Magnetic field response and chiral symmetry of time-reversal-invariant topological superconductors

We study the magnetic field response of Majorana-Kramers pairs of one-dimensional time-reversal (TR)-invariant topological superconductors (class DIII) with or without a coexisting chirality symmetry. In addition to explaining the anomalous magnetic field response of all the DIII class topological superconducting systems proposed in the literature, we provide a realistic route to engineer a ``true'' TR-invariant topological superconductor, i.e., one whose pair of Majorana bound states at each end is split by an applied Zeeman field in arbitrary direction. We also prove that quite generally the splitting of the Majorana bound states in a time-reversal-invariant topological superconductor by time-reversal breaking fields is highly anisotropic in spin space.