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.

Figure 1

(a) The low energy BdG spectrum as a function of chemical potential $\mu$. Each band is doubly degenerate, with 2 MBSs at each end of the nanowire in the $\left|\mu \right|<2{\alpha }_R$ topologically nontrivial regime. (b) Parametric plot of $\mathrm{Re}\left[\text{Det}\right(A\left(k\right)\left)\right]$ and $\mathrm{Im}\left[\text{Det}\right(A\left(k\right)\left)\right]$ as the quasimomentum $k$ is varied through the 1D Brillouin zone $(k\in [-\pi ,\pi \left]\right)$. The red curve $(\left|\mu \right|<2{\alpha }_R$, nontrivial regime) winds about the origin twice as $k$ is varied through the Brillouin zone yielding a chiral invariant $W=2$. Upon increasing the chemical potential, the blue curve $(\left|\mu \right|>2{\alpha }_R$ topologically trivial regime) winds about the origin 0 times indicating a decrease in the chiral invariant from $W=2$ to $W=0$.

Figure 2

(a) Winding curves for the bulk BdG Hamiltonian given by Eqs. (1) and (4) which preserves the chiral symmetry $(\mathit{b}\perp \widehat{a})$. The red, blue, and green curves yield chiral invariants $W=2,1,0$ connected by gap closures and topological phase transitions. (b) Low energy BdG quasiparticle spectrum corresponding to the winding curves in (a). Here $N$ counts the number of positive (negative) energy eigenvalues above (below) the Fermi energy. The chiral invariant counts the number of topologically protected modes at each end of the nanowire.

Figure 3

(a) Energy splitting of the MBSs in the presence of Zeeman splitting $V$ along $\widehat{y}$ (along the direction of spin orbit direction) as described in Eq. (4). (b) Energy splitting as a function of Zeeman field along three orthogonal directions ${\mathit{b}}_{1,2,3}$ described in the text on a log-log scale are found (from the slope) to split as $V$, $V^3$, and $V^5$, where $V$ is the magnitude of the Zeeman field. Thus, in the presence of next-nearest neighbor spin-orbit coupling described by Eq. (5), the MBSs split in every direction. Both axes are in units where the pairing potential ${\Delta }_1=2$.