Abstract
We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λϕ4 theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λϕ6 conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λϕ4 superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.
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Acknowledgments
It is a pleasure to thank Paolo Creminelli, Gabriel Cuomo, Luca Delacrétaz, Lorenzo Di Pietro, Paolo Glorioso, Hofie Hannesdottir, Oliver Janssen, Austin Joyce, and Riccardo Rattazzi for useful discussions. We are especially grateful to Austin Joyce for early collaboration, to Paolo Creminelli for prompting the analysis of the λϕ6 model in d = 3, and to Lorenzo Di Pietro for a question that inspired our study of the O(N) model. We also thank Jens Andersen, Gabriel Cuomo, Luca Delacrétaz and Sean Hartnoll for comments on a preliminary version of this work. AP is grateful to the audiences of the Sapienza University of Rome, ICTP, APC Paris, Saclay and Perimeter seminars, where part of this work was presented, for interesting comments and questions. The work of AP is supported by the grant DOE DE-SC0011941. LS is supported by the Centre National de la Recherche Scientifique (CNRS).
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Nicolis, A., Podo, A. & Santoni, L. The connection between nonzero density and spontaneous symmetry breaking for interacting scalars. J. High Energ. Phys. 2023, 200 (2023). https://doi.org/10.1007/JHEP09(2023)200
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DOI: https://doi.org/10.1007/JHEP09(2023)200