Spin ladder quantum simulators from spin-orbit-coupled quantum dot spin qubits

TL;DR

This work builds and analyzes a two-leg spin ladder formed from spin-orbit-coupled Ge hole quantum-dot qubits, revealing a rich spin Hamiltonian with AF Heisenberg exchange, Dzyaloshinskii–Moriya, and anisotropic exchange terms. It develops two complementary analytic routes: (i) a strong rung coupling limit that maps the ladder to effective spin-$1/2$ chains and shows how DM and anisotropic terms survive or can be engineered, including staggered DM realizations that connect to sine-Gordon physics; and (ii) a weak rung coupling limit that uses basis rotations and Abelian bosonization (with Luther-Emery methods) to derive phase diagrams under longitudinal and transverse magnetic fields. The results provide concrete guidance for Ge-hole experiments, showing how SOC strength and DM interactions shape commensurate-incommensurate transitions and how to identify SOC via low-energy phase diagrams. Overall, the paper establishes spin-orbit-coupled quantum dot ladders as a versatile platform for engineering exotic spin models, studying quantum phases, and enabling programmable quantum simulations and computations.

Abstract

Motivated by the recent Ge hole spin qubit experiments, we construct and study a two-leg spin ladder from a quantum dot array with spin-orbit couplings (SOCs), aiming to uncover the many-body phase diagrams and provide concrete guidance for the Ge hole spin qubit experiments. The spin ladder is described by an unprecedented, complex spin Hamiltonian, which contains antiferromagnetic Heisenberg exchange, Dzyaloshinskii-Moriya (DM), and anisotropic exchange interactions. We analyze the spin ladder Hamiltonian in two complementary situations, the strong rung coupling limit and the weak rung coupling limit. In the strong rung coupling limit, we systematically construct effective spin-1/2 chain models, connecting the well-studied one-dimensional spin models and providing a recipe for Hamiltonian engineering. It is worth emphasizing that effective DM interactions can be completely turned off while the microscopic DM interactions are generically inevitable. Moreover, the staggered DM interactions, which are not possible in the microscopic spin model, can also be realized in the effective spin-1/2 model. In the weak rung coupling limit, we employ Abelian bosonization and Luther-Emery fermionization, uncovering a multitude of phases. Several commensurate-incommensurate transitions are driven by both the longitudinal magnetic field and the DM interactions in the legs (chains). Remarkably, the low-energy phase diagrams show strong dependence in the DM interaction, providing a concrete way to identify the strength of SOC in the experiments. Our work bridges quantum many-body theory and spin qubit device physics, establishing spin ladders made of spin-orbit-coupled quantum dots as a promising platform for engineering exotic spin models, constructing quantum many-body states, and enabling programmable quantum computations.

Figures (6)

• Figure 1: Setup of the two-leg spin ladder model. The blue (red) dots represent the $s$ ($\tau$) spins. The horizontal bonds indicate the interactions within a leg; the vertical bonds (black dashed lines) indicate the rung interactions. We use arrows in the bonds to indicate the direction of the DM vectors in the model. Notably, the DM vectors of the leg and rung couplings are orthogonal, realizing a complex spin model as we discuss in the main text.
• Figure 2: The energy spectrum of $\hat{H}_n$ as a function of magnetic field in the isotropic AF Heisenberg limit.
• Figure 3: The energy spectrum of $\hat{H}_n$ [Eq. (\ref{['Eq:H_gen_rung']})] under a magnetic field along $\hat{x}$. The results here are identical to Fig. \ref{['Fig:AF_rung']}.
• Figure 4: The energy spectrum of $\hat{H}_n$ [Eq. (\ref{['Eq:H_gen_rung']})] under a magnetic field perpendicular to $\hat{x}$. (a) $\alpha=\pi/16$. (b) $\alpha=\pi/4$. (c) $\alpha=\pi/3$. (d) $\alpha=\pi/2$. Note that there are two degenerate energy levels that are independent of magnetic field in (d). The results show strong $\alpha$-dependence. The magnetic field induced level crossing is generically avoided for $\alpha\neq 0$.
• Figure 5: Spin ladder phase diagrams with $\eta=0$ or $\zeta=0$. (a) $\eta=0$, corresponding to zero DM interaction in the legs. Phase i is driven into phase ii for a sufficiently large $\zeta$. (b) $\zeta=0$, corresponding to zero magnetic field. Phase ii transits into phase i or iv for a sufficiently large $\eta$; phase iii is replaced by phase v in the large $\eta$ limit. The phases are summarized in Table \ref{['tab:phases']}. See main text for detailed discussions.