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Register a new userMagnetic Flux Periodicity in Second Order Topological Superconductors
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The magnetic flux periodicity of $frac{hc}{2e}$ is a well known manifestation of Cooper pairing in typical s-wave superconductors. In this paper we theoretically show that the flux periodicity of a two-dimensional second-order topological superconductor, which features zero-energy Majorana modes localized at the corners of the sample, is $frac{hc}{e}$ instead. We further show that the periodicity changes back to $frac{hc}{2e}$ at the transition to a topologically trivial superconductor, where the Majorana modes hybridize with the bulk states, demonstrating that the doubling of periodicity is a manifestation of the non-trivial topology of the state.
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We show that a two-dimensional semiconductor with Rashba spin-orbit coupling could be driven into the second-order topological superconducting phase when a mixed-pairing state is introduced. The superconducting order we consider involves only even-parity components and meanwhile breaks time-reversal symmetry. As a result, each corner of a square-shaped Rashba semiconductor would host one single Majorana zero mode in the second-order nontrivial phase. Starting from edge physics, we are able to determine the phase boundaries accurately. A simple criterion for the second-order phase is further established, which concerns the relative position between Fermi surfaces and nodal points of the superconducting order parameter. In the end, we propose two setups that may bring this mixed-pairing state into the Rashba semiconductor, followed by a brief discussion on the experimental feasibility of the two platforms.
Two-dimensional second-order topological superconductors host zero-dimensional Majorana bound states at their boundaries. In this work, focusing on rotation-invariant crystalline topological superconductors, we establish a bulk-boundary correspondence linking the presence of such Majorana bound states to bulk topological invariants introduced by Benalcazar et al. We thus establish when a topological crystalline superconductor protected by rotational symmetry displays second-order topological superconductivity. Our approach is based on stacked Dirac Hamiltonians, using which we relate transitions between topological phases to the transformation properties between adjacent gapped boundaries. We find that in addition to the bulk rotational invariants, the presence of Majorana boundary bound states in a given geometry depends on the interplay between weak topological invariants and the location of the rotation center relative to the lattice. We provide numerical examples for our predictions and discuss possible extensions of our approach.
We theoretically investigate the Floquet generation of second-order topological superconducting (SOTSC) phase in the high-temperature platform both in two-dimension (2D) and three-dimension (3D). Starting from a $d$-wave superconducting pairing gap, we periodically kick the mass term to engineer the dynamical SOTSC phase within a specific range of the strength of the drive. Under such dynamical breaking of time-reversal symmetry (TRS), we show the emergence of the textit{weak} SOTSC phase, harboring eight corner modes ie two zero-energy Majorana per corner, with vanishing Floquet quadrupole moment. On the other hand, our study interestingly indicates that upon the introduction of an explicit TRS breaking Zeeman field, the textit{weak} SOTSC phase can be transformed into textit{strong} SOTSC phase, hosting one zero-energy Majorana mode per corner, with quantized quadrupole moment. We also compute the Floquet Wannier spectra that further establishes the textit{weak} and textit{strong} nature of these phases. We numerically verify our protocol computing the exact Floquet operator in open boundary condition and then analytically validate our findings with the low energy effective theory (in the high-frequency limit). The above protocol is applicable for 3D as well where we find one dimensional (1D) hinge mode in the SOTSC phase. We then show that these corner modes are robust against moderate disorder and the topological invariants continue to exhibit quantized nature until disorder becomes substantially strong. The existence of zero-energy Majorana modes in these higher-order phases is guaranteed by the anti-unitary spectral symmetry.
We identify four types of higher-order topological semimetals or nodal superconductors (HOTS), hosting (i) flat zero-energy Fermi arcs at crystal hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only at finite momentum. Bulk-boundary correspondence relates the hinge states to the bulk topology protecting the nodal point or loop. We classify all HOTS for all tenfold-way classes with an order-two crystalline (anti-)symmetry, such as mirror, twofold rotation, or inversion.
Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of $e^2/h$.
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