Generalized Josephson effect with arbitrary periodicity in quantum magnets

TL;DR

This work demonstrates a generalized $Z_n$ Josephson effect in easy-plane spin-$\tfrac{1}{2}$ XXZ chains controlled by boundary phases, with the ground-state energy evolving periodically and the period growing linearly with system size. By deriving a low-energy edge-projected Hamiltonian and employing both Kato adiabatic projection and adiabatic-tracking methods, the authors reveal a universal energy–phase relation that persists across chain lengths and connect winding dynamics to phantom helices via a translationally invariant $P$-site hopping model in angular momentum space. The study further uncovers super-winding states among excited levels, showing larger periodicities than the ground state and detailing permutation cycles that arise after a full $2\pi$ rotation. These results extend fractional Josephson concepts to arbitrary integer periodicities in quantum magnets and suggest experimental realizations in cold-atom and solid-state spin chains, with implications for spin-based quantum devices and quantum batteries.

Abstract

Easy-plane quantum magnets are strikingly similar to superconductors, allowing for spin supercurrent and an effective superconducting phase stemming from their $U(1)$ rotation symmetry around the $z$-axis. We uncover a generalized fractional Josephson effect with a periodicity that increases linearly with system size in one-dimensional spin-$1/2$ chains at selected anisotropies and phase-fixing boundary fields. The effect combines arbitrary integer periodicities in a single system, exceeding the $4π$ and $8π$ periodicity of superconducting Josephson effects of Majorana zero modes and other exotic quasiparticles. We reveal a universal energy-phase relation and connect the effect to the recently discovered phantom helices.

Figures (5)

• Figure 1: $Z_n$ Josephson effect in the $XXZ$ Heisenberg chain, i.e., the adiabatic evolution of the ground state (red curve) where the first spin is fixed while the last spin is rotated by a phase $\phi$. The instantaneous eigenenergies are dotted blue lines in the energy-phase diagram, and the black stars mark the phantom helical states. The adiabatic curve passes through all the degenerate eigenspaces of phantom states. Panel (a) exemplifies the $Z_8$ effect for chain length $N=9$ and (b) the $Z_5$ effect for chain length $N=10$. Panel (c) and (d) show spin expectation values at selected rotation phases.
• Figure 2: Asymptotic universal energy-phase relation of the $Z_n$ Josephson effect at $\Delta = -|J|/2$ for chain lengths up to $N=20$. (a) Asymptotic universal curve for even chains after scaling the phase by the system's periodicity in Eq. (\ref{['eq:PeriodicityGS']}) and the energy by the maximally pumped energy. (b) Scaled curve including odd chain lengths. (c) Convergence of the time-reversal symmetric point's energy for odd chains when only rescaling the phase.
• Figure 3: The helical states after an integer number of rotations are the eigenstates of a translationally invariant $P$-site hopping model in angular momentum space in Eq. (\ref{['eq:RotarModel']}), whose hoppings follow the Fourier transform $\gamma$ of the universal energy-phase relation.
• Figure 4: Super-winding Josephson effect of selected excited states (red) for chain length $N=10$. (b) $Z_8$ periodicity for the third and eighth excited states, exceeding ground state periodicity. (a) and (c) spin expectation values at selected $\phi$.
• Figure 5: Scaling of the energy vs. the system size $N$ including the fit shown in the respective inset. Panel (a) shows the linear fit of the increase in the maximum pump of adiabatic energy w.r.t. the ground state energy, and the decrease of ground state energy is shown in panel (b). Panel (c) shows the $1/N$ fitting of center minima for the odd chains adiabatic curve.