PT symmetry, that is, a combined parity and time-reversal symmetry, is a key milestone for non-Hermitian systems exhibiting entirely real eigenenergy. In the present work, motivated by a recent experiment, we study PT symmetry of the time-evolution operator of nonunitary quantum walks. We present the explicit definition of PT symmetry by employing a concept of symmetry time frames. We provide a necessary and sufficient condition so that the time-evolution operator of the nonunitary quantum walk retains PT symmetry even when parameters of the model depend on position. It is also shown that there exist extra symmetries embedded in the time-evolution operator. Applying these results, we clarify that the nonunitary quantum walk in the experiment does have PT symmetry.

We study topological phases in one-dimensional open Floquet systems driven by chiral symmetric nonunitary time evolution. We derive a procedure to calculate topological numbers from nonunitary time-evolution operators with chiral symmetry. While the procedure has been applied to open Floquet systems described by nonunitary time-evolution operators, we give the microscopic foundation and clarify its validity. We construct a model of chiral symmetric nonunitary quantum walks classified into class BDI † or AIII, which is one of the enlarged symmetry classes for topological phases in open systems based on experiments of discrete-time quantum walks. Then we confirm that the topological numbers obtained from the derived procedure give correct predictions of the emergent edge states. We also show that the model retains PT symmetry in certain cases, and its dynamics is crucially affected by the presence or absence of PT symmetry.

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The angular momentum of an electron is characterized well by pseudospin with J = 3 2 in the presence of strong spin-orbit interactions. We study theoretically the Josephson effect of superconductors in which two such J = 3 2 electrons form a Cooper pair. Within even-parity symmetry class, pseudospin-quintet pairing states with J = 2 can exist as well as pseudospin-singlet states with J = 0. We focus especially on the Josephson selection rule among these even-parity superconductors. We find that the selection rule between quintet states is more severe than that between spin-triplet states formed by two S = 1 2 electrons. The effects of a pseudospin-active interface on the selection rule are discussed as well as those of odd-frequency Cooper pairs generated by pseudospindependent band structures.

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