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arXiv:2512.13242 · Physical Review Letters

Probing Ground-State Degeneracies of a Strongly Interacting Fermi-Hubbard Model with Superconducting Correlations

Sebastiaan L. D. ten Haaf, Sebastian Miles, Qingzhen Wang, A. Mert Bozkurt, Ivan Kulesh, Yining Zhang, Christian G. Prosko, Michael Wimmer, Srijit Goswami

Published Jul 16, 2026• 0 citations
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Abstract

The Fermi-Hubbard model describes a large variety of condensed matter systems with spinful fermions and strong interactions. On the other hand, the Kitaev chain model deals with noninteracting spinless fermions and produces robust ground-state degeneracies that give rise to Majorana bound states. In this Letter, we connect these two domains by experimentally studying small arrays of quantum dots (QDs) coupled via superconductors. Without a magnetic field, this system constitutes a Fermi-Hubbard model with intersite superconducting correlations. Using two coupled QDs, we find sweet spots in parameter space where we observe zero-bias conductance peaks that are robust against on-site perturbations of the QDs, similar to the behavior expected of a two-site Kitaev chain. This observation is consistent with the presence of Majorana Kramers pairs or Z_{3} parafermions in this minimal system. Extending to three sites, we find that the spinful system scales very differently compared to the spinless Kitaev chain. When the same sweet-spot conditions are satisfied, the ground state degeneracy of the full three-site system is lifted. The degeneracy can be restored by tuning the superconducting phase difference between the hybrid segments, but the robustness against on-site detunings is lost. Our findings are a first step in studying ground-state degeneracies in strongly interacting Fermi-Hubbard systems with superconducting correlations.

DOI
10.1103/nz6v-xxrv

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The Fermi-Hubbard model describes a large variety of condensed matter systems with spinful fermions and strong interactions. On the other hand, the Kitaev chain model deals with noninteracting spinless fermions and produces robust ground-state degeneracies that give rise to Majorana bound states. In this Letter, we connect these two domains by experimentally studying small arrays of quantum dots (QDs) coupled via superconductors. Without a magnetic field, this system constitutes a Fermi-Hubbard model with intersite superconducting correlations. Using two coupled QDs, we find sweet spots in parameter space where we observe zero-bias conductance peaks that are robust against on-site perturbations of the QDs, similar to the behavior expected of a two-site Kitaev chain. This observation is consistent with the presence of Majorana Kramers pairs or Z_{3} parafermions in this minimal system. Extending to three sites, we find that the spinful system scales very differently compared to the spinless Kitaev chain. When the same sweet-spot conditions are satisfied, the ground state degeneracy of the full three-site system is lifted. The degeneracy can be restored by tuning the superconducting phase difference between the hybrid segments, but the robustness against on-site detunings is lost. Our findings are a first step in studying ground-state degeneracies in strongly interacting Fermi-Hubbard systems with superconducting correlations.
We theoretically explore the emergence of strong zero modes in a two-site chain consisting of two quantum dots coupled due to a central dot that mediates electron hopping and singlet superconducting pairing. In the presence of time-reversal symmetry, the on-site Coulomb interaction leads to a three-fold ground-state degeneracy when tuning the system to a sweet spot as a function of the inter-dot couplings. This degeneracy is protected against changes of the dot energies in the same way as “poor man’s” Majorana bound states in short Kitaev chains. In the limit of strong interactions, this protection is maximal and the entire spectrum becomes triply degenerate, indicating the emergence of a “poor man’s” version of a strong zero mode. We explain the degeneracy and protection by constructing corresponding Majorana Kramers-pair operators and \mathbb{Z}_3 ℤ 3 -parafermion operators. The strong zero modes share many properties of Majorana bound states in short Kitaev chains, including the stability of zero-bias peaks in the conductance and the behavior upon coupling to an additional quantum dot. However, they can be distinguished through finite-bias spectroscopy and the exhibit a different behavior when scaling to longer chains.
Few-site implementations of the Kitaev chain offer a minimal platform to study the emergence and stability of Majorana bound states. Here, we realize two- and three-site chains in semiconducting quantum dots coupled via superconductors, and tune them to the sweet spot where zero-energy Majorana modes appear at the chain ends. We demonstrate control of the superconducting phase through both magnetic field and sweet-spot selection, and fully characterize the excitation spectrum under local and global perturbations. All spectral features are identified using the ideal Kitaev chain model. To assess Majorana localization, we couple the system to an additional quantum dot. The absence of energy splitting at the sweet spot is compatible with high-quality Majorana modes, despite the modest chain size.
Artificial Kitaev chains based on arrays of quantum dots are promising platforms for realizing Majorana Bound States (MBSs). In a two-site Kitaev chain, it is possible to find these non-Abelian zero-energy excitations at certain points in parameter space (sweet spots). These states, commonly referred to as Poor man's Majorana bound states (PMMs), are challenging to find and stabilize experimentally. In this work, we investigate the evolution of the sweet spots as we increase the number of sites of the Kitaev chain. To this end, we use the Bogoliubov-de Gennes representation to study the excitations of the system, and the scattering matrix and Green functions formalisms to calculate the zero-bias conductance. Our results show that the sweet spots evolve into a region that grows bigger and becomes gradually more protected as the number of sites NNN increases. Due to the protection of the MBSs, we refer to this region as a topological island. We obtain similar results by considering a realistic spinful model with finite magnetic fields in a chain of normal-superconducting quantum dots. For long chains, N≥20N \geq 20N≥20, we show the emergence of strictly zero-energy plateaus robust against disorder. Finally, we demonstrate that the topological islands can be observed by performing conductance measurements via a quantum dot side-coupled to the Kitaev chain. Our work shows that the fine-tuning required to create and detect PMMs in a 2-site Kitaev chain is significantly relaxed as the length of the chain increases and details how PMMs evolve into MBSs. Our results are consistent with experimental reports for 2 and 3-site chains.
Motivated by the recent experimental realization of a minimal Kitaev chain in quantum dot systems, we present our theoretical findings on the dynamics and fusion of MZMs at or near the “sweet spot” th = Δ (where the fermionic hopping th and superconducting coupling Δ are equal). We investigated the dynamics and fusion of MZMs using time-dependent real-space local density-of-states methods. The movement of Majoranas and the detection of fusion channels are crucial for topological quantum computations. Additionally, we discuss our recent discovery of exotic “multi-site” MZMs in Y-shaped Kitaev wires, which is important for the potential braiding of Majoranas in Y-junctions formed from arrays of quantum dots. Finally, we present results on ”non-trivial” fusion using canonical Kitaev wires at the sweet spot.
Majorana zero modes provide a route to fault-tolerant qubits by encoding information non-locally in fermion parity. Their sensitivity to noise is expected to decrease exponentially with increasing separation between the Majoranas, a suppression known as topological protection. Kitaev chains engineered in quantum dot-superconductor arrays provide a tunable platform in which separated Majorana zero modes can emerge at the ends of the chain, even in two-site chains. These minimal-chain modes are known as poor man's Majoranas and retain characteristic Majorana properties, including near-zero energy and equal electron-hole character, but have only limited protection. A key outstanding challenge is to move beyond identifying such modes in electrical transport measurements and achieve coherent qubit control in the time domain. Here, we demonstrate a Majorana parity qubit by realizing coherent coupling between two-site Kitaev chains. Since total fermion parity is conserved, the system separates into global even and odd parity manifolds. We observe coherent parity oscillations in both manifolds with equal oscillation frequencies at the Majorana sweet spot, as predicted for isolated Majorana zero modes. We further show that the oscillation frequency and coherence depend systematically on inter-chain coupling and quantum-dot detunings, in close agreement with our model for short, partially protected chains. Our results establish the first coherent control of a Majorana qubit, encoded in the fermion parity of Majorana zero modes in minimal Kitaev chains.
We investigate the Bose-Hubbard chain in the presence of nearest-neighbor pairing. The pairing term gives rise to an unusual gapped Z2{\mathbb{Z}}_{2}Z2​ Ising phase that has number fluctuation but no off-diagonal long range order. This phase has a strongly correlated many-body doubly degenerate ground state which is effectively a gap-protected macroscopic qubit. In the strongly interacting limit, the system can be mapped onto an anisotropic transverse spin chain, which in turn can be mapped to the better-known fermionic sister of the paired Bose-Hubbard chain: the Kitaev chain which hosts zero-energy Majorana bound states. While corresponding phases in the fermionic and bosonic systems have starkly different wave functions, they share identical energy spectra. We describe a possible cold-atom realization of the paired Bose-Hubbard model in a biased zigzag optical lattice with reservoir-induced pairing, opening a possible route towards experimental Kitaev chain spectroscopy.