Observation of magnetic quantum phase crossovers in a semiconductor spin ladder
Elizaveta Morozova, Xin Zhang, Utso Bhattacharya, Pablo Cova Fariña, Daniel Jirovec, Alexander Nico-Katz, Stefan D. Oosterhout, Sougato Bose, Giordano Scappucci, Menno Veldhorst, Eugene Demler, Lieven M. K. Vandersypen
cond-mat.mes-hallquant-phSubmitted Aug 18, 2026• Announced Aug 19, 2026
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.
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.
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.
Quasi-particle dynamics in interacting systems in the presence of disorder challenges the notion of internal thermalization, but proves difficult to investigate theoretically for large particle numbers. Engineered quantum systems may offer a viable alternative, as witnessed in experimental demonstrations in a variety of physical platforms, each with its own capabilities and limitations. Semiconductor gate-defined quantum dot arrays are of particular interest since they offer both a direct mapping of their Hamiltonian to Fermi-Hubbard and Heisenberg models and the in-situ tunability of (magnetic) interactions and onsite potentials. In this work, we use an array of germanium quantum dots to simulate the dynamics of both single-spin excitations (magnons) and two-spin excitations (triplons). We develop a methodology that combines digital spin qubit operations for state preparation and readout with analog evolution under the full system Hamiltonian. Using these techniques, we can reconstruct quantum walk plots for both magnons and triplons, and for various configurations of Heisenberg exchange couplings. We furthermore explore the effect of single-site disorder and its impact on the propagation of spin excitations. The obtained results can provide a basis for simulating disorder-based solid-state phenomena such as many-body localization.
We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below Ec, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band (N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale Ec, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above Ec of the same type as in N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
Many-body systems with strong interactions often exhibit macroscopic behavior markedly absent in single-particle or noninteracting limits. Such emergent phenomena are well exemplified in lattice Hubbard models, where the interplay between interactions, geometric frustration, and magnetic flux gives rise to rich physics. Superconducting qubits naturally enable analog quantum simulation of Bose-Hubbard models, while offering tunable parameters, site-resolved control, and rapid experimental repetition rates. Here, we study a superconducting-qubit device that realizes the Bose-Hubbard model on a triangular-ladder lattice. By tuning the magnitude and sign of couplings, we engineer a synthetic magnetic flux to characterize the resulting half-filling ground state for various parameter regimes. We measure observables analogous to current-current correlators and bond kinetic energies, finding signatures consistent with chiral superfluids, Meissner superfluids, and bond-ordered insulators. Our results establish superconducting circuits as a platform for robustly probing quantum phases of matter in frustrated Bose-Hubbard systems, even in strongly correlated and gapless regimes.
We consider a two-chain, spin-21 antiferromagnetic Heisenberg spin ladder in an external magnetic field H. The spin ladder is known to undergo second-order quantum phase transitions (QPTs) at two critical values Hc1 and Hc2 of the magnetic field. There are now known examples of strongly coupled (where the rung exchange interaction is much stronger than the nearest-neighbor intrachain exchange interaction) organic ladder compounds in which QPTs have been experimentally observed. In this paper, we investigate whether well-known bipartite entanglement meaures such as one-site von Neumann entropy, two-site von Neumann entropy, and concurrence develop special features close to the quantum critical points. As suggested by an earlier theorem, the first derivatives of the measures with respect to magnetic field are expected to diverge as $H\ensuremath{\rightarrow}{H}_{c1}$ and $H\ensuremath{\rightarrow}{H}_{c2}$. Based on numerical diagonalization data and a mapping of the strongly coupled ladder Hamiltonian onto the XXZ chain Hamiltonian, for which several analytical results are known, we find that the derivatives of the entanglement measures diverge as $H\ensuremath{\rightarrow}{H}_{c2}$ but remain finite as $H\ensuremath{\rightarrow}{H}_{c1}$. The reason for this discrepancy is analyzed. We further calculate two recently proposed quantum information theoretic measures, the reduced fidelity and reduced fidelity susceptibility, and show that these measures provide appropriate signatures of the QPTs occuring at the critical points H=Hc1 and H=Hc2.
The magnetization M(h) is used to identify three singlet quantum phases of the ladder with isotropic exchange interactions. The Dimer phase with frustrated F exchanges in rungs and legs has a first-order M(h) transition at 0 K from singlet to ferromagnetic at the saturation field hs. The Haldane-DAF phase with strong F exchange in rungs and net AF exchange between rungs has continuous M(h) and is adiabatically connected to the S=1 Heisenberg AF chain. The AF phase with strong F exchange in legs and net AF exchange between legs has continuous M(h) and is adiabatically connected to the spin-1/2 J1−J2 model with J1>0 and J2<0. All three singlet phases have finite gaps to the lowest triplet state.