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We study the superconducting state of multi-orbital spin-orbit coupled systems in the presence of an orbitally driven inversion asymmetry assuming that the inter-orbital attraction is the dominant pairing channel. Although the inversion symmetry is absent, we show that superconducting states that avoid mixing of spin-triplet and spin-singlet configurations are allowed, and remarkably, spintriplet states that are topologically nontrivial can be stabilized in a large portion of the phase diagram. The orbital-dependent spin-triplet pairing generally leads to topological superconductivity with point nodes that are protected by a nonvanishing winding number. We demonstrate that the disclosed topological phase can exhibit Lifshitz-type transitions upon different driving mechanisms and interactions, e.g., by tuning the strength of the atomic spin-orbit and inversion asymmetry couplings or by varying the doping and the amplitude of order parameter. Such distinctive signatures of the nodal phase manifest through an extraordinary reconstruction of the low-energy excitation spectra both in the bulk and at the edge of the superconductor.

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Odd-frequency Cooper pairs with chiral symmetry emerging at the edges of topological superconductors are a useful physical quantity for characterizing the topological properties of these materials. In this work, we show that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency, which is called a "spectral bulk-boundary correspondence," and can be evaluated from the spectral features of the bulk. The odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form ∼ 1/z (where z is a complex frequency) that reflects the presence of a topological surface Andreev bound state, whereas the amplitudes in the second category have the regular form ∼ z and are regarded as non-topological. We discuss the topological phase transition by using the coefficient in the latter category, which undergoes a power-law divergence at the topological phase transition point and is used to indicate the distance to the critical point. These concepts are established based on several concrete models, including a Rashba nanowire system that is promising for realizing Majorana fermions. arXiv:1809.05687v2 [cond-mat.supr-con]

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