We theoretically investigate how the spin susceptibility of a planar Josephson junction is affected when the system transits into the topological superconducting state. We show that the magnetic flux and magnetic field dependence of the spin susceptibility closely maps the phase diagram of the system. In the absence of an external magnetic flux the system can self-tune into the topological superconducting state by minimizing its free energy. Self-tuned topological transitions are accompanied by sharp peaks in the spin susceptibility, which can therefore be used as measurable fingerprints for detecting the topological superconducting state. Away from the peaks, the amplitude of the spin susceptibility can provide qualitative information about the relative size of the topological gap. The signatures in the spin susceptibility are robust, even in junctions with narrow superconducting leads, where critical current minima may no longer serve as an indication of topological phase transitions. The predicted results could be particularly relevant for future experiments on realization and detection of the topological superconducting state in planar Josephson junctions.

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• Majorana bound states
• Proximity effect
• Spintronics
• Superconductivity
• Topological phase transition
• Josephson junctions
• Topological materials
• Topological superconductors
• Two-dimensional electron system
• Bogoliubov-de Gennes equations
• Tight-binding model