Designing XY and Dzyaloshinskii--Moriya couplings in Majorana Cooper pair boxes

Abstract

We theoretically study how to design spin couplings in networks of Majorana Cooper pair boxes (MCBs) connected by multiple normal-metal leads. The inter-box interaction is generated by the conduction-electron-mediated Ruderman--Kittel--Kasuya--Yosida (RKKY) interaction. We show that the connectivity of Majoranas to the leads enables arbitrary types of couplings. As concrete examples, we show the realization of the XY exchange interaction and the Dzyaloshinskii--Moriya (DM) interaction, which are difficult to implement in previously proposed MCB-based schemes. The sign and magnitude of the couplings can be tuned continuously via gate-controlled tunneling amplitudes. These results establish MCBs as a versatile platform for engineered quantum spin systems.

Figures (7)

• Figure 1: Schematic setup for the two-spin system formed by two MCBs. The left (L) and right (R) boxes are MCBs with charging energy $E_{\rm c}^{\alpha}$ ($\alpha = {\rm L, R}$), where each of the four Majoranas (pink circles) is connected to leads (blue wires). Yellow double-headed arrows denote the tunnel coupling between the Majoranas and the leads. Panels (a) and (b) show the cases of infinite and finite leads, respectively.
• Figure 2: Schematic setups for realizing (a) the XY coupling and (b) the Heisenberg + DM coupling.
• Figure 3: Tunneling amplitudes realizing the XY coupling. $\tilde{\mathcal{T}}_{11}$ (solid line) and $\tilde{\mathcal{T}}_{22}$ (dotted line) as a function of $\tilde{J}$ obtained from Eqs. \ref{['XY_T11']} and \ref{['XY_T22']}, for $(\tilde{B}_x, \tilde{A}_y, \tilde{B}_y)=(1,-1,-1)$.
• Figure 4: Color maps representing the region in which a real solution $\mathcal{T}_{33}$ exists in Eq. \ref{['T33']}. The color bar indicates the value of $\mathcal{T}_{33}/A$.
• Figure 5: Site $N$ dependence of the isotropic spin susceptibility $\chi$ for infinite length. The black, red, and blue lines correspond to the numerical calculation results of the susceptibility for Fermi energies of 0 (at the center of the energy band), $-t$, and $-1.8t$ (around the bottom of the energy band), respectively.