Compressibility as a probe of quantum phase transitions in topological superconductors

While there have been recent reports of zero-energy modes in single-particle tunneling density of states, their identity as Majorana modes has not been unequivocally established thus far. We make predictions for the local compressibility kappa(loc), tuned by changing the chemical potential mu in a semiconducting nanowire with strong spin-orbit coupling and in a Zeeman field in proximity to a superconductor, which has been proposed as a candidate system for observing Majorana modes. We show that in the center of the wire, the topological phase transition is signaled by a divergence of kappa(loc) as a function of mu, an important diagnostic of the phase transition. We also find that a single strong impurity potential can lead to a local negative compressibility at the topological phase transition. The origin of such anomalous behavior can be traced to the formation of Andreev bound states close to topological phase transitions. Measurable by a gate-tunable scanning electron transistor, the compressibility includes contributions from both single-particle states and collective modes and is therefore a complementary probe from scanning tunneling spectroscopy, which is sensitive to only the single-particle density of states.