Traversable Wormholes with a Spontaneous Symmetry Breaking Scalar Field

The authors argue that a spherically symmetric traversable wormhole solution of the Einstein field equations can be supported by minimally coupled self-interacting scalar field, which allows a spontaneous symmetry breaking of the field around the wormhole throat. Two cases are studied by the authors: (i) the phantom wormhole solution of Bronnikov and (ii) a generalized Kiselev wormhole. The property of radial null geodesics is studied by authors and the metric can describe either a two-way or a one-way traversable wormhole depending on certain parameter ranges are showed. The scalar field exhibits spontaneous symmetry breaking (SSB) within the coordinate range where a wormhole throat forms and helps one suggest that SSB may act as a threshold for wormhole throat formation. The authors also computed the radius of the photon sphere, the Lyapunov exponent, the shadow radius, and the innermost stable circular orbits for the geometries.