Goldstone modes are massless particles resulting from spontaneous symmetry breaking. Although such modes are found in elementary particle physics as well as in condensed matter systems like superfluid helium, superconductors and magnons - structural Goldstone modes are rare. Epitaxial strain in thin films can induce structures and properties not accessible in bulk and has been intensively studied for (001)-oriented perovskite oxides. Here we predict Goldstone-like phonon modes in (111)-strained SrMnO3 by first-principles calculations. Under compressive strain the coupling between two in-plane rotational instabilities give rise to a Mexican hat shaped energy surface characteristic of a Goldstone mode. Conversely, large tensile strain induces in-plane polar instabilities with no directional preference, giving rise to a continuous polar ground state. Such phonon modes with U(1) symmetry could emulate structural condensed matter Higgs modes. The mass of this Higgs boson, given by the shape of the Mexican hat energy surface, can be tuned by strain through proper choice of substrate.
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