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arXiv:2508.08358 · Physical review. B./Physical review. B

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

Yang-Zhi Chou, S. Das Sarma

Published Dec 22, 2025• 1 citation
arXivHTMLPDF

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.

DOI
10.1103/q9g5-459v

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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.
Understanding collective phases of strongly correlated quantum magnets relies on theoretically tractable model systems with precise microscopic control. Antiferromagnetic spin ladders provide such a setting, hosting field-tunable gapped and gapless phases at half filling and unconventional pairing tendencies upon doping. Here, we realize a programmable Heisenberg spin ladder in a half-filled germanium quantum dot array featuring site-resolved, continuously tunable exchange interactions. Under a fixed magnetic field, we vary the rung and leg coupling to map the rung-singlet, canted antiferromagnetic, and fully polarized phases. Hamiltonian-learning protocols combining equilibrium and dynamical measurements quantitatively characterize the ladder, incorporating spin-orbit interactions to reproduce the observed crossover behavior. Measurements of higher-order spin correlators -- including four-point correlations inaccessible to conventional bulk probes -- reveal signatures of the underlying phase structure despite the finite size. Our results establish germanium quantum dot arrays as a controllable platform for quantum magnetism, opening routes to investigate unconventional superconductivity in doped ladders.
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In the last years, the electron spin in quantum dots has become a most promising candidate for the realization of qubits, the basic element for quantum computation. In quantum dot spin qubits the exchange interaction implements a fundamental two qubit gate, which originates in the Coulomb interaction and the Pauli principle. Compared to single qubit gates, the exchange based gates are much faster and easier to control locally, motivating the solely exchange energy based quantum computation. The control is based on the exponential sensitivity of the exchange energy on the interparticle distance. Manipulation then can proceed, for example, shifting the single particle states electrically or compressing them magnetically. The practical manipulation schemes require quantitative knowledge of the exchange energy. The spin-orbit interaction plays an important role in this nano-system. Indeed the usual exchange interaction changes its magnitude, while a new term, called anisotropic exchange, appears. In this thesis we propose an accurate effective spin Hamiltonian for modeling spin-orbit-induced spin dynamics in the presence of the magnetic field. We have also performed numerically exact calculations of the isotropic and anisotropic exchange coupling in realistic two-electron GaAs coupled quantum dots in the presence of both Dresselhaus and Bychkov-Rashba spin-orbit interactions. The numerics allow us to establish the goodness of our model and to see the limits of previous statements. Furthermore we find that in zero magnetic field the second-order spin-orbit effects are absent at all interdot couplings.
This article considers recent advances in the investigation of the thermal and magnetic properties of integrable spin ladder models and their applicability to the physics of strong coupling ladder compounds. For this class of compounds the rung coupling J ⊥ is much stronger than the coupling J ∥ along the ladder legs. The ground state properties of the integrable two-leg spin- and the mixed spin-() ladder models at zero temperature are analysed by means of the Thermodynamic Bethe Ansatz (TBA). Solving the TBA equations yields exact results for the critical fields and critical behaviour. The thermal and magnetic properties of the models are discussed in terms of the recently introduced High Temperature Expansion (HTE) method, which is reviewed in detail. In the strong coupling region the integrable spin- ladder model exhibits three quantum phases: (i) a gapped phase in the regime , (ii) a fully polarized phase for , and (iii) a Luttinger liquid magnetic phase in the regime H c1<H<H c2. The critical behaviour in the vicinity of the critical points H c1 and H c2 is of Pokrovsky-Talapov type. The temperature-dependent thermal and magnetic properties are directly evaluated from the exact free energy expression and compared to known experimental results for the strong coupling ladder compounds (5IAP)2CuBr4· 2H2O, Cu2(C5H12N2)2Cl4, (C5H12N)2CuBr4, BIP-BNO and [Cu2(C2\O 2)(C10H8N2)2)](NO3)2. Similar analysis of the mixed spin-() ladder model reveals a rich phase diagram, with a and a full saturation magnetization plateau within the strong antiferromagnetic rung coupling regime. For weak rung coupling, the fractional magnetization plateau is diminished and a new quantum phase transition occurs. The phase diagram can be directly deduced from the magnetization curve obtained from the exact result derived from the TBA and HTE. The results are applied to the mixed ferrimagnetic ladder compound PNNBNO. The thermodynamics of the spin-orbital model with different single-ion anisotropies is also discussed. For this model single-ion anisotropy can trigger different quantum phase transitions within the spin and orbital degrees of freedom, with magnetization plateaux arising from different spin and orbit Landé g-factors.
Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t\text{\ensuremath{-}}J$ models of SU(2)\mathrm{SU}(2)SU(2) spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by SU(2)3\mathrm{SU}{(2)}_{3}SU(2)3​ theories, which are quantum deformations of the SU(2)\mathrm{SU}(2)SU(2) algebra. Inspired by the physics of SU(2)\mathrm{SU}(2)SU(2) spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t\text{\ensuremath{-}}J$ models. Here, we aim to explore the combined effects of extended dimensionality and doping by studying ladders composed of coupled chains of interacting itinerant Fibonacci anyons. We show analytically that in the limit of strong rung couplings these models can be mapped onto effective 1D models. These effective models can either be gapped models of hole pairs, or gapless models described by $t\text{\ensuremath{-}}J$ (or modified $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$) chains of Fibonacci anyons, whose spectrum exhibits a fractionalization into charge and anyon degrees of freedom. The charge degrees of freedom are described by the hardcore boson spectra while the anyon sector is given by a chain of localized interacting anyons. By using exact diagonalizations for two-leg and three-leg ladders, we show that indeed the doped ladders show exactly the same behavior as that of $t\text{\ensuremath{-}}J$ chains. In the strong ferromagnetic rung limit, we can obtain a new model that hosts two different kinds of Fibonacci particles, which we denote as the heavy $\ensuremath{\tau}$'s and light $\ensuremath{\tau}$'s. These two particle types carry the same (non-Abelian) topological charge but different (Abelian) electric charges. Once again, we map the two-dimensional ladder onto an effective chain carrying these heavy and light $\ensuremath{\tau}$'s. We perform a finite size scaling analysis to show the appearance of gapless modes for certain anyon densities, whereas a topological gapped phase is suggested for another density regime.
Motivated by a recent experiment [L. F. Livi et al., Phys. Rev. Lett. 117, 220401 (2016)], we study the ground-state properties of interacting fermions in a one-dimensional optical lattice clock with spin-orbit coupling. As the electronic and the hyperfine-spin states in the clock-state manifolds can be treated as effective sites along distinct synthetic dimensions, the system can be considered as multiple two-leg ladders with uniform magnetic flux penetrating the plaquettes of each ladder. As the interorbital spin-exchange interactions in the clock-state manifolds couple individual ladders together, we show that exotic interaction-induced vortex states emerge in the coupled-ladder system, which compete with existing phases of decoupled ladders and lead to a rich phase diagram. Adopting the density matrix renormalization group approach, we map out the phase diagram, and investigate in detail the currents and the density-density correlations of the various phases. Our results reveal the impact of interactions on spin-orbit coupled systems, and are particularly relevant to the ongoing exploration of spin-orbit coupled optical lattice clocks.