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Topological defects (such as monopoles, vortex lines, or domain walls) marklocations where disparate choices of a broken symmetry vacuum elsewhere in thesystem lead to irreconcilable differences. They are energetically costly (theenergy density in their core reaches that of the prior symmetric vacuum) buttopologically stable (the whole manifold would have to be rearranged to get ridof the defect). We show how, in a paradigmatic model of a quantum phasetransition, a topological defect can be put in a non-local superposition, sothat - in a region large compared to the size of its core - the order parameterof the system is "undecided" by being in a quantum superposition of conflictingchoices of the broken symmetry. We demonstrate how to exhibit such a"Schr\"odinger kink" by devising a version of a double-slit experiment suitablefor topological defects. Coherence detectable in such experiments will besuppressed as a consequence of interaction with the environment. We analyzeenvironment-induced decoherence and discuss its role in symmetry breaking.

Topological Schr\"odinger cats: Non-local quantum superpositions of topological defects