic and numerical bootstrap for the long-range Ising model. J. High Energ. Phys. 2024, 136 (2024). https://doi.org/10.1007/JHEP03(2024)136
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DOI: https://doi.org/10.1007/JHEP03(2024)136