Theory of topological spin Josephson junctions

TL;DR

This paper investigates topological spin Josephson effects in a 1D spin chain model, revealing fractional periodicities and symmetry protections, with implications for various quantum spin systems.

Contribution

It introduces a spin chain model emulating topological superconducting junctions, analyzing fractional Josephson effects and symmetry protections both analytically and numerically.

Findings

Identification of $ ext{Z}_2$ and $ ext{Z}_4$ fractional spin Josephson effects.

Discovery of symmetry protections for Josephson periodicities.

Proposal for microwave cavity detection of spin Josephson effects.

Abstract

We study the spin transport through a 1D quantum Ising-XY-Ising spin link that emulates a topological superconducting-normal-superconducting structure via Jordan-Wigner (JW) transformation. We calculate, both analytically and numerically, the spectrum of spin Andreev bound states and the resulting $\mathbb{Z}_2$ fractional spin Josephson effect (JE) pertaining to the emerging Majorana JW fermions. Deep in the topological regime, we identify an effective time-reversal symmetry that leads to $\mathbb{Z}_4$ fractional spin JE in the $\textit{presence}$ of interactions within the junction. Moreover, we uncover a hidden inversion time-reversal symmetry that protects the $\mathbb{Z}_4$ periodicity in chains with an odd number of spins, even in the $\textit{absence}$ of interactions. We also analyze the entanglement between pairs of spins by evaluating the concurrence in the presence of spin current and highlight the effects of the JW Majorana states. We propose to use a microwave cavity setup for detecting the aforementioned JEs by dispersive readout methods and show that, surprisingly, the $\mathbb{Z}_2$ periodicity is immune to $\textit{any}$ local magnetic perturbations. Our results are relevant for a plethora of spin systems, such as trapped ions, photonic lattices, electron spins in quantum dots, or magnetic impurities on surfaces.