Compressibility as a probe of quantum phase transitions in topological superconductors

The non-Abelian statistics of Majorana fermions, their role in topological quantum computation, and the possibility of realizing them in condensed matter systems, has attracted considerable attention. While there have been recent reports of zero energy modes in single particle tunneling density of states, their identity as Majorana modes has so far not been unequivocally established. We make predictions for the local compressibility $\kappa_{\rm{loc}}$, 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, that 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 $\kappa_{\rm{loc}}$ as a function of $\mu$ which is an important diagnostic of the topological phase transition. We also find that a single strong impurity potential can lead to a local {\it 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 scanning electron transistor, the compressibility includes contributions from both single particle states and collective modes and is therefore a complimentary probe from scanning tunneling spectroscopy.