For Robin boundary conditions with -pi/2 < theta < 0, the total induced vacuum energy of a scalar field around a magnetic tube depends on the curvature coupling xi, while Dirichlet and Neumann cases do not.
Kibble, Some implications of a cosmological phase transition, Phys
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Dependence of scalar matter vacuum energy, induced by a magnetic topological defect, on the coupling to space-time curvature
For Robin boundary conditions with -pi/2 < theta < 0, the total induced vacuum energy of a scalar field around a magnetic tube depends on the curvature coupling xi, while Dirichlet and Neumann cases do not.