Explicit definition of PT\mathcal{PT} symmetry for non-unitary quantum walks with gain and loss

$\mathcal{PT}$ symmetry, that is, a combined parity and time-reversal symmetry is a key milestone for non-Hermite systems exhibiting entirely real eigenenergy. In the present work, motivated by a recent experiment, we study $\mathcal{PT}$ symmetry of the time-evolution operator of non-unitary quantum walks. We present the explicit definition of $\mathcal{PT}$ symmetry by employing a concept of symmetry time frames. We provide a necessary and sufficient condition so that the time-evolution operator of the non-unitary quantum walk retains $\mathcal{PT}$ 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 non-unitary quantum walk in the experiment does have $\mathcal{PT}$ symmetry.Comment: 14 pages, 8 figure

Josephson effect of superconductors with J=3/2J=3/2 electrons

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 such two $J=3/2$ electrons form a Cooper pair. Within even-parity symmetry class, pseudospin-quintet pairing states with $J=2$ can exist as well as pseudospin-singlet state 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 severer 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 pseudospin dependent band structures.Comment: 10 pages, 2 figure

Spin Susceptibility of a J=3/2 Superconductor

We discuss the spin susceptibility of superconductors in which a Cooper pair consists of two electrons having the angular momentum J=3/2 due to strong spin-orbit interactions. The susceptibility is calculated analytically for pseudospin quintet states in a cubic superconductor within the linear response to a Zeeman field. The susceptibility for $A_{1g}$ symmetry states is isotropic in real space. For $E_g$ and $T_{2g}$ symmetry cases, the results depend sensitively on choices of order parameter. The susceptibility is isotropic for a $T_{2g}$ symmetry state, whereas it becomes anisotropic for an $E_{g}$ symmetry state. We also find in a $T_{2g}$ state that the susceptibility tensor has off-diagonal elements.Comment: 11 pages, 3 figure

Bulk-edge correspondence in nonunitary Floquet systems with chiral 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 dagger 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